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1 8 / 1 -- Communcaton Networs II (Görg) -- Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP & MP) 4 Fnte State Marovan Processes 5 Analyss of Marovan servce systems 6 Queues for modelng communcaton networs 7 M/G/1 model 8 The model M/G/1/FCFS/NONPRE 9 The model M/G/1/FCFS/PRE

2 8 / 2 -- Communcaton Networs II (Görg) The Model M/G/1/FCFS/NONPRE; statc prortes e consder the model n fgure 2.7 from chapter 2 wth several queues and ndependent Posson arrvals for the ndvdual queues. Each of the queues receves a prorty number (l N). The schedulng s non-preemptve (NONPRE). That s, a newly arrvng job of any prorty never nterrupts the job n servce. Thus, ths s sutable for modelng of I/O traffc n real computers wth ther bac up memory, pacet swtchng and others. server Fgure 2.7 (see also chapter 2): Traffc Model

3 8 / 3 -- Communcaton Networs II (Görg) -- Jobs of prorty are also called jobs of type. e agan consder the statonary equlbrum state of the system. For ths system hgh prorty queues can be n a stable state even for a total load exceedng one,.e. > 1. The type random varables nter-arrval tme TA and servce tme T B are consdered to be statstcally ndependent and negatve-exponentally (for T A A ) and generally (G) (for T ) dstrbuted. B e ntroduce a new arrval rate λ, whch gves the sum of arrval rates of all prortes less than or equal to. 1. (8.1) A sum of several Posson processes produces agan a Posson process [Cha 1]. In addton, we defne the r-th moment of weghted servce tme dstrbuton of the jobs belongng to the set of the hghest prortes through ( r) 1 ( r) (8.2)

4 8 / 4 -- Communcaton Networs II (Görg) -- Now, we get the total traffc load produced by prortes 1 to as gven below: 1 1 (8.3) Thus N corresponds to the overall traffc load of the M/G/1 model. Also n ths case, t s possble to derve the generatng functon of the queue length dstrbuton and the LST of the watng tme dstrbuton. However, the dervaton s more complcated than n the case wthout prortes. Thus we lmt ourselves to the computaton of the mean watng tme. e observe a test job of prorty from ts arrval to departure and follow ts fate n the model: At the arrval of the test t job there are n watng jobs belongng to the prorty class (=1,2,...,N) plus the job n servce n case of a busy system. Obvously the test job cannot receve servce, f there are unserved ed jobs n the same or hgher prorty queue. e In addton to that t has to wat due to all the hgher prorty jobs (prorty 1 to -1), that arrve durng the watng tme of the test job.

5 8 / 5 -- Communcaton Networs II (Görg) --

6 8 / 6 -- Communcaton Networs II (Görg) -- The mean watng tme of the test job conssts of three components: 0 (8.4) 0 s the mean of the remanng servce tme of the job, that s beng served at arrval. s the mean of the so called vrtual watng tme, whch corresponds to the sum of servce tmes of all type 1 to jobs watng at arrval of the test job. The mean of ths number s gven by n 1 +n n. s the mean of the total servce tmes of all hgher (than ) )prorty yjobs that arrve durng the full watng tme of the test job. Let us calculate these three components one by one. 0: 2 For a sngle prorty system we have 0 () / 2. Extendng ths result for N prortes through weghted addton, we get N 1 () 2 1 () 2 0 NN 2 1 2

7 8 / 7 -- Communcaton Networs II (Görg) -- : Tang the fact, that we can group the jobs for summaton ndependent of ther actual servce order due to the from system and schedulng strategy ndependent Posson arrval process, we can wrte for 1 E( n) where E(n ) s the expected number of prorty jobs watng at arrval of the test job. As Lttle s law s applcable for each of the ndvdual d prortes separately, we can wrte E(n )= (and also L = V ). Thus, we get wth = 1 1

8 8 / 8 -- Communcaton Networs II (Görg) -- : The mean number of prorty jobs arrvng durng the watng tme of the test job s. Thus smlar to we get Ths leads to (8.5) N 1 ( 2 )

