Schedulability Bound of Weighted Round Robin Schedulers for Hard Real-Time Systems

Transcription

1 Schedulablty Bound of Weghted Round Robn Schedulers for Hard Real-Tme Systems Janja Wu, Jyh-Charn Lu, and We Zhao Department of Computer Scence, Texas A&M Unversty {janjaw, lu, Abstract We derve a parameterzed, closed-formed schedulablty bound for weghted round robn schedulers n hard real-tme computng systems. The schedulablty bound uses several parameters to represent mportant system behavors: (1) Number of tasks; (2) Normalzed deadlne that measures the tghtness of task deadlnes; (3) Tasks set workload burst-ness; (4) Overhead rato, whch measures the porton of overhead tme consumed n round robn operatons; (5) Normalzed token rotaton frequency, that measures the number of token rotatons n a tme nterval of the shortest relatve deadlne. Our work follows the network calculus representaton framework and s based on a generalzed workload and servce approxmaton models. We derve a closed form expresson of the utlzaton bound, so that one can analyze the mpact of partcular system parameters on the schedulablty bound and schedulablty bounds for specfc system confguratons can be easly obtaned by smply pluggng n proper parameters. 1

2 TABLE OF CONTENT 1. INTRODUCTION SYSTEM MODEL TASK, TASK SET, WORKLOAD, AND SERVICE FUNCTION TASK MODEL Workload Constrant Functon S-shaped Workload Constrant Functons Parameters of Task Set SCHEDULER MODEL Servce Constrant Functon Weghted Round Robn Scheduler Servce Constrant Functon of WRR Scheduler Parameters of Weght Round Robn Schedulers SCHEDULABILITY BOUND SCHEDULABILITY BOUND A LOWER BOUND OF SCHEDULABILITY BOUND SCHEDULABILITY BOUND OF WEIGHTED ROUND ROBIN SCHEDULERS PARAMETERIZED SCHEDULABILITY BOUND OF WRR SCHEDULER PARAMETER SENSITIVITY ANALYSIS OF THE WRR SCHEDULABILITY BOUND OPTIMAL TTRT SELECTION TO MAXIMIZE SCHEDULABILITY BOUND COMPARISON WITH EXISTING RESULTS COMPARISON WITH STATIC PRIORITY SCHEDULERS COMPARISON WITH TIMED TOKEN PROTOCOL FINAL REMARKS...28 REFERENCES...29 APPENDIX

3 1. INTRODUCTION Weghted round robn (WRR) schedulng dscplne has been mplemented n a broad range of msson crtcal computng and communcaton systems. In a WRR scheduler, tasks are performed n a cyclc order, n whch the tme a task can execute wthn each round s proportonal to the weght assgned to t. Weghted round robn schedulers have two major advantages: Ablty to mprove the system robustness, gven ther guarantee of a mnmum servce rate for each task. For weghted round robn schedulers, the maxmum amount of servce every task can receve n each round s upper-bounded by ts allocaton. As such, no task can consume more servce than what has been assgned. Ablty for mplementaton n a dstrbuted fashon. A WRR scheduler can be easly realzed n a dstrbuted envronment through a logc rng lke the tmed token protocol [2]. By passng a token along the rng, nodes can access resource n turn based on ther weght assgnments. It has been proven n [21] that wth proper weght assgnment, weghted round robn schedulers can provde deadlne guarantees for real-tme computng systems. Yet a major challenge s how to devse a low complexty schedulablty test, whch would guarantee the deadlne requrements, acheve hgh level of resource utlzaton, and s applcable to a broad range of system confguratons. Accordng to [22], a schedulablty test can be drect or ndrect. A drect schedulablty test explctly calculates the precse worst-case delay of each task n order to determne the permssblty of a new task. Despte ts accuracy, ths type of test has hgh run-tme cost n calculatng the worst case task delays, and therefore may not be sutable for on-lne admsson control. In contrast, an ndrect schedulablty test would use one or more system ndcators to determne the schedulablty of a new task wthout computng the worst case task delays drectly. The utlzaton based schedulablty test s the most common ndrect schedulablty test, n whch a new task can be admtted only f the total system utlzaton (as the system ndcator) s lower than a pre-determned bound. Utlzaton bound based schedulablty test s hghly desrable for large, 1

4 complex systems, because of ts extreme effcency and the ablty to provde a safe operaton margn by settng the system utlzaton bound to a value lower than the proven utlzaton bound [22]. Despte ts extreme smplcty n on-lne admsson control, lmted results have been obtaned on schedulablty bound of WRR schedulers. Most exstng work about WRR schedulers has focused on maxmzng the resource utlzaton through clever weght assgnments, wthout consderng the complexty and overhead of the schedulablty test algorthms. In many large-scale systems, t s desrable to trade some level of resource utlzaton for a smple and fast schedulablty test. To acheve ths goal, ths paper proposes to employ the general framework developed n [22] to analyze the schedulablty bound of the weghted round robn schedulers. We focus on a specal weght assgnment scheme n whch task weghts are assgned n proporton to ther resource demand rates. Ths weght assgnment scheme matches many resource allocaton requrements n practce,.e. assgnng larger weghts to users wth greater resource consumpton rates. We derve a closed-form expresson for the schedulablty bound of WRR schedulers wth normalzed weght assgnments. The schedulablty bound s parameterzed for number of tasks, the normalzed deadlne, whch measures the tghtness of deadlne assgnment, the tasks set workload burst-ness, the overhead rato, whch measures the porton of overhead tme consumed n round robn operatons per round, and the normalzed token rotaton frequency, whch measures the number of token rotatons n a tme nterval of the shortest relatve deadlne of the tasks. Unlke the relatvely developed knowledge body on the schedulablty bounds of statc prorty schedulers [1], [4], [5], [6], [7], [12], [15], [16], [17], [18], [19], [22], earlest deadlne frst schedulers [17], tmed token rng schedulers [2], [3], [11], [24], [25], [26], [27], [28], [29], [3], and ther varatons, no known bounds could be found for the weghted round robn schedulers n the lterature. Ths paper presents the frst systematc results of schedulablty bounds for weghted round robn schedulers, derved usng a formal modelng and optmzaton technque developed for general real-tme systems. Its closedform expresson allows one to easly analyze the mpact of system parameters over the schedulablty 2

