Iterative approximation of k-limited polling systems

Transcription

1 Queueng Syst (2007) 55: 6 78 DOI 0.007/s Iteratve approxmaton of k-lmted pollng systems M. van Vuuren E.M.M. Wnands Receved: May 2006 / Revsed: 5 December 2006 / Publshed onlne: 2 March 2007 Sprnger Scence+Busness Meda, LLC 2007 Abstract The present paper deals wth the problem of calculatng queue length dstrbutons n a pollng model wth (exhaustve) k-lmted servce under the assumpton of general arrval, servce and setup dstrbutons. The nterest for ths model s fueled by an applcaton n the feld of logstcs. Knowledge of the queue length dstrbutons s needed to operate the system properly. The mult-queue pollng system s decomposed nto sngle-queue vacaton systems wth k-lmted servce and state-dependent vacatons, for whch the vacaton dstrbutons are computed n an teratve approxmate manner. These vacaton models are analyzed va matrx-analytc technques. The accuracy of the approxmaton scheme s verfed by means of an extensve smulaton study. The developed approxmaton turns out to be accurate, robust and computatonally effcent. Keywords Pollng systems k-lmted servce Approxmaton Decomposton Mathematcs Subject Classfcaton (2000) 60K25 68M20 90B30 Ths research s supported by the Technology Foundaton STW, appled scence dvson of NWO and the technology programme of the Dutch Mnstry of Economc Affars. M. van Vuuren Department of Mathematcs and Computer Scence, Technsche Unverstet Endhoven, P.O. Box 53, 5600 MB Endhoven, The Netherlands e-mal: m.v.vuuren@tue.nl E.M.M. Wnands ( ) Department of Mathematcs and Computer Scence, Department of Technology Management, Technsche Unverstet Endhoven, P.O. Box 53, 5600 MB Endhoven, The Netherlands e-mal: e.m.m.wnands@tue.nl Introducton A typcal pollng system conssts of a number of queues, attended by a sngle server n a fxed order. There s a huge body of lterature on pollng systems that has developed snce the late 950s, when the papers [29, 30] concernng a patrollng reparman model for the Brtsh cotton ndustry were publshed. Pollng systems have a wde range of applcatons n communcaton, producton, transportaton and mantenance systems. Excellent surveys on pollng systems and ther applcatons may be found n [39, 4, 42] and n [28]. The vast majorty of the lterature s concerned wth the two tradtonal servce dscplnes, the exhaustve and gated polces. Exhaustve means that a queue must be empty before the server moves on, whereas n case of gated servce only those customers n the queue at the pollng start are served. Suggested references for readers who would lke to pursue ther study of the exhaustve and gated polces are [39, 4, 42]. The man drawback of these tradtonal polces s the nablty to prortze among the dfferent queues for mprovng total system performance. A more sophstcated servce strategy offerng ths possblty s the k-lmted servce strategy. Under ths k-lmted strategy the server contnues workng at a queue untl ether a predefned number of k customers s served or untl the queue becomes empty, whchever occurs frst. Note that the case k s equvalent to the exhaustve servce strategy. In many applcatons of pollng systems, the objectve functon typcally depends not only on the mean queue lengths, but on the complete margnal queue length dstrbutons (an llustratve applcaton s descrbed at the end of the present secton). The present paper, therefore, ams to study the margnal queue length dstrbutons n contnuous-tme pollng systems wth

2 62 Queueng Syst (2007) 55: 6 78 k-lmted servce under the assumpton of general arrval, servce and setup dstrbutons. To ths very day, not only hardly any exact results for pollng systems wth the k-lmted servce polcy have been obtaned [25, 36, 37, 47], but also ther dervatons gve lttle hope for extensons to more realstc systems. Ths defcency of exact results s due to the fact that the k-lmted servce dscplne does not satsfy a well-known branchng property ndependently ascertaned by [7] and [38]. Ths branchng property causes a strkng dchotomy n complexty across the analyss of varous pollng systems, where the k-lmted servce polcy s on the wrong sde of the borderlne mplyng that even mean queue lengths are n general not known. In the absence of exact results for the margnal queue length dstrbutons, people have resorted to numercal approaches, such as the power seres algorthm [2] and technques based on dscrete Fourer transforms [26]. The man dsadvantage of both methods s that tme and memory requrements are exponental functons of the number of queues. A feasble approxmate approach for the queue length dstrbuton n a k-lmted pollng system s the decomposton method, n whch the pollng system s decomposed n vacaton