M/G/1 POLLING SYSTEMS WITH RANDOM VISIT TIMES

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1 Probablty n the Engneerng and Informatonal Scences, 22, 2008, Prnted n the U.S.A. DOI: /S M/G/1 POLLING SYSTEMS WITH RANDOM VISIT TIMES M. VLASIOU Georga Insttute of Technology H. Mlton Stewart School of Industral & Systems Engneerng Atlanta, GA E-mal: vlasou@gatech.edu U. YECHIALI Department of Statstcs and Operatons Research School of Mathematcal Scences Raymond and Beverly Sackler Faculty of Exact Scences Tel Avv Unversty Tel Avv 69978, Israel E-mal: ury@post.tau.ac.l We consder a pollng system where a group of an nfnte number of servers vsts sequentally a set of queues. When vsted, each queue s attended for a random tme. Arrvals at each queue follow a Posson process, and the servce tme of each ndvdual customer s drawn from a general probablty dstrbuton functon. Thus, each of the queues comprsng the system s, n solaton, an M/G/1-type queue. A job that s not completed durng a vst wll have a new servce-tme requrement sampled from the servce-tme dstrbuton of the correspondng queue. To the best of our knowledge, ths artcle s the frst n whch an M/G/1-type pollng system s analyzed. For ths pollng model, we derve the probablty generatng functon and expected value of the queue lengths and the Laplace Steltjes transform and expected value of the sojourn tme of a customer. Moreover, we dentfy the polcy that maxmzes the throughput of the system per cycle and conclude that under the Hamltonan-tour approach, the optmal vstng order s ndependent of the number of customers present at the varous queues at the start of the cycle. 1. INTRODUCTION A typcal pollng system conssts of a number of queues, attended by a sngle server n a cyclc fashon. There s a huge body of lterature on pollng systems that has # 2008 Cambrdge Unversty Press /08 $

2 82 M. Vlasou and U. Yechal developed snce the late 1950s, when the work of Mack [12] and Mack, Murphy, and Webb [13] concernng a patrolng reparman model for the Brtsh cotton ndustry were publshed. Rather than gvng a partal overvew of the lterature, we refer the nterested reader to the followng books, surveys, and papers on pollng systems: Takag [17 19], Boxma and Groenendjk [5], Levy and Sd [11], Yechal [26], Borst [4], Elazar and Yechal [10], and Nakdmon and Yechal [14]. Pollng systems have been used as a central model for the analyss of a wde varety of applcatons n the areas of repar problems [12,13], telecommuncaton systems [9], road traffc control [15], computer networks [25], multple access protocols [3], multplexng schemes n ISDN [21], satellte systems [1], flexble manufacturng systems [24], and the lke. In many of these applcatons, as well as n most pollng models, t s customary to control the amount of servce gven to each queue durng the server s vst. Common servce polces are the exhaustve, gated, globally gated, and lmted regmes. Under the exhaustve regme, at each vst, the server attends the queue untl t becomes completely empty, and only then s the server allowed to move on. Under the gated regme, the only customers served durng a vst are the ones who are present when the server enters (polls) the queue, and customers arrvng when the queue s attended wll be served durng the next vst. The globally gated regme, ntroduced by Boxma, Levy and Yechal [6], s a modfcaton of the gated one: The only customers served durng a vst are those who are present at the begnnng of a cycle. Fnally, under the k-lmted servce dscplne, only a lmted number of jobs (at most k) are served at each server s vst to each queue. These servce polces mply that the duraton of the vst tme n a polled queue s a functon of the number of customers present there at a gven moment (such as the begnnng of the cycle or the moment the server enters the queue). In ths artcle, we analyze a pollng system that dffers n two ways from the classcal pollng model. Rather than consderng a sngle server provdng servce to customers at the varous queues, we assume that an nfnte number of servers are movng as a sngle group between the queues. Moreover, the servce polcy we study s ndependent of the queue length. We assume that the group of servers vsts each queue for an (possbly random) amount of tme that s ndependent of everythng else and that has a dstrbuton that mght vary per queue. We further assume that the arrval process of customers to each queue s Posson and that the servce-tme dstrbuton for customers n each queue s general. To the best of our knowledge, ths artcle s the frst n whch an M/G/1-type pollng system s analyzed. The specfc applcaton that rased our attenton and led us to ths model s n the feld of road traffc control. Pollng models for road traffc are typcally along the lnes of the classcal pollng system; namely they nvolve a sngle server rotatng around a number of queues. Other assumptons that are typcally beng made for such models nclude determnstc servce tmes (.e., the amount of tme that a car needs to pass a traffc lght after possbly standng n the queue) and determnstc vst tmes (.e., the tme the traffc lght remans green); see, for example, van der Hejden, van Harten,

