CS/ECE 715 Spring 2004 Homework 5 (Due date: March 16)

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1 CS/ECE 75 Sprig 004 Homework 5 (Due date: March 6) Problem 0 (For fu). M/G/ Queue with Radom-Sized Batch Arrivals. Cosider the M/G/ system with the differece that customers are arrivig i batches accordig to a Poisso process with rate λ. Each batch has customers, where has a give distributio ad is idepedet of customer service times. Show that the waitig time i queue is give by λx X W () ( ρ) ( ρ) where X deotes service time. [Solutio] The waitig time of a customer, W, cosists of three parts, the residual service time of the customer beig served R, the average service time of all customers waitig i queue ρw, ad the average waitig time of the customer for other customers who arrived i the same batch. where the mea residual service time is W R + ρw + W B () λx R (3) We derive the average waitig time of a customer for other customers that arrived i the same batch where W B j r j E{ W B batch has size j} (4) P j Probability a batch has size j (5) r j Proportio of customers arrivig i a batch of size j (6) We have Also sice the customer is equally likely to be i ay positio withi the batch Thus r j jp j P jp j j E{ W B batch is of size j} ( k )X -- j X j k (7) (8)

2 E{ W B } Substitutig i () we obtai (). Problem. I the M/G/ system show that jp j ( j )X j X ( ) (9) P( the system is empty) λx Average legth of time betwee busy periods -- λ X Average legth of busy period λx Average umber of customers served i a busy period λx where X is average service time. [Solutio] From Little's Theorem we have that P{ busy} λx. Therefore P{ idle} λx. The legth of a idle period is the remaiig iterarrival time. Therefore it has a expoetial distributio with parameter λ. Let B be the average legth of a busy period ad let I be the average legth of a idle period. By expressig the proportio of time the system is busy as B/(I+B) ad also as B X ( λx). From this the expressio period is evidet. ( λx) we obtai for the average umber of customers served i a busy λx Problem. Cosider a computer system with a CPU ad oe disk drive. After a burst at the CPU the job completes executio with probability 0. ad request a disk I/O with probability 0.9. The time of a sigle CPU burst is expoetially distributed with mea 0.0 s. The disk service time is broke up ito three phases: expoetially distributed seek time with mea 0.03 s, uiformly distributed latecy time with mea 0.0 s, ad a costat trasfer time equal to 0.0 s. After a service completio at the disk, the job always requires a CPU burst. The average arrival rate of jobs is 0.8 job/s ad the system does ot have eough mai memory to support multiprogrammig. Solve for the average respose time. I order to computer the mea ad the variace of the service time distributio, you may eed the followig results o radom sums. [Radom Sums] We have cosidered sums of N mutually idepedet radom variable whe N is a fixed costat. Here we are iterested i the case whe N itself is radom variable that is idepedet of

3 X k. Give a list X, X,, of mutually i.i.d. radom variables with distributio fuctio Fx ( ), mea EX [ ] ad variace Var[ X], cosider the radom sum T X + X + + X N. Here the pmf of the discrete radom variable p N ( ) is assumed to be give. For a fixed value N, the coditioal expectatio of T is easily obtaied ETN [ ] EX [ i ] The, usig the theorem of total expectatio, we get i E[ X] (0) Similarly, we have ET [ ] E[ X]p N ( ) EX [ ] p N ( ) EX [ ]EN [ ] () Var[ T] ET [ ] ( ET [ ]) E{ E[ T N ] } ( ET [ ]) () E{ Var[ X] + ( EX [ ]) } ( ET [ ]) Var[ X]EN [ ] + ( EX [ ]) Var[ N] [Solutio] From the give parameters, we have: λ 0.8, mea time of a CPU burst ET [ c ] 0.0, mea disk seek time ET [ s ] 0.03, mea disk latecy time ET [ l ] 0.0, ad mea trasfer time ET [ t ] 0.0. Besides, the job requests a disk with probability p 0.9. Sice ad are expoetially distributed, T c T s ET [ s ] ( ET [ s ]) [ ] ( ET [ c ]) 0 4 Sice is uiformly distributed i iterval (0,0.0), T l ET c (3) ET [ l ] (4) Sice T t is a costat time, ET [ t ] 0 4, the, the disk service time T d T s + T l + T t, we get ET [ d ] ET [ s ] + ET [ l ] + ET [ t ] 0.05 (5) ET [ d ] E[ ( T s + T l + T t ) ] (6) From the descriptio of the problem, the total service time T is it c + ( i )T d with probability p ( i ) ( p), i,,,. Thus ET [ ] p ( i ) ( p) [ ie[ T c ] + ( i )ET [ d ]] i ET [ c ] ip i ET [ d ] ip i 0 ET [ c ] + 9 ET [ d ] (7)