systems, for whch the vacaton dstrbutons are computed n an teratve approxmate manner. At each step n the teraton the mathematcal analyss focusses on one sngle queue, whereas the other queues n the system determne the length of the vacaton perod. Ths decomposton method s adopted by the present research as well. We have to remark that these decomposton methods seem to be applcable to a wde varety of queueng systems (see, e.g., [9, 8, 44, 45]). In the past, some systems related to the one of the present paper have been studed by the decomposton approach,.e., a k-lmted pollng system wth fnte buffers under the assumpton of Posson arrval processes [23] ora k-lmted pollng system n combnaton wth a reservaton mechansm [24]. The qualtatve observatons of these studes seem to carry over to the system of the present paper. The key observaton, whch s at the same tme the mathematcal motvaton of the present study, s the fact that t s extremely mportant to capture the correlatons among the dfferent queues, snce these correlatons have a sgnfcant mpact on the performance measures. Whereas [23] does not take these dependences nto account, [24] proposes to take a weghted sum of a completely uncorrelated and a perfectly correlated system n each step of the teraton by usng a pre-defned mxng probablty. Although the method of [24] clearly outperforms the procedure that gnores the correlatons, ths procedure s unable to compensate for correlatons n systems wth only two queues and s also dffcult to apply for systems wth more than two queues. That s, snce the qualty of the procedure strongly depends on the mxng probablty, t s rather complcated to fnd an expresson of ths probablty provdng accurate results over the entre range of parameters. Further, the procedure of [24] s based on generatng functons, the numercal determnaton of zeros and the numercal nverson of characterstc functons, consderably ncreasng the computatonal tme of the algorthm. Fnally, due to specal features of the protocol studed n [24] the correlatons between the queue lengths are relatvely small compared to our system (e.g., n case all queues have a servce lmt of the correlatons vansh), whch makes the approach of [24] well suted for that partcular protocol. Therefore, the goal of the present study s the development of a computatonally effcent teratve approxmaton method for the margnal queue length dstrbutons n the k-lmted pollng model. The man challenge can be found n the estmaton of the correlatons between the queue lengths n each step of the teratve algorthm. The vast majorty of the lterature on pollng systems s devoted to delay fgures, whle almost no attenton has been gven to the analyss of such correlatons. By usng the recently developed mean value analyss for pollng systems of [48] asthestartng pont, [32] derves heavy-traffc asymptotcs for the covarances between successve vst tmes n pollng systems wth mxtures of gated and exhaustve servce under the assumpton of Posson arrvals. Subsequently, [32] proposes smple closed-form approxmatons of these covarances for stable systems,.e., wth load less than one. However, to the best of our knowledge no results are known for the correlatons among queues n pollng systems wth k-lmted servce. The key deas of the approach undertaken n the present paper for pollng systems wth k-lmted servce are as follows:. The dependence between the queue under consderaton and the other queues s taken nto account by the ntroducton of condtonal vacatons (also called ntervst perods),.e., the length of the ntervst perod s postvely correlated to the length of the precedng vst perod. 2. The mutual dependences of the other queues are approxmated va standard probablstc arguments and the condtonal ntervst perods. The man contrbuton of the present paper s the development of a novel teratve approxmaton scheme for k-lmted pollng systems wth general arrval, servce and setup dstrbutons. The algorthm developed n the present paper only needs nformaton on the frst two moments of all dstrbutons. The accuracy of the approxmaton scheme s verfed by means of an extensve smulaton study. The approxmaton scheme turns out to be robust and computatonally effcent, whle the dfferences between the exact and approxmate values are small wthn a reasonable margn. In partcular, the tme complexty s only polynomal n the number of queues and the servce lmts. The man

