Real-Time Scheduling Analysis

Transcription

1 DOT/FAA/AR-05/7 Real-Tme Scheulg Aalyss Offce of Avato Research a Developmet Washgto, D.C. 059 November 005 Fal Report Ths ocumet s avalable to the U.S. publc through the Natoal Techcal Iformato Servce (NTIS), Sprgfel, Vrga 6. U.S. Departmet of Trasportato Feeral Avato Amstrato

2 NOTICE Ths ocumet s ssemate uer the sposorshp of the U.S. Departmet of Trasportato the terest of formato exchage. The Ute States Govermet assumes o lablty for the cotets or use thereof. The Ute States Govermet oes ot eorse proucts or maufacturers. Trae or maufacturers ames appear here solely because they are cosere essetal to the objectve of ths report. Ths ocumet oes ot costtute FAA certfcato polcy. Cosult your local FAA arcraft certfcato offce as to ts use. Ths report s avalable at the Feeral Avato Amstrato Wllam J. Hughes Techcal Ceters Full-Text Techcal Reports page: actlbrary.tc.faa.gov Aobe Acrobat portable ocumet format (PDF).

3 . Report No. DOT/FAA/AR-05/7 4. Ttle a Subttle REAL-TIME SCHEDULING ANALYSIS. Govermet Accesso No.. Recpets Catalog No. 5. Report Date Techcal Report Documetato Page November Performg Orgazato Coe 7. Author(s) Joseph Leug a Harog Zhao 9. Performg Orgazato Name a Aress Departmet of Computer Scece New Jersey Isttute of Techology Newark, NJ 070. Sposorg Agecy Name a Aress U.S. Departmet of Trasportato Feeral Avato Amstrato Offce of Avato Research a Developmet Washgto, DC Supplemetary Notes 8. Performg Orgazato Report No. 0. Work Ut No. (TRAIS). Cotract or Grat No. NJIT/WOW 0/C/AW/NJIT Amemet. Type of Report a Pero Covere 4. Sposorg Agecy Coe AIR-0 The Feeral Avato Amstrato Arport a Arcraft Safety R&D Dvso COTR was Charles Klgore. 6. Abstract Ths project was cocere wth scheulg aalyss of real-tme tasks. It cosste of two major tasks. The frst task explore a reporte the ustry approaches to scheulg real-tme tasks a the tools they use the verfcato of temporal correctess. To carry out ths task, a questoare was esge a set to a umber of ustry people who are volve evelopg software for real-tme systems. Ther resposes were aalyze a coclusos were raw. The seco task cosste of evelopg scheulg algorthms a temporal verfcato tools for a moel of peroc, real-tme tasks. A optmal scheulg algorthm, calle Deale-Mootoc-wth-Lmte-Prorty-Levels, was evelope for a system wth a sgle processor a a lmte umber of prorty levels. A proceure to eterme f a gve set of peroc, real-tme tasks s feasble o oe processor wth m prorty levels, where m s less tha the umber of tasks, was also evelope. Two heurstcs for a multprocessor system wth a lmte umber of prorty levels were gve. Atoally, a cojecture o the processor utlzato bou, U(), below whch a set of ut-executo-tme tasks s always scheulable was prove. Whle a complete proof of the cojecture has ot bee accomplshe, t has bee emostrate that t s val for several specal cases. 7. Key Wors Peroc, real-tme tasks; Rate-mootoc a ealemootoc algorthms; Valato proceure; Real-tme scheulg 9. Securty Classf. (of ths report) Uclassfe 0. Securty Classf. (of ths page) Uclassfe 8. Dstrbuto Statemet Ths ocumet s avalable to the publc through the Natoal Techcal Iformato Servce (NTIS) Sprgfel, Vrga 6.. No. of Pages 5. Prce Form DOT F (8-7) Reproucto of complete page authorze

4 TABLE OF CONTENTS Page EXECUTIVE SUMMARY v. INTRODUCTION -. Purpose a Backgrou -. Report Overvew -5. Usg Ths Report -5. SURVEY OF INDUSTRY APPROACHES -. TUTORIAL ON DIFFERENT SCHEDULING APPROACHES - 4. DESCRIPTION OF SCHEDULING MODEL 4-5. RESULTS AND FUTURE WORK 5-5. Results 5-5. Future Work 5-6. REFERENCES 6- APPENDICES A Iustry Survey a Resposes B The Research Project Detals Wth Assumptos C The Implemetato of the Algorthm DM-LPL

5 LIST OF FIGURES Fgure Page - Dgtal Cotroller - - Software Cotrol Structure of a Flght Cotroller - - Ar Traffc a Flght Cotrol Herarchy - v

6 LIST OF ACRONYMS A/D ATC D/A DM DM-LPL EDF FCFS FF FFDU I/O NP RM RMFF RMNF RTSA TS Aalog to gtal cotroller Ar Traffc Cotrol (System) Dgtal to aalog cotroller Deale-Mootoc (Algorthm) Deale-Mootoc-wth-Lmte-Prorty-Level (Algorthm) Earlest-Deale-Frst (Algorthm) Frst-come-frst-serve Frst-Ft (Algorthm) Frst-Ft-Decreasg-Utlzato (Algorthm) Iput/Output (Actvtes) Noetermstc Polyomal Rate-Mootoc (Algorthm) Rate-Mootoc-Frst-Ft (Algorthm) Rate-Mootoc-Next-Ft (Algorthm) Real-Tme Scheulg Aalyss Task system v/v

7 EXECUTIVE SUMMARY Real-tme (computg, commucato, a formato) systems have become creasgly mportat every ay lfe. A real-tme system s requre to complete ts work a elver ts servces o a tmely bass. Examples of real-tme systems clue gtal cotrol, comma a cotrol, sgal processg, a telecommucato systems. Such systems prove mportat a useful servces to socety o a aly bass. For example, they cotrol the ege a brakes o cars; regulate traffc lghts; scheule a motor the takeoff a lag of arcraft; eable arcraft cotrol system; motor a regulate boly fuctos (e.g., bloo pressure a pulse); a prove up-to-ate facal formato. May real-tme systems are embee sesors a actuators a fucto as gtal cotrollers. Typcally, these type of applcatos, sgals arrve perocally at fxe peros. Whe the sgals arrve, they must be processe before the arrval of the ext batch of sgals. Real-tme tasks ca be classfe as har real-tme or soft real-tme. Har real-tme tasks are those that requre a strct aherece to eale costrats, or else the cosequece s sastrous. By cotrast, soft real-tme tasks are those that o ot requre a strct aherece to eale costrats, but t s esrable to o so. Most real-tme systems have a combato of both har a soft real-tme tasks. Har real-tme tasks create aother meso verfyg ther correctess. Not oly o ther logcal correctess ee to be verfe (.e., the program mplemets exactly the thgs t s suppose to o), ther temporal correctess must also be verfe (.e., all eales are met). A har real-tme task must be both logcally a temporally correct for t to be usable. Sce a har real-tme task s execute perocally urg ts etre operatoal tme a sce the pero of every task s very small compare to the urato of ts operato, the scheule ca be regare as a fte scheule for all practcal purposes. Verfyg the temporal correctess of a fte scheule s a challegg problem sce there are ftely may eales to check. Oe of the ma goals of ths project s to evelop tools to solve ths verfcato problem. Ths project cossts of two major jobs. The frst job was to explore a report the ustry approaches to scheulg real-tme tasks a the tools they use the verfcato of temporal correctess. A questoare was evelope a set to a umber of ustry represetatves who are volve evelopg software for real-tme systems. Base o ther resposes, some coclusos were raw, whch are escrbe ths report. The seco job cosste of evelopg scheulg algorthms a temporal verfcato tools for a moel of peroc, real-tme tasks. A optmal scheulg algorthm, calle Deale- Mootoc-wth-Lmte-Prorty-Levels, was evelope for a system wth a sgle processor a a lmte umber of prorty levels. As a byprouct of the work o the Deale-Mootocwth-Lmte-Prorty-Levels algorthm, a proceure to eterme f a gve set of peroc, realtme tasks s feasble o oe processor wth m prorty levels, where m s less tha the umber of tasks, was also evelope. Ths report begs wth a summary of the ustry survey results, the the three approaches that were use to scheule a real-tme task system are scusse: () Clock-Drve, () Processor- Sharg, a () Prorty-Drve. It was reasoe that the Prorty-Drve approach s far v

8 superor to the Clock-Drve a Processor-Sharg approaches. The report the revews the lterature o Prorty-Drve scheulg algorthms, whch ca be ve to two categores: Dyamc-Prorty a Fxe-Prorty. Whle Dyamc-Prorty scheulg algorthms are more effectve tha Fxe-Prorty scheulg algorthms, they are rarely use practce because of the overhea volve. Therefore, the report cocetrates o Fxe-Prorty scheulg algorthms. The Deale Mootoc algorthm s a optmal Fxe-Prorty scheulg algorthm for oe processor. Ufortuately, the algorthm assumes that the umber of prortes s the same as the umber of real-tme tasks. I practce, oe ca oly have a lmte umber of prortes, say m, supporte by a system. Uer ths scearo, the Deale Mootoc algorthm fals to be optmal, a as a result of the work to f a optmal scheulg algorthm, the Deale- Mootoc-wth-Lmte-Prorty-Levels algorthm was evelope, alog wth a proceure to check f a gve set of real-tme tasks s feasble o oe processor wth m prorty levels. The same problem was explore for multprocessor systems. It was emostrate that fg a optmal assgmet s strogly oetermstc polyomal (NP)-har, whch s tatamout to showg that there s o effcet algorthm to solve ths problem. Motvate by the computatoal complexty, several heurstcs for solvg ths problem are suggeste. The mmum processor utlzato, U(), for a set of ut-executo-tme tasks, was also stue. U() s the threshol for the total processor utlzato of the tasks, below whch they are always scheulable. It s cojecture that U( ) Some specal cases of ths cojecture are prove, but a complete proof fale to be solvable. Some other jobs were plae for ths research effort (.e., stuy of fault-tolerat ssues, whch are cocere wth scheulablty aalyss whe there are tme losses ue to traset harware or software falures, a stuy of CPU scheulg couple wth I/O actvtes). However, ue to tme costrats, sgfcat progress was ot mae these areas. v

9 . INTRODUCTION.. PURPOSE AND BACKGROUND. Real-tme computg, commucato, a formato systems have become creasgly mportat every ay lfe. A real-tme system s requre to complete ts work a elver ts servces o a tmely bass. Examples of real-tme systems clue gtal cotrol, comma a cotrol, sgal processg, a telecommucato systems. Such systems prove mportat a useful servces to socety o a aly bass. For example, they cotrol the ege a brakes o cars; regulate traffc lghts; scheule a motor the takeoff a lag of arcraft; eable arcraft cotrol systems; motor a regulate boly fuctos (e.g., bloo pressure a pulse); a prove up-to-ate facal formato. May real-tme systems are embee sesors a actuators a fucto as gtal cotrollers. A example of a gtal cotroller, take from Lu [] s show fgure -. The term plat the fgure refers to a cotrolle system (such as a ege, a brake, a arcraft, or a patet), A/D refers to aalog-to-gtal coverter, a D/A refers to gtal-to-aalog coverter. The state of the plat s motore by sesors a ca be chage by actuators. The real-tme (computg) system estmates from the sesor reags the state of the plat, y(t), at tme t a computes a cotrolle output, u(t), base o the fferece betwee the curret state a the esre state (calle referece put the fgure), r(t). Ths computato s calle the cotrollaw computato the fgure. The output geerate by the cotrol-law computato actvates the actuators, whch brg the plat closer to the esre state. cotroller referece put r(t) A/D A/D r k y k Cotrol-law computato u k D/A y(t) u(t) Sesor Plat Actuator FIGURE -. DIGITAL CONTROLLER I fgure -, r(t) a y(t) are sample perocally every samplg pero, T uts of tme. Therefore, the cotrol-law computato ees to be oe perocally every T uts of tme. For each sample ata, the computato must be complete wth T uts of tme, or else t wll be erase by the ext sample ata. Each computato s farly etermstc the sese that the maxmum executo tme ca be estmate farly accurately. -

10 A plat typcally has more tha oe state varable; e.g., the rotato spee a temperature of a ege. Therefore, t s cotrolle by multple sesors a by multple actuators. Because fferet state varables may have fferet yamcs, the samplg peros may be fferet. As a example, also take from Lu [], fgure - shows the software structure of a flght cotroller. The plat s a helcopter, whch has three velocty compoets: forwar, se-slp, a alttue rates, whch together are calle collectve the fgure. It also has three rotatoal (agular) veloctes, referre to as roll, ptch, a yaw. The system uses three samplg rates: 80, 90, a 0 Hz;.e., the samplg peros are /80, /90, a /0 secos, respectvely. Do the followg each /80 secos cycle: Valate sesor ata a select ata source; the presece of falures, recofgure the system. Do the followg 0-Hz avocs tasks, each every sx cycles: keyboar put a moe selecto ata ormalzato a coorate trasformato trackg referece upate Do the followg 0-Hz computatos, each oce every sx cycles: cotrol laws of the outer ptch-cotrol loop cotrol laws of the outer roll-cotrol loop cotrol laws of the outer yaw- a collectve-cotrol loop Do the followg 90-Hz computatos oce every two cycles, usg outputs prouce by 0-Hz computatos a avocs tasks as put: cotrol laws of the er ptch-cotrol loop cotrol laws of the er roll- a collectve-cotrol loop Compute the cotrol laws of the er yaw-cotrol loop, usg outputs prouce by 90-Hz cotrol-law computatos as put. Output commas. Carry out bult--test. Wat utl the begg of the ext cycle. FIGURE -. SOFTWARE CONTROL STRUCTURE OF A FLIGHT CONTROLLER The above cotroller cotrols oly flght yamcs. The cotrol system o boar a arcraft s coserably more complex. It typcally cotas may other equally crtcal subsystems (e.g., ar let, fuel, hyraulc, a at-ce cotrollers) a may ocrtcal subsystems (e.g., compartmet lghtg a temperature cotrollers). So, ato to the flght cotrol-law computatos, the system also computes the cotrol laws of these subsystems. Cotrollers a complex motor a cotrol system are typcally orgaze herarchcally. Oe or more gtal cotrollers at the lowest level rectly cotrol the physcal plat. Each output of a -

11 hgher-level cotroller s a referece put of oe or more lower-level cotrollers. Oe or more of the hgher-level cotroller terfaces wth the operator(s). Fgure -, also take from Lu [], shows the herarchy of flght cotrol, avocs, a ar traffc cotrol (ATC) systems. The ATC system s at the hghest level. It regulates the flow of flghts to each estato arport. It oes so by assgg to each arcraft a arrval tme at each meterg fx e route to the estato: The arcraft s suppose to arrve at the meterg fx at the assge arrval tme. At ay tme whle flght, the assge arrval tme to the ext meterg fx s a referece put to the oboar flght maagemet system. The flght maagemet system chooses a tme-referece flght path that brgs the arcraft to the ext meterg fx at the assge arrval tme. The cruse spee, tur raus, esce/asce rates, a so forth requre to follow the chose tme-referece flght path are the referece puts to the flght cotroller at the lowest level of the cotrol herarchy. respose commas Operator-system terface from sesors State estmator Ar-traffc cotrol avgato vrtual plat State estmator Flght maagemet vrtual plat State estmator flght cotrol Ar ata physcal plat FIGURE -. AIR TRAFFIC AND FLIGHT CONTROL HIERARCHY -

12 Real-tme tasks ca be classfe as har or soft. Har real-tme tasks are those that requre a strct aherece to eale costrats, or else the cosequece s sastrous. A example of a har real-tme task s the flght cotroller show fgure -. By cotrast, soft real-tme tasks are those that o ot requre a strct aherece to eale costrats, but t s esrable to o so. A example of a soft real-tme task s the cotroller that cotrols the compartmet lghtg a temperature a arcraft. Most real-tme systems have a combato of both har a soft realtme tasks. Har real-tme tasks create aother meso valatg ther correctess. Not oly o ther logcal correctess ee to be verfe (.e., the program mplemets exactly the thgs t s suppose to), ther temporal correctess must also be verfe (.e., all eales are met). A har real-tme task must be both logcally a temporally correct for t to be usable. Sce a har realtme task s execute perocally urg ts etre operatoal tme a sce the pero of every task s very small compare to the urato of ts operato, the scheule ca be regare as a fte scheule for all practcal purposes. Verfyg the temporal correctess of a fte scheule s a challegg problem sce there are ftely may eales to check. Oe of the ma goals of ths project s to evelop tools to help solve ths verfcato problem. Sce the ma cocer of ths report s har real-tme tasks, the term har real-tme task wll smply be calle a real-tme task throughout ths report. Ths project cosste of two major jobs. The frst job was to explore a report the ustry approaches to scheulg real-tme tasks a the tools they use the verfcato of temporal correctess. A questoare was evelope a set to a umber of ustry represetatves who are volve evelopg software for real-tme systems. Base o ther resposes, some coclusos were raw, these are escrbe ths report. The seco job cosste of evelopg scheulg algorthms a temporal verfcato tools for a moel of peroc, real-tme tasks. A optmal scheulg algorthm, calle Deale- Mootoc-wth-Lmte-Prorty-Levels (DM-LPL), was evelope for a system wth a sgle processor a a lmte umber of prorty levels. As a byprouct of the DM-LPL algorthm, a proceure to eterme f a gve set of peroc, real-tme tasks s feasble o oe processor wth m prorty levels, where m s less tha the umber of tasks, was also evelope. A peroc, real-tme task, T, s characterze by the quaruple (s, e,, p ), where s s the tal request tme, e s the executo tme, s the relatve eale, a p s the pero. I ths characterzato, T makes a tal request at tme s, a thereafter at tmes s + kp, k,, The k-th request requres e uts of executo tme, a t must be complete o later tha the eale s +(k-)p +. A real-tme task system cossts of peroc, real-tme tasks, a s eote by TS ({T, {s },{e }, { }, {p }). A scheule S for a real-tme task system TS s sa to be val f the eale of each request of each task s met. Sce the scheule s fte, checkg f the scheule s val s a otrval problem. TS s feasble f there s a val scheule for t. TS s scheulable by a partcular scheulg algorthm f the scheulg algorthm prouces a val scheule for t. A scheulg algorthm s sa to be optmal f every feasble task system s scheulable by the scheulg algorthm. -4

13 . REPORT OVERVIEW. Ths report begs wth a summary of the ustry survey results. Three approaches that have bee use to scheule a real-tme task system are () Clock-Drve, () Processor-Sharg, a () Prorty-Drve. It was reasoe that the Prorty-Drve approach was far superor to the Clock-Drve a Processor-Sharg approaches. The report the revews the lterature o Prorty-Drve scheulg algorthms, whch ca be ve to two categores: Dyamc- Prorty a Fxe-Prorty. Whle Dyamc-Prorty scheulg algorthms are more effectve tha Fxe-Prorty scheulg algorthms, they are rarely use practce because of the overhea volve. Therefore, the report cocetrates o Fxe-Prorty scheulg algorthms. The remag portos of the report focus o specfc scheulg moel. Leug a Whtehea have show that the Deale Mootoc algorthm s a optmal Fxe-Prorty scheulg algorthm for oe processor. Ufortuately, the algorthm assumes that the umber of prortes s the same as the umber of real-tme tasks. I practce, oe ca oly have a lmte umber of prortes, say m, supporte by a system. Uer ths scearo, the Deale Mootoc (DM) algorthm fals to be optmal, a the optmal scheulg algorthm DM-LPL was evelope, alog wth a separate proceure to check f a gve set of real-tme tasks s feasble o oe processor wth m prorty levels. The same problem s explore for multprocessor systems. It s emostrate that fg a optmal assgmet s strogly oetermstc polyomal (NP)-har, whch s tatamout to showg that there s o effcet algorthm to solve ths problem. A problem Q s NP-har f all problems the NP-class are reucble to Q. Motvate by the computatoal complexty, several heurstcs for solvg ths problem are suggeste. The mmum processor utlzato, U(), for a set of ut-executo-tme tasks s also stue. U() s the threshol for the total processor utlzato of the tasks, below whch they are always scheulable. It s cojecture that U( ) Some specal cases of ths cojecture are prove, but tme was ot avalable to perform the complete proof. Some other tasks were plae for ths research effort (.e., stuy of fault-tolerat ssues, whch are cocere wth scheulablty aalyss whe there are tme losses ue to traset harware or software falures, a stuy of cetral processg ut (CPU) scheulg couple wth put/output (I/O) actvtes). However, ue to tme costrats, sgfcat progress was ot mae these areas. These rema topcs to be cosere future research.. USING THIS REPORT. There are a umber of potetal uses for ths report. Sce ths research task falls more to the category of basc research tha may of the Feeral Avato Amstrato (FAA) Software a Dgtal System Safety Project research a evelopmet tatves, some explaato of -5