9 8 / 9 -- Communcaton Networs II (Görg) -- The fnal expresson s obtaned by mathematcal nducton N () 2 () 2 1 NN ( )( ) ( )( ) (8.5a) As we are nterested n the cdf of the watng tme, we also state the 2nd moment as well. e use the followng abbrevatons: () 2 () 1 NN 2 1 ( ) () 3 () 2 () 2 () 2 NN NN 2 31 ( ) 2 ( 1 ) under the condton N <1 we get the frst and second moments of the watng tme to be () 1 () 2 1 ( 1 ) ( 2 ) ( 1 ) ( 2 ) 2 3 ( 1 ) ( 1 ) (8.6) (8.7) (8.8) (8.9)

10 8 / Communcaton Networs II (Görg) -- Themeansystemtmeofaprorty tme a job s V (8.10) and the mean number of type jobs n the system s [Lt 1] L V (8.11) The probablty blt for an dle system s P{ dle} 1 N (8.12) Ths result holds for any servce tme dstrbuton.

11 8 / Communcaton Networs II (Görg) -- Fgure 8.1: Normalzed mean watng tme of prorty jobs as a functon of the traffc load N for the model M/M/1/FCFS/NONPRE. It holds 1 = 2 = = and =/N wth =. The dashed curve holds for the model M/M/1/FCFS. a) N=2 b) N=5

12 8 / Communcaton Networs II (Görg) -- Fgure 8.1 shows two examples (N=2 and d5) for the model M/M/1/FCFS/NONPRE wth dentcal servce tme dstrbuton for all jobs and wth unformly dstrbuted traffc load over all the prortes ( = N /N). The results for the FCFS model wthout prortes (Fgure 5.9) s also shown for comparson. In contrast to fgure 7.5, where the type of the job was dentfed accordng to ts servce tme t, all the servce tme duratons occur for each of the prortes n ths system. e see that for the least mportant prorty =N, even at smallest total traffc load N, the nstablty level s reached whereas for the most mportant prorty =1 1even for N >1 stll fnte mean watng tmes preval. Ths behavor s due to the fact that the lowest prorty receves hgh preference and reaches hgh mean watng tmes only for the case where ts own traffc load 1 (n ths example 1 = N /N) approaches 1. However, for the computaton of the moments of watng tme for a saturated arrval process N > 1, results whch are also shown n fgure 8.1, we have to nclude addtonal condton equatons (8.5) and (8.9).

13 8 / Communcaton Networs II (Görg) -- A statonary equlbrum s possble n the model n Fgure 2.7 only for the prortes wth <1 [Cob 1], [Ja 1]. Only the jobs n these prortes experence fnte watng tme. Jobs n prortes > have to wat for an nfnte duraton wth probablty 1. For the computaton of moments of watng tme, we frst fnd out the number N, satsfyng the condton (-1) <1,.e., we determne the prorty, whose traffc load saturates the system. Then, for the computatons usng equatons (8.7) and (8.8), should be lmted n such a way that =1. All the traffc loads n prortes wth (<N) must be set equal to zero. Ths approach results from the method of dervaton of the formulas to compute moments of the watng tme. Unfortunately, ths lmtaton s often omtted n lterature.

14 8 / Communcaton Networs II (Görg) The varance of the watng tme of jobs n prorty can be calculated () 1 wth from 2 (2) 2. (8.13) Fgure 8.2 shows the standard devaton normalzed w. r. t. expected servce tme, / for the same example n fgure 8.1 wth N=5 5 prortes. Fgure 8.2: Normalzed standard devaton for the Model M/M/1/FCFS/NONPRE; Parameter as n Fgure 8.1

15 8 / Communcaton Networs II (Görg) -- Introducton of prortes always ncreases the varance and the hgher moments of the watng tme dstrbuton of all the jobs havng prortes hgher than =1 compared to the system wthout prortes. Explct reverse transformaton of the LST of watng tme dstrbuton of types >1 s not possble even for the smplest model M/M/1/FCFS/NONPRE. The possblty of approxmatng the watng tme dstrbuton through a hyperexponental p dstrbuton n such a way that at least the most mportant moments match remans. The other opton s to get the dstrbuton through smulaton. In the model descrbed below, the overheads for swtchng from one job to another are consdered n such a way that they get added to the servce tme of the job.