5 bounds, and to effcently and effectvely fnd the schedulablty bounds for specfc system confguratons through smple substtuton of parameter values. The hghly versatle modelng and optmzaton method presented n ths paper can be talored for analyss of other types of real-tme systems. The rest of the paper s organzed as follows. Secton 2 ntroduces our general system model. Secton 3 dscusses the schedulablty bound analyss methodologes and derves a general bound result for arbtrary schedulers. In Secton 4, a closed-form schedulablty bound s derved and analyzed for WRR scheduler. Detaled comparsons of the newly derved bound wth the known results of other schedulers are provded n Secton 5, followed by conclusons n Secton SYSTEM MODEL The closed-form expresson of the schedulablty bound for WRR based real-tme schedulers s derved by applyng a constraned optmzaton technque to the workload-servce models proposed n [22]. To make ths paper self contaned, we ntroduce the generalzed system model n ths secton. The detals on dervaton of the schedulablty bound for WRR schedulers usng the general model wll be gven n the next secton. 2.1 Task, Task Set, Workload, and Servce Functon Ths paper consders sngle processor systems. We use Γ = { T1, T2,..., T n } to denote a task set, where T s the th task. When the context s clear, we may omt ndex n the subsequent dscussons. Each task s composed of a sequence of jobs. The worst-case executon tme of a job s called the job sze, whch s measured n second. A job can start ts executon after ts release tme, t r, and must be fnshed by ts absolute deadlne t d = t r + D where D s called relatve deadlne. For a job, the tme elapsed from the release tme t r to the completon tme t f s called the delay of the job, and the worst-case (.e., largest) 3

6 delay of all jobs n a task s denoted by d *. Wthn a task, the jobs have the same relatve deadlne, but may not necessarly be the same sze. Jobs wthn a task are executed n a frst come, frst served order. To characterze the resource demand of task T analytcally, we defne f() t, the workload functon for T, as follows, f ( t) = the summaton of the szes of all the jobs from T n [, t]. (2.1) Smlarly, to characterze the actual processor tme receved by task T, we defne g() t, the servce functon for T, as follows, gt ( ) = the total executon tme rendered to jobs of task Tdurng [, t]. (2.2) Based on the defntons of d *, f() t and g() t, the followng worst case delay formula can be easly derved [7]: ( ( τ τ )) * d t f t g t = sup nf ( ) ( + ). (2.3) From [8] and [22], we know that n (2.3) nf ( τ f( t) g( t τ )) and thus * d s the worst case delay of all the jobs from task T. + s the delay of the jobs arrvng at tme t A basc functon of the schedulablty test algorthm s to determne whether the followng nequalty holds: * d D. (2.4) One may want to use (2.3) to calculate d * and then compare the result wth D to test the schedulablty. However, ths method may not be sutable for onlne operaton because the exact forms of f () t and g() t may not be avalable when schedulablty test s made. Furthermore, even f f () t and g() t 4

7 are avalable, they are often too cumbersome to handle. A practcal soluton s usng some alternatve forms of f () t and g() t that can be obtaned durng schedulablty test. 2.2 Task Model Workload Constrant Functon Much work has been performed to fnd alternatves to f () t n order to model task workload for delay analyss. For example, a typcal alternatve s the workload constrant functon F( I ) ntroduced n [13] and [14] (under the name of workload curve). F( I ) s sad to be a workload constrant functon for task T f for any I t, f () t f( t I) F( I). (2.5) It s obvous that, gven any F that satsfes (2.5), F( I ) s an upper bound of total sze of jobs that can be released n any tme wndow [ t I, t]. We use I n (2.5) because F s defned on the doman of tme ntervals, whle f () t s defned on the doman of tme. Note that (2.5) defnes a group of functons, and for analyss convenence, we wll focus on those whch are non-decreasng and satsfy F () = S-shaped Workload Constrant Functons In ths paper, we consder a specal class of F, called s-shaped workload constrant functon. As ts name suggests, an s-shaped workload constrant functon conssts of segmented peces, and resembles a starcase. The values of an s-shaped workload constrant functon ncrease only at border ponts of segments. We assume that the segment length S s fxed and the ncrements may not be dentcal for the frst L segments where L s a parameter n the functon. 5

8 F() I L = 4 C C C 3 C 4 C 1 C 2 S 2S 3S 4S 5S I Fgure 1. An Example S-Shaped Workload Constrant Functon Formally, an s-shaped workload constrant functon can be expressed as follows: F( I) = h j C h L j = 1 L j C + ( a L) C h > L j = 1, (2.6) where h = I / S, j C s the ncrement at the begnnng of the j th segment, and C s the constant ncrement after the L th segment. Fgure 1 shows an example of the s-shaped workload constrant functon. When L = 1, an s-shaped constrant functon reduces to the classcal perodc task model F( I) = I / P C. In ths paper, we assume 1 2 L C C C C L. (2.7) Parameters of Task Set To characterze the task set n the sense of ts real-tme demand, we ntroduce several parameters. These parameters wll be used n the derved schedulablty result Workload Rate 6

9 Recall that the goal of ths paper s to derve a bound based schedulablty test, such as the utlzaton based method. To do so, we need to generalze the classcal utlzaton defned for perodc tasks to accommodate other non-perodc tasks usng the workload constrant functon defned above. Generally speakng, utlzaton s the resource consumpton rate n a measurng tme wndow. For perodc systems, task perod s typcally used as the length of the measurng wndow. Ths approach s not applcable to non-perodc tasks snce one may not have a well defned "perod". As such, n [1] and [4], the authors proposed to use the task relatve deadlne as the length of the measurng wndow. Whle ths choce s smple and convenent for some cases, we fnd that t s too restrctve for the desgn of a versatle utlzaton bound analyss system. To relax the constrants, we propose to defne the length of the measurng wndow as a lnear scale of the relatve deadlne. That s, the length of the measurng wndow wll be expressed as θ D, where θ > s called the scalng parameter and D s the relatve deadlne of the task. To avod confuson wth notatons from other lteratures, we refer to ths generalzed utlzaton as the scaled workload rate, and formally express t as follows: F( θ D) W ( θ ) =, (2.8) θ D and the task set workload rate as follows: n W( θ, Γ ) = W( θ ). (2.9) = 1 When the context of dscusson s clear, the term scaled may be omtted. Snce F ( θ D ) s an upper bound of the sze of jobs that can be released n any tme wndow of length θ, W ( θ, Γ ) can be treated as an upper bound of the job releasng rate averaged n a wndow of length D θ D. Introducng θ nto the modelng process parameterzes the utlzaton measurement. For example, when θ = 1, (2.9) reduces to 7

10 the defnton provded n [1] and [4]. Ths parameterzed measurement of utlzaton enables flexble representaton of dfferent schedulng and workload scenaros, and more mportantly, leads to a unform analyss system of schedulablty bounds Normalzed Deadlne To capture the tghtness of the task deadlne requrements of dfferent systems, we defne the normalzed deadlne k for T as follows: k D / S =, (2.1) where D s the relatve deadlne of task T and S s the segment length n the s-shaped workload constrant functon defned n (2.6). We follow the conventon that for = 1, 2,..., n k = k, (2.11) k can be vewed as the deadlne usng S as the measurement unt, and t characterzes tghtness of the deadlne requrements. The smaller the k, the more dffcult t s to schedule the task Burst-ness To characterze the burst-ness of dfferent tasks we ntroduce burst-ness parameter μ for T as: FS ( )/ S μ =, (2.12) F( k S)/( k S) and the burst-ness for the task set as: μ = max ( μ ). (2.13) = 1, 2,..., n By (2.7), one can notce that μ 1. (2.14) 8