14 how varous reaers may use the report s prove. The tee auece for ths report s certfcato authortes, ustry represetatves, a researchers. A bref summary of how each auece ca use ths report s lste below. Certfcato Authortes. Certfcato authortes wll prmarly beeft from the summary of the ustry survey, the tutoral of the fferet scheulg approaches, a the research results (sectos,, a 5). Atoally, certfcato authortes mght esre to browse secto 4 a appex B for formato purposes, realzg that these sectos are research-focuse a wll requre sgfcat work before they ca be mplemete a actual arcraft project. Iustry Represetatve. The ustry ca beeft from ths etre report but shoul realze that secto 4 a appex B are at a research stage. The proofs wll requre verfcato by a qualfe epeet etty before they ca be mplemete a actual avato project. It s also lkely that the ustry woul ee to evelop tools to help mplemet the algorthms of ths report to a usable format. Researchers. As metoe before a scusse throughout ths report, ths research effort s really the begg of what ees to be oe before mplemetg the algorthms. Researchers wll lkely beeft most from secto 4 a appex B, a wll lkely wat to bul upo these atoal work. Secto 5 proves specfc formato about where the future research ees to go. -6

15 . SURVEY OF INDUSTRY APPROACHES. A Real-Tme Scheulg Aalyss (RTSA) Questoare was set out to ustry represetatves who are volve evelopg software for real-tme systems. The questoare s show appex A. Fftee questoares were reture a are tabulate appex A. Of the 5 respoets, the majorty () of them work for avocs or ege cotrol evelopers or arcraft or ege maufacturers. The majorty ether verfy/test real-tme scheulg performace, perform RTSA o avato system projects, or evelop real-tme operatg systems (RTOS) that support RTSA. O the whole, the respoets ha the approprate backgrou to aswer ths questoare. The resposes a questos are summarze below. Questo A.. aske, What type of evets are typcally use to trgger tme-crtcal fuctos of your real-tme system (e.g., terrupts, ata message queres, ata refresh rates, (plot) user put, chage of state, certa cotos, splay refresh rates, etc.)? Iterrupts were metoe by te respoets as the ma evets that are typcally use to trgger the tme-crtcal fuctos of ther real-tme systems. Ths fts well wth the DM algorthm, whch s essetally terruptrve. Questo A.. aske, What are typcal performace requremets that your real-tme system must meet? The majorty of the respoets metoe that crtcal tasks must meet har realtme eale costrats. The respose tmes metoe are from a few mllsecos to hures of mllsecos. Ths justfes the stuy of scheulg aalyss of har real-tme tasks, whch s the ma topc ths project. Questo A.. aske, Where are your performace requremets for tme-crtcal fuctos typcally efe (e.g., system requremets or terface cotrol ocumets, software requremets ocumet)? System requremets, terface requremets, a software requremets are the most popular resposes. It appears that performace requremets for tmecrtcal fuctos are typcally efe those ocumets. Questo A..4 aske, How o you stgush tme-crtcal fuctos from other fuctos your applcato? The majorty of the respoets aswer that tme-crtcal fuctos are explctly state the requremet. Questo A..5 aske, Do your tme-crtcal fuctos have epeeces o harware or share harware evces (cetral processg ut, memory, ata buses, I/O ports, queues, etc.) wth other software fuctos of your applcato or other applcatos reset the system? If yes, please expla. The aswers were mxe. Some say that there are o epeeces, whle others say that there are. The results are coclusve. Questo A..6 aske, What are some mechasms that your applcato (software a harware) uses to esure that tme-crtcal trggers get hale at the approprate prorty a ga the relevat resources to esure that your performace requremets are acheve? Prorty levels assge to terrupts were metoe by several people as the mechasm use to -

16 scheule tme-crtcal tasks. Some people metoe that the computers they use have oly oe prorty level, such as the PowerPC. Ths fts well wth the moel that the system has a lmte umber (m) of prorty levels (as propose for ths research project). I ths case, m. Questo A..7 aske, What type of revews, aalyses a testg oes your team use to esure that tme-crtcal fuctos wll satsfy ther performace requremets, especally worst-case coto scearos? The majorty of respoets metoe that they ca obta worst-case executo tme by aalyzg the coe. As for valato, most respoets use emulator or some a hoc approach to test. Ths s rsky because a emulator ca oly show that t wll work most of the tme. It oes ot show that t wll work all of the tme. There s a efte ee for a formal valato proceure whch gves a guaratee that t wll work all the tme. Questo A.. aske, What approaches to message passg have your projects utlze? The aswers are so fferet that t was ffcult to raw ay coclusos. It seems that message passg mechasm s a fucto of the harware a operatg systems use by the orgazato. Ths explas the verse aswers. Questo A.. aske, Do your messages commucate wth each other? If yes, please expla how. The majorty aswere that messages o ot commucate wth each other. Questo A.4. aske, What type of processors have you use for your systems? The majorty of respoets curretly use Itel processors; however, PowerPC seems to be gag mometum. A small umber of respoets use Motorola or TI chps. By far, the largest umber use Itel famly of processors. Questo A.4. aske, Have you fou ay peculartes wth ay of the processors that affect the real-tme scheulg aalyss? If yes, please expla the peculartes a how they were aresse. The aswers to ths questo vare sgfcatly. Some pote out that the lack of multple terrupt prortes PowerPC makes t ffcult to scheule real-tme tasks. Ths cofrms the hypothess of ths research effort that more prorty levels make scheulg easer. Some metoe that the cache memory makes t ffcult to aalyze the worst-case rug tme, sce the executo tme epes o the ht rato of the cache. Some metoe that the ppele processor also makes t ffcult to estmate the worst-case rug tme. Questo A.4. aske, Do your systems use a sgle processor or multple processors? If multple processors, how s the system fuctoalty strbute a hale across processors? Ne respoets sa that they use a sgle processor whle sx respoets sa that they use multple processors. Oe respoet metoe that they use both sgle a multple processors. It seems that they are about evely ve, wth the sgle processor havg a slght ege. Questo A.5. aske, What scheulg algorthms/approaches have you use to scheule your system tasks at ru tme? Please match the algorthm (e.g., preemptve prorty, rou rob, etc.) wth the system type (e.g., splay, commucato, avgato, etc.). The majorty respoe that they use pre-emptve prorty scheulg algorthm, of whch the DM algorthm s a member. Oe respoet metoe that they use Rate Mootoc (RM) Aalyss. -

17 Questo A.5. aske, If you use prorty scheulg, how may prortes levels were assge? How was prorty verso avoe? How the umber of prorty levels compare to the umber of processes? The majorty of the respoets sa that they use to 0 levels. Oe ca coclue that the umber of prorty levels s relatvely small, compare to the umber of real-tme tasks the system. Questo A.5. aske, What k of scheulg problems have you ecoutere multtaskg systems a how were they aresse? A far umber of respoets ot commet o ths questo. Therefore, t was ot possble to raw ay val coclusos. Questo A.5.4 aske, Have you use real-tme operatg systems to support your scheule guaratees? If yes, what k of operatg systems have you use a what k of scheulg challeges have you ecoutere? Most respoets sa that they ot use real-tme operatg systems to support ther scheule guaratees. For the few who sa they, they use -house propretary systems. It seems that there s a learg curve here. If the tools are mae avalable to them free of charge, they may fact use these tools the future. Questo A.5.5 aske, Do you verfy what ata gets umpe, ue to prorty settgs a fuctos gettg preempte? If yes, how oes t affect your system? Most respoets reple No or N/A. Therefore, there was suffcet ata to raw ay coclusos. Questo A.5.6 aske, Do you use tools to assst the real-tme scheulg aalyss? If yes, what k of tools? How are the outputs of these tools verfe? Most respoets reple No or that they use emulators a smulators. Ths ca create problems sce these are ot rgorous a formal aalyses. Questo A.5.7 aske, What tres commercal avato systems o you thk wll challege the curret scheulg approaches (.e., may lea to the ee for ew scheulg algorthms)? Some sa that multple threa real-tme eale scheulg aalyss wll be the future tres that challege the curret scheulg approaches. Some sa that the esre to reuse, the esre to hert cofece from reuse, a the esre to use oevelopmetal tems wll be the major challeges to the curret scheulg approaches. These commets pot to the mportace of a theory of scheulg o multple processors. Questo A.6. aske, After system evelopmet, o you verfy that eales are met a scheulg aalyss assumptos are correct? If yes, please expla how. The majorty of the respoets sa that they verfe that eales are met a scheulg aalyss assumptos are correct after system evelopmet. Questo A.6. aske, I what areas of tmg verfcato or valato have you ecoutere problems a how were they aresse? The aswers were so verse that t was ffcult to raw ay coclusos. It seems that the problems ecoutere s hghly epeet o the specfc problems a the harware or software use the compay. Questo A.7. aske, Does your testg allow for faults? If yes, please expla. Most respoets sa that ther testg allow for faults. Ths s mostly hale by jectg faults to -

18 the system a checkg to see how the system respos to the faults. Resposes cate that o worst-case aalyss s oe;.e., t s mostly oe a a hoc maer. Questo A.8. aske, I your opo, what are the major ssues regarg RTSA a ts verfcato? Resposes vare sgfcatly a are summarze below. Cofrmato of tmg ssues uer all foreseeable crcumstaces s a major ssue regarg RTSA a ts verfcato. Testg s ffcult whe mofcatos a/or chages are mae. The tools are very expesve a ot always avalable. The aalyss tes to be tutve a lacks formal aalyss. The operatg systems are so geeral that they are of lttle use ealg wth real-tme systems. From the resposes of the questoare, the followg coclusos were raw: There s a ee for scheulg aalyss a verfcato the avocs ustry. The curret practce s by a hoc methos. Tools are selom use ether because they are expesve a ot avalable, or the operatg systems are for geeral purpose a ot usable for real-tme systems. The tre s towars multprocessor systems. Software evelopers o test for fault tolerace, but the ma metho use s by meas of fault jecto whch s rather a hoc. It was coclue that evelopg efe approaches a algorthms for scheulg, eale verfcato, a fault tolerace wll sgfcatly help the avocs ustry. Furthermore, these theores shoul be mplemete to a software tool sute that ca be mae avalable to ayoe who esres to use t. As more a more people use these tools (whch may ee to be qualfe), future systems wll be less error-proe a easy to mata a mofy. -4

19 . TUTORIAL ON DIFFERENT SCHEDULING APPROACHES. Whether a set of real-tme tasks ca meet all ther eales epes o the characterstcs of the tasks (e.g., peros a executo tmes) a the scheulg algorthms use. Scheulg algorthms ca be classfe as pre-emptve a o-pre-emptve. I o-pre-emptve scheulg, a task oce starte must be execute to completo wthout ay terruptos. By cotrast, pre-emptve scheulg permts suspeso of a task before t completes, to allow for executo of aother more crtcal task. The suspee task ca resume executo later o from the pot of suspeso. Whle pre-emptve scheulg curs more system overhea (e.g., cotext swtchg tme ue to pre-emptos) tha o-pre-emptve scheulg, t has the avatage that processor utlzato (the percetage tme that the processor s executg tasks) s sgfcatly hgher tha o-pre-emptve scheulg. For ths reaso, most of the scheulg algorthms presete the lterature are pre-emptve scheulg algorthms. There are three major approaches esgg pre-emptve scheulg algorthms for real-tme tasks: Clock-Drve, Processor-Sharg, a Prorty-Drve. Each approach s escrbe below. The Clock-Drve approach s the olest metho use to scheule real-tme tasks. I ths metho, a scheule s hacrafte a store memory before the system s put operato. At ru tme, tasks are scheule accorg to the scheulg table. After the scheuler spatches a task, t wll set the harware tmer to geerate a terrupt at the ext task swtchg tme. The scheuler wll the go to sleep utl the tmer expres. Ths process s repeate throughout the whole operato. The Clock-Drve approach has several savatages that reer t uesrable to use: () t requres a far amout of memory to store the scheulg table; () a slght chage task parameters (e.g., executo tme a pero) requres a complete chage of the scheulg table, whch ca be very tme-cosumg, a () ths approach s ot aaptve to ay chage at ru tme. For example, f a system fault occurs or a task rus for less (or more) tme tha precte, t s ot clear how the scheulg ecsos ca be aapte to respo to the chage. The Processor-Sharg approach s to assg a fracto of a processor to each task, epeg o the utlzato factor (executo tme ve by pero) of the task. Sce a processor caot be use by more tha oe task at the same tme, the processor sharg s approxmate by vg a tme terval to smaller tme slces a gvg each task a amout proportoal to the fracto of processor assge to the task. For example, f a task s assge 0.5 of a processor, the the task woul receve 5 percet of the tme slces the tme terval. The tme slce has to be mae very small to obta a close approxmato of processor sharg. But whe the tme slce s very small, a sgfcat amout of tme wll be spet cotext swtchg. Ths s a major rawback of the Processor-Sharg approach. I the Prorty-Drve approach, each task s assge a prorty. At ru tme, the reay task that has the hghest prorty wll receve the processor for executo. Prortes ca be assge at ru tme (Dyamc-Prorty) or fxe at the begg before the operato starts (Fxe-Prorty). Fxe-Prorty scheulg algorthms cur far less system overhea (cotext swtchg tme) -

20 tha Dyamc-Prorty scheulg algorthms, sce the scheuler oes ot ee to eterme the prorty of a task at ru tme. Furthermore, Fxe-Prorty scheulg algorthms ca be mplemete at the harware level by attachg the prorty of a task to the harware terrupt level. O the other ha, processor utlzato uer Fxe-Prorty scheulg algorthms s usually ot as hgh as Dyamc-Prorty scheulg algorthms. It s kow that Fxe-Prorty scheulg algorthms may yel a processor utlzato as low as 70 percet, whle Dyamc- Prorty scheulg algorthms may yel a processor utlzato as hgh as 00 percet. Oe of the most well-kow Dyamc-Prorty scheulg algorthm s the Earlest-Deale- Frst (EDF) algorthm, whch assgs the hghest prorty to the task whose eale s closest to the curret tme. It s kow that EDF s optmal for oe processor [ a ], the sese that ay set of tasks that ca be feasbly scheule by ay Dyamc-Prorty scheulg algorthms ca also be feasbly scheule by EDF. However, EDF s ot optmal for two or more processors [4]. At the preset tme, o scheulg algorthm s kow to be optmal for two or more processors. The two most well kow Fxe-Prorty scheulg algorthms are the Rate-Mootoc (RM) a Deale-Mootoc (DM) algorthms [ a 5]. RM assgs the hghest prorty to the task wth the smallest pero (or equvaletly, the hghest request rate), whle DM assgs the hghest prorty to the task wth the smallest relatve eale. It shoul be ote that DM a RM are etcal f the relatve eale of each task s etcal to ts pero. Leug a Whtehea [5] have show that DM s optmal for oe processor, the sese that ay set of tasks that ca be feasbly scheule by ay Fxe-Prorty scheulg algorthms ca also be feasbly scheule by DM. Lu a Layla [] have show that RM s optmal whe the relatve eale of each task coces wth ts pero; t fals to be optmal f the eale of some task s ot etcal to ts pero. Both DM a RM fal to be optmal for two or more processors [6]. At the preset tme, o scheulg algorthm s kow to be optmal for two or more processors. The followg ocumets are specfcally recommee to further escrbe the scheulg approaches, a also see refereces -6. S. Nataraja, e., (995), Imprecse a Approxmate Computato, Kluwer, Bosto. A.M. Va Tlborg a G.M. Koob, es., Fouatos of Real-Tme Computg: Scheulg a Resources Maagemet, Kluwer, Bosto. -

21 4. DESCRIPTION OF SCHEDULING MODEL. The research topcs ths project were plae to be () prorty assgmet, () multprocessor scheulg, () fault tolerat ssue, a (4) I/O actvtes. However, because of tme lmtatos, oly the frst two topcs were stue. I ths report, the scheulg moel for the frst two topcs s efe. A peroc, real-tme task, T, s characterze by a quaruple (s, e,, a p ), where s s the tal request tme, e s the executo tme, s the relatve eale, a p s the pero. I ths characterzato, T makes a tal request at tme s, a thereafter at tmes s + kp, k,,... The k th request requres e uts of executo tme a t must be complete o later tha the eale s + (k-)p +. A real-tme task system cossts of peroc, real-tme tasks, a s eote by TS ({ T }, { s }, { e }, { }, { p }). A scheule S for a real-tme task system TS s sa to be val f the eale of each request of each task s met. Sce the scheule s fte, checkg f the scheule s val s a o-trval problem. TS s sa to be feasble f there s a val scheule for t. TS s sa to be scheulable by a partcular scheulg algorthm f the scheulg algorthm prouces a val scheule for t. A scheulg algorthm s sa to be optmal f every feasble task system s scheulable by the scheulg algorthm. Wth respect to the above moel, there are several mportat questos whose aswers are essetal valatg the temporal correctess of a real-tme task system. Frst, how oes oe eterme f a real-tme task system s feasble? Seco, how oes oe eterme f a real-tme task system s scheulable by a partcular scheulg algorthm? Thr, what are the optmal scheulg algorthms? By efto, a real-tme task system s feasble f a oly f t s scheulable by a optmal scheulg algorthm. Thus, these three questos are terrelate. There are several mportat assumptos assocate wth ths moel. Frst, e s assume to be the maxmum executo tme requre by T. At ru tme, t s assume that T ever requres more tha e uts of executo tme at each request, although t coul use less tme. Seco, t s assume that cotext swtchg tme s eglgble. If ths s ot a val assumpto, e must be ajuste to accout for the tme loss ue to cotext swtchg. Thr, the mmum tme lapse betwee two cosecutve requests of T s p. At ru tme, the tme lapse betwee two cosecutve requests s at least p ; t coul be more tha p, but ot less. Fourth, the relatve eale of each request s. At ru tme, the relatve eale of each request ca be loger tha but ot shorter. These assumptos must be strctly ahere to orer for the theory to work. 4-/4-

22 5. RESULTS AND FUTURE WORK. 5. RESULTS. Ths project cosste of two major jobs. The frst job was to explore a report the ustry approaches to scheulg real-tme tasks a the tools they use the verfcato of temporal correctess. A questoare was evelope a set to a umber of ustry represetatves who were volve evelopg software for real-tme systems. From the resposes of the questoare, the followg coclusos ca be raw: There s a ee for scheulg aalyss a verfcato the avocs ustry. The curret practce s by a hoc methos. Tools are selom use ether because they are expesve a ot avalable, or the operatg systems are for geeral purpose a ot usable for real-tme systems. The tre s towars multprocessor systems. Software evelopers o test for fault tolerace, but the ma metho use s by meas of fault jecto whch s rather a hoc. It was coclue that evelopg efe approaches a algorthms for scheulg, eale verfcato, a fault tolerace wll sgfcatly help the avocs ustry. Furthermore, these theores shoul be mplemete to a software tool sute that ca be mae avalable to ayoe who esres to use t. As more a more people use these tools (whch may ee to be qualfe), future systems wll be less error-proe a easy to mata a mofy. The seco job cosste of evelopg scheulg algorthms a temporal verfcato tools for a moel of peroc, real-tme tasks. The report bega wth a scusso of the three approaches that have bee use to scheule a real-tme task system: () Clock-Drve, () Processor-Sharg, a () Prorty-Drve. It was reasoe that the Prorty-Drve approach s far superor to the Clock-Drve a Processor- Sharg approaches. The report the revewe the lterature o Prorty-Drve scheulg algorthms, whch ca be ve to two categores: Dyamc-Prorty a Fxe-Prorty. Whle Dyamc-Prorty scheulg algorthms are more effectve tha Fxe-Prorty scheulg algorthms, they are rarely use practce because of the overhea volve. Therefore, the report cocetrate o Fxe-Prorty scheulg algorthms. Prorty-Drve scheulg s probably the most approprate approach scheulg peroc, real-tme tasks. I ths project, Fxe-Prorty scheulg algorthms for computg systems wth lmte prorty levels were stue. A proceure to test f a gve prorty assgmet s val (.e., all eales are met) was evelope. Furthermore, a optmal prorty assgmet algorthm, DM-LPL, for oe processor was gve. The algorthm was mplemete usg the C laguage a s show appex C. 5-