3 Queueng Syst (2007) 55: buldng block of ths algorthm s a k-lmted servce vacaton model wth state-dependent vacatons, whch has not been studed before n the open lterature. In ths vacaton model, the vacaton length depends on the length of the precedng vst perod to the queue. As a spn-off, we present an exact analyss for ths vacaton model wth the help of matrx-analytc technques. A fnal word on the applcablty of the algorthm s that t can also be used as approxmaton for the exhaustve dscplne by takng a large value of the servce lmts. Therefore, our algorthm can also be seen as extenson of [3] for the exhaustve pollng system wth Posson arrvals. The remander of the present secton s devoted to the applcaton that led us to ths model. Although n the past the k-lmted strategy proved ts mert n communcaton systems (see, e.g., [3, 7]), the specfc applcaton that rased our attenton s n the feld of logstcs. In many stochastc mult-product sngle-capacty make-to-stock producton systems consderable setup tmes are ncurred,.e., the socalled stochastc economc lot schedulng problem (SELSP) [46]. The presence of these setup tmes n combnaton wth the stochastc envronment are the key complcatng factors of the SELSP. On the one hand, one ams for short cycle lengths, and thus frequent producton opportuntes for the varous products, n order to be able to react to the stochastcty n the system. On the other hand, short cycle lengths wll ncrease the setup frequency, whch has a negatve nfluence on the amount of capacty avalable for producton. Consequently, ths effect wll hnder the tmely fulfllment of demand. In the context of the SELSP, the exhaustve servce dscplne has been studed under the assumpton of Posson demand processes by [4, 5]. A major drawback of ths exhaustve polcy s that one sngle product, for whch a hgh demand arrves n a certan perod of tme, may occupy the machne for qute a whle. The mpacts of ths phenomenon on the other products are stock outs, hghly varable cycle lengths and hgh costs. The k-lmted polcy crcumvents ths drawback and offers the possblty to the manager to control both the setup frequences and the cycle lengths. The optmal base-stock levels n ths system can be obtaned by solvng standard newsboy problems for whch the complete queue length dstrbutons n (k-lmted) pollng systems are requred. For more nformaton on newsboy problems, see, e.g., [50]. Moreover, n many telecommuncaton systems the sngle most mportant performance measure s often not an aggregate measure lke the mean watng tme, rather the probablty that the delay exceeds a predefned threshold. In vew of both the descrbed producton settng and the dmensonng of a telecommuncaton network, the mportance of an accurate approxmaton of the complete queue length dstrbuton, as obtaned n the present paper, s evdent. The rest of the present paper s organzed as follows. Secton 2 gves, besdes the ntroducton of the model and further notaton, a hgh-level vew of the approxmaton scheme. In Sect. 3 the approxmatons for the mean and the varance of the condtonal ntervst perod are presented. Buldng on these results, Sect. 4 analyses a k-lmted vacaton model wth state-dependent vacatons. Secton 5 contans an overvew of the teratve procedure to calculate the performance measures of nterest. An extensve numercal study to test the accuracy of the approxmaton algorthm s presented n the penultmate secton. Fnally, the last secton descrbes the man conclusons of the present research and ndcates some possble drectons for further research. 2 Model descrpton and notaton We consder a system wth one sngle server for N 2 queues, n whch there s nfnte buffer capacty for each queue. The server vsts and serves the queues n a fxed cyclc order. We ndex the queues by, =, 2,...,N,n the order of the server movement. When vstng queue, =, 2,...,N, the server contnues workng at ths queue untl ether a predefned number of k customers s served or untl the queue becomes empty, whchever occurs frst. Notce that k = amounts to the standard exhaustve servce polcy. Customers arrve at all queues accordng to ndependent processes, of whch the mean and second moment are denoted by E[A ] and E[A 2 ], =, 2,...,N, respectvely. The servce tmes at queue are ndependent, dentcally dstrbuted random varables wth mean E[B ] and second moment E[B 2 ], =, 2,...,N. When the server starts servce at queue, a setup tme S s ncurred of whch the frst and second moment are denoted by E[S ] and E[S 2], =, 2,...,N, respectvely. These setup tmes are dentcally dstrbuted random varables, ndependent of any other event nvolved. In partcular, they are ndependent of the servce tmes. The mean total setup tme E[S] n a cycle s gven by E[S]= N E[S ]. = The occupancy rate ρ at queue s defned by ρ = E[B ] E[A ], and the total occupancy rate ρ s gven by ρ = N = ρ. Note that the occupaton rates do not nclude the setup tmes. Hence, especally for small values of the servce lmts k the effectve load on the system s consderably hgher.