3 M/G/1 POLLING SYSTEMS 83 and Ebben [22]. Although these models provde farly good approxmatons of realty, such assumptons fal to capture the varaton both n servce tmes and n vst tmes. Cars do not need the same amount of tme to cross a segment of the road: The ones standng ahead n the queue wll nevtably need less tme and those that arrve whle the queue s empty and the traffc lght s stll green wll not even requre the addtonal tme ncurred by acceleraton. Moreover, recent developments n the technology of traffc lghts has led to the desgn of traffc lghts that do not turn green unless a queue s formed, and they turn red ether when the queue s empty or after a maxmum amount of tme, whch mght also vary wthn a day. As a result, n ths artcle we provde a framework for studyng road traffc control under less restrctve assumptons. We propose an nfnte-server pollng system, whch models the behavor of traffc: Whle the traffc lght s green, all cars present n the queue or approachng the traffc lght proceed (receve servce) and the tme they need to complete servce s assumed to be a random varable followng a general dstrbuton. Furthermore, we assume that the tme the traffc lght s green (vst tme) s random, although our results are drectly applcable n the case of determnstc vst tmes or, more generally, n the case that the vst tmes follow a dscrete dstrbuton takng postve values. A common approxmaton to road traffc s to consder the traffc as flud passng through the road. Ths approxmaton s farly accurate when the traffc s relatvely hgh. Mathematcally, hgh traffc can be modeled by assumng that the arrval rate of customers at each of the queues tends to nfnty. The study of such a model provdes nsghts at the queue length (and thus the congeston of a juncton) under a heavy load. In ths artcle though we do not study the evoluton of the system under a heavy load. We assume that the arrval rate at each queue s fxed. Ths assumpton s usually made for the standard pollng systems and provdes a reasonable approxmaton to normal traffc condtons. The rest of the artcle s organzed as follows. Secton 2 ntroduces the model, gves further notaton, and descrbes formally the evoluton of the system. In Secton 3 we compute recursvely the frst moment and the probablty generatng functon of the queue length dstrbutons at a pollng nstant. Later, n Secton 4 we derve the mean and the Laplace Steltjes transform of the sojourn tme of a customer arrvng at queue, and we show how these expressons smplfy n the specal case where both the servce tme and the vst tme at queue are exponentally dstrbuted. Based on the results derved up to that pont, n Secton 5 we gve some numercal results. Specfcally, we examne numercally the effect of the frst two moments of the vst and servce tmes on the sojourn tme of an arbtrary customer. These numercal results ndcate that there s an optmal value for the mean vst tme to the varous queues that mnmzes the mean sojourn tme of an arbtrary customer. In Secton 6 we nvestgate how we can optmze the vst order of the servers at the varous queues so that the expected throughput of the system s maxmzed. It emerges that even when consderng semdynamc control polces, n whch the group of servers plans a new route for each cycle, the optmal vstng order that maxmzes the expected throughput per cycle s fxed for all cycles. In other

4 84 M. Vlasou and U. Yechal words, because of the nfnte number of servers, nformaton regardng the queue lengths of all queues at the begnnng of a cycle has no effect on the choce of the optmal strategy. 2. MODEL DESCRIPTION AND NOTATION We consder a pollng system wth N 5 2 nfnte-buffer queues attended by a group of ample number of servers that vsts the queues n a fxed cyclc fashon. We ndex the queues by ¼ 1, 2,..., N n the order of the servers movement. We wll refer to the pollng nstant of queue as the moment when the servers enter that queue. When vstng queue, the group of servers contnues workng at ths queue for V unts of tme and acts there as an M/G/1 queue. We assume that the vst tmes are ndependent and dentcally dstrbuted (..d.) random varables. Customers arrve at all queues accordng to ndependent homogeneous Posson processes wth rate l for queue. After completng ther servce tme, customers leave the system. The servce tme of each ndvdual customer at queue s denoted by B. It s assumed that all servce tmes n one queue are..d. random varables, whch are mutually ndependent of all servce tmes at any other queue. At the end of a vst to queue, the group of servers moves to queue þ 1, ncurrng a swtch-over tme D and a realzaton of V þ1 s drawn. We assume that fd g s a sequence of ndependent random varables. The total swtch-over tme durng a full cycle s D ¼ P N ¼1 D, and the length of a full cycle s denoted by the random varable C. We assume that all random varables so far are mutually ndependent. Durng the vst tme of the group of servers to queue, a customer present at queue at the pollng nstant of that queue wll successfully complete hs servce wth probablty p (V ) ¼ P[B 4V j V ]. We assume that f the servce of a customer of queue s not completed durng a sngle vst, then at the next vst, a new servce tme wll be drawn from the servce tme dstrbuton of B for that partcular customer. For a generc random varable Y, we denote ts frst two moments by E[Y ] and E[Y 2 ], respectvely. Thus, for example, E[V ] s the mean vst tme of the servers at queue. By conventon, P =j Y ¼ P N ¼1 Y, and smlarly for the product operator. =j All further notaton wll be ntroduced when t s frst used Law of Moton Let X j,, j ¼ 1, 2,..., N, denote the number of customers n queue j at the moment when queue s polled and let A j (t) denote the number of Posson arrvals to queue j durng a tme nterval of length t. The law of moton descrbng the evoluton of j j the system when the server moves from queue to queue þ 1 connects X þ1 to X and s gven by X j þ1 ¼ Xj þ A j,1(v ) þ A j,2 (D ), j = Bnom(X,1 p (2.1) (V )) þ Posson(L (V )) þ A (D ), j ¼,

5 M/G/1 POLLING SYSTEMS 85 where for all k, A j,k (t) s an..d. copy of A j (t), Bnom(n, p) s a bnomal random varable wth parameters n and p, and Posson(L (t)) s a Posson random varable wth rate L (t) ¼ l ð t 0 P[B. y] dy: Note that from (2.1) we see that for all j, the random varables X j are ndependent of V and D, whch s evdent, consderng that the number of customers n a queue at the begnnng of a vst does not depend on the length of the upcomng vst tme or swtch-over tme. The relaton for j = s straghtforward. The number of customers at queue j at the pollng nstant of queue þ 1 equals the number of customers that were there at pollng nstant of queue plus all customers that arrved durng the vst tme of queue and the swtch-over tme from queue to queue þ 1. For j ¼, the relaton s more nvolved. When the servers start pollng queue, they encounter X customers. After V tme unts, only a bnomal number of customers out of the ntal X s stll present. The probablty that a sngle customer does not complete hs servce after V tme unts s 1 2 p (V ) ¼ P[B. V j V ]. In addton, there s a stream of new arrvals to queue. The number of customers present at tme t n an M/G/1 queue (startng wth zero customers at tme t ¼ 0) s Posson dstrbuted wth rate L (t), as s gven earler; see Takács [16]. The last term on the rghthand sde of (2.1) ncorporates the customers that arrved at the queue durng the swtch-over tme from queue to queue þ 1. We wll employ ths relaton to derve the mean queue length and the probablty generatng functon of all queues at a pollng nstant. 3. QUEUE LENGTHS AT POLLING INSTANTS One of the man tools used n the analyss of pollng systems s the dervaton of a set of multdmensonal probablty generatng functons of the number of jobs present n the varous queues at a pollng nstant of queue. The common method s to derve the probablty generatng functon of a gven queue at some pollng nstant n terms of the probablty generatng functon of the same queue at the prevous pollng nstant. Then, from the set of N (mplct) dependent equatons of the unknown probablty generatng functons, one can obtan expressons that allow for numercal calculaton of the mean queue length at each queue. These equatons smplfy sgnfcantly for several cases of the dstrbuton of the vst tmes. In ths secton, we use the law of moton (buffer occupancy), whch s gven by (2.1) and apply ths technque to compute recursvely the frst moment and the probablty generatng functon of the queue length dstrbutons at a pollng nstant.