23 For multprocessors, the problem of fg the mmum umber of processors wth m prorty levels to scheule a set of tasks was show to be NP-har. Two heurstcs were prove, FF a FFDU, for ths problem. The specal elvery case where each task s executo tme s oe ut was also cosere. Uer ths moel, a attempt was mae to evelop a utlzato threshol, U(), below whch a set of tasks s always scheulable. For the ulmte prorty levels, t was / cojecture that U ( ) +, whch s better tha the bou of ( ) gve by Lu a Layla []. Ths cojecture was prove for two specal cases. 5. FUTURE WORK. There are four areas where future work shoul be performe: Develop a full proof of the cojecture starte ths report. The specal case whe a are arbtrary has bee prove. It remas to be show that the cojecture s val whe, 4,, - are also arbtrary. Perform a epeet verfcato of the algorthms the report, as well as a verfcato of the full proof. The results obtae ths report shoul be revewe by a epeet expert scheulg theory. Perform a stuy of fault-tolerat ssues, whch are cocere wth scheulablty aalyss whe there are tme losses ue to traset harware/software falures. A major ssue ths area s to characterze the worst-case scearo ue to tme losses. Perform a stuy of cetral processg ut (CPU) scheulg couple wth put/output (I/O) actvtes. The ma ssue ths area s to couple pre-emptve scheulg (CPU scheulg) wth o-pre-emptve scheulg (I/O actvtes). 5-

24 6. REFERENCES.. Lu, J.W.S., Real-Tme Systems, Pretce Hall, New Jersey, Labetoulle, J., Some Theorems o Real-Tme Scheulg, Computer Archtecture a Networks, E. Gelebe a R. Mahl, es., North-Holla, Amsteram, Lu, C.L., a Layla, J.W., Scheulg Algorthms for Multprogrammg a Har Real-Tme Evromet, J. of ACM, Vol. 0, 97, pp Leug, J.Y-T., A New Algorthm for Scheulg Peroc, Real-Tme Tasks, Algorthmca, Vol. 4, 989, pp Leug, J.Y-T. a Whtehea, J., O the Complexty of Fxe-Prorty Scheulg of Peroc, Real-Tme Tasks, Performace Evaluato, Vol., 98, pp Dhall, S.K. a Lu, C.L., O a Real-Tme Scheulg Problem, Operatos Research, Vol. 6, 978, pp /6-

25 APPENDIX A INDUSTRY SURVEY AND RESPONSES The followg s the Real-Tme Scheulg Aalyss (RTSA) Questoare Aalyss A. BACKGROUND QUESTION. A.. BACKGROUND QUESTION #. What k of orgazato o you work for? Avocs or ege cotrol eveloper Arcraft or ege maufacturer Commucatos, avgato, or survellace system eveloper for ar traffc maagemet Software tool eveloper Thr party software eveloper (e.g., operatg system or lbrary) Cosultat Feeral Avato Amstrato Other govermet agecy (please specfy): Other, (please specfy): Of the 5 surveys respog, the breakow was as follows: 8 Avocs or ege cotrol eveloper 4 Arcraft or ege maufacturers Software tool eveloper Cosultat Other, (please specfy): Software Verfcato Co. A.. BACKGROUND QUESTION #. What s your role relevat to RTSA? (Check all that apply) I perform RTSA o avato system projects I evelop real-tme operatg systems (RTOS) that support RTSA I use tools to perform RTSA I verfy/test real-tme scheulg performace I am a FAA egeer who approves complace to DO-78B I am a Desgate Egeerg Represetatve (DER) who approves complace to DO-78B Other, (please specfy): Of the 6 surveys respog, wth may vuals resposble for more tha oe role, the breakow was as follows:. 6 I perform RTSA o avato system projects I use tools to perform RTSA 9 I verfy/test real-tme scheulg performace A-

26 5 I am a Desgate Egeerg Represetatve (DER) who approves complace To DO-78B A. REAL-TIME SYSTEM DEVELOPMENT. A.. REAL-TIME SYSTEM DEVELOPMENT QUESTION A. What type of evets are typcally use to trgger tme-crtcal fuctos of your real-tme systems (e.g., terrupts, ata message queres, ata refresh rates, (plot) user put, chage of state, certa cotos, splay refresh rates, etc.)? Of the 5 surveys respog, ther aswers were as follows:. Combato of fxe scheule (.e., clock tme) plus certa cotos (for example, tracke sgal weakess swtches moe from trackg to acqusto). Data puts, ata refresh rates, ata queres, splay refresh rates. I the OS all of the above are supporte 4. Elapse tme, message upate ue 5. Iterrupts 6. Iterrupts, RTOS messages, RTOS semaphores or other smlar mechasms. 7. Fxe tme terval scheulg terrupts oly 8. RTSA s terrupt rve by tme-to-go couters. We use a executve route to establsh prortes. Our Ru Tme Executve was wrtte I-house. 9. Iterrupts a chage of state 0. Table rve scheuler base upo statcally bult table by a exteral, qualfe tool. The table cotas formato regarg I/O tmg (whe to se ata a whe to rea ata), as well as process scheulg formato. Tme-base terrupts. Harware terrupts, sesor/harware puts, arframe computer puts, etc. Typcally we use terrupts. I some staces t s a exteral put, such as comg ata, from aother evce that causes a terrupt to be geerate. 4. Icomg ata messages (terrupt or polle), tmer exprato, user put (Key press, toggle swtch, etc), sesor state chage (terrupts or polle). A-

27 5. Iterrupts are use for the tato of the real-tme processg. Aalyss of Questo A Real-Tme System Developmet Iterrupts were metoe by 0 respoets as the ma evets that are typcally use to trgger tme-crtcal fuctos of ther real-tme systems. A.. REAL-TIME SYSTEM DEVELOPMENT QUESTION B. What are typcal performace requremets that your real-tme system must meet? Of the 5 surveys respog, ther aswers were as follows:. There are harware a software eales. For the software, three epeet bts ee to be assemble a elvere every 0.6 mcrosecos.. Some requremets ee to meet mllseco respose tmes/toleraces; others have multple seco resposes wth mllseco toleraces.. The clock tcks at khz. All scheulg operatos are rve by the tckg clock. User frame tmes are typcally Hz 4. Servo loop closure wthout overshoot or sgfcat lag. Servce ata bus a access formato from the ata bus accorg to scheule. Determe fault state base o successve out of rage put for tme frame. 5. Crtcal task must meet ther real-tme ea-le scheule actvtes. 6. Sample rates excess of 50 Hhz. For DSP applcatos. Multple 8 KHz. Sample rate auo chaels. 7. Level A assurace that all fuctos are complete before the ext terrupt. 8. Max m lmts o put/output evets, ga phase marg o cotrol loops, terato rates a trasport elay tmes o selecte fuctos 9. Throughput margs of 50% were requre by govermet cotract. There were smlar harware reserve marg requremets o memory a I/O 0. There s a KHz terrupt a several I/O evces terrupt. All arcraft I/O clues trasmsso a jtter requremets. Typcal ata rates rage from Hz to 80 Hz.. 80 Hz upate rate A-

28 . The harware/computer shall operate wthout averse affect o the ege or arcraft, such as lost of ege thrust, averse crease or ecrease of ege thrust or cause flght shut ow. 4. As a mltary applcato our performace requremets are maly base off of our 55 bus rates. Thus we have certa message rates that our system must mata for example rates of 50Hz or 5Hz etc. I ato we may terface to a exteral evce that requres ata to be trasferre to t wth a few mllsecos after the ata has chage. 5. Dgtal sgal processg comg ata rate (rao IF): 00 ksam Rao push-to-talk to RF o: < 50 ms; user put to splay upate: < 00 m Sesor chage to splay upate: < 00 ms 6. Performace requremets that are most crtcal clue the tme that the system etects a ege threshol exceeace (such as overspee; uer spee; excessve ege temperature; etc.) utl the fuel flow s termate to shut the ege ow. Ths s usually o the orer of a few mllsecos. A.. REAL-TIME SYSTEM DEVELOPMENT QUESTION C. Where are your performace requremets for tme-crtcal fuctos typcally efe? (e.g., system requremets or terface cotrol ocumets, software requremets ocumet) Of the 5 surveys respog, ther aswers were as follows:. System requremets a terface requremets. Tme crtcal requremets may show up our system requremets, software requremets or software esg (low level requremets). System requremets 4. Software Requremets Documet 5. Software requremets Documets 6. System requremets, terface cotrol ocumets, software requremets ocumet 7. All the above 8. Ours were specfe by govermet cotract- typcally some verso of the TADSTAND 9. Through system esg aalyss or from smlarty to prevous systems, we ca usually stgush tme-crtcal fuctos. A-4

29 0. System requremets a terface cotrol ocumets. O occaso, lower level performace requremets may appear the software requremets ocumet. Harware ocumets also cota performace requremets as they perta to the support of the I/O a O/S.. Systems requremets. Iterface cotrol ocumet, computer harware operato requremets a software requremet ocumets. Crtcal tme wth max lateces woul be efe our System a Sub-System Specfcato (.e., our system requremets) ocumet. 55 message rates woul be efe our Iterface Cotrol Documets a some tmg woul be ocumete our Software Requremets. 4. For ter-process commucatos tmg. System requremets for put to output tmgs a user terface performace, software requremet 5. These are typcally efe the software requremets ocumet- wth some beg efe the system requremets ocumet. A..4 REAL-TIME SYSTEM DEVELOPMENT QUESTION D. How o you stgush tme-crtcal fuctos from other fuctos your applcato? Of the 5 surveys respog, ther aswers were as follows:. All requremets must be met evetually. Some eales are obvously easy to meet; others wll requre some effort a/or have some rsk assocate wth them. Perhaps t woul be helpful to mage all requremets havg a eale to be flle, a f a recpet of a tem (ata, for example) really oes t care by whe somethg s accomplshe, they shoul say so. NASA/JPL sometmes uses a tme costrat etwork, where tmg s state wth respect to other evets, rather tha wth respect to tme eales. For example, some aspects of a flght mght ot be requre to be elvere utl the flght arrves at the gate.. By the tme elemet of the requremet (.e., must respo wth 00 ms, or must o x after y +/- z secos.. Tme crtcal formato s embee the requremet 4. Error reportg has a uque etfer for each malfucto 5. By creatg software or system requremets specfyg the tmg 6. Baly A-5

30 7. I beleve your questo refers to arbtrato oe by the Executve. Levels of prorty were establshe a the Executve route carre them out. Prortes were establshe urg the software esg phase followg aalyss of the requremets. 8. Through system esg aalyss or from smlarty to prevous systems, we ca usually stgush tme-crtcal fuctos. 9. From a O/S perspectve, all fuctos are tme crtcal. That s, they have specfc tmg costrats place upo them that must be hoore. The etermato of who s more tme crtcal tha others s a system archtecture exercse. 0. We o t. We o t. Tme-crtcal fuctos woul be ocumete as such. Lste software esg escrpto, otato fucto commet heaer 4. All fuctos the applcato are tme-rve a are treate as tme-crtcal- wth the excepto of the low-level trasmsso a recepto of seral ata. Ths seral commucato s terrupt-rve a wll be servce whe ata s avalable from the host computer to whch the electroc cotrol ut s commucatg or whe ew ata ees to be set to the host computer. The processg a respog to complete messages the system s also tme-rve (a etermato s mae for whether a complete message has bee receve that ees to be acte o). Cosequetly, stgushg from the tme-crtcal fuctos a those that are ot tme-crtcal s oe by ecatg separate memory resources to the seral. A..5 REAL-TIME SYSTEM DEVELOPMENT QUESTION E. Do your tme-crtcal fuctos have epeeces o harware or share harware evces (cetral processg ut, memory, ata buses, I/O ports, queues, etc.) wth other software fuctos of your applcato or other applcatos reset the system? If yes, please expla. Of the 5 surveys respog, ther aswers were as follows:. Multple software ettes ca be complete at oe tme; kowlege about whch oes are complete s use ecg whch ettes get resource use assge to them.. Sometmes, mght have mult-processor system that must meet ata bus tmg protocol. Mght be a harware evet that trggers the start of a tme crtcal fucto.. The applcato has a epeecy o the RTOS. The RTOS has a epeecy o the system a applcatos, e.g. cache usage, memory cofguratos (coheret or ocoheret etc). A-6

31 4. No 5. Yes. A system clock (couter/tmer) s use for all software relate tmg fuctos. The RTS ows the system clock. A harware watch og s also use, whch must be upate o a peroc bass 6. Yes, A/Ds a D/As for RF a auo samplg 7. No 8. It epes- harware lmtatos that were kow were part of the esg. These epeeces were usually ote the System a/or Software Requremets Specfcato. They were occasoally scovere urg the Requremets Aalyss phase- at great embarrassmet. 9. Yes. Geerally the terrupt servcg fuctos must share harware evces a memory wth other applcatos. 0. Typcally the aswer s yes, whe lookg at I/O harware evces. However, our harware esgers use queues, state maches a other methos to abstract as much tmg crtcal stuff away from the software, as possble.. No. Yes. There s memory loaer software that s actvate oly whe certa harware a software cotos are set.. Yes, we must receve ata from oe subsystem a wth a specfe amout of tme pass the formato oto aother subsystem some case usg the put ata to calculate the ata that s to be output. 4. Tme-crtcal fuctos typcally share CPU, memory a other resources wth other fuctos 5. Yes, the tme-crtcal fuctos o have harware epeeces o I/O evces. A..6 REAL-TIME SYSTEM DEVELOPMENT QUESTION F. What are some mechasms that your applcato (software a harware) uses to esure that tme-crtcal trggers get hale at the approprate prorty a ga the relevat resources to esure that your performace requremets are acheve? Of the 5 surveys respog, ther aswers were as follows: A-7

32 . A look-up table s prove (all at oce) wth put escrbg the tasks curretly complete. The cotet of the look-up table tells whch software task s to receve the processg resources ext.. We o t use RTOS, so we wrte software to acheve esre resposes (whe possble).. Iterrupt lockg s use, OS/kerel lockg, semaphores a prorty hertace, work queues for work eferrals, etc. 4. Hgh Iterrupt prorty, also tme crtcal coe has har vectors to terrupt servce routes that caot be mofe by a ru tme operato. We are cocere about software mateace actvtes ag ew prorty sets wth hgher prortes for less mportat processes. We ocumet heavly the prorty ratoale. 5. Task prortes a terrupt prortes. 6. They are gve the hghest prorty level wth the RTOS. Or else they are hale outse of the RTOS wth epeet terrupts 7. By restrctg to the oe terrupt, a sstg o completo of tme crtcal tasks. 8. Careful Itegrato testg a thorough V&V was our preferre path. Demostrato, aalyss, whatever was approprate. 9. Iterrupt a task prortes are assge approprate values 0. All resources are efe statcally at bul tme by a qualfe tool. Processes are assge specfc resources. Ths esures the resources wll be reay whe eee, a permts the O/S to etect ay resource usage volatos (.e., parttog volatos).. Check o process completo a harware watchog. Our software uses watch og motor a certa CPU terrupt halers.. As we use a PowerPC wth oly oe exteral terrupt the tme crtcal ssues have to be esge to the system. There s NO way to prortze terrupts our system so the top level esg must take ths ssue to accout. Software must esure all terrupt halers are as quck as possble. 4. Harware tmers, prortze harware a software terrupts, pre-emptable a prortze tasks. No (or very low terato cout) loops rou-rob systems. 5. There are o prorty mechasms a terrupt occurs at a pre-efe terval a ths tates real-tme processg. The software verfes that the processg from the prevous terval has complete. If t has ot, the frst tme the aomaly occurs, t s smply A-8

33 recore a processg for the ext terval s tate rectly after the completo of the tasks from the prevous terval. If two such overrus occur a row, the the applcato smply executes a fte loop, whch wll cause the ege to shut ow. I ato, there s a harware watchog screte output that ees to get strobe each terval. If the watchog s NOT strobe urg each terval, the the harware wll cause the ege to shut ow our software. A..7 REAL-TIME SYSTEM DEVELOPMENT QUESTION G. What type of revews, aalyses a testg oes your team use to esure that tme-crtcal fuctos wll satsfy ther performace requremets, especally worst-case coto scearos? Of the 5 surveys respog, ther aswers were as follows:. We kow the structos a the structo executo tmes.. Worst case tmg aalyss, overloae resource tests. Nee a two ay essay to aswer ths. I the past we aalyze the paths of the RTOS a eterme what the relatoshp to tme was wth puts. Thgs have got a lot trcker sce the. We are curretly og ths as a stuy for a parttoe RTOS that we are certfyg. 4. Revews are requremets base revews. Aalyss clues calculato of worst case safety marg for tmg, for memory usage, a for stack usage. Verfcato tests use emulators that track the percetage of tme spet fferet areas of the software, so we ca get a goo estmate of how much tmg marg remas the testg stage. 5. Ut test are ru o all relate software moules. Itegrato tests are ru to verfy fuctoalty the system. Bult--tests check fuctoalty each tme the ut s powere up. 6. Requremets, Desg a coe revews are coucte. A worst-case throughput aalyss s coucte. 7. Two aspects oe at the requremet level, oe at the mplemetato level. 8. Requremets are peer revewe for wth our prouct ssues a revewe wth ata source / sk supplers for exteral stuff. Coe moules are all path teste a executo tme motore. Each moule s assge ts logest ru tme. The total of these tmes for all moules that execute betwee tmer terrupts are summe a must be less tha the mmum terrupt tme. 9. For terrupts, eurace testg wth approprate motors allows us to eterme the effectve worst-case tmg as well as the average cotos. A-9

34 0. Automate test cases are evelope that exercse the lmts of the system. A rogue partto s use to stress the parttog aspects of the O/S.. Tmg measuremets. All of them. Software requremets/esg revew, test reaess revew, coe/esg revew, ut tests, tegrato tests, software/harware system tests. The systems team uses MatrxX to output equatos. These equatos are the moele a smulate usg MatLab whch also oes boe plots. The software team utlzes a tool (WVew) to verfy that the tasks are beg scheule as requre a pre-empte as requre. 4. E to e testg of LRU puts a outputs. Test cases esge to assure full loag o ata puts 5. The worst-case terval tme s measure a recore. The fact that the system s so tme-etermstc combe wth the above halg of overru cotos assures that whle we are executg the ege cotrol software, we wll be meetg our tme-crtcal fuctos. A. MESSAGE PASSING. A.. MESSAGE PASSING QUESTION A. What approaches to message passg have your projects utlze? Of the 5 surveys respog, ther aswers were as follows:. Wrte to harware buffers that are swappe o a fxe tme.. Typcally we ve use ual port ram for multply processor systems, we o t use multthreae executves.. Message passg lbrary s prove the RTOS. Prorty a FIFO base. Semaphores of varous types are also use. (Bary, Coutg a MUTEX) 4. We pass messages oe of two ways. The preferre metho s to place the ata packet geeral purpose regsters, a call the servce route wth the kowlege that the servce route wll look for specfc ata specfc regsters. A alterate way s to pass a poter to the ata specfc to the message to the servce route. 5. Evets flags; reezvous; ata passe through share memory, a through share buffers owe by the RTS. A-0

35 6. RTOS base messages 7. Do t o t 8. Or was Ufamlar wth the term- o t uersta the questo. Our commucatos software was ether terally evelope coe to o ata halg wth the processor system ut formatte to oe of the popular protocols such as ARINC 49, RS, RS4 MIL STD 558, etc. 9. Share memory accesses usg semaphores, ouble bufferg, a terrupt blockg have all bee use 0. Peroc ter-process messages a malboxes. Noe. ARINC. Iformato passe teral o the same processor s ofte passe va message queues. Iformato passe to other processors s passe ual-port memory. 4. Share memory a semaphores, malboxes, queues 5. A ot va message passg mechasms. The major compoets the system commucate largely va memory terfaces A.. MESSAGE PASSING QUESTION B. Do your messages commucate wth each other? If yes, please expla how: Of the 5 surveys respog, ther aswers were as follows:. Processg ettes commucate wth each other accorg to the followg paragm Fast but umb vs. smart but slow. If the messages are smple, the seer commucates rectly wth the recpet. If the message s ot smple, t goes to a processg ut wth more processg capablty, whch ca fgure out what s to be oe, a form all tereste other processg. No. Tasks commucate wth terrupt routes a other tasks. Messages are just the carrers. 4. No 5. No A-

36 6. No. They oly pass formato to other tasks. 7. Stll ot clear how to aswer- we have seral gtal commucatos a tratoal aalog /screte I/O a follow a popular ustry staar. Hashakg, party etc. are use as requre. 8. No 9. No 0. ARINC 65 protocol. No, our messages teral to our system te to go oe recto. Tasks a processes commucate wth each other usg messages, messages o t commucate wth Each other.. N/A A.4 PROCESSOR TYPES. A.4. PROCESSOR TYPES QUESTION A. What types of processors have you use for your systems? Of the 5 surveys respog, ther aswers were as follows:. Custom CPUs for the fast but umb, a commercal CPUs for smart but slow.. 805, 68, TMS400, a we are eyeg power PC style processors for future proucts. PowerPC. / Itel X eght bt mcrocotrollers 5. Itel 486DX4, 486DX, 486Dx, 486SX, 8086, 680, 8085 DSPs, TMS0C67, Itel x86 class processors 8086, 8086ex, & 80486DX Curretly, home grow 7. TI9900 (I m a really OLD guy), Itel 80C86, 80C86EX, 68HC, Motorola 8000 a 80040(memory s hazy- probably wrog), TI 0C0 DSP (as I sa, I m a ol guy) 8. Itel 8086, Motorola 68HC, Motorola 68HC6, Motorola 68XX a PowerPC40 A-