4 64 Queueng Syst (2007) 55: 6 78 The cycle length C of queue, =, 2,...,N, s defned as the tme between two successve arrvals of the server at ths queue. It s well-known that the mean cycle length s ndependent of the queue nvolved and s gven by E[C]= E[S] ρ. () Ths dentty can be proved by observng that the amount of work arrvng durng a cycle should on average equal the amount of work departng durng a cycle,.e., ρe[c]=e[c] E[S]. (2) Unfortunately, hgher moments of the cycle length are analytcally ntractable and, certanly, depend on the queue nvolved. The vst perod V of queue, =, 2,...,N, s the tme the server spends servcng customers at queue excludng setup tme. Snce the server s workng a fracton ρ of the tme on queue, the mean of a vst perod of queue reads E[V ]=ρ E[C], =, 2,...,N. (3) Subsequently, the ntervst perod I of queue, the tme between a departure epoch of the server from queue and ts subsequent arrval to ths queue, s defned as I := C V, =, 2,...,N. A necessary and suffcent stablty condton reads here (see [6], for a rgorous proof n the specal case of Posson arrvals) ρ + E[S] max <. (4),2,...,N E[A ]k If the system s stable, (4) may be rewrtten by usng () as follows E[C] E[A ] <k, =, 2,...,N. In words, ths means that for a stable system the average number of type- customers arrvng n a cycle s smaller than the servce lmt k,.e., the maxmum number of type- customers served n a cycle. Throughout the present paper, the assumpton s made that stablty condton (4) sfulflled. Our man nterest s n L, the queue length at queue at an arbtrary pont n tme, =, 2,...,N.Themanresult of the present paper s the development of an teratve scheme to approxmate the complete dstrbuton of L.For the specal case of Posson arrvals, our results for the queue length dstrbuton can be readly translated nto results for the dstrbuton of the customer delay va the dstrbutonal form of Lttle s law [20]. We contnue the present secton wth a hgh-level descrpton of our approxmaton method. The key approxmaton dea s that we decompose the orgnal k-lmted pollng system wth N queues nto a set of N separate k-lmted sngle-queue models wth vacatons. At each step n the teraton the mathematcal analyss focusses on one sngle queue, whereas the other queues n the system determne the length of the vacaton perod (ntervst perod) of queue, =, 2,...,N. The bottleneck n ths approxmaton s the dervaton of the dstrbuton of the ntervst perod, whch wll be done n an teratve way. If we assume that the dstrbuton of the ntervst perod s known n step n of the teraton, the dstrbuton of the vst perod n step n + s derved by means of a queueng analyss for the k-lmted sngle-queue model wth vacatons (see Sect. 4). On ts turn, the latter dstrbuton can be used to compute the dstrbuton of the length of the ntervst perod n step n + (see Sect. 3). Snce t s more lkely that a long (short) vst perod s followed by a long (short) ntervst perod, condtonal ntervst perods are ntroduced. That s, the length of an ntervst perod s assumed to be postvely correlated to the number of customers served n the precedng vst perod. The subsequent two sectons am to answer the followng questons:. What are the frst two moments of an ntervst perod for queue gven that l = 0,,...,k customers are served n queue n the precedng vst perod (see Sect. 3). 2. What s the dstrbuton of a vst perod for queue gven the frst two moments of the condtonal ntervst perods (see Sect. 4). 3 Intervst perod The present secton computes the frst two moments of an ntervst perod for queue gven that l = 0,,...,k customers are served n queue n the precedng vst perod. The nput of the present secton are the statonary probabltes π (l) that l customers are served durng ths vst perod of queue. These probabltes follow from the analyss of the vacaton model n the prevous teraton step as expounded n Sect. 4. For presentaton reasons, we omt throughout ths secton the superscrpt n n all random varables denotng the correspondng teraton step n. 3. Frst moments The ntervst perod of a queue s obvously postvely correlated to the precedng vst perod of queue, =, 2,...,N. Therefore, we ntroduce so-called condtonal vst perods V (l), ntervst perods I (l) and cycles C (l)