6 86 M. Vlasou and U. Yechal 3.1. Mean Queue length From (2.1) we have the followng relaton for the mean queue length of queue j at two consecutve pollng nstants: E[X j þ1 ] ¼ E[Xj ] þ l je[v ] þ l j E[D ], j = (1 p )E[X ] þ E[L (3.1) (V )] þ l E[D ], j ¼, where p ¼ P[B 4V ] ¼ E[ p (V )]. Summng (3.1) over, we obtan p j E[X j j ] ¼ l X j E[V ] þ E[L j (V j )] þ l j E[D]: (3.2) =j Indeed, n steady state, the mean number of jobs n queue j at a pollng nstant equals the fracton of jobs (1 2 p j )E[X j j] left behnd at the end of the prevous vst, plus the mean number of arrvals durng the cycle tme out of queue j, whch s l j ( P =j E[V ] þ E[D]), plus the mean number of customers n a M/G/1 queue at tme V j. The mean queue length of queue j at the pollng nstant of queue s easly derved from (3.1), yeldng E[X j ] ¼ E[Xj j ](1 p X 1 X 1 j) þ E[L j (V j )] þ l j E[V k ] þ l j E[D k ]: (3.3) k¼jþ1 For example, suppose that B j s exponentally dstrbuted wth parameter m j. Then ð Vj L j (V j ) ¼ l j e mjy dy ¼ l j (1 e m jv j ): 0 m j Thus, E[L j (V j )] ¼ l j (1 2 E[e 2m jv j ])/m j. So, n partcular, f V j s also exponentally dstrbuted wth parameter g j, then we have that E[L j (V j )] ¼ l j /(g j þ m j ), and the mean queue length of each queue can now easly be computed recursvely from (3.3). k¼j 3.2. Recursve Relaton for the Generatng Functon Defne the generatng functon of the queue length of all queues at pollng nstants of queue as G (z) ¼ E[ Q N X j¼1 z j j]. Then, from (2.1), we have that Y G þ1 (z) ¼ E z Xj þa j,1(v )þa j,2 (D ) j z Bnom(X,1 p (V ))þposson(l (V ))þa (D ) : (3.4) j= By condtonng on the vector (X 1,..., X N ), on V, and on D, (4.1) becomes " G þ1 (z) ¼ E Y " z Xj j E Y # " # z A j,1(v ) j j V E YN z A j,2(d ) j j D j= j= j¼1 h h # E z Bnom(X,1 p (V )) j X,V E z Posson(L (V )) j V : (3.5)

7 Snce the number of arrvals at any queue durng a fxed amount of tme s ndependent of the number of arrvals at any other queue durng the same gven perod, we have that " # " # E YN z A j(d ) j j D ¼ x ¼ E YN z A j(x) j j¼1 M/G/1 POLLING SYSTEMS 87 ¼ YN j¼1 ¼ YN j¼1 j¼1 h E z A j(x) j ¼ YN X 1 z n j j¼1 n¼0 e l jx(1 z j ) : (l j x) n n! e l jx Therefore, we have " # E YN j¼1 z A j(d ) j j D P N ¼ e D l j¼1 j(1 z j ) : (3.6) Lkewse, we obtan that " # E Y j= z A j(v ) j j V P ¼ e V l j= j(1 z j ) : (3.7) Moreover, h E z Bnom(X,1 p (V )) n other words, h E z Bnom(X,1 p (V )) j X ¼ k, V ¼ x h ¼ E z Bnom(k,1 p (x)) ¼ Xk z ¼0 k (1 p (x)) p (x) k ¼ ( p (x) þ z [1 p (x)]) k ; j X, V ¼ ( p (V ) þ z [1 p (V )]) X : (3.8) For the last term of the rght-hand sde of (3.5) we have that h h E z Posson(L (V )) j V ¼ x ¼ E z Posson(L (x)) ¼ X1 z n n¼0 (L (x)) n e L(x) ¼ e L (x)(1 z ), n!