11 μ s rato between the workload rate n a tme wndow S, and that n k S. Larger μ means more bursty workload. 2.3 Scheduler Model Recall that g defned n (2.2) characterzes the amount of servces a task may receve va a WRR scheduler. That s, g reflects the behavor of the scheduler. However, we cannot practcally use the form of g as defned n (2.2) whch was ponted out n Secton 2.1. Thus, n ths secton, we wll start modelng the scheduler by consderng the alternatves of g. We wll then formally defne the WRR scheduler and present an analytcal model for t Servce Constrant Functon A common alternatve to g() t s the generalzed servce constrant ntroduced n [8], [9], and [1] (under the name of servce curve). GI ( ) s sad to be a generalzed servce constrant functon f for any t, there exsts I t that preserves the property gt () f( t I) + GI ( ). (2.15) Typcally, we assume that GI ( ) s non-decreasng and G(). (2.15) means that for any t, we can fnd I, where I t, such that 1) all the jobs released n [, t I ] have been served, and 2) for jobs released n [ t I, t], at least G(I) amount of jobs have been served, as llustrated n Fgure 2. jobs released and served n ths nterval at least G(I ) of jobs released and served n ths nterval t - I t Fgure 2: Components n The Generalzed Servce Constrant Functon 9

12 Weghted Round Robn Scheduler Under the WRR schedulng dscplne, task servces are tme-multplexed n a cyclc, round robn fashon. A token s rotated n the cycle and a task can execute only f has the token. After recevng the token, task T can run for up to H tme unt where H s the tme allocated to a task. Typcally, H s calculated as ( ) H = O TTRT τ, (2-16) where O, O 1, s the weght of task T, and the TTRT s the target token rotaton tme, whch s the desred tme to complete one round of token rotaton, and τ s the tme overhead for token rotaton and other round robn operatons n each round, (e.g. the context swtchng cost n a sngle processor system or the propagaton delay n a dstrbuted system). Generally speakng, τ can be expressed as τ = n τ, (2-17) where n s the number of tasks n the system and τ s a overhead constant whch models the cost of context swtch or/and token propagaton delay. To use the WRR schedulng dscplne to schedule hard real-tme computng tasks, weghts O must be properly allocated. In ths paper, we wll show that an effectve weght assgnment scheme s based on the normalzed weght assgnments: O W (1) = W (1, Γ ), (2-18) where W(1) = F( D) / D s the workload of task T and W(1, Γ ) = W(1) s the workload rate of the task set defned n (2.9). Intutvely, the weghts of the tasks are assgned n proporton to ther n = 1 1

13 workload rates. Ths weght assgnment scheme matches many resource allocaton requrements n practce,.e. assgnng larger weghts to users havng greater resource consumpton rates Servce Constrant Functon of WRR Scheduler For the WRR scheduler, we have the followng result on ts servce constrant functon. Theorem 1. For WRR schedulers, a servce constrant functon for task T, s gven by I G( I) = n H. nτ + H j= 1 j (2-19) Proof: See Appendx. GI () I Fgure 3. An Example Servce Constrant Functon of Weghted Round Robn Scheduler We can make the followng observaton over (2-19): The servce constrant functons are of perodc shapes,.e. the values of the servce constrant n functon ncrease only at multples of the perod n + H 1 j at the amount of H for task τ j= T. Ths feature can be attrbuted to the fact that a WRR scheduler grants servces to each task T up to H amount of tme n each round, whose length s equal to + H. An example n τ j= 1 j servce constrant functon s gven n Fgure 3. 11

14 The value of the servce constrant functon s a decreasng functon of overhead constantτ. But an ncreasng functon of assgned weght H. Both are ntutve snce the lower the overhead and/or the bgger the weght, the more the servce tme for the task. When the overhead constant τ s zero and TTRT, the scheduler becomes the well-known Generalzed Process Sharng (GPS) system and the servce constrant functon reduces to G( I) = O I where O s the weght of task T Parameters of Weght Round Robn Schedulers To derve schedulablty bounds that can acheve hgh system resource utlzaton and s easly applcable to varous system settngs, t s mportant to select proper system parameters to capture the dynamcs of the system confguratons. We wll ntroduce two parameters,.e. overhead rato and normalzed token rotaton frequency for ths purpose. To measure the porton of tme consumed n round robn operatons relatve to the length of rotaton, we defne the overhead rato α as follows: τ TTRT α =, (2.2) where τ s the overhead constant defned n (2-17). To capture the effect of token rotaton speed on the schedulablty bound, we defne the second system parameter normalzed token rotaton frequency as follows: γ = D / TTRT mn, (2.21) where Dmn = mn( D ), (2.22) 12

15 and we assume that Dmn TTRT. (2.23) γ s the number of rounds the token rotates wthn a tme nterval of length D mn. The larger the γ, the faster the token rotates. 3 SCHEDULABILITY BOUND Wth workload and servce constrant functons defned n (2.5) and (2.15) as chosen alternatves of f () t and g() t for modelng task workload and scheduler servce, a general result on the schedulablty test usng F( I ) and GI ( ) s derved n [22], whch can be stated as n the followng theorem. Theorem 2: A task s schedulable f for any I FI ( ) GI ( + D). (3.1) where D s the relatve deadlne of the task. Though one can use (3.1) for each task to decde ts schedulablty test, t may be tme consumng, snce (3.1) needs to be checked for all I. An alternatve s the utlzaton based schedulablty test. 3.1 Schedulablty Bound For a gven system, we say that W ( θ ) s a schedulablty bound f an arbtrary task set Γ s schedulable when the followng condton holds: W( θ, ) W ( θ ) Γ <. (3.2) 13

16 The challenge s how to derve W * ( θ ) for a broad range of workload patterns and schedulng dscplnes. Let the space of all task sets be denoted as Ω,.e., Ω = {Γ}. Ω can be parttoned nto two subsets, Ω s and Ω ns, where Ω s = {Γ Γ s schedulable} (3.3) and Ω ns = {Γ Γ s not schedulable}. (3.4) W * ( θ ) s a lower bound of the workload rate of these task sets that belong to Ω ns. That s ( W θ ) W * ( θ) nf (, Γ ). (3.5) Γ Ω ns Fgure 4 llustrates ths concept. W ( θ, Γ) Ω ns W * ( θ ) Ω s Fgure 4. Illustraton of Schedulablty Bound 3.2 A Lower Bound of Schedulablty Bound In practce, t s often very dffcult to obtan an exact expresson of Ω ns. Instead, t may be desrable to use a substtuton of Ω ns n (3.5) for schedulablty analyss. Specfcally, we want to derve the schedulablty bound usng the followng nequty: 14