16 8 / Communcaton Networs II (Görg) Mnmum common mean watng tme n model M/G/1/FCFS/NONPRE There are applcaton examples, where a group of N dfferent job types can exst that are prncpally raned equally. As an example, we are only nterested n the goal that the mean watng tme of the jobs n group ( 1 ) N s as small as possble. The r-th moment of the watng tme dstrbuton correspondng to jobs n ths group s defned through () r N N 1 N () r N For r=1 we also wrte ths n the form. (8.14) In cases where the servce tme s not nown but the mean of the group the job belongs to s nown n advance, t s possble to allocate external prortes n such a way that the mean watng tme s nfluenced. As shown n [Con 1], [Ja 1] the overall mean watng tme (and also the overall mean system tme) become mnmal, f the prortes are assgned accordng to the ncreasng order of the mean servce tme of the group. That means the prortes are allocated n such a way that the condton < +1 s always satsfed.

17 8 / Communcaton Networs II (Görg) -- If we assume a lnear cost factor g for jobs of type (=1,2,...,N) then the average cost per job,.e., the mean watng cost s gven by N N 1 N 1 g g (8.15) In order to mae sure that ths quantty becomes a mnmum the followng condton needs to be satsfed: 1 2 N L L g g g g 1 2 (1 stands for the hghest prorty). N (8.16) For a general evaluaton functon g (t) the prortes are best assgned [Ol 1] for the types and j n such a way that the condton gven below s satsfed g ( t ) dp ( TB t ) gj ( t ) dpj ( TB t ) 0 0 (8.17) tdp ( T t) tdp ( T t) 0 B 0 j B Type jobs should have a hgher prorty than type j jobs.

18 8 / Communcaton Networs II (Görg) -- If we schedule the jobs accordng to FCFS wthout external prortes then we get the frst and second moments of the watng tme to be, see also equatons (7.24) and (7.28) of the M/G/1 system. () 1 FCFS FCFS () 2 N N N 2 ( 1 ) (8.18) N N 2 FCFS 31 ( ) () 2 2 FCFS () 3 N (8.19) th ths, the comparson between dfferent prorty allocatons s made possble, that can be used to select the prortes n an optmal way.

19 8 / Communcaton Networs II (Görg) The model M/G/1//SJF /SJF revsted t Now, we derve the results n 7.4 n a smple manner, usng the results for the non-preemptve external prorty example. e now assume that the servce tme of all the jobs are nown at arrval. After each completon the shortest job n the queue s selected for processng as the next. For smplcty we assume dscrete servce tmes: g (=1,2,...) are the probabltes, that a job needs Q unts of servce tme. g s the arrval rate of jobs that need exactly unts of duraton Q. Ths system can be represented as shown n fgure 2.7, as the arrvals n the ndvdual types also happen to be Posson dstrbuted due to the addtve nature of the Posson process, wth g n place of. By replacng through g and through Q g n equaton (8.4), we get the mean watng tme of a job wth unts of servce tme where H () 2 2 ( 1 H )( 1 H1 ) ( Q) g 1

20 8 / Communcaton Networs II (Görg) -- For the lmtng case of contnuous servce tme dstrbuton, we get the result by replacng Q through the dfferental tme element and g through the servce tme densty functon. For Q0, we get the condtonal watng tme of jobs, that have a servce tme of t, as gven n equaton 7.36.a.

21 8 / Communcaton Networs II (Görg) --

22 8 / Communcaton Networs II (Görg) --

23 8 / Communcaton Networs II (Görg) --

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