37 9. Itel x86 famly (86, 486, Petum I, II, III) 0. Itel 80960, Motorola , 6800 Power PC. Curretly we are usg the PowerPC for our cotrollg processor. Prevously we use the Itel 960MC (8-bt), XAG49 (6-bt) 8086 (6-bt), 8086 (-bt), TI a Motorola DSP (6, 4, a -bt) 4. PowerPC 60e, PowerPC 555, 960, Itel x86 A.4. PROCESSOR TYPES QUESTION B. Have you fou ay peculartes wth ay of the processors that affect the real-tme scheulg aalyss? If yes, please expla the peculartes a how they were aresse: Of the 5 surveys respog, ther aswers were as follows:. Yes, o t etals are t realy avalable. Memory maagemet uts a cache memory make a fferece. We have fou that the ucertaty the umber of structo cycles to vector to a terrupt servce route has le us to utlze exteral harware clocks for tmg certa cases 4. Great care must be use whe usg o-maskable terrupts 5. No 6. Slghtly off topc a log tme ago a Texas Istrumets processor whch clame a asychroous reset capablty was show to have a sychroous wow whe t ot reset. We chage processors. 7. We were og the work urg a tme whe smulato tools t exst, a whe they - t was usually too rch for our pocketbook. The geeral approach was to let t fly for may tmes the expecte msso tme a fully exercse all moes of the applcato coe a pray we ha tmg margs by lookg at what was felt to be a worst case ata movg scearo. I m certa those tools have evelope a are more afforable toay. A-

38 8. Dfferet processors have fferet mechasms for assgg terrupt prortes a for terrupt maskg 9. Not that I am aware of 0. Cachg makes tmg measuremets usg bus aalyzers a ghtmare. Do t kow. The lmtato of the PowerPC (ot a pecularty), as I see t, s that t oly allows for oe exteral terrupt to the processor. As we rely o multple terrupts, our system harware has to esg exteral terrupt cotrols to hale the multple terrupts but t s mpossble to prortze the terrupts sce the PowerPC accepts oly oe exteral terrupt. The Itel 960MC hale multple terrupts that coul easly be prortze. The terrupt lmtato s aresse at esg attemptg to esure that the terrupt wll be space out a the the software works to esure the halers are as quck as possble.. Istructo ppeles. Avo software tmg loops 4. No A.4. PROCESSOR TYPES QUESTION C. Do your systems use a sgle processor or multple processors? If multple processors, how s the system fuctoalty strbute a hale across processors? Of the 5 surveys respog, ther aswers were as follows:. Multple. Mmze the amout of commucato that must flow betwee processors. Try to have a preomat recto for commucato: A wth respect to B s maly A ses to B.. Both sgle a multple processor proucts. We establsh fuctoalty resposbltes up frot a o t ajust real tme. Our curret work uses a sgle processor. May years ago a multprocessor system was use wth share memory. 4. Sgle 5. Multple. Fuctoalty s allocate at the System ocumet level. Iter-processor commucato s by ual port share memory or through a RS seral lk or a sychroous lk. 6. Typcally sgle processor. However, oe was a mult-processor system that commucate va Dual Port RAM. A ICD was create to escrbe ths commucato. A-4

39 7. Reuat sgle 8. Our applcatos were geerally sgle processor. We o some multple processor work, but t was usually wth some form of share ual port regsters or RAM. We felt teracto eee to be mmze to mtgate cotrol problems where oe ut mght jam aother oe. Uambguous sychrozato was occasoally a problem. 9. Sometmes multple processors are use. Usually the fuctoalty s strbute accorg to hgh level fucto a re-uses coseratos. 0. Multple processors, however, each s treate as a epeet processor. That s, they work as feerate LRUs tercoecte by a avocs bus.. Sgle. It s a ual chael operato wth each chael havg the same software. Each chael s executg urg operato wth oe chael beg cotrol a actve whle the other chael s the sta-by moe.. Our system cotas multple processors but each processor hales a separate fucto. Oe processor s the cotroller, aother processor hales splays, aother processor commucates to recorg evces, etc. Iformato betwee the processors s commucate through ual-port share memory. 4. Multple. Dsplay/keyboar, commucatos I/O, sesor I/O, hgh level rao fuctos, low level rao fuctos, DSP Seral ata lk s typcally use betwee processors. 5. The system s a sgle processor archtecture A.5 SCHEDULING. A.5. SCHEDULING QUESTION A. What scheulg algorthms/approaches have you use to scheule your system tasks at ru tme? Please match the algorthm (e.g., preemptve prorty, rou rob, etc.) wth the system type (e.g., splay, commucato, avgato, etc.) Scheulg algorthm/approach System type Of the 5 surveys respog, ther aswers were as follows:. Combato of scheule establshe avace wth varyg stuato-epeet prorty wth o pre-empto, (but also wth crtcal sectos). commucato A-5

40 . Do ot use real tme scheulers. Rou rob wth tme-outs Peroc tasks, wth messagg a semaphores Iterrupt rve Prorty Overheat etecto systems Temperature motorg system FADEC Cotrol systems 4. Preemptve prorty Dsplay Preemptve prorty Navgato - DME 5. Preemptve prorty Map Dsplay Preemptve prorty Cooperatve mult-taskg Smple sequecer Commucato Commucato/Dsplay Commucatos/DSP 6. Strct lst scheule Ege cotrol 7. Preemptve prorty Navgato 8. Preemptve prorty Autoomous cetral cotrol of a UAV 9. Preemptve prorty Navgato 0. Rate Mootoc Aalyss (RMA) All. Our systems utlze very smple terrupt-base scheulers for effcecy a safety reasos. There are o prortes process ru whe voke; completo s checke usg Boolea flags. A-6

41 . Foregrou/backgrou wth mor frame watch og motor Ege cotrol. Preemptve Prorty Scheulg System Cotroller (Commucato a Navgato) 4. Preemptve Commucatos (rao) Sgle Ma Loop Wth Iterrupts User-terface, avgato A.5. SCHEDULING QUESTION B. If you use prorty scheulg, how may prortes levels were assge? How was prorty verso avoe? How umber of prorty levels compare to umber of processes? Of the 5 surveys respog, ther aswers were as follows:. Each process ha ts ow prorty level, that s, we really use process etfcato rectly, rather tha a classfcato accorg to prorty. Furthermore, epeg upo the stuato at ha, fferet processes were chose for resource allocato. That s, oe stuato where tasks A, B a C were all reay to ru, process B woul be chose, but aother stuato, process C woul be chose. There was a look-up table whch was prove wth a aress costructe of the Booleas for each ruable task, true for each of the tasks that were reay to ru, a other stuato bts of terest. The cotet of the look-up table was the etfcato of the task that was to be ru that stuato. Each task ha at most a very small crtcal secto. O completo of each task, the stuato s etecte a supple as put to the look-up table. Each task s very short.. N/A. Temperature system was evet rve. Each task has a uque prorty. 5 task system 4. Three prorty levels, prorty verso was elmate wth usg prorty regsters talze a subsequetly left aloe. There are sx processes. 5. O oe partcular project: levels. Prorty verso was avoe by careful software esg. Some tasks share the same prorty umber; (rou rob effect); Total tasks: levels a we ever ecoutere prorty verso 7. I recall there were at least three levels, possbly as may as fve- our applcatos were ty- less tha 00K les of HOL coe. A-7

42 prorty levels are typcal. Prorty verso s avoe by lmtg the amout of resource sharg betwee wely sparate prorty tasks. 9. N/A 0. N/A. Te prorty levels are assge. Each rate group s assge ts ow prorty level. Message queues are ot elete a o-blockg queues are efe.. Four tasks at fferet prorty levels. Desg hgh prorty tasks o ot wat o low prorty tasks. N/A A.5. SCHEDULING QUESTION C. What k of scheulg problems have you ecoutere multtaskg systems a how were they aresse? Of the 5 surveys respog, ther aswers were as follows:. Oe (more structos eee a tme terval) was solve by creasg the clock spee o the processor. Oe (coul t fgure out whch task to ru ext examg successve status bts, because status bts chage so quckly) was solve by latchg status to a aress regster for a look-up table. Oe (prorty verso) was aresse by aggressvely prug the cotet of crtcal sectos.. N/A. Workg wth urato tme for peroc elays cause us to chage the mplemetato to work wth Absolute tme for elays (elay utl) Problem solve OS by cremetg tme a o-preemptable block. 4. N/A 5. Not eough bawth for the processor (ths s a commo problem). Rates of tasks are ajuste as well as shftg task work loas. 6. Noe 7. Crashes were pafully exame wth support from ebuggers a a logc aalyzer. Most troubles were trace to mproper mapulato of the memory regster stack rather tha some uocumete harware feature. A-8

43 8. I ve ecoutere ealy embrace problems whch were aresse by a re-esg of the affecte tasks. 9. Do t kow 0. Mor frames overru, whch causes a watch og terrupt thus cause a software reset. We have see task overrus. Whe ths occurs sometmes a coe problem s fou a fxe. If ee more s scheule the ca be complete the allocate tme all fuctos the task are evaluate to eterme whch fuctos ca be oe at a slower rate or ve up slghtly fferet.. No commet. N/A A.5.4 SCHEDULING QUESTION D. Have you use real-tme operatg systems to support your scheule guaratees? If yes, what k of operatg systems have you use a what k of scheulg challeges have you ecoutere? Real-tme operatg system type Scheulg challeges Of the 5 surveys respog, ther aswers were as follows:. NO. Home grow oly. Preemptve prorty More robust RTS meas slower RTS 4. Not really, we typcally use C laguage asserts to etect ther occurrece a the esg them out. 5. NA 6. I left the fel before usg commercal RTOS 7. Preemptve mult-taskg Iterrupt latecy, excessve overhea 8. I-house propretary, usg RMA. 9. Operatg system usg Aa or assembly Keepg from cause a mor frame overru A-9

44 0. VxWorks 5.4 Ma challege s to terface to our custom harware A.5.5 SCHEDULING QUESTION E. Do you verfy what ata gets umpe, ue to prorty settgs a fuctos gettg preempte? If yes, how oes t affect your system? Of the 5 surveys respog, ther aswers were as follows:. N/A. We have verfe cotext swtchg urg our work o certfcato of the VxWorks/Cert OS. Preempte fuctos because of prorty settgs are oly elaye. I our esgs, preempto geerally happes whe we are tryg to smultaeously trasmt o a ARINC 49 bus whle we are recevg ata. Ths meas fuctos that come up whle the trasmtter s o wll have to wat utl t s off to complete. We guaratee that ths wll happe less tha 00 msec. 4. Yes. If the problem s severe eough the the system s eclare val a fuctoalty s remove 5. No 6. N/A 7. Not sure I uersta where the questo s gog. 8. Yes. The mpact vares. I some cases the loss of ata s acceptable a other cases t s absolutely crtcal that o ata s lost. 9. Do t kow 0. N/A. No. Buffer szes a terrupt/task prortes are selecte to assure that o ata are lost. N/A A-0

45 A.5.6 SCHEDULING QUESTION F. Do you use tools to assst the real-tme scheulg aalyss? If yes, what k of tools? How are the outputs of these tools verfe? Scheulg Tool Type Approach to Verfyg Tool Output Of the 5 surveys respog, ther aswers were as follows:. revew of source coe, test of executable o platform. Emulators a smulators. Noe yet 4. Emulator We ca sert test patters o port ps that moel oe or two of the moule etry pots, the motor them wth a osclloscope to valate the emulator calculatos. 5. Real-tme tmg a aalyss tools assste by ebugger tools Tmg measuremets a aalyss of ata. 6. MS Excel We just use MS Excel for aalyss. We the attempt to create a esg that accommoates the worst case. Ths has worke well so far. 7. Not sure I uersta where the questo s gog. 8. No 9. I-house propretary. Qualfe tools, va DO-78B 0. Scope a ARINC bus aalyzer Motor certa varables software. WVew Off the shelf tool - t s ot verfe house. NO. No tools were utlze RTSA A-

46 A.5.7 SCHEDULING QUESTION G. What tres commercal avato systems o you thk wll challege the curret scheulg approaches (.e., may lea to the ee for ew scheulg algorthms)? Of the 5 surveys respog, ther aswers were as follows:. Desre to reuse, esre to hert cofece from re-use, esre to use o-evelopmetal tems. Do t kow. Partto base OS s, Evet base scheulg 4. A vrtual exploso the use of very hgh spee ata buses wll be very challegg. 5. Multple threa real-tme ea le scheulg aalyss 6. Hgh spee ata busses 7. The regulatory evromet 8. More splay a I/O requremets 9. Do t kow. 0. Parttog (ARINC-65). Do t kow. The move to 78B wll challege our exstg approach. Thus we are lookg to off the shelf RTOSs that hale these challeges. It seems that ata protecto a proof of coe coverage wll be bgger challeges the the scheulg algorthms.. Fly-by-wre, automate lag, 4. Collso avoace A.6 TIMING. A.6. TIMING QUESTION A. After system evelopmet, o you verfy that eales are met a scheulg aalyss assumptos are correct? If yes, please expla how. A-

47 Of the 5 surveys respog, ther aswers were as follows:. Yes. Test all the paths through the logc, whch we, usg test puts a a logc aalyzer. Also, a commucato system, the achevable bt error rate vs. sgal-toose rato s calculable. We coul cotrol the sgal a the ose, a we use e-toe bt error rate testg over very log peros of tme, a coul observe performace to be that precte by theory.. Yes, by test a aalyss. Not yet, we leave ths to our customers 4. Yes, we verfy at the boar level usg fuctoal tests, we corporate flght tests wth parameter evaluato, a we measure urg software verfcato a valato. 5. Yes. Real-tme checks are cotually mae the software. Status coes are store NVRAM for after the fact vewg. 6. All eales specfe as system or software requremets are verfe. Less formal aalyss s ot always verfe. 7. As metoe above, I m a ol guy a we were rather prmtve the 970s a980s. 8. Yes. Software motors are serte to verfy scheulg performace 9. Yes, va a combato of system/fucto testg (base upo system requremets) a also staar software testg. 0. Yes, tmg aalyzers. Yes. Per our software requremets, we revew the tmg aalyss ata.. Yes, several methos are use for verfcato. Rug specal system scrpts a osclloscope s use to verfy tmg of certa evets. 55 Bus captures are oe to esure specfc ata s set out at the requre tme. Iputs are stmulate a latecy to the assocate output s measure to verfy t s wth tolerace. I ato, spectos are oe to aalyze that the coe s performg certa operatos at a specfe tme f the prevous methos caot be use to verfy ths.. Yes. Apply ata at maxmum rate to LRU puts, check for tmely a approprate respose at LRU outputs. Error messages emtte o buffer overflow 4. Yes. The crtcal tmg fuctos are verfe by exterally recorg the tme from the put tato to the requre acto. Ths recorg s oe exteral to the electroc cotrol ut by puttg stmul to the ut a measurg the acto from the system. A-

48 A.6. TIMING QUESTION B. I what areas of tmg verfcato or valato have you ecoutere problems a how were they aresse? Of the 5 surveys respog, ther aswers were as follows:. Tmg verfcato for us was much easer usg test equpmet tha by usg software self-motorg tools. Wrtg to a logc-aalyzer accessble output regster at kow pots the software ca be helpful.. Bggest problems have bee area of system respose tmes followg applcato of power. Most ofte meas of aressg problem has bee to she tasks eee to be performe before system resposes are supple (e.g., BIT).. Other area s tme base ata processg systems where ueve work loa has resulte some ma loop overrus, soluto has bee to reallocate tasks to eve processg out. 4. Tmg s har a taskg preemptve OS the presece of cachg 5. Problems have occurre measuremet ucertates the emulator. We scovere that we ca ot rely o sap shot measuremets; we ee to allow a log elapse tme measuremet to get the most accurate pcture. 6. If a abormalty occurs, there s ever eough ata to help aalyss. If the problem s very termttet the specal mofe software may be eee to aalyze the problem. 7. People ot og what they shoul 8. I thk you re askg a questo that may be a o-braer because we tee to be rather coservatve our esgs. It just was t part of our culture to press the evelope sce we lacke aequate smulato tools that shoul be part of a esger s kt halg tmg ssues. 9. Throughput problems have lea to a re-balacg of tasks a prortes 0. Tmg problems typcally appear because of shortcomgs the orgal requremets.. Istrumetato requres access to the aress a ata buses; ths s ot possble proucto (seale box) harware.. Caot test every possble falure moe or coto. It gets eve harer to test multple falure moes/cotos.. Verfyg the rate of ata exchage betwee processors has bee challegg. Specal ebug coe s left a bul to be use for verfcato A-4

49 4. Ukow 5. Noe A.7 FAULT-TOLERANT. Does your testg allow for faults? If yes, please expla. Of the 5 surveys respog, ther aswers were as follows:. Stuato bts the what to o ext look-up table clue error cotos havg bee sgale. Error etecto a error halers appear the system at several levels. We cotrolle the commucato put, as well as the sgal-to-ose rato, a we coul ject faults may places to test resposes to faults.. Do t uersta the questo. N/A 4. Our testg corporates fault jecto at tmes to evaluate falure mechasm motors. For example, we tur off the watchog toggle a evaluate the mpact t has o tmg. We also wrte patches that allow elay loops to exte to fty whle og othg. Ths helps us evaluate the effectveess of our motors a our protecto schemes. 5. Yes. A persstece cout of faults s matae a motore cotuously 6. C laguage asserts are set up the coe to catch tmg faults that occur urg ru-tme. These cause a pre-efe software fault that ca be etecte a trace exteral to the system. 7. Exteral to the processor system yes. See above all paths commets 8. Our faults were exterally serte through the I/O, but geerally we gore ths area by creatg a evromet where program executo was assume by a Watchog Tmer that was upate at some regular terval from the Executve. Beg a UAV ha ts avatages. 9. Yes. I some cases, thgs such as task slppage are completely acceptable 0. Fault jecto testg s clue as a part of robustess testg. Yes. Whe we try to accommoate t by usg other sesor ata f coto s averse we swtch to the sta-by chael. No A-5

50 . Ukow 4. No A.8 OPINION ABOUT RTSA AND ITS VERIFICATION. I your opo, what are the major ssues regarg RTSA a ts verfcato? Of the 5 surveys respog, ther aswers were as follows:. The greatest ffculty I have experece s from sources of requremets who beleve what they wat ca be ha suffcetly stataeously that they o ot exame ther ees, a are therefore ot able to establsh what ther requremets are. The seco largest source of ffculty s from software teams that o ot wat to have to meet real tme requremets, a te to clam that whatever the task, ther coe a ther favorte operatg system s bou to be goo eough. Ths gets reflecte to the team maager who caot force the programmers to use the approprate tools.. Cofrmato of tmg ssues uer all foreseeable crcumstaces. The OS, the harware, the evce rvers, are tryg to abstract the performace of the uerlyg applcato so that t appears to ru o a vrtual processg system. Ths vrtual processg system s tryg to optmze performace by globalzg formato, e.g. cache (structo a ata), ppelg, speculatve structo scheulg, share resources (e.g. memory). At the same tme the OS s tryg ts best to keep throughput hgh usg bufferg, terrupt rve rvers, asychroous perpherals, bus soopg etc. Ths makes tme etermsm very har. 4. The major ssue s ot the frst mplemetato of the real tme system. It s the mofcatos a chages that occur later as the system evolves, pcks up more features a fuctos. Regresso testg s suppose to scover all those ae aomales, but t oes t always. 5. The tools are very expesve a ot always avalable; ore s trag for the tools avalable 6. Poor early aalyss a esg. Poor sttutoalzato of worst case throughput aalyss 7. Wthout a oubt- the FAA s the ONLY ssue. 8. It tes to be somewhat tutve a has the potetal for beg sub-optmal 9. Operatg systems are typcally rve by erve or mplemetato requremets (.e., low level). Ay system requremets (.e., hgh level) leve agast a OS are so geeral a broa, that they are of lttle value. They really cat be teste (other tha to say, yup, A-6