17 Γ Ω* ( W θ ) * W ( θ) nf (, Γ ). (3.6) In order to mantan the correctness of the schedulablty bound, Ω * must satsfy the followng constrant: * Ωns Ω. (3.7) That s, Ω * must contan all elements n Ω ns so that the derved bound wll be no hgher than that obtaned wth orgnal Ω ns. Fgure 5 llustrates ths concept. W * ( θ ) Ω ns W ( θ, Γ) * Ω Ω s Fgure 5. Relatonshp Between Ω ns and Ω *. If the Ω * s selected properly, the bound obtaned wth the substtuton could be close to the tght schedulablty bound obtaned wth Ω ns, f not the same. An nterestng queston s how to select Ω * that can work for dfferent type of tasks and schedulers. Clearly, we do not want the expresson of Ω * to depend on f and g, snce they are dffcult to obtan and handle. Instead, t s desrable to defne Ω * usng F and G. It presents an nterestng and challengng task to perform a full nvestgaton of the optons of Ω * and ther effects on the resultng schedulablty bound. Ths nvestgaton s yet to be seen. Nevertheless, we notce that the followng defnton works well for WRR schedulers: { T such that F( I) G ( I D) } * Ω = Γ Γ > +. (3.8) 15

18 That s, Ω * s the collecton of task set whch has at least one task not satsfyng (3.1). The followng Theorem proves that substtutng Ω ns wth Ω * s feasble. Theorem 3. Gven a collecton of task set Ω, a schedulablty bound wth scalng parameter θ s gven by: Γ Ω* ( W θ ) * W ( θ) = nf (, Γ ). (3.9) where * Ω and W ( θ, ) Γ are defned n (3.8) and (2.9) respectvely. Proof: By a close comparson of (3.9) and (3.5), we know that we just need to prove the followng: * Ωns Ω. (3.1) Assume (3.1) does not hold, then there must exst a non-schedulable task set Γ ' such that * Γ ' Ω. However, by (3.8) we know that for all, =1, 2,..., n, F( I) G ( I + D). By Theorem 2, we know that Γ ' s schedulable. Ths contradcts the fact that Γ ' s non-schedulable. Then the Theorem follows. Q.E.D. In the followng secton, we wll derve W for weghted round robn schedulers usng Theorem 3. 4 SCHEDULABILITY BOUND OF WEIGHTED ROUND ROBIN SCHEDULERS In ths secton, we wll analyze the schedulablty bound of WRR scheduler usng the general system model and methodology ntroduced n the prevous sectons. 16

19 4.1 Parameterzed Schedulablty Bound of WRR Scheduler Based on the task parameters defned n (2.1) and (2.13), the scheduler parameters defned n (2.2) and (2.21), we derve our schedulablty bound by substtutng the servce constrant functons of WRR scheduler derved n (2-19) nto Theorem 3 and solvng the resultng optmzaton problem. The result s stated formally n the next Theorem Theorem 4. A lower bound of schedulablty bound wth scalng parameter round robn scheduler wth normalzed weght assgnment, and s-shaped tasks s gven by θ = k / k for weghted W * ( k / k) = 1 ( 1 nα ) 1 mn( 1, k ) 1/ γ + 1 μ. (4-1) Proof: See Appendx. Q.E.D. (4-1) s a parameterzed bound result. By substtutng specfc values of these parameters nto (4-1), one can obtan the dfferent schedulablty bound results. To llustrate how the bound result can be used n practcal systems, we ntroduce the followng two examples. Example 1: Consder a smple real-tme robot controller who s responsble for the routng control of three data samplng robots, A, B, and C. The three robots are drvng at dfferent speeds and the controller communcates wth the robot at a dfferent frequency,.e. once every 2., 5., and 2. seconds, for A, B, and C, respectvely. For the three robots, the controller takes.25,.3, and.65 seconds to fnsh the route selecton and communcaton. The route selecton and communcaton must be fnshed wthn 4., 3., and 2. seconds for A, B, and C to avod robot damages. The controller uses a WRR schedulng dscplne wth target token rotaton tme of 2. seconds per round. There s a cost of.2 seconds per context swtchng (changng the robot to be served). The weghts for robots are assgned n normalzed fashon usng (2-18). Now we need to decde whether the controller can fnsh all the routng tasks wthn ther deadlnes. 17

20 From the above descrpton, we know there are three perodc tasks { T, T, T } Γ= where F( I) = I / P C (4-2) and C =.25 P = 4. D = C =.3 P = 5. D = C =.65 P = 2. D = (4-3) For ths set of tasks, by (2.2), (2.21), (2.1), and (2.12), we have α =.1, γ = 2, k = 1, and μ = 1. To decde whether the task set s schedulable or not, we must calculate the total system workload rate as follows: W F( D) C C C (4-4) D P P P (1, Γ ) = = + + = + + =.155. = By substtutng n = 3, α =.1, γ = 2, k = 1, and μ = 1 nto (4-1), we have a schedulablty bound of.66. Snce.155<.66, we conclude that task set s schedulable. Example 2: Now we consder a more complex scenaro of the above robot controller. The controller needs to send routng commands to robot A, B, C every 4., 5., and 2. seconds. The frst tme the controller communcates wth the robots takes longer and requres 1., 1.2, 1.3 seconds for A, B, and C, respectvely. After the frst tme, the tme reduces to.5,.6, and.65 seconds. Ths tme, the routng and communcaton for robot A, B, C must be fnshed wthn 8., 1., and 4. seconds (ncludng the frst tme). Agan, the controller uses a WRR dscplne wth normalzed weght assgnments and TTRT=2, τ =.2. We need to decde whether the controller can serve the three robots whle guaranteeng ther deadlnes. 18

21 From the above descrpton, we know that we can model the communcaton and routng selecton wth three s-shaped tasks: { T, T, T } Γ= where C I P F ( I) = 1 C + ( I / P 1 ) C I > P, (4.5) and C = 1. C =.5 P = 4. D = C = 1.2 C =.6 P = 5. D = 1. C = 1.3 C =.65 P = 2. D = 4.. (4-6) To decde whether the task set s schedulable or not, we should calculate the total system workload rate as follows: W F( D) C C C (4-7) D D D D (1, Γ ) = = + + = + + =.17. = For ths set of tasks, by (2.2), (2.21), (2.1), and (2.12), we have α =.1, γ = 4, k = 2, and μ = 1.5 By substtutng n = 3, α =.1, γ = 4, k = 2, and μ = 1.5 nto (4-1), we obtan a schedulablty bound of.53. Snce.53 >.17, we conclude that the task set s schedulable. The above two examples llustrate how to use the newly derved schedulablty bound for schedulablty analyss. In the next secton, we wll dscuss n detal how the dfferent parameters affect the schedulablty bound as well as a comparson wth the well known schedulablty bound of the rate monotonc scheduler. 19