8 88 whch yelds that M. Vlasou and U. Yechal h E z Posson(L (V )) j V ¼ e L (V )(1 z ) : (3.9) Substtutng (3.6) (3.9) nto (3.5), we obtan " G þ1 (z)¼e Y P P # z Xj j e V N j= j(1 z j ) D e l j¼1 j(1 z j ) (p (V )þz [1 p (V )]) X e L (V )(1 z ) : j= (3.10) Recall that for all j, the random varables X j are ndependent of V and D. Therefore, (3.10) becomes P N G þ1 (z)¼e e D l j¼1 j(1 z j ) " E Y P # z Xj j e V j= j(1 z j ) (p (V )þz [1 p (V )]) X e L (V )(1 z ) : j= (3.11) Consequently, G þ1 (z)¼ ~D X N j¼1 l j (1 z j )! h P E e V l j= j(1 z j ) e L (V )(1 z ) G (z 1,z 2,...,z 1,p (V ) þ[1 p (V )]z,z þ1,...,z N ), (3.12) where D (s) ¼ E[e 2sD ] denotes the Laplace Steltjes transform of the random varable D. Evdently, f V follows a dscrete dstrbuton, the above expresson smplfes sgnfcantly. Note that the mean queue length at a pollng nstant (3.3) can also be obtaned by dfferentatng (3.12). Remark 1: Applyng smlar technques, we can also derve the probablty generatng functon of the number of customers at the end of a vst at queue þ 1. If we denote by Y j the number of customers n queue j ¼ 1,..., N at the moment when the servce at queue ¼ 1,..., N s completed, then the law of moton descrbng the evoluton of the system s gven by Y j þ1 ¼ Yj þ A j(d ) þ A j (V þ1 ), j = þ 1 Bnom(Y j þ A (3.13) j(d ), 1 p j (V j )) þ Posson(L j (V j ), j ¼ þ 1: Also note that the expected value of Y j can be easly computed from (3.3) by observng that for all j =, Y j ¼ X j þ A j (V ), whereas for the th queue, we have that Y ¼ Bnom(X,12p (V )) þ Posson(L (V )).

9 M/G/1 POLLING SYSTEMS SOJOURN TIME Let the sojourn tme of a customer at queue be denoted by S. We compute ts expected value (and thus, by Lttle s law, also the mean queue length of queue at an arbtrary moment), and we derve the Laplace Steltjes transform of S. As stated earler, for each queue we assume that f the servce of a customer s not completed durng a vst, then for the next vst at that queue, a new servce tme wll be resampled for the same customer from the servce-tme dstrbuton of that queue Mean Sojourn Tme Recall that the cycle tme s gven by C ¼ PN ¼1 (V þ D ). In order to derve the mean sojourn tme of a customer arrvng at queue, we wll need some further notaton. Denote by V res the resdual vst tme of the group of servers at queue and by C / the cycle tme except the tme spent servng queue (.e., C / ¼ C 2 V ). Smlarly, C res / represents the resdual cycle tme excludng the vst tme of queue ; that s, C res / measures the length of tme from a random moment after leavng queue untl the next pollng nstant of queue. Furthermore, let fc m g be a famly of..d. random varables dstrbuted lke C and let N be a (shfted) geometrc random varable wth success probablty p ¼ E[ p (V )] ¼ P[B 4V ] (.e., P[N ¼ n] ¼ (1 2 p ) n p ), for all nteger n 5 0. One should observe here that N þ 1 s a stoppng tme, as t s the frst tme when the servce tme of a customer n queue s less than or equal to the vst tme at that queue; that s, N þ 1 ¼ nffk : B,k 4 V,k g, where B,k and V,k are..d. copes of B and V respectvely. Smlarly, a second ndex s added to a random varable, every tme that we explctly need to ndcate that an ndependent copy s consdered. Then the sojourn tme of a customer at queue s gven by 8 B,0 (arrval durng V and B,0 4V,0 res) V,0 res S ¼ þ PN >< C m þ C = þ B,N þ1 (arrval durng V,0 and B,0. V,0 res) C= res þ PN >: C m þ B,N þ1 (arrval durng C = ): (4.1) Note that the probablty of an arrval occurrng durng the vst tme of queue s E[V ]/E[C] (.e., the expected vst tme of queue over the expected cycle tme) and smlarly for the other two events. Therefore, from (4.1) we obtan that the

10 90 M. Vlasou and U. Yechal expected sojourn tme of a customer of queue s gven by E[S ] ¼ E[V ] E[C] P[B 4 V res ]E[B j B 4 V res ] " þ E[V ] E[C] P[B. V res ]E V,0 res þ XN C m þ C = þ B,N þ1 j B,0 5 V res,0 þ E[C " # =] E[C] E Cres = þ XN C m þ B,N þ1 : (4.2) # In order to compute the second condtonal expectaton appearng at the rght-hand sde of the above equaton, we thnk as follows. For N cycles, the servce of the customer s not completed durng that vst because for every vst B. V, whle at the (N þ 1)st vst, the servce s completed wthn that cycle. Therefore, defne C m ¼ C =,m þ mn (B,m, V,m ) and observe that " # " # C m ¼ E XN C m þ C = þ B : E XN þ1 Thus, " # E V,0 res þ XN C m þ C = þ B,N þ1 j B,0 5 V,0 res ¼ E[V res " # j B 5 V res ] þ E XN þ1 C m ¼ E[V res j B 5 V res ¼ E[V res ] þ E[N þ 1]E[ C m ] j B 5 V res ] þ E[N þ 1](E[C = ] þ E[ mn (B,V )]), (4.3) where n the second equalty we used Wald s equaton.

11 For the thrd condtonal expectaton appearng at the rght-hand sde of (4.2), we have that " # E C= res þ XN C m þ B,N þ1 M/G/1 POLLING SYSTEMS 91 ¼ E[C2 = ] " # 2E[C = ] þ E XN (C =,m þ V,m ) þ B,N þ1 ¼ E[C2 = ] " # " # 2E[C = ] þ E XN C =,m þ E XN þ1 mn (B,m,V,m ) ¼ E[C2 = ] 2E[C = ] þ E[N ]E[C = ] þ E[N þ 1]E[ mn (B,V )]: (4.4) Summarzng the above, we have that E[S ] ¼ E[V ] E[C] P[B 4 V res ]E[B j B 4 V res ] þ E[V ] E[C] P[B. V res ](E[V res j B. V res ] þ E[N þ 1](E[C = ] þ E[ mn (B,V )])) þ E[C =] E[C= 2 ]! E[C] 2E[C = ] þ E[N ]E[C = ] þ E[N þ 1]E[ mn (B,V )] : (4.5) In Secton 5 we wll llustrate through an example the effect of the frst two moments of the vst tme and the servce tme on the mean sojourn tme of an arbtrary customer The Laplace Steltjes Transform We now derve the Laplace Steltjes transform of the sojourn tme of a customer of queue. We frst rewrte (4.1) n terms of the Laplace Steltjes transforms of all varables nvolved (cf. (4.5)), and thus we get that E[e ss ] ¼ E[V ] E[C] P B 4 V res E e sb þ E[V ] E[C] P B. V res E e sv res þ E[C h =] E[C] E e scres = E " j B 4 V res e sp N C =,m j B. V res # " E E "e sp N þ1 # C m e sp N þ1 mn (B,m,V,m ) # : (4.6)