51 t oes that). Cosequetly, the resultg ocumetato for a OS cossts of very etale software requremets -- wthout ay real system requremets. DO-78B a the FAA/JAA seem to have a ffcult tme wth ths realty. I have attee SOI auts were much scusso was spet o the requremets, so they were so etale a t trace up to hgh level requremets. Ths s a fact of lfe, whe ealg wth platform/fouato/utlty software. 0. Do t kow. I thk the ssues for the future maly have to o wth the verfcato ot so much the actual scheulg. Better o-trusve tools wth more vsblty to the scheulg aspects. Possbly the processors themselves ee bul- hooks/fuctoalty that ca be tappe to.. No commet A.9 ADDITIONAL INFORMATION. What other real-tme scheulg experece or ssues woul you lke to share? Of the 5 surveys respog, ther aswers were as follows:. I chare a sesso at a FAA Data coferece (sposore by ASD/Carol Ur/Felx Rausch) a tre to scuss requremets. A vocabulary for commucato of requremets betwee system users a system evelopers seems to be eee.. Noe. Nee more tme to thk about ths? 4. Noe 5. Noe 6. Noe 7. Noe II. Woul you be tereste partcpatg FAA-sposore efforts to aress real-tme scheulg aalyss (e.g., partcpate a tervew, partcpate evelopmet of polcy or guace, etc.)? Of the 5 surveys respog, ther aswers were as follows: 6 yes 8 o Maybe A-7

52 III: If you sa yes to II, what s your area of terest? Of the surveys respog, yes ther aswers were as follows:. ATC systems. We certfe Aa RTS s, VxWorks/Cert, a BSP a cotue certfcato work wth a Parttoe Itegrate Moular Archtecture. We are partcularly tereste solvg tme aalyss problem. Rght ow we push the problem back to the user of the RTOS, but I expect we wll be aske may more questos about the cotrbuto that the OS makes to the tmg of the applcato.. We have thought about ths a lot, a whle we ca prouce a lot of ata we o t kow what format ths shoul be to be useful to the user. 4. Real tme cotrol systems a etworkg ssues 5. Software evelopmet. 6. Verfcato a certfcato 7. Obvously- Drve-By-Wre techology trasfer from the automotve ustry 8. I thk Cots a parttoe operatg systems are gettg more & more atteto I lke to lear more about these IV. If you sa yes to II, how ca we cotact you? No atoal formato was prove by the respoers. A-8

53 APPENDIX B THE RESEARCH PROJECT DETAILS WITH ASSUMPTIONS I ths appex, the kow results are escrbe sectos B. a B.. The ew results obtae ths research effort are escrbe sectos B. through B.6. B. DYNAMIC-PRIORITY SCHEDULING DISCIPLINE. As metoe secto, the Earlest-Deale-Frst (EDF) algorthm s optmal o oe processor wth respect to the Dyamc-Prorty scheulg scple [B- a B-]. Ths meas that ay real-tme task system (TS) scheulable o oe processor by ay Dyamc-Prorty scheulg algorthm s also scheulable by EDF. It also meas that a real-tme TS s feasble o oe processor wth respect to Dyamc-Prorty scheulg scple f, a oly f, t s scheulable by EDF. Sce EDF s optmal for oe processor, there s o compellg reaso to coser other Dyamc-Prorty scheulg algorthms. Therefore, ths effort s restrcte to EDF oly. EDF ca be mplemete by matag a queue of actve tasks (.e., tasks that have mae a request but have ot yet fshe executo) arrage asceg orer of the eales of the requests of the tasks. Wheever the processor becomes free for assgmet, the task at the hea of the queue wll be assge to the processor. Whe a ew request arrves, ts eale wll be compare wth the eale of the task that s curretly executg, a f the eale of the ewly arrve request s closer to the curret tme, t wll receve the processor. The task that was executg before wll be pre-empte a put back the queue. EDF s usually mplemete by software because of the operatos volve. Ths makes t ot as appealg as Fxe-Prorty scheulg algorthms, sce the cotext swtchg tme s hgher tha those of Fxe-Prorty scheulg algorthms. O the other ha, EDF yels a hgher processor utlzato tha Fxe- Prorty scheulg algorthms. Table B- shows a real-tme TS whose EDF scheule s show fgure B-. TABLE B-. A REAL-TIME TASK SYSTEM (EXAMPLE ) T s e p T T 5 T T T T T T T T T T T T FIGURE B-. EARLIEST-DEADLINE-FIRST SCHEDULE OF THE TASK SYSTEM IN TABLE B- B-

54 The questo of etermg f a real-tme task system (TS) s scheulable o oe processor by EDF s cosere. Ths s tatamout to etermg f the scheule prouce by EDF s val. As the ext theorem [B-] shows, some smple cases of ths questo ca be eterme effcetly. Theorem : Let TS ({ T }, { s }, { e }, { }, { p }) be a real-tme TS cosstg of e peroc, real-tme tasks. The () s a suffcet coto for TS to be scheulable by EDF a () e p s a ecessary coto for TS to be scheulable by EDF. I the specal case where p for each, to be scheulable by EDF. e p s both a ecessary a suffcet coto for TS Theorem gves a smple test for scheulablty the specal case where. The TS s scheulable f, a oly f, e p. p for each There s o smple test for scheulablty whe e p s ot equal to >, oe ca safely say that t s ot scheulable. O the other ha, f caot coclue that t s scheulable. Smlarly, f scheulable. O the other ha, f e e p for some. If e p, oe, oe ca safely say that t s >, oe caot coclue that t s ot scheulable. Whle there s o smple test for the geeral case, oe ca stll test scheulablty by checkg f the eale of each request of each task the EDF scheule s met. For ths metho to work, oe ees to establsh a a pror tme-bou for whch oe ees to costruct the EDF scheule. If the tal request tmes of all tasks are etcal, a obvous tme-bou woul be P least commo multple of { p,..., p }. Ths s because all tasks smultaeously make a request at ther tal request tme, a the aga smultaeously make a request at P tme uts later. The EDF scheule wll be cyclc wth a cycle legth equal to P. Thus, f the EDF scheule s val for a pero of P tme uts, t wll be val for ay legth of tme. O the other ha, f the tal request tmes of the tasks are ot etcal, t s ot clear that such a tme-bou ecessarly exsts. Leug a Merrll [B-] showe that such a tme-bou ee exsts a s gve by s+ P, where s max{ s,..., s }. I the followg scussos, m{ s,..., s } 0 s assume. B-

55 Let S be the EDF scheule of the real-tme TS. Defe the cofgurato of S at tme t, eote by C S (t), to be the -tuple (e,t,, e,t ), where e,t s the amout of tme for whch has execute sce ts last request up utl tme t, a prove the followg theorem. et, s uefe f t < s. Leug a Merrll [B-] Theorem : Let S be the EDF scheule of a real-tme TS. TS s scheulable by EDF o oe processor f, a oly f, () all eales the terval [0, t ] are met S, where t s+ P, a () C ( t ) C ( t ), where t t P. S S T A algorthm to eterme f a real-tme TS s scheulable by EDF o oe processor cossts of costructg a EDF scheule S a checkg f all eales the terval [0, t ] are met S a CS( t) CS( t). By Theorem, TS s scheulable by EDF f, a oly f, both cotos are satsfe. Note that the rug tme of the above algorthm s a expoetal fucto of the put parameters, p,..., p. Oe woers whether there are more effcet algorthms, e.g., algorthms whose rug tme s a polyomal fucto of the put parameters. Ufortuately, Leug a Merrll [B-] showe that t s ulkely that such a algorthm coul be fou, as the ext theorem shows. Theorem : The problem of ecg f a real-tme TS s scheulable by EDF o oe processor s oetermstc polyomal (NP)-complete. NP-complete problems are a class of otorous computatoal problems. Ths class of problems has the property that f ay problem the class has a polyomal-tme algorthm, the every problem the class woul have a polyomal-tme algorthm. At the preset tme, oe of the NP-complete problems ca be solve polyomal tme. Sce ths class cotas may otorous problems (such as travelg salesma problem, Hamltoa path problem, etc.), whch ha bee stue for more tha a cetury, t s wely cojecture that oe of the NP-complete problems ca be solve polyomal tme. The reaer s referre to the excellet book by Garey a Johso [B-4] for a scusso of the cocept a mplcatos of NP-completeess a NP-haress (whch wll be scusse later ths theorem, as well as theorems 4, 7, 0, a ). Leug [B-5] showe that EDF s ot optmal for m > processors. At the preset tme, o smple algorthm s kow to be optmal for more tha oe processor. Lawler a Martel [B-6] use the ea of etwork flow to costruct a val scheule o m processors f the TS s ee feasble o m processors. At ru tme, the scheuler must scheule tasks accorg to the scheulg table. Ths s essetally the Clock-Drve approach, whch, as scusse secto, s ot esrable. There are two geeral approaches scheulg o m > processors: the global approach a the partto approach. I the global approach, the m processors are treate as a pool, a a actve task s assge to a avalable processor f there s oe. Otherwse, t wll be put to a B-

56 watg queue utl a processor becomes avalable. The watg queue s orere by the prortes of the tasks. By cotrast, the partto approach parttos the set of tasks to m groups, wth each group of tasks assge to a processor. Tasks assge to a processor ca oly be execute by that processor. The partto approach s geerally preferre over the global approach because of the ease processor maagemet a the avalablty of optmal algorthms for oe processor. The followg scussos are restrcte to the partto approach oly. The ma goal multprocessor scheulg s to partto a set of real-tme tasks to the smallest umber of groups such that each group s scheulable by EDF o oe processor. Ufortuately, ths problem s also NP-har, as show by Leug a Whtehea [B-7]. (Note: A NP-har problem s at least as har as a NP-complete problem, a possbly harer. At the preset tme, there are o kow polyomal-tme algorthms to solve ether a NP-complete or a NP-har problem. It s wely cojecture that oe of these ca be solve polyomal tme.) Theorem 4: The problem of parttog a set of real-tme tasks to the smallest umber of groups such that each group s scheulable by EDF o oe processor s NP-har. Theorem 4 suggests that t s extremely ulkely to solve ths problem polyomal tme. Thus, there s a ee to evelop fast heurstcs that wll yel ear-optmal solutos. I the lterature, there seems to be a proouce absece of fast heurstcs for the geeral case. Ths s probably ue to the fact that t s tme-cosumg to check f a set of tasks s scheulable o oe processor by EDF. However, for the specal case where each task has ts relatve eale etcal to ts pero, there s a smple test for scheulablty; a set of tasks s scheulable o oe processor by EDF f, a oly f, e p. Oe woul expect that fast heurstcs exst for ths specal case. As t turs out, ths specal case ca be moele as a b packg problem. I the b packg problem, a fte collecto of ut-capacty bs a a lst of peces wth szes betwee 0 a are gve. The goal s to pack the peces to a mmum umber of bs so that o b cotas peces wth szes totalg more tha. The b packg problem s kow to be NPhar, but there are umerous effectve heurstcs for t; see Coffma, Garey, a Johso [B-8] for a survey. Oe ca moel the task partto problem as a b packg problem as follows. Each processor s vewe as a ut-capacty b. The tasks are treate as peces wth szes gve e by p. The problem of parttog the tasks to a mmum umber of groups so that each group s scheulable by EDF s equvalet to packg the peces to a mmum umber of bs so that o b cotas peces wth szes totalg more tha. B. FIXED-PRIORITY SCHEDULING DISCIPLINE. As metoe secto, the Deale Mootoc (DM) algorthm s optmal o oe processor wth respect to the Fxe-Prorty scheulg scple [B-7]. Ths meas that ay real-tme TS scheulable o oe processor by ay Fxe-Prorty scheulg algorthm s also scheulable by DM. It also meas that a real-tme TS s feasble o oe processor wth respect to Fxe-Prorty scheulg scple f, a oly f, t s scheulable by DM. Sce DM s optmal o oe B-4

57 processor, there s o compellg reaso to coser other Fxe-Prorty scheulg algorthms. For ths reaso, the stuy wll be restrcte to DM oly. DM assgs the hghest prorty to the task wth the smallest relatve eale a the lowest prorty to the task wth the largest relatve eale. Whe the relatve eale of each task s etcal to ts pero, DM coverges to the Rate-Mootoc (RM) algorthm ue to Lu a Layla [B-]. DM a RM ca be mplemete by attachg the prorty of the task to the harware terrupt level;.e., the task wth the hghest prorty s assge to the hghest level terrupt. Scheulg s mplemete by harware terrupt, a cotext swtchg s oe the terrupt halg route. Thus, the overhea volve scheulg ca be kept to a mmum. Table B- shows a real-tme TS whose DM scheule s show fgure B-. TABLE B-. ANOTHER REAL-TIME TASK SYSTEM (EXAMPLE ) T s e p T 0 T 4 4 T T T T T T T T T T T T T T T T T T T T T FIGURE B-. DEADLINE-MONOTONIC SCHEDULE OF THE TASK SYSTEM IN TABLE B- The questo of etermg f a real-tme TS s scheulable o oe processor by DM s cosere. Ths s tatamout to etermg f the scheule prouce by DM s val. Lu a Layla [B-] gave a effectve proceure for ths, as the ext theorem shows. Theorem 5: The scheule prouce by DM s val f the eale of the frst request of each task s met whe all tasks make ther frst request smultaeously at the same tme. The proceure to eterme f a real-tme TS s scheulable by DM cossts of costructg a scheule from tme 0 whe all tasks make ther frst request a checkg f the eale of the frst request of each task s met. The TS s scheulable by DM f the eale of each task s met. Ths s a suffcet coto for all stuatos, eve f the tal request tmes ( s ) of the tasks are ot etcal. So far, the stuy ha bee assumg that tasks make requests perocally. I some stuatos, tasks may make requests at raom tmes, but t s guaratee that two cosecutve requests of the same task, say T, are separate by a mmum tme terval, say p. These k of tasks wll be calle sporac tasks, whle tasks efe earler wll be calle peroc tasks. Note that DM s also optmal a Theorem 5 s also applcable for sporac tasks. B-5

58 Lu a Layla [B-] gave a suffcecy test for a real-tme TS to be scheulable by DM (or RM). Ther suffcecy test assumes that the relatve eale of each task s etcal to ts pero. (Uer ths assumpto, DM a RM are etcal.) As the followg theorem shows, ths gves a test to check scheulablty faster tha the above metho. Theorem 6: Let TS be a real-tme TS cosstg of real-tme tasks, where each task s relatve e / eale s etcal to ts pero. TS s scheulable by DM (or RM) f p ( ). e A smple test for scheulablty cossts of comparg the sum of the utlzato factors ( ) p of / the tasks agast the value ( ). If the sum of the utlzato factors s less tha or equal / to ( ), the the real-tme TS s scheulable by DM (or RM). Otherwse, there s o / coclusve evece; t may or may ot be scheulable. The fucto ( ) s a ecreasg fucto of. Whe approaches fty, the fucto approaches l Leug a Whtehea [B-7] showe that DM s ot optmal for m > processors. At the preset tme, o smple algorthm s kow to be optmal for more tha oe processor. Researchers have cosere the partto approach scheulg o multprocessor systems. Ufortuately, theorem 7 was show by Leug a Whtehea [B-7]. Theorem 7: The problem of parttog a set of real-tme tasks to a mmum umber of groups so that each group s scheulable by DM o oe processor s NP-har. Theorem 7 shows that t s extremely ulkely that a fast algorthm exsts to scheule a set of real-tme tasks o a mmum umber of processors. Motvate by the computatoal complexty, Dhall a Lu [B-9] cosere two heurstcs, the Rate-Mootoc-Next-Ft (RMNF) a the Rate-Mootoc-Frst-Ft (RMFF) algorthms, both of them are aapte from b packg heurstcs. They showe that the worst-case bous for RMNF a RMFF are.67 4, respectvely. Specfcally, they showe the followg theorem. + a ( ) Theorem 8: Let N RMNF a N RMFF be the umbers of processors requre to feasbly scheule a set of tasks by the RMNF a RMFF algorthms, respectvely, a let N OPT be the mmum NRMNF N / RMFF 4 umber requre. The, as N OPT approaches fty,.4.67 a. B. FIXED-PRIORITY SCHEDULING WITH LIMITED PRIORITY LEVELS. NOPT NOPT / (+ ) I secto B., t was assume that the system has as may prorty levels as the umber of realtme tasks. Cosequetly, each task ca be assge a stct prorty. I practce, the umber of prorty levels a computer system s very lmte a s far exceee by the umber of realtme tasks. Thus, several tasks ee be mappe to the same prorty level. I the remaer of ths secto, t s assume that there are m prorty levels to whch tasks are mappe, where m <. Whe tasks wth the same prorty make a request, t s assume that they are B-6

59 scheule a frst-come, frst-serve (FCFS) maer. Ths s usually how the harware servce terrupts. Two mportat questos aturally arse. Frst, how oes oe eterme f a gve prorty assgmet prouces a val scheule? Note that several tasks at the same prorty level may smultaeously make a request, a t s ot clear how to characterze the worst-case scearo. Seco, what s a optmal prorty assgmet? Here, a optmal prorty assgmet meas that t ca always scheule ay set of tasks that ca be feasbly scheule o m prorty levels. Pertag to the frst questo, theorem 5 (state secto B.) s stll applcable to ths moel, but oe more coto ees to be satsfe as well. The atoal coto s that tasks at the same prorty level are servce the reverse orer of ther relatve eales;.e., the task wth the largest relatve eale s servce frst a the task wth the smallest relatve eale s servce last. Ths s because tasks at the same prorty level may make requests the reverse orer of ther relatve eales. Sce they are scheule a FCFS maer, ths represets the worst-case scearo. Thus, the followg theorem exsts. Theorem 9: The scheule prouce by a gve prorty assgmet s val f the eale of the frst request of each task s met whe all tasks make ther frst request smultaeously at the same tme, wth the stpulato that tasks at the same prorty level are servce the reverse orer of ther relatve eales (.e., tasks wth the largest relatve eale servce frst). A alteratve metho s to carry out tme-ema aalyss, as escrbe referece B-0. Let G eote a prorty assgmet, a let G eote the set of tasks havg prorty, m. By coveto, assume that prorty s the hghest prorty a prorty m s the lowest. Suppose there are k tasks G. Use G,j to eote the j-th task the set. Wthout loss of geeralty, assume that the relatve eale of G,j s less tha or equal to that of G,j +, for each j k. For a gve task Ta G, suppose t makes a request at tme 0, alog wth all tasks at a equal or hgher prorty tha T a. The, the total tme ema wa ( t) of ths request of Ta, alog wth all tasks at a equal or hgher prorty tha Ta the tme terval [0, t) s: j< a t wa( t) el + p e, for 0 l l < t Tl G Tl Gj Task T a ca meet ts eale f, a oly f, there s a tme stat t, 0 < t a, such that w () a t t. Thus, all oe ees to o s to check those tme stats, t, whch are tegral multples of the peros of some tasks belogg to Gj, j <, to see f there s a tme stat t such that w () a t t. It s easy to see that for the task set G, f G, ca meet ts eale, the all other tasks G ca also meet ts eale. Therefore, to show that the scheule prouce by a gve prorty assgmet s val, all oe ees to show s that ca meet ts eale usg the tmeema aalyss, for all,,..., m. G, B-7

60 The rug tme of the above proceure s a polyomal fucto of m, a max{ }. It s a pseuo-polyomal tme algorthm. A pseuo-polyomal tme algorthm s oe that rus polyomal tme wth respect to the sze of the put, prove that all teger parameters are represete uary (base ) otato. (Note: ormally, teger parameters are represete bary (base ) otato. The effect of represetg tegers uary s to flate the sze of the put so that a expoetal-tme algorthm looks lke polyomal.) Aga, pertag to the seco questo, recall that DM assgs the hghest prorty to the task wth the smallest relatve eale a that DM s optmal whe there are as may system prorty levels as the umber of tasks. Oe ca aapt DM to systems wth lmte prorty levels as follows. Assg the hghest prorty to the task wth the smallest relatve eale. Keep o assgg the same prorty to the task wth the ext smallest relatve eale utl t s feasble (accorg to the tme-ema aalyss gve above), at whch pot assg the ext prorty level to the task. Ths assgmet s calle the Deale-Mootoc-wth-Lmte-Prorty-Level (DM-LPL) assgmet. Show below s algorthm DM-LPL. It tres to assg m prorty levels to tasks, where m <. If the resultg assgmet s ot val, t wll output Not scheulable. I the algorthm, G eotes the set of tasks wth prorty level, m; t s assume that prorty level s the hghest prorty a prorty level m s the lowest. Sort the jobs asceg orer of ther relatve eales;.e.,.... Let G for all, m.. For j to G G { Tj}. Use tme-ema aalyss to check f G, ca meet ts eale. If G, caot meet ts eale, the G G - { T j }. If + > m, the output Not Scheulable a ext, else G+ G+ { T j } Use tme ema aalyss to check f G +, ca meet ts eale. If G +, caot meet ts eale, the output Not Scheulable a ext. Else +. Output G, G,... Gm. The rug tme of algorthm DM-LPL s a polyomal fucto of, m, a max{ }. It s a pseuo-polyomal tme algorthm. The algorthm was mplemete C laguage. The source coe has bee teste usg a Torao computer. I referece to ths algorthm, see the mplemetato show appex C. B-8