22 4.2 Parameter Senstvty Analyss of the WRR Schedulablty Bound By (4-1), we can make the followng observatons on the senstvty of the schedulablty bound on the dfferent system parameters: For gven values of n, k, μ, and α, the schedulablty bound ncreases wth the token rotaton frequency. Thus, the faster the token rotates, the better chance a task set can be scheduled. Fgure 6 llustrates ths trend for the case of α =. For gven values of n, k, μ, and α, the schedulablty bound decreases wth the task set burst-ness μ. Increasng of μ mples a more bursty workload, whch tends to be more dffcult to schedule than less bursty ones. The schedulablty bound s maxmzed when μ =1, whch corresponds to perodc tasks. For gven values of n, k, γ, and μ, the schedulablty bound ncreases when overhead rato α decreases. The schedulablty bound s maxmzed when α =, whch s an deal case and means that there s no overhead n round robn operatons. For gven values of k, γ, μ, and α, the schedulablty bound decreases when the number of tasks n the system ncrease and approaches zero when n 1/ α whch means all processor tme wll be devoted to round robn operatons. For perodc tasks wth relatve deadlnes equal to perods,.e., D =P, and token rotaton frequency γ = 2, (token rotates twce per D mn nterval), the schedulablty bound s 67%. When the normalzed deadlne ncreases from 1. to 1., the schedulablty bound remans at 67%. Ths s llustrated as a seral of ponts n Fgure 7. Note that ths does not mply that k has no effect on the schedulablty test, snce the workload rate s measured dfferently. For perodc tasks wth relatve deadlnes equal to ther perod (D =P ), and token rotaton frequency γ = 1, (token rotates at least once per D mn nterval), the schedulablty bound s 5%. Ths s hghlghted on Fgure 8 as a pont. 2

23 Fgure 6. Schedulablty Bound Wth k = 1 Fgure 7. Schedulablty Bound Wth γ = 2 21

24 Schedulablty Bound k 1 μ = 1, α = % γ 1 2 Fgure 8. Schedulablty Bound Wth μ = Optmal TTRT Selecton to Maxmze Schedulablty Bound In a WRR system, for a gven task set, parameters n, k and μ are fxed. In order acheve a hgher schedulablty bound, one can adjust TTRT. By (2.2) and (2.21), we know that settng larger TTRT may reduce operaton overhead rato but at the same tme leadng to lower rotaton frequency. Thus, TTRT should be adjusted n a way that balances the operaton overhead rato and the token rotaton frequency. The followng theorem gves the value of TTRT that maxmzes the schedulablty bound. Theorem 4. The schedulablty bound of a WRR scheduler, for a gven set of tasks, s maxmzed when the value of TTRT s selected as follows: TTRT = D / γ *, (4-8) mn where 22

25 1 f nτ / Dmn 1/3 f nτ / Dmn < 1/ 3 and γ* = Dmn / nτ 1 + 1, (4-9) Z ( Dmn / nτ 1 1 ) <Z ( Dmn / nτ ) Dmn / nτ otherwse and 1 nτ x. (4-1) 1/ x+ 1 D ( ) = 1 Z x mn When the TTRT takes the optmal value, the schedulablty bound s gven by 1 nτ 1 W * ( k / k) = 1 γ * mn( 1, k ). (4-11) 1/ γ * + 1 D μ mn Proof: See Appendx. Q.E.D. By a careful observaton of Theorem 4, we notce that the optmal TTRT value that maxmzes the schedulablty bound only depends on rato nτ / Dmn. Ths rato s the percentage of tme consumed n round robn operatons wth a wndow of length D mn. When rato n Dmn τ / 1/3, the optmal TTRT equals to D mn and when n Dmn τ / < 1/3, the optmal TTRT value would be ether D Dmn / nτ mn or D Dmn / nτ mn. Fgure 9 plots the trend of the maxmum schedulablty bound of WRR wth the optmal TTRT selecton for the case of k = 1 and μ = 1. It s clear from the Fgure 9 that, when the optmal value of TTRT s used, the schedulablty bound of WRR s a monotonc decreasng functon of nτ / Dmn. The 23

26 rate of decrease s low for small values of nτ / Dmn, say, less than.1, and the rate gradually ncreases nτ / D becomes larger. When nτ / Dmn approaches 1., the schedulablty bound s very close when mn to %. Schedulablty Bound nτ /D mn Fgure 9. Schedulablty Bound wth the Optmal TTRT Selecton 5. COMPARISON WITH EXISTING RESULTS We now compare the newly derved schedulablty bound wth the results of statc prorty scheduler and the tmed token rng scheduler. 5.1 Comparson wth Statc Prorty Schedulers In ths secton, we wll compare the newly derved schedulablty bound wth those of the statc prorty scheduler. For the sake of smplcty, we focus on perodc tasks and assume that overhead rato α of the weghted round robn schedule s. 24

27 a). Consder a set of perodc tasks wth deadlnes equal to ther perods. A schedulablty bound for the rate monotonc scheduler s 69% [17]. For the same task system and WRR scheduler, we know that μ = 1 and k = 1. By substtutng these parameters nto (4-1), we have a schedulablty bound of γ /( γ + 1). The curve of ths functon n plotted n Fgure 1. As can be seen from Fgure 1, the bound of the weghted round robn scheduler s lower than the bound of statc prorty schedulers when γ < When γ = 2.26, the weghted sound robn scheduler acheves the same 69% bound. When γ > 2.26, WRR out-performs the rate monotonc scheduler n term of hgher schedulablty bound. Ths phenomenon can be explaned by the nature of the two types of schedulers. Statc prorty scheduler renders servce to a job only f no hgher prorty job s watng to be executed, even though completng the lower prorty job frst may avod mssng a deadlne. In other words, statc prorty scheduler may allocate more servce tme to hgh prorty tasks than what s needed to guarantee the deadlnes, whle WRR scheduler assgns task servce tme n proportonal to ts workload rate whch avods overallocaton of servce tme to certan tasks. As such, WRR scheduler can out-perform statc prorty scheduler under certan stuaton. b). For a set of perodc tasks wth deadlnes beng half as long as ther perods, a schedulablty bound for the rate monotonc scheduler s 5% [16]. For the same task system wth the WRR scheduler, we know that μ = 1, and k=1/2. By substtutng these parameters nto (4-1), we have a bound of γ /( 2( γ + 1) ). Snce ( ) γ / 2( γ + 1) 1/2, we know that the rate monotonc scheduler out-performs the WRR scheduler n ths case. However, when we ncrease token rotaton frequency γ, the schedulablty bound of WRR scheduler s approachng the 5% bound and attans t when γ. c). For a set of perodc tasks wth deadlnes beng twce as long as ther perods, a schedulablty bound for the rate monotonc scheduler s 81% [16]. For the same task system wth the WRR 25