12 92 M. Vlasou and U. Yechal We rewrte several of the terms appearng above as follows. The dstrbuton functon of V res s gven by yeldng Smlarly, we have that whch mples that P[V res 4 x] ¼ 1 E[V ] P[B. V res ] ¼ 1 E[V ] ð 1 P[C res = 4 x] ¼ 1 E[C = ] 0 ð x 0 P[V. y] dy, P[B. x]p[v. x] dx: (4.7) ð x 0 P[C =. y] dy, h E e scres = ¼ 1 ec = (s), (4.8) se[c = ] where C / denotes the Laplace Steltjes transform of the random varable C /. Moreover, E e sp N þ1 C m ¼ X1 n¼0 ¼ X1 ¼ n¼0 E e sp nþ1 C m (1 p ) n p E[e s C ] nþ1 (1 p ) n p p E[e s C ] 1 (1 p )E[e s C ], (4.9) where C ¼ C / þ mn(b, V ). Lkewse, we have that h E e sp N C p =,m ¼ 1 (1 p )E[e sc (4.10) = ] and E e sp N þ1 mn (B,m,V,m ) p E[e s mn (B,V ) ] ¼ 1 (1 p )E[e s mn (B,V ) ] : (4.11)

13 Substtutng (4.7) (4.11) nto (4.6), we have that the Laplace Steltjes transform of the sojourn tme of a customer of queue s gven by E[e ss ] ¼ 1 E[C] E e sb j B 4 V res ð 1 P[B x]p[v. x] dx þ 1 E[C] ð 1 0 res E e sv 0 j B. V res p E e s C 1 (1 p )E e s C P[B. x]p[v. x] dx þ 1 ec = (s) se[c] M/G/1 POLLING SYSTEMS 93 p 1 (1 p )ec = (s) p E e s mn (B,V ) 1 (1 p )E½e s mn (B,V ) Š : (4.12) Clearly, from the above expresson, one can retreve (4.5) for the mean sojourn tme of a customer of queue. The transforms appearng n (4.12) mght be cumbersome to compute when the servce tmes or the vst tmes are generally dstrbuted. However, when both B and V follow a phase-type dstrbuton, all transforms can be computed explctly snce the class of phase-type dstrbutons s closed under fnte mnma. Phase-type dstrbutons are wdely used n computatons. The class of phase-type dstrbutons s dense (n the sense of weak convergence) n the class of all dstrbutons on (0, 1) (cf. [2, Props. 1 and 2]). As an example, we wll derve the Laplace Steltjes transform of the sojourn tme of a customer of queue, as well as ts mean, n the case that both the vst tme and the servce tme at queue are exponentally dstrbuted A Specal Case Let the servce tme and the vst tme at queue be exponentally dstrbuted wth rates m and g, respectvely. Then all terms appearng n (4.5) can be easly computed n terms of m and g. For example, E[B j B 4 V res ] ¼ 1 g þ m and P[B. V res ] ¼ g /(g þ m ). Thus, (4.5) becomes E[S ] ¼ 1 m 1 g E[C] g þ m g þ m þ 1 g 1 þ g þ 1 E[C = ] þ 1 g E[C] g þ m g þ m m g þ m þ E[C =] E[C= 2 ] E[C] 2E[C = ] þ g E[C = ] þ g! 1 þ 1 m m g þ m

14 94 M. Vlasou and U. Yechal or E[S ] ¼ (g E[C = ] þ 1) 2 þ E[C2 = ] g m E[C] 2E[C] : Smlarly, (4.12) reduces to E[e ss ] ¼ 1 g þ m m E[C] g þ m þ s g (g þ m ) þ 1 g þ m m E[e s C ] 1 E[C] g þ m þ s g þ m g E[e s C ] g þ m þ 1 ec = (s) m se[c] g þ m g ec = (s) m (g þ m )=(g þ m þ s) g þ m g (g þ m )=(g þ m þ s) : Snce E[e s C ] ¼ ~C = (s)e[e s mn (B,V ) ], we have that the prevous expresson reduces to E[e ss ] ¼ 1 1 m E[C] g þ m þ s þ 1 1 m ec = (s) E[C] g þ m þ s g þ m þ s g ec = (s) þ 1 ec = (s) m m se[c] g þ m g ec = (s) m þ s : Smlar expressons can be easly derved n the case that both the vst tmes and the servce tmes follow some phase-type dstrbuton, such as Gamma, hyperexponental, or Coxan dstrbutons. g 5. NUMERICAL RESULTS Ths secton s devoted to some numercal results. In partcular, we want to examne numercally the effect of the frst two moments of the vst and servce tmes on the sojourn tme of an arbtrary customer. In all examples, we make the followng assumptons. We consder a pollng system wth two queues. The arrval rate at the frst queue s l 1 ¼ 0.8, and at the second queue, t s l 2 ¼ 0.5. The servce tme and the vst tme at the frst queue are exponentally dstrbuted wth rates m 1 ¼ 1 and g 1 ¼ 1, respectvely. Moreover, the total mean swtch-over tme s taken to be E[D] ¼ 0.5, and ts second moment s assumed to be zero. In all fgures that follow, we plot the mean sojourn tme of an arbtrary customer, whch s estmated by (l 1 E[S 1 ] þ l 2 E[S 2 ])/(l 1 þ l 2 ).