61 DM-LPL wll be prove to be optmal the sese that ay TS feasble o oe processor wth m system prorty levels s also scheulable by DM-LPL. Before provg the theorem, the ext lemma wll be prove frst. Lemma shows that f a TS s feasble, there s always a prorty assgmet such that tasks wth small relatve eales have equal or hgher prorty tha tasks wth large relatve eales. Lemma : If TS s feasble o oe processor wth m system prorty levels, the there s a val prorty assgmet such that tasks wth small relatve eales have equal or hgher prorty tha tasks wth large relatve eales. I other wors, there s o task wth a large relatve eale havg a hgher prorty tha a task wth a small relatve eale. Proof. Sce TS s feasble, there must be a val prorty assgmet, say G, for the set of tasks. If G satsfes the property of the lemma, the the lemma s prove. Otherwse, there must be two tasks, T a a T b wth a < b such that T b G k a Ta G k +. Create a ew prorty assgmet G such that Tb has the same prorty as T a a G s stll val. Repeatg ths argumet wll fally prove Lemma. G { G, G,... G m } s efe as: Gk Gk { Tb}, Gk+ Gk+ { T b }, a for all others, k, k +, G. G wll be show to be a val prorty assgmet ext. G Note that the oly prorty chage G s Tb; the prorty of all other tasks rema the same as before. Sce a lower prorty s assge to T b G, t s clear that all tasks havg equal or hgher prorty as Tb G ca stll meet ther eales. Smlarly, all tasks havg lower prorty tha T a G ca also meet ther eales, sce the prorty chage of T b wll ot affect ther operato. Therefore, t remas to prove that the tasks G k + ca meet ther eales uer the ew prorty assgmet. To prove ths, t s suffcet to show that the task ca stll meet ts eale. Note that G + s the same task as G k +,. Let ths task be T c. k, G k +, Sce T c ca meet ts eale uer G, there must be a tme stat t, followg equalty hols 0 < t c, such that the j< k+ t c( ) + p T Gk+ T Gj w t e e t Uer G, the maxmum respose tme for Tc s j< k+ t c p T Gk+ T G j c w ( t ) e + e, for 0 < t B-9

62 t By assumpto, c a < b p a. Moreover, p b, sce t c p b. Thus, the maxmum respose tme for T c at tme t s j< k+ t c( ) + p T Gk+ T G j w t e e j< k+ t e eb p e T Gk+ T G j + + j< k+ t t e p e b b p T Gk+ T G j + + e + j< k+ T Gk+ T G j w ( t ) t c. c t p e e But ths meas that T c ca meet ts eale uer G. Theorem 0: DM-LPL s a optmal prorty assgmet for oe processor. Proof. Let TS be a feasble task system o oe processor a let G be a val prorty assgmet for TS. By Lemma, t ca be assume that G tasks wth small relatve eales have equal or hgher prorty tha tasks wth large relatve eales. If G s etcal to the prorty assgmet obtae by DM-LPL, the the lemma s prove. Otherwse, there must be a task T b G k+ that ca be feasbly assge to G k. Wthout loss of geeralty, assume that T b s G k +,. Let Gk, be Ta. Sce T b ca be feasbly assge to G k, there must be a tme stat t, 0 < t a, such that wa( t) + eb t. Costruct a ew prorty assgmet G { G, G,... G m } such that Gk Gk { Tb}, Gk+ Gk+ { Tb}, a for all other k, k +, G G. Oe ca clam that G s also a val prorty assgmet. It s easy to see that all tasks G, k, k+, ca stll meet ther eales, sce G G for k, k +. Suppose G k +, s Tc. If t ca be prove that both T a a T c ca meet ther eales uer G, the G s also val. By assumpto, T b ca be feasbly assge to G k. Therefore, uer the ew assgmet G, Ta ca stll meet ts eale. Sce T b ca meet ts eale uer G, there must be a tme stat t, < t, such that 0 b B-0

63 j< k+ t b( ) + p T Gk+ T G j w t e e t j< k+ t e eb p e T Gk+ T G j + + For T c, the maxmum respose tme uer G s j< k+ t c p T Gk+ T G j c. w ( t ) e + e for 0 < t t Sce t b pb, p b. Lettg t t, wc( t) wb( t) t. Sce t b c, Tc ca meet ts eale uer the ew prorty assgmet G. B.4 PRIORITY ASSIGNMENT ON MULTIPROCESSORS WITH LIMITED PRIORITY LEVELS. Theorem 6 shows that the processor utlzato s at least 0.69 whe the computg system has fte prorty levels. Whe the computg system has lmte prorty levels, however, the processor utlzato ca be qute poor. Coser tasks wth the followg characterstcs. All tasks have ther relatve eales etcal to ther peros. Task has ts executo tme much smaller tha ts pero;.e., e << p. Task has ts executo tme equal to the pero of the frst task (.e., e p ), but s much smaller tha ts ow pero (.e., e << p ). Smlarly, task has ts executo tme equal to the pero of the seco task (.e., e p ), but s much smaller tha ts ow pero (.e., e << p ). The remag tasks have executo tmes a peros followg the same patter. Sce e + e+ > p for all <, o two tasks ca be assge the same prorty level. Therefore, m processors are eee, where each processor has exactly m prorty levels. Notce that the processor utlzato each processor s very low. O the other ha, f the computg system has fte prorty levels, all tasks ca be scheule o oe processor by DM. The above example reveals that the processor utlzato ca be qute low for some pathologcal stuatos. It s ulkely that ths k of stuato occurs frequetly practce. The above example also suggests that more processors are ecessary whe computg systems wth lmte prorty levels are use, compare wth systems that have fte prorty levels. I ths subsecto, the problem of scheulg tasks o multprocessors wth lmte prorty levels s cosere. The goal s to f the mmum umber of processors ecessary to scheule tasks. Such a algorthm wll be calle a optmal algorthm. B-

64 I a search for a optmal algorthm, a atural caate s the greey algorthm, whch works as follows. Sort the tasks asceg orer of ther relatve eales. Startg wth the frst task, scheule as may tasks as possble by the DM-LPL algorthm oto the curret processor, utl a task that caot be feasbly scheule by DM-LPL s ecoutere, at whch pot scheule the task oto a ew processor (whch ow becomes the curret processor). The above process s repeate utl all tasks have bee scheule. Oe woers f the above greey algorthm s optmal. Ufortuately, the aswer s egatve. Coser the followg sx tasks, all of whch have ther relatve eales etcal to ther peros a all have ther tal requests mae at tme 0. I the followg, each task T s represete by a orere par ( e, ) : T (, 5), T (, 6), T (, 9), T 4 (5, 0), T 5 (6, 6), a T 6 (, 0). Suppose each processor has two prorty levels. The greey algorthm yels three processors, wth P : G {T, T }, G { T }; for P : G {T 4 }, G {T 5 }; a for P : G {T 6 }. However, these sx tasks ca be scheule o two processors such that P : G {T, T }, G {T 5 }; for P : G {T }, G {T 4, T 6 }. As t turs out, the problem of fg the mmum umber of processors s NP-har. By a smple mofcato of the proof referece B-7, the followg theorem ca be prove. Theorem : The problem of fg the mmum umber of processors wth m prorty levels to scheule a set of tasks s NP-har. Theorem suggests that there s o effcet algorthm to scheule a set of tasks o the mmum umber of processors. Motvate by the computatoal complexty of the problem, two heurstcs, calle Frst-Ft (FF) a Frst-Ft-Decreasg-Utlzato (FFDU), respectvely, are cosere. The FF algorthm sorts the tasks asceg orer of ther relatve eales, e whle the FFDU algorthm sorts the tasks asceg orer of ther utlzato factors ( ). p Both algorthms try to scheule the ext task to the lowest-exe processor by the DM-LPL algorthm. Algorthm FF: Sort the tasks asceg orer of ther eales. Let k be the largest ex of the processor to whch tasks have bee assge. Itally, k. For each processor P, let L m be the lowest prorty ex (hghest prorty level) to whch tasks have bee assge. Itally, L for each. Let eote all the tasks assge to prorty ex j o P. Itally, P G j for each a j. G j B-

65 For j to Assg task T j as follows. ( s processor ex). Whle T j has ot bee assge Use tme-ema aalyss to test f T j ca be assge to. If T j ca be assge to Else f Output k. G L Assg T j to. G L L + m a Tj ca be assge to G L + L L +. Assg T j tog. L Else f < k +. Else k k +, k. G L. It s a pseuo- The rug tme of FF s a polyomal fucto of m,, a polyomal tme algorthm. max{ } Algorthm FFDU: Sort the tasks esceg orer of ther utlzato factors. Let k be the largest ex of the processor to whch tasks have bee assge. Itally, k. For each processor P, let G eote all the tasks assge to P. Itally, G for each. For j to Assg task T j as follows. ( s processor ex). Whle T j has ot bee assge Use the DM-LPL algorthm to test f T j ca be assge to P. If T j ca be assge P Assg T j to. G L Else f < k +. Else k k +, k. Output k.. It s a pseuo- The rug tme of FFDU s a polyomal fucto of m, a polyomal tme algorthm. max{ } The worst-case bous for FF a FFDU are ot kow yet. B-

66 B.5 UNIT-EXECUTION-TIME TASK SYSTEMS: UNLIMITED PRIORITY LEVELS. I the prevous secto, t was show that processor utlzato ca be qute low whe the computg system has lmte prorty levels. The stuato coul be better f each task has etcal executo tme, say ut. Ut-executo-tme task systems are terestg ts ow rght, sce t moels bus commucato where each packet of formato takes ut tme to se. Here, the commucato bus s the processor. I ths secto, the case where the computg systems have ulmte prorty levels s cosere. A set of real-tme tasks, where each task has ut of executo tme (.e., e for each ) a the relatve eale of each task s etcal to ts pero (.e., p for each ), s cosere. Thus, each task s completely characterze by ts relatve eale (or equvaletly, ts pero), whch s assume to be a teger. Assume that the tasks have bee sorte asceg orer of ther relatve eales;.e.,.... The goal s to f a utlzato bou for tasks, U(), such that f the tasks have total utlzato less tha or equal to U(), the they are always scheulable o a sgle processor by DM. If the task set has total utlzato larger tha U(), the t may or may ot be scheulable by DM, epeg o the relatoshps amog ther relatve eales. It s cojecture that: U ( ) A full proof has ot bee obtae. Two specal cases as escrbe appex B.5. a B.5. were prove. Defto : A task set s sa to perfectly utlze the processor wth respect to a prorty assgmet f t s scheulable, but a ecrease the largest pero wll make the task set uscheulable uer the same prorty assgmet. B.5. PERFECT INSTANCE WHEN IS ARBITRARY. Gve the umber of tasks a T (, ) where, costruct T (, ), T (, ),, T (, ) as follows. For, s the maxmum teger such that ( 4) + + t t + ( + 4) for t < (B-) To efe - a, efe two varables t a t, whch wll also be referre throughout the report hereforth. B-4

67 Let t be the maxmum teger such that t t 5 + t t + 5 for t < t (B-) Let t be the mmum teger such that t t + t t < + for t < t (B-) If t k + for some k, the t + a t. Otherwse, f t t+, the t t + t + 4; else t t+, let - t a t. For coveece, the costructe stace wll be calle the perfect stace for a. The costructe perfect stace wll be show to have several propertes, a the the stace wll be show to perfectly utlze the processor.. For, s ot a multple of. Proof: Suppose k a, the by efto + ( + 4 ). Sce ( + 4), oe ca always crease by a stll satsfy equato B-. But ths cotracts the fact that s the maxmum teger satsfyg equato B-. For, by costructo, ether t or t+. I the former case, t caot be a multple of, because otherwse the same cotracto as above wll arse. I the latter case, t +, whch meas that t t +. By equatos B- a B-, t+ t. However, ths hols oly whe t +, s ot a multple of. Therefore, both cases, - s ot a multple of.. satsfes the followg equato: ( ) r for some r such that r (B-4) Proof: It has alreay bee show that s ot a multple of. So, assume that k+ r for some tegers k a r such that r. Accorg to the costructo, satsfes equato B-. Therefore, k + + k + r, whch B-5

68 ( ) r mples that. Throughout ths secto, whe r s metoe, by efault, t meas r as efe above.. For each, ether + + or + +. Hece, for. + r + + Proof: For r., let k + r for some tegers k a r such that + If r, the. It s easy to show that + satsfes equato B- for +, a t s ee the maxmum teger that satsfes equato B-. Therefore, + + a r+ r +. + If r, the +. Smlarly, oe ca show that + s the maxmum teger satsfyg equato B- for +. So + + a r +. I summary, for, + r + + for (B-5) 4. Ether t, or t, or t. Ether t or t +. Proof: Let terms of. k +r for some tegers k a r <. It s esrable to express t r. The, ( ) ( ) t Oe ca easly show that t + 5 for t < - a t > -. Therefore, by the efto of t, t -. t t > + 5for B-6

69 r <. + ( ) ( ) As above, t ca be show that - s the maxmum teger satsfyg equato B-. Hece, by efto t - ths case. r >. + ( ) ( ) Smlarly, t ca be show that t - ths case. Next, express t terms of. r. ( ) + ( ) r >. ( ) Usg the same argumet as for t a by the efto of t, t ca be show that t whe r a t + whe r >. Accorg to the costructo of, f t t+, the t. I summary, tables B- a B-4 show the relatoshps betwee -, a. B-7

70 TABLE B-. RELATIONSHIP BETWEEN -,, AND WHEN IS EVEN t t - r - - r r TABLE B-4. RELATIONSHIP BETWEEN -,, AND WHEN IS ODD t t r r r r Ether k or k + r for some tegers k a r, where r. Proof: There are three cases epeg o the relatoshp betwee a. (). From tables B- a B-4, - happes whe r f s eve or whe r f s o. I ether case, r. Therefore r r. () k + r a. From tables B- a B-4, r,.e., k +. Therefore, k +. If, the r 0; otherwse, r. () + ( k + r) + (k+ ) + (r + ). If r, the r 0; otherwse, f s eve, the + r a r 4, else + r a r. 6. The perfect stace T, T,, T costructe as above perfectly utlzes the processor. Proof: Some characterzatos about the eale mootoc scheule of the perfect stace wll be gve frst. The clam follows rectly from these characterzatos a the efto of perfectly utlzato. B-8

71 a. There s o le tme before t. It s eough to prove that at ay tme t < t, the umber of requests R from T, T,, T - s greater tha or equal to t. There are two cases epeg o t. t. If, at tme t, R -. Sce, t R. For, R ( ) ( ) ( 4) < t < +.. By costructo, t R. It has alreay bee show ether + + or + + for. So, t < + ths case. Therefore, at tme t, T+, T+,..., T each makes oe request, whle T,..., T each makes exactly two requests. Hece, the total umber of requests + + R ( ) ( ) ( ) It s obvous that t R. b. All requests from T,, T - urg the pero [0, t ) have bee execute at t. From the costructo, < t < < for < <. So, at tme t, T,..., T all make exactly two requests, a T makes oe request. Therefore, the total umber of requests mae by T, T,, T - s t t R ( ) whch s equal to t. Therefore, all requests from T,..., T have bee execute. c. Noe of T,..., T wll make ay request urg the pero [t, -]. Frst, < t, -. So T,, T - have all mae ther seco requests before t. O the other ha, t < <. Therefore, T,, T - wll ot make ther thr requests from t utl.. There s o le tme urg the pero [t, -] a there s o request from T, T,..., T at -. Therefore, the frst request from T wll be execute at - a fshe just before ts eale. B-9

72 By (c), T,,T - wll ot make ay request urg the pero [t, -), ether wll these tasks make ay request at -. Therefore, t remas to show that the requests from T, T, a T - wll occupy the pero [t, -) a wll ot make ay requests at -. It has bee show that t s ot a multple of a t <. So, ether T or T wll make a request at t. For T-, there are two cases epeg o the relatoshp betwee t a - : t +. The, t - < -. Therefore, T - wll ot make a request at t. Hece, the frst request of T wll be execute at t a the clam s true. t. Task T makes the seco request at t a the request s execute rght away. Aga, There are the followg cases: t a t. The,,, a. From tables B- a B-4, r. Therefore, t + ( k + ) + k. So, T must make a request at t+. T wll ot make a request before. Therefore, at tmes a, the processor wll execute the requests from T a T, respectvely. t a t +. The,, +, a +. T wll ot make a request at t+ sce s ot a multple of, or wll t make a request at + because t has bee prove that s ether a multple of or a multple of plus a teger greater tha. T makes a request at. Therefore, at tmes a, the processor wll execute the requests from T - a T, respectvely. t a t +. The,, +, r a + t + ( k + ) (k+ ). + B-0

73 So, T wll make a request at -, a t wll ot make a request at a +. T makes a request at. Therefore, at tmes -, -, a, the processor wll execute the requests from T -, T, a T, respectvely. Clam : Gve, the perfect stace gets the mmum utlzato whe -. Proof: For 5, the clam ca be prove by just eumeratg all task sets. Tables B-5 a B-6 show the perfect staces for 4 a 5, respectvely. For, the clam s obvously true because there s oly oe possble value of, whch s. Therefore, assume that 6 the followg. TABLE B-5. PERFECT TASK SET INSTANCES WHEN 4 4 U TABLE B-6. PERFECT TASK SET INSTANCES WHEN U For each a Let ( ) r, let the correspog for some r. U ( ) r (B-6) B-

74 It wll be prove that U ( ) attas ts mmum whe by showg that () the mmum of U ( ) occurs whe > a () U ( + ) < U) ( for.. The mmum of U ( ) occurs whe >. From tables B- a B-4, Applyg the above equalty to equato B-6 a removg the floor fucto, U ( ) r ( + ) + For coveece, let f( ) (B-7) ( + ) + The U ( ) f ( ). Next, t wll be prove that ( ) f s a ecreasg fucto a that U ( ) < f ( ). B-

75 a. f ( ) ecreases wth respect to. f ( r) ( ) r ( + ) ( ) r r + + ( ) r+ ( r ) ( ) + 4 < + + ( ) ( ) + + ( ) ( ) + < ( ) ( ) ( + ) ( 5) ( ) ( ) + 4 < 0 ( ) ( 5) ( ) I summary, for ay, f ( ) strctly ecreases whe creases. Whe, f ( ) has the mmum value. b. U ( ) < f ( ) U( ). The proof s by showg that f ( ) U( ) > 0. It has bee show that: U ( ) (B-8) Let. It s esrable to remove the floor fucto f ( ) (see equato B-7). There are two cases, epeg o whether s o or eve. s eve. The,. By equato B-, + a r. Applyg to equato B-7, B-

76 f ( ) f ( ) ( ) ( + ) (B-9) Let S ( + )( + )( + ) ( )( ) ( + )( + )( + )( )( ) For, let S + ( + ). f S s a ecreasg fucto wth respect to. Whe, S. + + Equato B-9 mus B-8 gves: ( ) S U( ) ( )( ) ( + )( + )( + )( )( ) > ( 4) ( + )( + )( + )( )( ) ( + )( ) > 0 S ( )( )( )( )( ) S B-4

77 s o. The +. By equato B-, a r. Applyg to equato B-7, f ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( + )( + ) (B-0) Let S + + ( ) For ( + )( + )( ) ( )( ) ( + )( + )( )( )( ) 4, let S + ( + )( + ). S s a ecreasg fucto wth respect to. Whe 4, Equato B-0 mus B-8 gves: ( ) + S4 + ( + )( + ) ( + ) ( + ). f U( ) S S 4 ( 5) > S S S 4 ( ) ( 5) + ( + )( + ) > 0 ( )( )( )( )( ) To summarze, t was frst prove that f ( ) s a ecreasg fucto wth respect to. So for ay, f( ) > f ( ). O the other ha, B-5

78 ( ) > ( ) for ay. Therefore, U ( ) > f for ay. It U f ( ) was ext prove that U ( ) < f ( ) mmum of U must occur whe. Combg them together, the.. U( + ) < U( ) for <. Let +, a correspog to r, - a, r, -, a are efe. Base o tables B- a B-4, the relatoshp betwee, r, -,, a r, -, are show tables B-7 a B-8. The tables are bult as follows. Frst, because the case > s beg cosere, by equato B-, a r. Whe s crease by, +,, a r r-. From tables B- a B-4, the expresso of - a oly chages whe r chages at some values. So oly these specal values of r a r wll be exame. Gve r a, epeg o the party of, check ether table B- or B-4 to f the correspog - a for r. If - a are obtae from table B-, the get - a from table B-4, a f - a are obtae from table B-4, the get - a from table B-. Look at etres of the table accorg to the relatoshp betwee r a stea of r a. U( ) s expresse by equato B-6, a U( +) ca be expresse the followg equato: U( + ) ( ) Let H ( ) ( ) + +. (B-) Let a. k k Let g () k + + k < 6 a.. The 0 f < + k f + < + f + 0 f < + 4 k f + 4 < + 4 f + 4 B-6