28 scheduler, we know that μ = 1, and k=2. By substtutng these parameters nto (4-1), we have a bound of γ /( γ + 1). It s easy to see that when γ = 4.32, the WRR scheduler acheves the same bound as the rate monotonc scheduler and out-performs t when γ further ncreases. 1.9 Schedulablty Bound % γ Fgure 1. Schedulablty Bound Wth μ = 1 Based on the above analyss, we know that for the perodc tasks, weghted round robn scheduler can acheve the same or even hgher schedulablty bound than the rate monotonc schedulers. We should note that the above analyss focuses on a very smple case n whch tasks are of a perodc shape and token rotaton overhead s neglgble. However, smlar trends stll hold for more complex cases. The fact that weghted round robn schedulers can solate ll-behaved tasks and can acheve the same or hgher schedulablty bound than the well-known rate monotonc scheduler makes ths type of scheduler applcable to practcal real-tme systems. 26

29 5.2 Comparson wth Tmed Token Protocol A close related varant of the weghted round robn scheduler s the tmed token rng scheduler used n FDDI networks [2], [3], [24], [25], [26], [27], [28], [29], [3]. In a typcal FDDI tmed token rng system, there are n communcaton nodes connected nto a rng. Each node has two types of packets: real-tme and non real-tme. Real-tme packets have hard deadlnes, e.g. packets must be sent before ther deadlnes, whle non real-tme packets do not have deadlne requrements. Smlar to the weghted round robn scheduler, a token s rotated among the nodes n the system and the desred tme to fnsh one round of token rotaton s denoted as TTRT. Upon recevng the token, a node wll frst send ts real-tme packets up to the allocated unts of tme. Each node also keeps track the last token rotaton tme, denoted by TTR. If the token arrves earler n the last round,.e. TTR < TTRT, then t wll send ts non real-tme packets up to the TTRT-TRT amount of tme. The ratonal behnd ths s to "steal" the unused tme slots n the last token rotaton. Due to the nterference of the non real-tme packets, the servce avalable to each node may be less, compared wth what s provded by a weghted round robn scheduler. In turn, the schedulablty bound wll be lower. It has been proven n [3] that for perodc tasks wth relatve deadlne equals to ther perods and normalzed weght assgnment scheme wth token rotaton at lease twce for any nterval of length of mnmum task perods, the tmed token rng scheduler has a schedulablty bound of (1 nα ) / 3 1. For ths system, by (2.21), (2.1), and (2.12), we know that k=1, μ =1, and γ =2. By substtutng them nto (4-1), we know that a schedulablty bound of the weghted round robn scheduler s 2(1 nα ) / 3, twce of the tmed token rng. Fgure 11 llustrates ths dfference for the case of n=1. 1 In [3], the authors assumes the scheduler spend a const amount of tme n round robn operatons and derved a bound of (1 α ) / 3 where α = τ / TTRT. Ths bound converts to (1 nα ) / 3 n our new defnton of α = τ / TTRT snce τ = nτ. 27

30 Schedulablty Bound Overhead Rato α Fgure 11. Schedulablty Bound Comparson Between WRR and Tmed Token Rng Scheduler 6. FINAL REMARKS In summary, n ths paper, we derve a closed-form expresson for the schedulablty bound of WRR schedulers wth normalzed weght assgnment scheme for s-shaped tasks. The schedulablty bound s parameterzed for round robn overhead rato α, normalzed token rotaton rato γ, the normalzed deadlne k, and the task set workload burst-ness μ. From the general result, one can easly obtan schedulablty bounds for specfc system confguraton by smple plug-n of proper parameters. Our work reported here s the frst that systematcally derves the schedulablty bound for WRR systems. Our results are general and can be appled to a wde range of systems. Nevertheless, these results are prelmnary and can be easly extended to other type of task workloads and weght assgnment scheme. 28