15 In Fgures 1 and 2 we nvestgate the effect of the frst two moments of the servce tme at the second queue on the mean sojourn tme of an arbtrary customer. For these plots, the vst tme at the second queue s consdered to be exponentally dstrbuted wth rate g 2 ¼ 3/2. For varous values of the squared coeffcent of varaton of the servce tme at the second queue, whch s denoted by c 2 B 2, we plot n Fgure 1 the mean sojourn tme of an arbtrary customer versus the mean servce tme E[B 2 ]. The squared coeffcent of varaton of the servce tme s chosen to be comparable to the squared coeffcent of varaton of the (exponentally dstrbuted) vst tme, whch s equal to 1. In Fgure 2, we plot the mean sojourn tme of an arbtrary customer versus c 2 B 2 for three values of E[B 2 ], whch agan are chosen to be comparable to E[V 2 ]. For each case of c 2 B 2, we ft a mxed Erlang or hyperexponental dstrbuton to E[B 2 ] and c 2 B 2, dependng on whether the squared coeffcent of varaton s less or greater than 1; see, e.g., Tjms [20]. So, f 1/n 4 c 2 B 2 4 1/(n 2 1) for some n ¼ 2, 3,..., then the mean and squared coeffcent of varaton of the mxed-erlang dstrbuton G(x) ¼ p M/G/1 POLLING SYSTEMS 95 Xn 2 zx 1 e j¼0 þ (1 p)! (zx) j j! Xn 1 zx 1 e j¼0! (zx) j, x 5 0, j! matches wth E[B 2 ] and c 2 B2, provded the parameters p and z are chosen as p ¼ 1 qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 1 þ c 2 nc 2 B 2 n(1 þ c 2 B 2 ) n 2 c 2 B 2, z ¼ n p B 2 E[B 2 ] : On the other hand, f c 2 B 2. 1, then the mean and squared coeffcent of varaton of the hyperexponental dstrbuton G(x) ¼ p(1 e z 1x ) þ q(1 e z 2x ), x 5 0, match wth E[B 2 ] and c 2 B2, provded the parameters z 1, z 2, p, and q are chosen as s ffffffffffffffffffffffffffff! p ¼ þ c 2 B 2 1 c 2, q ¼ 1 p, B 2 þ 1 z 1 ¼ 2p E[B 2 ], z 2 ¼ 2q E[B 2 ] : As s evdent from the plot n Fgure 1, the expected sojourn tme of an arbtrary customer ncreases as the mean servce tme at the second queue ncreases. Moreover, the rate at whch t ncreases s almost lnear as c 2 B 2 grows and the effect of the second moment s less pronounced than the effect of E[B 2 ].

16 96 M. Vlasou and U. Yechal FIGURE 1. Mean sojourn tme of an arbtrary customer aganst the mean servce tme E[B 2 ]. FIGURE 2. Mean sojourn tme of an arbtrary customer aganst the squared coeffcent of varaton of the servce tme B 2.

17 M/G/1 POLLING SYSTEMS 97 In Fgure 2, one observes that the mean sojourn tme of an arbtrary customer decreases as the squared coeffcent of varaton of the servce tme ncreases, contrary to the case for the M/G/1 queue. Ths result s due to the fact that the servce tme of a customer that dd not complete hs servce durng one vst tme s resampled for the followng vst tme. Therefore, the larger the varablty n the servce tmes, the larger s the probablty that durng the next vst tme, ths partcular customer wll complete hs servce. Recall that the mean vst tme at the second queue s equal to 2/3 and observe that n the case that E[B 2 ] s less than E[V 2 ], the effect of the second moment of the servce tme on the mean sojourn tme of an arbtrary customer s almost neglgble. In Fgures 3 and 4 we now nvestgate the effects of the frst two moments of the vst tme at the second queue on the mean sojourn tme of an arbtrary customer. For these plots, we now take the servce tme at the second queue to be exponentally dstrbuted wth rate m 2 ¼ 3/2. For varous values of the squared coeffcent of varaton of the vst tme at the second queue, whch s denoted by c 2 V 2, we plot n Fgure 3 the mean sojourn tme of an arbtrary customer versus the mean vst tme E[V 2 ]. As earler, the squared coeffcent of varaton of the vst tme s chosen to be comparable wth the squared coeffcent of varaton of the (exponentally dstrbuted) servce tme, whch s equal to 1. In Fgure 4, we plot the mean sojourn tme of an arbtrary customer versus c 2 V 2 for three values of E[V 2 ], whch agan are chosen to be comparable wth E[B 2 ]. The plot n Fgure 3 s nterestng. Evdently, when E[V 2 ] s sgnfcantly smaller than E[B 2 ], only a very small number of customers wll be served durng a vst. As the FIGURE 3. Mean sojourn tme of an arbtrary customer aganst the mean vst tme E[V 2 ].