79 f + + f f + g () + f f for all other For <, r > >. From table B-8,. Therefore, H 0. Equato B-6 mus B- gves: U ( ) U ( ) g () H ( + ) + 0 ( + ) + + > 0 a TABLE B-7. THE CHANGES OF - AND WHEN IS EVEN AND IS INCREASED BY r - r ( + ) ( ) others B-7

80 TABLE B-8. THE CHANGES OF - AND WHEN INCREASED BY IS ODD AND IS r - r ( + ) ( ) others < 6 b.. The 0 f < + k f + < + f + 0 f < + 4 k f + 4. f + + f + g () + + f for all other + + r,, + a. Therefore, +. ( ) ( )( ) H + + B-8

81 4 U ( ) U ( ) g () H ( + ) + H ( + ) + + H ( + ) ( + ) + ( )( ) ( )(+ ) ( ) 7 9 ( )( ) ( )( + ) ( )( )( )( + ) > 0. The f + + f g () f + + f for all other r. Note that -4 s a eve umber a s a teger. So, must also be eve. Checkg the correspog etres 4 table B-7, +,, a. B-9

82 5 ( ) ( ) ( )( ) H U ( ) U ( ) g () H ( + ) + H ( + ) + + H ( + ) ( + ) + ( 4)( ) ( )( ) ( ) ( 4)( ) ( 4)( ) ( )( ) 5 7 ( )( ) ( ) ( )( ) ( )( ) > 0. The f + + f f + 4 g () + f f for all other r > a r. Note that -5 s a o umber a ( + ) + 5 s a teger. So, must also be o. Checkg the correspog etres of table B-8, + a. Therefore, H 0. B-0

83 U ( ) U ( ) g () H ( + ) ( + ) + 0 ( + ) > 0 c. <. k 0 f < + f + g () 0 f < + 4 k f f f + 0 for all other Equato B-6 mus B- gves: U ( ) U ( + ) g () H ( + ) ( + ) H ( + ) + H (B-) Depeg o, H takes o fferet values. There are four cases: +. The r. Because - s a eve umber a s a teger, must be eve. From table B-7,, B-

84 , a +. Hece, H. Applyg to equato B- gves: U ( ) U ( + ) + ( ) (+ ) > 0 ( )( )( ) +. The r. Because - s a o umber a s a teger, must be eve. From table B-8,,, a to equato B- gves: H + +. Hece, ( ) U ( ) U ( + ) + ( + ) > 0 ( )( + )( ) +. Applyg < <. The a. So H 0. U ( ) U ( + ) 0> 0 ( + ). The r. From tables B-7 a B-8,,, a. Therefore, equato B- gves: H ( ) U ( ) U ( + ) ( )( )( ) +. Applyg to + ( )( + ) + ( ) > 0 B-

85 Clam : Gve the umber of tasks a, the Perfect Istace for a costructe above yels the mmum utlzato of tasks that perfectly utlze the processor amog all those task sets that have the same a as the Perfect Istace. Proof: For 9, the clam ca be prove by eumerato. So, assume that 0. The 0. Let the perfect stace for a be TS { T (, ), T (, ),..., T (, )}. It wll be prove by cotracto that TS has the mmum utlzato amog all task set staces that have the same a as TS. Suppose there s aother task set TS { T (, ), T (, ),..., T (, )} j that perfectly utlzes the processor a a, but has a smaller utlzato. The there must be some task Tj, j, such that <. Let T be such a task that has the mmum eale. It wll be show that j {,..., },, a case by case.. {,..., }. Proof. Suppose o the cotrary, there s a,, such that >. Usg tme ema aalyss, t ca be prove that: + + <. + U U j j + j j j j + > ( + + ) + + j j There are two cases:. The, U U ( + + ). By costructo of, + Ether +. U U > ( + + ) > Or + a +. U U > ( + + ) > B-

86 For all other, + + U U > > j.. j j j Proof. Suppose o the cotrary. By costructo, ether t or t+. It wll be prove that uer both cases, U U > 0, where U a U are the total utlzato of TS a TS, respectvely. t. For j -, j < t < j. So T j makes exactly two requests before tme t. The total umber of requests from T, T, T - before t s t R + ( ) +. By equato B-, R t. Noe of T, T,, T -, T +,, T - wll make ther thr requests at tme t. Because > t, T wll ot make a request at tme t. Thus, the frst request of T wll be execute. Ths meas that +. For j, by assumpto, j j. U U + ( + ) ( + ) + By tables B- a B-4, oe ca f that for each par of a such that, the rght-ha se of the above equalty s postve,.e., U U > 0. t t +. It wll be show that ths case t s mpossble that >. By tables B- a B-4, t < j, where j. So at tme t, task T j makes exactly two requests. By equato B-, all requests from T,, T - wll be execute before t, a o request s release at t by task T j, where j, because t < j. It has bee prove that t s ot a multple of. Thus, T wll ot make a request at t. If >, T wll ot make the seco request at t. I orer to be a task set perfectly utlzg the processor, the frst request of T must be execute at t. Therefore,, whch s a cotracto to the assumpto that >. B-4

87 .. Suppose,.e., T s the oly task such that epeg o the relatoshp betwee a. <. There are two cases to coser. By assumpto, a for j. So for j. O the j other ha, j < j <. So, for all j, task T j makes exactly two requests before tme t, a t wll ot make the thr request before tme. It has bee show that all those requests by T, T,..., T ca be execute before tme, a T wll ot make a request at. So the frst request of T wll be execute at tme. However, ths meas that, whch cotracts the assumpto that <. +. It s kow that. To prove the clam, there are two cases to coser, epeg o whether > or. If >, the T wll ot make ts thr request before +. So, usg the same argumet for the case, t ca be show that, whch cotracts the assumpto that <. Coser ow the case. The proof s qute volve. The ea s to show that the task set TS ca be obtae from TS by a seres of mofcatos of a for. Each step ca be escrbe as follows: Startg from, f <, the ecrease to a f ecessary, crease to make the mofe task set to cotue perfectly utlzg the processor. Let the mofe be ( ), the ew task set be TS, a U be the total utlzato of TS. Let S eote the maxmum ex such that <. By assumpto, S. After S steps, the stace TS s obtae. Let U eote the total utlzato of the perfect stace. I the followg, t wll be prove that U < U, whch s a cotracto. For coveece let U U a () +. j The proof s by ucto o - the ex of the task beses each step. T that s chage at B-5

88 Iucto hypotheses: There are three hypotheses:. For S, ether ( ) ( ) +, or () ( ) + ; < ( ).. For S, ( ), a + r ( r) () (B-). U < U for S. Base case:.. () () + or () () + ; < (). By assumpto, < () +. To prove the remaers, there are two cases epeg o r. a. r. The +. The umber of requests of T before tme () + s. Whe s chage to, whch s equal to, the umber of requests by the ew task stace TS before T orer to make the ew stace perfectly utlze the processor, tme s,.e., oe more tha that of T the orgal stace. I be crease by at least. However, because ees to + s a multple of, at tme, T wll make aother request. Therefore, ees to be crease by at least. I fact, ees to be crease by exactly. Because the pero of ay other task has ot bee chage, 4 j -, > So, t makes exactly two requests before tme j +, whch s the same as the orgal task set. The requests of task T before + 4 s stll the same --. Therefore, (). b. r <. The +. The umber of requests of T before () + s. Whe s chage to, whch s equal to, the umber of requests by the ew task stace TS before T T j B-6

89 s more tha that of ew stace perfectly utlzes the processor, T at least. It s eough that + j the orgal stace. I orer to make the ees to be crease by s crease by exactly. Because s ot a multple of, T wll ot make ay request at tme. The pero of ay other tasks T, 4 j, s ot chage. So T makes exactly two requests before tme +, whch s the same as the orgal task set. Therefore, () () a + r ( r) () j. It has alreay bee show that f r, the () + 4 ; otherwse, () +. It s obvous that (). I both cases, () ca be wrtte the followg way: + r ( r) () U < U. Sce oly the eales of T a T are mofe, the fferece betwee the total utlzato of the ew stace U s U U U a that of the orgal stace ( ) ( ). () () + It has bee show that ether + a () + 4, or + a () +. It s easy to show that both cases, U U > 0. Therefore, U < U. Iucto step: Suppose the hypotheses are true for, 4,, -, where S. Now there s the task set TS. Oe ees to chage to, a chage ( ) to ( ) f ecessary. The hypotheses ee to be prove to be true for. Frst, t wll be prove that < ( ). Suppose ( ). It wll be show that there s a cotracto to the assumpto that TS has a smaller utlzato tha the perfect stace. Whe s ecrease to, the umber of requests of T before tme ( ) has ot chage,.e., t makes two requests. B-7

90 So there s o ee to crease ( ), whch meas that ( ) ( ). Therefore, U < U. By ucto, U < U. Thus, U < U. Furthermore, for all < j, j. Therefore, ( ). So the ecrease of to j wll j ot make ay chage of ( j ),.e., ( j) ( ). However, ths ca oly make the total utlzato crease. Therefore, t must be true that j j. Ths meas that TS TS a U < U U, whch s a cotracto. Now the remaer part of the hypotheses wll be prove. There are two cases to coser, a... a. ( ) ( ) + or () ( ) + ; a < ( ). It has alreay bee prove that < ( ). By ucto, ( ). So task T makes exactly three requests before tme ( ), whch s oe more request tha that of T the task set TS. So ( ) must be crease by at least. If ( ) s a multple of, ( ) must be crease by at least. O the other ha, t ca be show that ( ) + for j. Task wll ot make ts fourth request at ( ) or ( ) +. Therefore, ( ) oly ee to be crease by exactly, or by f ( ) s a multple of. Thus, ( ) ( ) + or ( ) ( ) +. It has bee prove that < ( ). By ucto, ( ) satsfes equato B- for. Together wth equato B-, t ca be show that ( ). b. () a + r ( r) () j It s kow that ether ( ) ( ) + or ( ) ( ) +, epeg o whether ( ) s a multple of. By ucto, ( ) satsfes equato B- for. Therefore, j T j + r ( r) () From the equato, oe ca show that ( ). B-8

91 c. U < U. It has alreay bee prove that < ( ) <. So ( ( ) ) / a ( ) /. It has alreay bee prove that () ( ) +. Therefore,.. U U ( + ) ( + ) () ( ) > ( + ) ( + ) ( ) ( ) + ( ) ( ) ( ) + ( ( ) )( ( ) + ) ( ) 6 ( ) ( ( ) )( ( ) + ) 0 It s kow that S a S. So S. What remas to be prove for ths case s U < U. Lke the prevous case, t ca be prove that < ( ). To prove the remag hypotheses, there are two cases to coser: a. ( ) <. If s chage to, the umber of requests by T - creases by before tme ( ). ( ) must be crease by at least. If ( ) + s a multple of, the ( ) ees to be crease by at least. It wll be show that oe ees to crease by exactly or. It s ecessary to esure that other tasks T j, j, oes ot make a extra request after ( ) s crease by or. If ( ), the ( ) + a T j wll ot make ts fourth request before. Thus, ( ). As before, t ca be show that U < U by smple calculato. Therefore, U < U. Otherwse, ( ). I ths case, T wll ot make a request at tme, sce + r (whch s mple by the assumpto that + a table B-). Smlarly, T j wll ot make ts fourth request before. Thus, ( ) a oe ca show that U < U. B-9

92 b. ( ). I ths case, f - s chage to -, t s ot true that oe oly ees to crease ( ) by or as before, because after ( ) s crease by or, T, a T wll make ther fourth requests. It may happe that whle these requests are beg execute, T 4, also make requests urg ths pero. So there s o chace to execute the frst request of T before fshg other requests wth hgher prorty. I orer to prove the hypotheses, oe caot o the smple calculato ths tme. Oe ees to compare U a U term by term. ca be estmate base o the ucto. ( ) ( r) + ( ) + ( r) ( r) + + r + r ( ) ( r+ ) + + ( + ) ( r + ) ( ) ( r r + ) ( r+ ) + ( r+ ) + (+ ) r ( ) B-40

93 ( r+ ) + ( )( ) + ( + r ) ( ) ( + ) (+ ) r ( ) ( ) + r+ ( ) [( + ) ][( + ) r ] ( ) ( ) [( + ) ][( + ) ] ( + )(+ ) U U ( ) ( ) > ( ) + ( ) + (0 ) > ( ) + ( ) + (0 ) + + ( ) ( ) > ( ) + ( ) > ( ) ( ) + > + ( ) ( ) [( ) r] [( ) r]( ) > + ( ) [ ] ( 6) ( )( ) > [( ) ( + )] ( + )( + ) ( 6) ( )( ) > 0 whe 0 It has just bee prove that U < US. Note that TSS TS, so U < U. B-4

94 B.5. PERFECT INSTANCE WHEN AND ARE ARBITRARY. I the followg, a stace of tasks for gve a wll be costructe. For coveece, the costructe stace wll be calle the perfect stace for,, a. It wll frst be show that the stace perfectly utlzes the processor. It s the prove that the stace costructe for - has the mmum total utlzato amog all task staces that have tasks a perfectly utlze the processor. Costructo of Perfect Istaces. Gve the umber of tasks, T (, ) a T (, ), where -, a, costruct T (, ),..., T (, ) as follows:. For -, let be the maxmum teger such that + + ( + 6) t t t + + ( + 6) for t (B-4). I orer to efe - a, efe the frst two varables t a t Let t be the maxmum teger such that t t t + + ( 7) t t t + + ( 7) for t t (B-5) Let t be the mmum teger such that t t t + + ( 5) t t t < + + ( 5) for t < t (B-6) If t k + for some k, the t + a - t. Otherwse, t k + for ay k. If t t +, let - t + t ; else t t +, let - t a t. It wll frst be prove that the costructe perfect stace has several propertes. It wll the be prove that ths stace perfectly utlzes the processor. Clam : For -, s ether a multple of or a multple of. Proof. Coser frst the case -. Let k +r k + r, where k a k are tegers, 0 r a 0 r -. It wll be prove that both r a r are greater tha 0; f ot, the there are two cases to coser. B-4

95 + + a. Ether r 0 or r 0. Suppose that r 0, the + a. Therefore, equato B-4 also hols for whe t +. But by efto, s the largest teger that satsfes equato B-4. Ths s a cotracto. The same cocluso ca be reache for the case r b. r r 0, because, + a +. Therefore, equato B-4, also hols whe t +, aga, a cotracto. If -, by the costructo, ether - t or - t +. I the former case, t caot be a multple of, because otherwse the same cotracto ca be obtae as before. I the latter case, - t +, whch meas that t t +. By equatos B-5 a B-6, t+ t+ t t + +. However, ths equato hols oly whe t + s ot a multple of or. Therefore, both cases, - s ot a multple of or. Clam 4: + + x for, where x or or. Proof. Let k + r k + r, where r a r -. Smlarly, let + k (+) + r (+) k (+) + r (+),where r (+) - a r (+). There are three cases to coser: r a r -. + The + a +. Let t +. It s easy to show that equato B- 4 hols for +, but for ay t > t, equato B-4 oes ot hol. Therefore, + +. r - a r, or r - a r -. + Suppose that r - a r -, the + a +. Let t +. Oe ca show that equato B-4 hols for +, but for ay t > t, equato B-4 oes ot hol. Therefore, + + ths case. For the case r - a r -, the proof s the same. r - a r, or r - a r -. Usg the same argumet as above, oe ca prove that + +. Clam 5: Let k + r k + r, where r - a r -. Express t terms of. The expresso epes o the value of r a r, a t s show tables B-9 to B-6. B-4

96 r r. See table B-9. TABLE B-9. VALUE OF t WHEN r r 4 5 or 6 or r a r. See table B-0. TABLE B-0. VALUE OF t WHEN r AND r r a r. See table B-. TABLE B-. VALUE OF t WHEN r AND r a r a r r a r >. If 5 a r, the t 7; otherwse, see table B-. TABLE B-. VALUE OF t WHEN r AND r > + r or or r r r a r. If r, the t 5; otherwse, t - 4. r a r. If r r, the t 5; otherwse t -. B-44

97 r a r >, or r > a r respectvely.. See tables B- a B-4, TABLE B-. VALUE OF t WHEN r AND r > r r r TABLE B-4. VALUE OF t WHEN r > AND r r r r r > a r. See table B-5. TABLE B-5. VALUE OF t WHEN r > AND r + r a + r a > r or r r > a r >. See table B-6. TABLE B-6. VALUE OF t WHEN r > AND r > + r + r + r + r + r + r To check the correctess of the values of t for fferet cases, t s suffcet to show that each case t + + 7) t t ( B-45

98 t t t + ( 7) + t + > + + 7), t+ t+ ( all of whch ca be easly verfe. Clam 6: Let k + r k + r, where r - a r -. Depeg o r a r, the value of t s show tables B-7 to B-. r r. See table B-7. TABLE B-7. VALUE OF t WHEN r r r a r. See table B-8. TABLE B-8. VALUE OF t WHEN r AND r r r 4 r a r >. See table B-9. TABLE B-9. VALUE OF t WHEN r AND r > + r + r + r + 4 r B-46

99 r a r. t -. r a r. t. r a r >, or r > a r respectvely.. See tables B-0 a B-, TABLE B-0. VALUE OF t WHEN r AND r > r r r TABLE B-. VALUE OF t WHEN r > AND r r r r r > a r. See table B-. TABLE B-. VALUE OF t WHEN r > AND r + r or r r r > a r >. See table B-. B-47

100 TABLE B-. VALUE OF t WHEN r > AND r > + r + r + r + r + r + R Clam 7: Base o t a t, - a are costructe as show tables B-4 to B-6. r r. See table B-4. TABLE B-4. VALUE OF - AND WHEN r r 4 5 or r a r. See table B-5. TABLE B-5. VALUE OF - AND WHEN r AND r B-48

101 r a r. See table B-6. TABLE B-6. VALUE OF - AND WHEN r AND r a r a r r a r. If 5 a r, the - -7 a -; otherwse, see table B-7. TABLE B-7. VALUE OF - AND WHEN r AND r + r or r + 4 r + 5 r r a r. If r, the - t -5 a t ; otherwse, - -. r a r. If r r, the a ; otherwse, t t + a hece - t + -. r a r >, or r > a r respectvely.. See tables B-8 a B-9, TABLE B-8. VALUE OF - AND WHEN r AND r > + r r r - - B-49

102 TABLE B-9. VALUE OF - AND WHEN r > AND r + r r r - - r > a r. See table B-0. TABLE B-0. VALUE OF - AND WHEN r > AND r + r a + r a + > r r + 4 r r > a r >. See table B-. TABLE B-. VALUE OF - AND WHEN r > AND r > + r + - r + - r r + + r r Clam 8: - s ot a multple of or. Proof. There are three cases to coser: - t,.e., t +. By costructo of, ths case happes whe t k + for some k. From the tables clam 7, oe ca see that ths case happes oly whe r > a r >, where t +. Let k + r. The t k +r +. Sce r, B-50

103 + r + <. Therefore, (k + ) < t < (k + ), whch mples that t caot be a multple of. Smlarly, oe ca prove that t s ot a multple of. - t -,.e., t. t t t t Suppose that t - s a multple of but ot, the a. Hece, equato B-6 also hols for t -. However, by efto, t s the mmum teger that satsfes equato B-6. Ths s a cotracto. Usg a smlar argumet, oe ca prove that - caot be a multple of. - t -,.e., t -. I ths case, -, whch meas that t t +. By equato B-5 a B-6, oe must t t have a t t. Thus, - or t - caot be a multple of or. Theorem : The perfect stace T, T,..., T costructe above perfectly utlzes the processor. Below are some lemmas about the DM scheule for the perfect stace. The theorem follows rectly from these lemmas a the efto of perfect utlzato. Lemma : For the DM scheule of the perfect stace, there s o le tme before t. Proof. It s eough to prove that at ay tme t < t, the umber of requests R from T,..., T -, T - s greater tha or equal to t. There are two cases epeg o t. t. t j for j. So T,, T - each makes exactly oe request before t. O the other ha, k t k for k. So T,..., T - each makes exactly two requests. Therefore, the total umber of requests before tme t from T,..., T - s t t R By costructo, t R. < t < +. t < j for j +. So T +,,T - each makes exactly oe request before t. O the other ha, j < t j for j. So T,..., T each makes exactly two requests. Therefore, at tme t, the total umber of requests from T,..., T - s t t R ( ) By the costructo of +, t R. Lemma : There s o suspeg request from T,..., T -, T - at t. B-5