31 REFERENCES [1] T. Abdelzaher and C. Lu, Schedulablty analyss and utlzaton bounds for hghly scalable real-tme servces, Proc. 7th Real-Tme Technology and Applcatons Symposum, Tape, Tawan, 21, pp [2] G. Agrawal, B. Chen, and W. Zhao, Local synchronous capacty allocaton schemes for guaranteeng message deadlnes wth the tmed token protocol, n Proc. IEEE INFOCOM'93, San Francsco, CA, Mar. 1993, pp [3] G. Agrawal, B. Chen, W. Zhao, and S. Davar, Guaranteeng synchronous message deadlnes wth the tmed token protocol, IEEE Trans. Computers, vol. 43, no. 3, pp , Mar [4] B. Andersson, Statc-prorty schedulng on multprocessors, Ph.D. dssertaton, Dept. Computer Engneerng., Chalmers Unv. of Technology, Göteborg, Sweden, 23. [5] T. P. Baker, Multprocessor EDF and deadlne monotonc schedulablty analyss, Proc. 24th IEEE Int. Real-Tme Systems Symposum, Cancun, Mexco, 23, pp [6] E. Bn, Schedulablty analyss of perodc fxed prorty systems, IEEE Trans. Computers, vol. 53, no.11, pp , Nov. 24. [7] E. Bn, G. C. Buttazzo, and G. Buttazzo, Rate monotonc analyss: the hyperbolc bound, IEEE Trans. Computers, vol 52, no. 7, pp , Jul. 23. [8] J. Y. Le Boudec and P. Thran, Network Calculus, a Theory of Determnstc Queung Systems for the Internet, New York: Sprnger-Verlag, 21. [9] S. Chang, On determnstc traffc regulaton and servce guarantee: a systematc approach by flterng, IEEE Trans. Inform. Theory, vol. 44, pp , Aug [1] S. Chang, Performance Guarantees n Communcaton Networks, New York: Sprnger-Verlag, 2. [11] B. Chen, G. Agrawal, and W. Zhao, Optmal synchronous capacty allocaton for hard real-tme communcatons wth the tmed token protocol, Proc. IEEE Real-Tme Systems Symposum (RTSS'92), Phoenx, AZ, Dec. 1992, pp [12] D. Chen, A. K. Mok, and T.-W. Kuo, Utlzaton bound revsted, IEEE Trans. Computers, vol. 52, no. 3, pp , Mar. 23. [13] R. L. Cruz, A calculus for network delay, part I: network elements n solaton, IEEE Transactons on Informaton Theory, 37(1), pp , Jan [14] R. L. Cruz, A calculus for network delay, part II: network elements n solaton, IEEE Transactons on Informaton Theory, 37(1), pp , Jan [15] J. P. Lehoczky, Fxed prorty schedulng of perodc task sets wth arbtrary deadlnes, n Proc. 11th IEEE Real Tme Systems Symposum, Pscataway, NJ, 199, pp [16] J. P. Lehoczky and L. Sha, Performance of real-tme bus schedulng algorthms, ACM SIGMETRICS Performance Evaluaton Revew, vol. 14, no. 1, May 1986, pp [17] C. L. Lu and J. W. Layland, Schedulng algorthms for multprogrammng n a hard-real-tme envronment, J. ACM, vol. 2, no. 1, pp , Jan [18] X. Lu and T. Abdelzaher, "On Non-Utlzaton Bounds for Arbtrary Fxed Prorty Polces," Proc. 12th IEEE Real-Tme and Embedded Technology and Applcatons Symposum (RTAS'6), CA, Apr. 26, pp [19] D. Oh and T. P. Bakker, Utlzaton bounds for n-processor rate monotone schedulng wth statc processor assgnment, Real Tme Systems J., vol. 15, no. 1, pp , Nov [2] A. Parekh and R. G. Gallager, A generalzed processor sharng approach to flow control n ntegrated servces networks: the sngle node case, IEEE/ACM Trans. Networkng, vol. 1, no. 3, pp , Jun [21] A. Raha, N. Malcolm, We Zhao, "Hard real-tme communcatons wth weghted round robn servce n ATM local area networks," ceccs, Frst IEEE Internatonal Conference on Engneerng of Complex Computer Systems (ICECCS'95), pp , Aug [22] J. Wu, J.-C. Lu and W. Zhao, On schedulablty bounds of statc prorty schedulers, Proc. 11th IEEE Real-Tme and Embedded Technology and Applcatons Symposum, San Francsco, CA, Mar, 25, pp [23] D. Xuan, C. L, R. Bettat, J. Chen, and W. Zhao, Utlzaton-based admsson control for real-tme applcatons, Proc. 2 Int. Conf. Parallel Processng, Toronto, Canada, 2, pp [24] S. Zhang and A. Burns, An optmal synchronous bandwdth allocaton scheme for guaranteeng synchronous message deadlnes wth the tmed-token MAC protocol, IEEE/ACM Trans. Networkng, vol. 3, no. 6, pp , [25] S. Zhang and A. Burns, Tmng propertes of the tmed token MAC protocol, Proc. 6th Internatonal Conference on Computer Communcatons and Networks, Sep, 1997, pp [26] S. Zhang and A. Burns, and T.-H. Cheng, Cycle-tme propertes of the tmed token medum access control protocol, IEEE Transactons on Computers, vol.51, no.11, pp , Nov. 22. [27] S. Zhang, A. Burns, J. Chen, and E. S. Lee, Hard real-tme communcaton wth the tmed token protocol: current state and challengng problems, Real-Tme System, vol. 27, no. 3, Sep. 24, pp [28] S. Zhang, A. Burns, A. Mehaoua, E. S. Lee, and H. Yang, Testng the schedulablty of synchronous traffc for the tmed token medum access control protocol,, Real-Tme Systems, vol.22, no.3, pp , May

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33 APPENDIX Theorem 3. For WRR schedulers, a servce constrant functon for task T, s I G( I) = n H. τ + H j= 1 j (A-1) Proof. We prove ths theorem based on the defnton of servce constrant functon. Let t be an arbtrary tme nstant. If at tme t, all the jobs from task T have been served, then we can let s =t and (2.15) s true. Now we focus the case that at tme t, task T has backlog. Let s be the last tme before t such that T dd not backlog. That s to say, at tme s, we have g () s = f () s. (A-2) n In tme nterval [ s, t ], the scheduler served at least I / ( τ + H j= 1 j ) length of H each round. In other words, task servce. Formally, we have rounds wth a servng tme n T receved at least I / ( τ + H j= 1 j ) H j amount of I g() t g() s n H. τ + H j= 1 j (A-3) By substtutng (A-2) nto(a-3), we have I g() t f() s n H. τ + H j= 1 j (A-4) 31

34 By comparng (A-4) wth (2.15), we know that the theorem follows. Q.E.D. Theorem 4. A lower bound of schedulablty bound wth scalng parameter round robn scheduler wth normalzed weght assgnment, and s-shaped tasks s gven by θ = k / k for weghted k = ( α ) ( ) * 1 1 W ( / k) 1 n mn 1, k 1/ γ + 1 μ. (A-5) Proof. By Theorem 3, we known that a servce constrant functon provded by task T by a WRR scheduler s: I G( I) = n H. τ + H j= 1 j (A-6) For normalzed weght assgnment, by (2-18) and (2-16), we have H W (1) = ( TTRT τ ) W (1, Γ) (A-7) By substtutng (A-7) nto (A-6), we have the servce constrant functon as I W (1) G ( I) = ( TTRT τ ). TTRT W (1, Γ) (A-8) By Theorem 2, we know that a schedulablty bound for WRR wth scalng parameter θ = k / k s ( ) * W ( k / k) = nf W( k / k, Γ ). (A-9) Γ Ω* where 32

35 { T such that F( I) G ( I D) } * Ω = Γ Γ > +. (A-1) By substtutng (A-8) nto (A-1) and rearrange the resultng equaton, we know that a schedulablty bound s ( ) * W ( k / k) = nf W( k / k, Γ ). (A-11) Γ Ω* where * I + D (1) W Ω = Γ T Γ such that W(1, Γ ) > ( TTRT τ ) TTRT. (A-12) F ( I) * Γ Ω, let us defne I + D TTRT τ Z () = mn I W(1) TTRT. (A-13) F ( I ) Then by (A-12), we have = 1, 2,..., n ( Z ) W(1, Γ ) > mn ( ). (A-14) By substtutng (2.2) and (2.2) nto (A-13), we have I + D I D F( D) Z () ( 1 n ) mn TTRT + = α I. (A-15) I + D F( I ) D TTRT I + D I + D Snce 1 TTRT + TTRT, we have 33

36 I + D ( ) () ( 1 ) mn TTRT I + D F D Z nα I. (A-16) I + D ( ) + 1 F I D TTRT It s easy to verfy that I + D TTRT = =. (A-17) I + D / γ TTRT I + D Dmn TTRT TTRT where have D mn and γ are defned n (2.22) and (2.21), respectvely. By substtutng (A-17) nto (A-16), we 1 I + D F( D) Z () ( 1 nα ) mni 1 1/ γ F( I ) D +. (A-18) Now let I = ms + ω where ω < S, S s the segment length of the s-shaped workload constrant functon defned n (2.6), and m s a non-negatve nteger. By (2.6), we have F( I) F(( m+ 1) S ). (A-19) By substtutng (A-19) nto (A-18) and rearrangng t, we have 1 ms + ω + D F( D) Z () ( 1 nα ) mni 1/ γ 1 F( ( m 1) S) D. (A-2) + + Snce ω, we have 34