18 98 M. Vlasou and U. Yechal FIGURE 4. Mean sojourn tme of an arbtrary customer aganst the squared coeffcent of varaton of the vst tme V 2. mean vst tme ncreases, more customers are served durng a vst and the mean sojourn tme of an arbtrary customer s reduced. However, as the mean vst tme contnues to ncrease, ths trend s reversed after the mean sojourn tme of an arbtrary customer reaches a global mnmum. In other words, there s an optmal value for the mean vst tme to some queue that mnmzes the mean sojourn tme of an arbtrary customer; beyond that value, the mean sojourn tme of an arbtrary customer ncreases at an almost lnear rate. Ths ndcates that the pollng system under consderaton can be optmzed n expectaton by controllng the vst tme to each queue. In the followng secton, we wll develop a polcy that mnmzes the mean sojourn tme of an arbtrary customer n the system. The plot n Fgure 1 s also nterestng. As s explaned earler, we ft ether a mxture of Erlang dstrbutons or a hyperexponental dstrbuton to each par of the frst two moments of the vst tme, dependng on the value of the squared coeffcent of varaton. For every value of c 2 V 2, we obtan a dfferent vst tme dstrbuton. Note that the jump n Fgure 1 occurs when the dstrbuton we ft to the frst two moments of the vst tme shfts from a mxture of Erlang dstrbutons to a hyperexponental dstrbuton. Ths ndcates that the shape of the vst tme dstrbuton s mportant; for example, hyperexponental dstrbutons are always unmodal, whch s not the case for mxed Erlang dstrbutons. Consequently, the frst two moments cannot capture suffcently the effect of the vst tme dstrbuton on the mean sojourn tme of an arbtrary customer; one needs to know the exact dstrbuton.

19 6. DYNAMIC CONTROL OF SERVERS VISITS M/G/1 POLLING SYSTEMS 99 A basc queston that arses when plannng effcent pollng systems concerns the order of vsts performed by the servers. As t s suggested by Fgure 3, the pollng system we are consderng can be optmzed n some way so that the mean sojourn tme of an arbtrary customer s mnmzed. Rather than dentfyng the value of the mnmum mean sojourn tme for the cyclc processng (vstng) order consdered so far, we frst nvestgate whether there exsts a fxed statc order that the servers vst the varous queues so that the mean (weghted) sojourn tme of an arbtrary customer s mnmzed. As s evdent from (4.5), the mean sojourn tme of a customer of queue does not depend on the order the queues have been vsted and, thus, nether does the weghted sum thereof. Snce the mean sojourn tme of an arbtrary customer remans unaffected when alterng the processng order, t does not consttute a practcal performance measure of our system. An appealng approach that leads to a smple and tractable rule s to develop a semdynamc control scheme. The dea s to dspatch the group of servers to perform Hamltonan tours, each tour beng possbly dfferent from the prevous one, dependng on the state of the system at the begnnng of the tour, so as to optmze some performance measure. An adequate performance measure s the throughput of the system, namely the number of customers served per cycle, as the throughput can be measured per cycle, and the sojourn tme of a customer spans over a random number of cycles. The goal s to maxmze the throughput of the system for each cycle. Specfcally, suppose that at the begnnng of a cycle n ¼ (n 1, n 2,..., n N ) s the state of the system, where n s the number of jobs watng n queue. We wll compute the expected value of the number of customers served per cycle under a specfc processng order and, consequently, dentfy the optmal processng order per cycle. The followng theorem summarzes the result. THEOREM 1: For the Hamltonan-tour approach, the optmal vstng order s ndependent of the number of customers present at the varous queues at the start of the cycle and s gven by the ndex-type rule l p E[V ] þ E[D ] n the sense that the throughput s maxmzed f and only f the vstng order s arranged accordng to an ncreasng sequence of ths rule. Before proceedng wth the proof, we pont out that havng more nformaton regardng the system, such as the number of customers present at all queues, has no effect on the optmal strategy; thus, t does not mprove the performance of the system. Ths stands n contrast to many other pollng systems where, typcally, more nformaton regardng a system leads to decsons that ncrease the effcency

20 100 M. Vlasou and U. Yechal of the system; see, for example, [7, 8, 26]. Ths concluson stems from the fact that no matter what cycle order s used, the number of customers served n a cycle among those ntally present wll have the same dstrbuton and wll have no effect on the number of others served (or the number recevng partal servce) durng the cycle. In other words, what happens to those ntally present at the begnnng of a tour s unaffected by what orderng s used. PROOF: The proof follows from an nterchange argument. Consder the processng order p 0 ¼ (1, 2,..., N ). Denote by u the throughput of queue under ths processng order (.e., the number of customers of queue that are served durng a cycle) and denote by u the total throughput of the system (.e., the sum of all u ). Gven n (.e., the state of the system at the begnnng of a cycle), we wll compute the expected value of u. The number of customers served after completng a vst at queue s equal to the porton of customers that where present at pollng nstant and successfully completed ther servce plus the number of customers that arrved durng the vst tme of queue and completed ther servce wthn that vst. In other words,!! X 1 u ¼ Bnom n þ A (V k þ D k ), p (V ) þ A (V ) Posson(L (V )Þ: As a result, E[u ] ¼! X 1 n þ l (E[V k ] þ E[D k ]) p þ l E[V ] E[L (V )], whch yelds where E[u] ¼ c þ XN X 1 l p (E[V k ] þ E[D k ]), (6.1) ¼1 c ¼ XN ¼1 (n p þ l E[V ] E[L (V )]): Observe that the constant c that appears n (6.1) does not depend on p 0, whereas the second term at the rght-hand sde of (6.1) does. Consder now the processng order p 1 ¼ (1, 2,..., j 2 1, j þ 1, j, j þ 2,..., N ), where the vst order of queues j and j þ 1 s nterchanged and denote by u 0 and u 0 the throughput of queue and of the whole system under p 1, respectvely. We promptly