104 Proof. Oe ca show that the umber of requests from T,...,T - urg the pero [0, t ) s equal to t. Lemma 4: Noe of T 4,..., T - wll make a request urg the pero [t, -]. Proof. Because t j < < + < j before t a wll ot make ts thr request before -. for 4 j -, Tj makes exactly two requests Lemma 5: There s o le tme urg the pero [t, -) a there s o request from T, T, T, a T - at -. Therefore, the frst request from T wll be execute at -. Proof. Let t t -. To prove that there s o le tme from t to -, t s suffcet to show that at t the umber of requests from all tasks except T s greater tha or equal to t. Coser frst t. At t the requests from T a T are t a t, respectvely. Task T j, j -, has two requests. Together wth T,, T -, there are -6 requests. The total umber of requests by T,..., T - s t t t < ( 5) + + R. If t - or t +, oe ca use a smlar argumet. t t R ( 6) + +. By the efto of t, It wll ow be show that there s o request release at -. It has alreay bee show that - s ot a multple of or, so ether T or T makes a request at -. Sce ether < - < or +, T wll ot make ts thr or fourth request at -. By costructo, ether - - or - < - < -. So T - wll ot make ts seco or thr request at tme -. Combg Lemmas,, a 4, the processor s busy from tme 0 to -, all requests from T,, T - have bee execute before tme -, a there s o request release at tme -. So the frst request of T wll be execute at tme -. By efto, the perfect stace perfectly utlzes the processor. Thus, theorem s prove. + B.5. PERFECT INSTANCE WHEN >. Frst, the followg lemma s very easy to verfy. + Lemma 6: Gve a >, let the perfect stace for,, a be TS { T (, ), T (, ), T (, ),..., T (, )}, the +. + Clam 9: Gve a >, let the perfect stace for,, a be TS { T (, ), T (, ), T (, ),..., T (, )}, a the total utlzato be U(). Let B-5

105 TS { T ( ) T ( ) T ( )..., T (,,, +,,,, )} +, a ts total utlzato be U( + ), the U( ) U( +). be the perfect stace for,, a The proof s by showg that U( ) U( + ) > 0 for every possble a. The cases are lste table B-. TABLE B-. THE CASES OF AND DEPENDING ON AND + 4 [, + + ] 4 + [, ] [, ] * 6 ** ** 7 ** ** 5 * * 8 ** ** 7 ** 9 * * * 0 ** ** 4 9 * * * * 0 ** ** 9 + * * * * * 0 ** 9 + * * * * * * [, ] * * * * * * * * * * * * * * * * * * * * * * * * 7 Note: * mpossble cases because. ** mpossble cases because both a are tegers. Other etres are the umbers of lemmas that prove the correspog cases. From llemma 6, t follows that +, r +, r + a r r -. Let S ) ( a S ( + ) ( + ). The + + U ( ) U ( + ) S S. ( + ) The values of, 4, epe o the values of a. Sce S a S epe o the values of, ther values epe o a as well. The clam wll be prove for each case. For all cases, the ea s the same: From a, oe ca obta r, r a r. Applyg clam 7, t s easy to calculate S. To calculate S, oe ees to f a for 4 -. Specfcally, oe ees to express a terms of. As t has bee show clam 4, + s usually greater tha by. Thus, +-. But t has bee prove that caot be a multple of or. Therefore, oe ees to o some correctos to ths formula to jump over those multples of or. Itally, s crease by to get +, so + -. At some pot. To jump over, the formula becomes +-. After that s cotuously crease by to +, utl at aother pot where - or -, the the formula s chage aga. These pots (the ces of the tasks, -) where the formula s correcte vares wth a. B-5

106 ( ) Lemma 7: U( ) U( +) whe + <. 6 Proof. + < r, + 6 < r, a + < r. By table B- clam 7, - -, a. Therefore, S 0. To calculate S, frst coser the case <. + < <. + f + + f + + f + + f + + f + f + + f f + + f f + Base o the above expressos, the cases whe a take fferet values are lste table B-. TABLE B-. AND WHEN ( ) + < S 5 ( ) ( ) ( +) U ( ) U ( + ) + + 0> 0 5 ( ) 6 ( ) Now coser the case where. B-54

107 + f + + f f + f + + f + + f + + f or f + Base o the above expressos, the cases whe a take fferet values are lste table B-4. TABLE B-4. AND WHEN ( ) S ( ) ( ) U( ) U( + ) (( + ) ( )) 0> 0. ( + ) Lemma 8: U( ) U( +) whe a Proof. r, r, a (+). Therefore, S , a r. By table B- clam 7, + f + + f + + f + + f + + f 4 or or B-55

108 + f + + f f + + f Base o the above expressos, the cases whe a take fferet values are lste table B-5. + TABLE B-5. AND WHEN S ( ) ( + + ) + U ( ) U ( + ) + ( + ) ( + + ) 0> 0. ( ) + 4 Lemma 9: U( ) U( +) whe a Proof. r, r, +, a r. By table B- clam 7, - - +, + +4, -, a Therefore, + S (+ ) + f + + f + + f + + f f or B-56

109 + f + + f f + + f Base o the above expressos, the cases whe a take fferet values are lste table B-6. + TABLE B-6. AND WHEN S + ( + ) ( ) U ( ) U ( + ) + ( + + ) + > 0. ( ) ( ) ( ) ( ) + 4 Lemma 0: U( ) U( +) whe a Proof. r, r, +, a r. By tables B-9 a B- clam 7, - - +, + +4, a Therefore, S (+ )(+ 4) + f + + f + + f + + f + + f + f + + f f + + f B-57

110 Base o the above expressos, the cases whe a take fferet values are lste table B-7. + TABLE B-7. AND WHEN S + ( + ) ( ) U ( ) U ( + ) > 0. ( ) ( ) ( ) ( )( 4) + 4 Lemma : U( ) U( +) whe a or + or Proof. Frst, calculate S f + + f + + f + + f + f + + f f + + f Base o the above expressos, the cases whe a take fferet values are lste table B TABLE B-8. AND WHEN AND OR + OR B-58

111 S ( + ) To calculate S, coser the followg three cases: r, r, +, a r. By tables B-9 a B- clam 7, - - +, + +4, a Therefore, S (+ )(+ 4) U ( ) U ( + ) > 0. ( ) ( ) ( )( 4) r, r, +, a r. By tables B-9 a B- clam 7, - -, + +4, a Therefore, S U ( ) U ( + ) + + ( + + ) > 0. ( ) ( ) r, r, +, a r. By table B-9 clam 7, a Therefore, S 0. U ( ) U ( + ) + + 0> 0. ( ) ( ) + Lemma : U( ) U( +) whe a Proof. r, 4 r <, a r r <. By table B-9 clam 7, a - -. Therefore, S 0. B-59

112 .. + f + + f + + f + + f + f + + f + + f f + If -, the a. If - +, the a +. S + ( + ) U ( ) U ( + ) 0> 0. ( + ) ( + ).. + f + + f + + f + f + + f + + f or + + f + If -, the a. If - +, the a +. S ( + ) B-60

113 U ( ) U ( + ) 0> 0. ( + ) ( + ) f f f f + + f + + f + + f + If - +, the a +. S ( + )( + ) U ( ) U ( + ) 0> 0. ( + ) ( + ) ( + ) f + + f + + f + + f + + f f + If - +, the a. S ( + ) U ( ) U ( + ) 0> 0. ( + ) ( + ) B-6

114 f + + f + + f + + f + If, + f + + f + + Otherwse, + f + + f f + If - +, the a. If - +, the a +. S + ( + ) U ( ) U ( + ) 0> 0. ( + ) ( + ) f + + f + + f + or + + f + + f + + f f + B-6

115 If - +, the a. If - +, the a +. S + ( +) U ( ) U ( + ) 0> 0 ( + ) ( + ) f + + f + + f + + f + f + + f f + + f If - +, the a. If - +, the a +. For all other - + or - +,. S + U ( ) U ( + ) + ( + ) 0> 0. ( ) Lemma : U( ) U( +) whe or or + a +. + Proof. r >, 4 r <, a a - -. Therefore, S 0. r r <. By table B-9 clam 7, B-6

116 + f + + f + + f + + f + + f f + If - +, the a. If - +, the a +. For all other - + or - +,. S + U ( ) U ( + ) + ( + ) 0> 0. ( ) Lemma 4: U( ) U( +) whe a. + Proof. r, r,, a + r. By clam 7, - + a - +. Therefore, S 0. + f + + f f or + f + + f + + Base o the above expressos, the cases whe a take fferet values are lste table B-9. TABLE B-9. AND WHEN AND B-64

117 S ( )( ) U ( ) U ( + ) + 0> 0. ( ) ( )( ) + Lemma 5: U( ) U( +) whe a Proof. r, r, a r. By table B-9 clam 7, +,, a. Therefore, + S 4 + ( )(+ ) f + + f + + f + + f + f + + f + + f + Whe,, a. S ( )( ). + U ( ) U ( + ) ( ) > 0. 4 ( + ) ( )( ) ( )(+ ) f + + f + + f B-65

118 + f + + f + + f Whe,, a. S ( )( ) + U ( ) U ( + ) ( ) > 0. 4 ( + ) ( )( ) ( )(+ ) + Lemma 6: U( ) U( +) whe Proof. r r, +, a r. By clam 7, +, + + 4, - - +, a Therefore, S 0. + f + + f f or + f + + f + + f f or Base o the above expressos, the cases whe a take fferet values are lste table B-40. TABLE B-40. AND WHEN S ( + ) ( + ) ( + )( + ) ( ). U ( ) U ( + ) + ( ) 0> 0. ( ) ( )( ) ( ) B-66

119 Lemma 7: U( ) U( +) whe or a +. Proof. Frst calculate S. + f + + f + + f f or + f + + f f f or Base o the above expressos, the cases whe a take fferet values are lste table B-4. TABLE B-4. AND WHEN AND + S or ( ) ( ) ( + ). Now calculate S.. a r, r, +, a r. By tables B- a B-9 clam 7, +, + + 4, a - +. Therefore, S ( + ) (+ )( + ) U ( ) U ( + ) > 0. ( ) ( ) ( )( 4) B-67

120 . + a r, r, +, a r. By table B- clam 7, - +, + +4, -, a Therefore, S ( + ) ( + ) (+ ) U ( ) U ( + ) > 0. ( ) ( ) ( ) Lemma 8: U( ) U( +) whe Proof. r r, +, a r. By table B- clam 7, - - +, + +4, -, a Therefore, S + ( + ) ( + ) (+ ) + f + + f f + f + + f + + f + + f Base o the above expressos, the cases whe a take fferet values are lste table B-4. S TABLE B-4. AND WHEN ( + ) ( + ) ( + )( + ) ( ) B-68

121 U ( ) U ( + ) + ( ) + > 0. ( ) ( )( ) ( ) ( ) Lemma 9: U( ) U( +) whe a. Proof. + f + + f + + f + + f + S 0. r, r a r. By tables B-9 a B-0 clam 7, oe ca get table B TABLE B-4. -,, -, AND WHEN AND - r - r S 4 + ( )(+ ) U ( ) U ( + ) ( ) 0> 0. ( + ) ( )(+ ) +.. r, r, a r. By tables B-9 a B-0 clam 7, oe ca get table B TABLE B-44. -,, -, AND WHEN AND - r - r B-69

122 S ( )( + ) + U ( ) U ( + ) ( ) > 0. ( + ) ( )( + ) +.. r, r, a r. By tables B-9 a B-0 clam 7, oe ca get table B TABLE B-45. -,, -, AND WHEN AND - r - r S ( )(+ ) U ( ) U ( + ) 0> 0. ( + ) ( )(+ ) r, r, a r. By tables B-9 a B-0 clam 7, oe ca get table B TABLE B-46. -,, -, AND WHEN AND - r - r S ( ) U ( ) U ( + ) > 0 ( + ) ( ) r, r a r. By tables B-9 a B-0 clam 7, oe ca get table B-47. B-70

123 TABLE B-47. -,, - AND WHEN + AND - r - r S ( ) U ( ) U ( + ) > 0. ( + ) ( ) + + Lemma 0: U( ) U( +) whe or or or a. Proof. Frst coser S. + f + + f + + f + + f + + f + If +, the + a +. S + + ( + )( + ) Now coser S r r, +, a r. By tables B-9 a B-0 clam 7, - - +, + +4, a Therefore, S ( + ) (+ 4)(+ ) U ( ) U ( + ) + ( + + ) + + > 0. ( ) ( )( ) ( 4)( ) B-7

124 r r, +, a r. By clam 7, - - -, + +4, - -, a +. Therefore, S ( + ) ( + ) ( + )( + ). U ( ) U ( + ) + ( + + ) + + > 0. ( ) ( )( ) ( )( ) r r, +, a r. By clam 7, -, a -. Therefore, S 0. U ( ) U ( + ) + ( + + ) 0> 0. ( ) ( )( ) r r, +, a r 5, -. Therefore, S 0.. By clam 7, - U ( ) U ( + ) + ( + + ) 0> 0. ( ) ( )( ) + Lemma : U( ) U( +) whe a Proof. r, r, +, a r 5 +, a - +. Therefore, S 0.. By clam 7, - + f + + f + + f f + + f f + B-7

125 Base o the above expressos, the cases whe a take fferet values are lste the table B TABLE B-48. AND WHEN AND S ( + ) ( + ) ( + ). U ( ) U ( + ) + + 0> 0. ( ) ( ) Lemma : U( ) U( +) whe a. + Proof. r, r, +, a r. By table B-9 clam 7,, +, - -, a +. Therefore, S 0. + f + + f f or + f + + f + + Base o the above expressos, the cases whe a take fferet values are lste the table B-49. TABLE B-49. AND WHEN AND S ( )( ) U ( ) U ( + ) + 0> 0. ( ) ( )( ) B-7

126 Lemma : U( ) U( +) whe + a -. + Proof. 4 r, r,, a r +. By table B-9 clam 7, a -. Therefore, S 0. + f + + f f or + f + + f + + Base o the above expressos, the cases whe a take fferet values are lste the table B-50. TABLE B-50. AND WHEN + AND S ( )( ) U ( ) U ( + ) + 0> 0. ( ) ( )( ) + Lemma 4: U( ) U( +) whe a Proof. It ca be show that r, r <, a a -. Therefore, S 0. r <. By clam 7, - + f + + f + + f + + f + + f f + B-74

127 Base o the above expressos, the cases whe a take fferet values are lste the table B-5. + TABLE B-5. AND WHEN AND S ( + ) U ( ) U ( + ) + + 0> 0. ( ) ( ) Lemma 5: U( ) U( +) whe + <. Proof. Oe ca show that r <, r < a r <. By clam 7, - - a - -. Therefore, S 0. Frst, suppose that <. The + f + + f + + f + + f + + f f + If - +, the a. If - +, the a +. For all other - + or - +,. S + U ( ) U ( + ) ( ) 0> 0. ( + ) + B-75

128 Now, coser the case where. + f + + f + + f + + f + + f + If - +, the a +. For all other - +,. S + + ( + )( + ) U ( ) U ( + ) 0> 0. ( + ) ( + )( + ) Lemma 6: U( ) U( +) whe -. Proof. r r a r. By clam 7, - - a - -. Therefore, S 0. + f + + f + f + + f If -, the - a I -. S. ( )( ) U ( ) U ( + ) + 0> 0. ( ) ( )( ) B-76

129 Lemma 7: U( ) U( +) whe -. Proof. r, r a r. The values of -,, -, a are lste table B-5. TABLE B-5. -,, -, AND WHEN - r - r S. (+ ) For -, + a + -. S 0. U ( ) U ( + ) > 0. ( ) ( ) B.6 UNIT-EXECUTION-TIME TASK SYSTEMS: LIMITED PRIORITY LEVELS. It was plae that the same problem be stue as secto B.5, wth the assumpto that each processor has m prorty levels. Ufortuately, because of lmte tme, o sgfcat progress was mae. It was, however, cojecture that the same threshol hols for computg systems wth m prorty levels, where m <. B.7 REFERENCES. B-. B-. B-. B-4. B-5. B-6. B-7. Labetoulle, J., Some Theorems o Real-Tme Scheulg, Computer Archtecture a Networks, E. Gelebe a R. Mahl, es., North-Holla, Amsteram, 974. Lu, C.L. a Layla, J.W., Scheulg Algorthms for Multprogrammg a Har Real-Tme Evromet, J. of ACM, Vol. 0, 97, pp Leug, J.Y-T. a Merrll, M.L., A Note o Preemptve Scheulg of Peroc, Real- Tme Tasks, Iformato Processg Letters, Vol., 980, pp Garey, M.R. a Johso, D.S., Computers a Itractablty: A Gue to the Theory of NP-Completeess, Freema, New York, 979. Leug, J.Y-T., A New Algorthm for Scheulg Peroc, Real-Tme Tasks, Algorthmca, Vol. 4, 989, pp Lawler, E.L. a Martel, C.U., Scheulg Perocally Occurrg Tasks o Multple Processors, Iformato Processg Letters, Vol., 98, pp. 9-. Leug, J.Y-T. a Whtehea, J., O the Complexty of Fxe-Prorty Scheulg of Peroc, Real-Tme Tasks, Performace Evaluato, Vol., 98, pp B-77

130 B-8. B-9. Coffma, Jr., E.G., Garey, M.R., a Johso, D.S., Approxmato Algorthms for B Packg: A Survey, Approxmato Algorthms for NP-har Problems, D. Hochbaum e., PWS Publshg Compay, 996. Dhall, S.K. a Lu, C.L., O a Real-Tme Scheulg Problem, Operatos Research, Vol. 6, 978, pp B-0. Lu, J.W.S., Real-Tme Systems, Pretce Hall, New Jersey, 000. B-78

131 APPENDIX C THE IMPLEMENTATION OF THE ALGORITHM DM-LPL /****************** greey-sgle.h *********************/ #fef _GREEDY_SINGLE_H_ #efe _GREEDY_SINGLE_H_ #clue <stlb.h> #clue <strg.h> #clue <math.h> #clue <lmts.h> #efe INFINITY INT_MAX /*Prorty s the hghest*/ typeef struct Task{ t exe; t eale; t pero; t prorty; } Task; t greey_sgle(t um_task, t um_prorty, Task *task_sys); #ef /****************** greey-sgle.c Fucto: t greey-sgle(t um_task, t um_prorty, Task *task_sys) put: um_task: umber of tasks um_prorty: umber-of-prorte levels task_sys: poter to the tasks array output: The prorty assgmet to tasks Algorthm:. sort the tasks o-ecreasg orer of eale. startg wth prorty level a the frst task, repeat for each task: -: try to assg the curret task to the curret prorty level -: f - fals, f there s o task assge the curret prorty level, the the task system s ot scheulable, retur - else f there are avalable prortes, C-

132 crease the curret prorty level by else there s ot eough prortes, retur 0 -: f - succees, let the ext task be the curret task.. all tasks are assge prortes, retur Harog Zhao 07/06/00 ******************/ #clue greey-sgle.h /********************************************************/ t compare( cost vo *task, cost vo *task); /* Compare eales: retur value (eale of task - eale of task) **/ /********************************************************/ /*******************Ma program ************************/ t greey_sgle(t um_task, t um_prorty, Task *task_sys) { t ; t retur_val; t cur_prorty; t cur_exe_sum; t cur_frst_task; /**************************************/ /** Sort accorg to eale ***/ /**************************************/ qsort((vo *)task_sys, um_task, szeof(task), compare); /**************************************/ /********** assg prorty a greey way **********/ /*********************************************/ cur_prorty ; C-

133 /* the task wth the smallest eale the curret level */ cur_frst_task 0 ; task_sys[cur_frst_task].prorty cur_prorty; /* assg the frst task wth hghest prorty */ cur_exe_sum task_sys[cur_frst_task].exe ; /* the total executo tme of the tasks the curret prorty level */ ; /* assg prorty to task,... */ whle ( < um_task){ t tme; /* try to assg curret task to curret prorty */ task_sys[].prorty INFINITY; /* talze */ cur_exe_sum cur_exe_sum + task_sys[].exe; /* verfy whether the smallest task ths prorty s stll scheulable */ for(tme cur_exe_sum; tme < task_sys[cur_frst_task].eale; tme ++){ t requests cur_exe_sum; t j; for(j 0; j < ; j++){ f(task_sys[j].prorty < cur_prorty) requests requests + task_sys[j].exe * cel(tme*.0/task_sys[j].pero); } f(tme requests){ task_sys[].prorty cur_prorty; break; } } f(task_sys[].prorty INFINITY){ /* task ca t be assge to the curret level */ f( cur_frst_task) C-

134 } break; /* s the frst task the curret level. So the tasks are ot scheulable. */ f(cur_prorty < um_prorty) { /* try to assg to the ext level*/ cur_prorty cur_prorty +; cur_frst_task ; cur_exe_sum 0; } else /* ot eough prorty level */ break; } else{ /* assg the ext task */ +; } f( < um_task) { f( cur_frst_task) retur_val -; /* ot scheulable */ else retur_val 0; /* ot eough prorty levels */ } else retur_val ; /* all tasks are assge prortes. */ retur (retur_val); } t compare( cost vo *task, cost vo *task) { retur (((Task *)task)->eale - ((Task * )task)->eale); } C-4