4 ET [ ] p ( i ) ( p)et [ () i ] i p ( i ) ( p) ie[ T c ] ii ( )ET [ c ]ET [ d ] ( i ) ( + + ET [ d ]) i -- ( ET [ 9 c ] ET [ c ]ET [ d ]) ip i + --E[ T 9 c ]ET [ d ] + 0. ET [ d ] i p i i i (8) time 0679 Variace: Var[ T] ET [ ] ( ET [ ]) So, ρ λet [ ] 0.44, the average respose ER [ ] ca be obtaied: EN [ ] λe[ T ] ER [ ] ET [ ] (9) λ ( ρ) Problem 3. Persos arrive at a Xerox machie accordig to a Poisso process with rate oe per miute. The umber of copies to be made by each perso is uiformly distributed betwee ad 0. Each copy requires 3 sec. Fid the average waitig time i queue whe: [Solutio] (a) Each perso uses the machie o a first-come first-serve basis. (b) Persos with o more tha copies to make are give preemptive priority over other persos. (a) λ 60 per secod EX ( ) 6.5 secods EX ( ) secods T E( X) + λex ( ) [ ( λe( X) )] 0.48 secods. W T E( X) 3.98 secods (b) Preemptive Queueig λ -----, λ, λ EX ( ).5, EX ( ) 47.5 λ R EX ( )

5 R R + --λ EX ( ).8875 EX ( T )( ρ ) + R EX ( , T )( ρ ρ ) + R ρ ( ρ )( ρ ρ ) 3.84 W T EX ( ) 0.038, W T EX ( ) 4.34 (Assume a customer with lower priority is put back ito the queue oce the service is preempted by a customer with higher priority.) Problem 4. Priority Systems with Multiple Servers. Cosider the priority systems assumig that there are m servers ad that all priority classes have expoetially distributed service times with commo mea µ. (a) Cosider the opreemptive system. Show that the average queueig time is give by R W k ( ρ ρ k )( ρ ρ k ) with R, the mea residual time, is give by (0) where P Q P R Q () mµ is the steady-state probability of queueig give by the Erlag C formula. [Here ρ i λ i ( mµ ) ad ρ ρ i.] i (b) Cosider the preemptive resume system. Argue that W ( k), defied as the average time i queue averaged over the first k priority classes, is the same as for a M/M/m system with arrival rate λ + λ + + λ k ad mea service time µ. Use Little s Theorem to show that the average time i queue of a k th priority class customer ca be obtaied recursively from W W ( ) k k W k ---- W, (3) λ ( k) λ i k W ( k ) λ i k 3,,, i i [Erlag C formula] () 5

6 I a M/M/m system, the probability that a arrival will fid all servers busy is [Solutio] (a) The same derivatio as i the otes applies for, i.e., W k R ( ρ ρ k )( ρ ρ k ) (6) where ρ i λ i ( mµ ), ad R is the mea residual service time. Thik of the system as beig comprised of a servig sectio ad a waitig sectio. The residual service time is just the time util ay of the customers i the waitig sectio ca eter the servig sectio. Thus, the residual service time of a customer is zero if the customer eters service immediately because there is a free server at the time of arrival, ad is otherwise equal to the time betwee the customer s arrival, ad the first subsequet service completio. Usig the memoryless property of the expoetial distributio it is see that p P Q P{ Queueig} 0 ( mρ) m , where p is (4) m! ( ρ) 0 R P Q E{ Residual service time queueig occurs} P Q ( mµ ) (7) (b) The waitig time of classes,, k is ot iflueced by the presece of classes (k+),...,. All priority classes have the same service time distributio, thus, iterchagig the order of service does ot chage the average waitig time,. We have p 0 m ( mρ) ( mρ) m! m! ( ρ) 0 W k (5) W ( k) Average waitig time for the M/M/m system with rate ( λ + + λ k ) By Little s theorem we have Average umber i queue of class k Average umber i queue of classes to k Average umber i queue of classes to k- ad the desired result follows. (8) (9) 6

Queuing Theory. Basic properties, Markovian models, Networks of queues, General service time distributions, Finite source models, Multiserver queues

Queuing Theory. Basic properties, Markovian models, Networks of queues, General service time distributions, Finite source models, Multiserver queues Queuig Theory Basic properties, Markovia models, Networks of queues, Geeral service time distributios, Fiite source models, Multiserver queues Chapter 8 Kedall s Notatio for Queuig Systems A/B/X/Y/Z: A