37 1 ms + D F( D) Z () ( 1 nα ) mni 1/ γ 1 F( ( m 1) S) D. (A-21) + + By (2.6) and (2.7), t can be verfed that (( + 1) ) ( ) F m S F S ( m+ 1) S S. (A-22) By substtutng (A-22) nto (A-21) and rearrangng t, we get 1 ms + D F( D) Z () ( 1 nα ) mni 1/ γ 1 ( m 1) F( S) D. (A-23) + + By defnton of s-shaped workload constrant functon n (2.6) and k n (2.11), we know that F( D) = F( ks ) = F( k S ). (A-24) By substtutng (A-24) nto (A-23), we have 1 ms + D F( k S) Z () ( 1 nα ) mni 1/ γ 1 ( m 1) F( S) ks. (A-25) + + By substtutng (2.1) nto (A-25) and rearrangng the resultng nequalty, we have 1 m+ k F( k S) Z () ( 1 nα ) mni 1/ γ 1 ( m+ 1) F( S) k. (A-26) + Rewrte (A-26) nto 1 1 F( k S) m+ k Z () ( 1 nα ) mn m=, 1, 2,... 1/ γ + 1 k F S m + 1. (A-27) ( ) 35

38 It can be verfed that m+ k mn m=, 1, 2,... mn 1, m + 1 ( k ). (A-28) By substtutng (A-28) nto (A-27), we have 1 1 F( k S) Z () ( 1 nα ) mn 1, k k 1/ γ + 1 F S ( ) ( ). (A-29) By (2.6) and (2.7), we have F( k S) k k =. (A-3) F( S ) μ μ By substtutng (A-3) nto (A-29), we get 1 k Z () ( 1 nα ) mn 1, k 1/ γ + 1 kμ ( ). (A-31) By substtutng (A-31) nto (A-14), we have 1 k W(1, Γ) ( 1 nα ) mn 1, k 1/ γ + 1 kμ ( ). (A-32) By (2.6), we know that n F( D) k n F( k S) k n k k k W ( 1, Γ) = = = W, W, = 1 = 1 = 1 Γ = Γ D k k S k k k k.(a-33) By substtutng (A-33) nto (A-32) and rearrange the resultng nequalty, we have 36

39 W k / k, Γ 1 nα 1 1 mn 1, 1/ γ + 1 μ ( ) ( ) ( k ). (A-34) Snce (A-34) s true * Γ Ω, we know that 1 1 nf * ( W( k / k, Γ )) ( 1 nα ) mn ( 1, k ) Γ Ω 1/ γ + 1 μ. (A-35) Then by substtutng (A-35) nto (A-9), we get * 1 1 W ( / k) 1 n mn 1, k 1/ γ + 1 μ k ( α ) ( ). (A-36) Comparng (A-36) wth (A-5), we have the theorem proven. Q.E.D. When the TTRT takes the optmal value, the schedulablty bound s gven by 1 nτ 1 W * ( k / k) = 1 γ * mn( 1, k ). (1-37) 1/ γ * + 1 D μ mn Theorem 4. The schedulablty bound of a WRR scheduler, for a gven set of tasks, s maxmzed when the value of TTRT s selected as follows: TTRT = D / γ *, (A-38) mn where 37

40 1 f nτ / Dmn 1/3 f nτ / Dmn < 1/ 3 and γ* = Dmn / nτ 1 + 1, (A-39) Z ( Dmn / nτ 1 1 ) <Z ( Dmn / nτ ) Dmn / nτ otherwse and 1 nτ x. (A-4) 1/ x+ 1 D ( ) = 1 Z x mn When the TTRT takes the optmal value, the schedulablty bound s gven by 1 nτ 1 W * ( k / k) = 1 γ * mn( 1, k ). (A-41) 1/ γ * + 1 D μ mn Proof: By (4-1), we know that a schedulablty bound s W * ( k / k) = 1 ( 1 nα ) 1 mn( 1, k ) 1/ γ + 1 μ. (A-42) Let us defne ω = D γ TTRT. (A-43) mn where γ s defned n (2.21). By (2.21), we have ω. (A-44) By substtutng (A-43) and (2.21) nto (A-42), we have 38

41 * 1 nτ 1 W ( k / k) = 1 mn 1, k 1/ γ + 1 ( Dmn ω) / γ μ ( ). (A-45) It s easy to see that (A-45) wll be maxmzed when ω = and the maxmum wll be 1 nτ 1 W * ( k / k) = 1 γ mn( 1, k ). (A-46) 1/ γ + 1 D μ mn Now, we need to fnd a value of γ, γ = 1, 2,..., that maxmzes (A-46). Clearly, the value of γ that maxmzes (A-46) wll also maxmze Z 1 nτ γ. (A-47) 1/ x+ 1 D ( γ ) = 1 mn We can rewrte (A-47) as follows: Z 1 nτ nτ = 1 (1 + ) ( γ + 1). (A-48) γ + 1 D D ( γ) mn mn An equvalent form of (A-48) s gven by: Z nτ nτ D 1 = 1+ 2 ( + 1) + ( γ + 1). (A-49) Dmn Dmn nτ γ + 1 mn ( γ) Snce Dmn TTRT nτ, by calculatng the dervaton of (A-49) over γ, we know that (A-49) wll be maxmzed when γ = Dmn / nτ But snce γ can only take postve nteger value, we know that (A-49) wll attan ts maxmum ether at ( D nτ ) γ = max 1, mn / + 1 1, (A-5) 39

42 or γ1 = Dmn / nτ (A-51) That s, the optmal value of TTRT that maxmzes the schedulablty bound s: TTRT mn 1 ( ) ( ) Dmn / γ f Z γ Z γ1 =. (A-52) D / γ otherwse It s easy to verfy that when Dmn / nτ 3, γ = γ1 = 1 and thus (A-52) s equvalent to TTRT = D / γ *, (A-53) mn where 1 f Dmn / nτ 3 γ * = Dmn / γ f Dmn / nτ > 3 and Z γ <Z γ1 Dmn / γ1 otherwse ( ) ( ). (A-54) By substtutng (A-54) nto (A-46), we know that the maxmum schedulablty bound 1 nτ 1 W * ( k / k) = 1 γ * mn( 1, k ). (A-55) 1/ γ * + 1 D μ mn s attaned when TTRT = D / γ * mn. Hence, the theorem follows. Q.E.D. 4