21 M/G/1 POLLING SYSTEMS 101 have that E[u ] ¼ E[u] 0 for all = j, j þ 1 and that!! E[u 0 j ] ¼ n X j 1 j þ l j (E[V k ] þ E[D k ]) þ E[V jþ1 ] þ E[D jþ1 ] Thus, þ l j E[V j ] E[L j (V j )],! E[u 0 jþ1 ] ¼ n X j 1 jþ1 þ l jþ1 ðe[v k ] þ E[D k ] Þ E[u 0 ] ¼ c þ þ l jþ1 E[V jþ1 ] E[L jþ1 (V jþ1 )]: þ l j p j X =j, jþ1 X j 1 p jþ1 X 1 l p (E[V k ] þ E[D k ]) (E[V k ] þ E[D k ]) þ E[V jþ1 ] þ E[D jþ1 ] X j 1 þ l jþ1 p jþ1 (E[V k ] þ E[D k ]): Therefore, we have that E[u]4E[u 0 ] f and only f or X j 1 X j l j p j (E[V k ] þ E[D k ]) þ l jþ1 p jþ1 (E[V k ] þ E[D k ]) X j 1 4 l j p j (E[V k ] þ E[D k ]) þ l j p j (E[V jþ1 ] þ E[D jþ1 ]) X j 1 þ l jþ1 p jþ1 (E[V k ] þ E[D k ]) l jþ1 p jþ1 (E[V j ] þ E[D j ]) 4 l j p j (E[V jþ1 ] þ E[D jþ1 ]): In other words, we get that the optmal processng order s by vstng the queues accordng to an ncreasng order of l p /(E[V ] þ E[D ]). B Roughly stated, ths rule arranges the vst order accordng to the rato between new arrvals per unt tme that wll successfully complete ther servce (.e., l p ) and the mean duraton of a vst there (.e., E[V ] þ E[D ]). It s ntutvely clear that f the mean vst and swtch tme for a queue are relatvely long, then one should vst ths queue early on. In ths way, the number of customers at the other queues durng ths! p j

22 102 M. Vlasou and U. Yechal cycle wll also be relatvely hgh, and as a result, the throughput wll be ncreased snce all customers are served smultaneously by an nfnte number of servers. Ths s an extremely smple rule, whch can be drectly mplemented. Moreover, suppose that, for one reason or another, the objectve s to mnmze the throughput of the system for each cycle. Then the ndex rule that determnes the order of vsts to the queues s smply reversed: The servers complete a Hamltonan tour arranged n a decreasng order of l p /(E[V ] þ E[D ]). Observe that, under ths strategy, the servers also vst the queues that are empty at the begnnng of the cycle. One expects that the throughput of the system n the long run s mproved when these queues are not vsted wthn a cycle; namely t mght be more effcent to avod queues that are empty at the begnnng of the cycle n order to allow them to buld up. Accordng to the way the system s desgned, even f the servers do not vst a queue that at the begnnng of the cycle was empty, the swtch tme assocated wth ths queue (.e., the tme to swtch from ths queue to the followng one) s stll ncurred. Therefore, as the number of queues that wll not be vsted n a cycle grows, the servers spend an ncreasng amount of tme beng essentally dle (as they swtch between queues). A possbly more effcent system desgn s as follows. Rather than envson the group of servers movng from one queue to another, we can thnk of a central pont to whch the servers always return after each completon of a vst to a queue. The return tme to the central pont after vstng queue s denoted by R. The servers depart from that central pont and move to the followng queue that wll be served. The total tme from the moment the servers leave the central pont untl they enter queue s denoted by E. Accordng to ths desgn, the total tme to go from queue to queue j s gven by R þ E j for any = j. The queston that arses s whether there exsts a semdynamc control of ths system. As earler, t emerges that a Hamltonan-tour approach leads to a statc processng order accordng to an ndex rule. THEOREM 2: For the pollng system wth a central pont, the Hamltonan-tour approach leads to a fxed optmal vstng order, whch s ndependent of the number of customers present at the varous queues at the start of the cycle. The throughput of the system for each cycle s maxmzed f and only f the vstng order s arranged accordng to an ncreasng sequence of the ndex-type rule l p E[E ] þ E[V ] þ E[R ] : (6.2) PROOF: As earler, let n ¼ (n 1, n 2,..., n N ) be the state of the system at the start of the tour and denote by L the number of nonempty queues, 0, L 4 N, at the begnnng of the cycle. The throughput of queue durng a Hamltonan cycle that vsts only the

23 M/G/1 POLLING SYSTEMS 103 nonempty queues accordng to the order p 0 ¼ (1,2,...,L) s gven by! X 1 u ¼ Bnom n þ A (E k þ V k þ R k ) þ E!, p (V ) þ A (V ) Posson(L (V )) : Consequently,!! X 1 E[u ] ¼ n þ l ðe[e k ] þ E[V k ] þ E[R k ] ÞþE[E ] þ l E[V ] E[L (V )], whch yelds p where E[u] ¼ c 0 þ XN X 1 l p (E[E k ] þ E[V k ] þ E[R k ]), (6.3) ¼1 c 0 ¼ XN ¼1 ((n þ l E[E ])p þ l E[V ] E[L (V )]): Applyng an nterchange argument, we have that the optmal processng order s constructed by an ncreasng sequence of the ndex rule gven by (6.2). B As earler, the optmal tour does not depend on the number of customers present at the begnnng of the cycle. Ths s a drect consequence of the fxed vst tmes and the underlyng M/G/1 process at each queue. Index rules appear regularly when optmzng pollng systems. Browne and Yechal [7,8] were the frst to obtan dynamc control polces for sngle-server systems under the exhaustve, gated, or mxed-servce regmes. The mechancs of the system are as descrbed here: At the begnnng of each cycle, the server decdes on a new Hamltonan tour and vsts the channels accordngly. The authors showed that f the objectve s to optmze the cycle duraton under these polces, then an ndex-type rule apples, whch s smlar to the one descrbed here. The man dfference s that the ndex rule that s optmal for these polces depends on the state of the system at the begnnng of a cycle, contrary to the results obtaned for the fxed-vst-tme polcy studed n ths artcle. The result derved by Browne and Yechal [7,8] s a surprsng result, as the ndex rule does not nclude the servce tmes at the varous channels. It s also surprsng that the same ndex rule holds for both the gated and the exhaustve dscplnes although the duraton of a cycle startng from the same state s dfferent for the two regmes. For a further dscusson on other types of ndex-rule polces, see Yechal [26], van der Wal and Yechal [23], and references theren.

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