5 Queueng Syst (2007) 55: condtoned on the number of customers D = l served n the vst perod of queue, l = 0,,...,k. The mean condtonal cycle lengths may be approxmated by usng approxmate balance equatons for C (l) as proposed by [22], (ρ ρ )E[C (l)]+le[b ] E[C (l)] E[S], =, 2,...,N, l= 0,,...,k, (5) whch equate the amount of work arrvng (left hand sde) and the amount of work departng durng condtonal cycles (rght hand sde). The balance equaton (5) s obvously an approxmaton, snce t assumes balance wthn each condtonal cycle whch may not hold. Notce the smlarty wth the exact balance equaton for the uncondtonal cycle length, for whch work-n s equal to work-out. Solvng (5) results n E[C (l)] l E[B ]+E[S], ρ + ρ =, 2,...,N, l= 0,,...,k. We extend the approxmaton of [22] by multplyng the ndvdual values E[C (l)] wth a scalng factor c R n such a way that the correct uncondtonal cycle length as gven by () s mantaned,.e., E[C] c = k l=0 π, =, 2,...,N, (l)e[c (l)] where π (l) are obtaned va the analyss of the vacaton model n the prevous teraton step (see Sect. 4). Ths scalng obvously facltates the convergence and stablty of the algorthm. Then, the mean condtonal ntervst perods I ( ) can be approxmated n the followng way, E[I (l)] E[C (l)] l E[B ], =, 2,...,N, l= 0,,...,k. (6) Fnally, we defne a condtonal vst perod V j (l) as the length of the vst perod of queue j gven that n the precedng vst to queue precsely l customers are served, l = 0,,...,k. The mean of ths random varable reads E[V j (l)] ρ j E[C (l)], =, 2,...,N, l= 0,,...,k, j = +,...,N,,...,, (7) whch completes the analyss of the condtonal frst moments. We have to remark that the approxmatons of the present subsecton only compensate for the correlatons between the vst perod and the mmedately followng ntervst perod. Although t s not nconcevable that one may come up wth more sophstcated approxmatons, the numercal evaluaton of Sect. 6 shows that our approxmatons are stll very effectve n capturng the correlatons among the queues. 3.2 Second moments The goal of the present subsecton s the development of an approxmaton for the varance of the condtonal ntervst perods I ( ). The startng pont of our analyss are the uncondtonal ntervst perods I. Snce the setup tmes are assumed to be uncorrelated (see Sect. 2), the varance of such an uncondtonal ntervst perod I s gven by Var[I ]= Var[V j ]+ j j + 2 j k>j k Var[S j ] Cov[V j,v k ]+ j k Cov[S j,v k ], (8) where the latter two summatons nclude all the covarances among the varous vst perods and among the setup tmes, respectvely, wthn an ntervst perod of queue. Therefore, the > sgn n ths summaton means that queue k s vsted after queue j n ths ntervst perod. The terms Var[V j ] n the rght-hand sde of (8) represent the varance of an uncondtonal vst perods V j of queue j. The second moment of such a vst perod can be approxmated as follows. Condtonng on the number of customers served durng the vst perod of ths queue and gnorng the correlatons between the length of the servce tmes and the number of customers served durng the vst perod yelds k E[V 2 ]= k π (l)e[v 2 (l)] π (l)(le[b 2 ] l=0 l=0 + l(l )E[B ] 2 ), =, 2,...,N, wth the remark that the probabltes π ( ) are stll unknown at ths stage. These probabltes are obtaned from the analyss of the vacaton model n the prevous teraton step, see Sect. 4. Now, the varance of V can be obtaned va standard probablstc arguments. Snce the terms Var[S j ] are assumed to be nput of the system (see Sect. 2), one does not need to approxmate them. By defnton, the covarance terms Cov[V j,v k ] appearng n (8) can be rewrtten as Cov[V j,v k ]=E[V j V k ] E[V j ]E[V k ], where the terms E[V j ] and E[V k ] follow from (3). To compute the unknown quantty E[V j V k ], we condton on the

6 66 Queueng Syst (2007) 55: 6 78 number D j of customers served n queue j durng the last vst perod as follows E[V j V k ]= k j E[V j V k D j = l]π j (l) l=0 k j l=0 le[b j ]E[V k j (l)]π j (l), where π j (l) follow from the analyss of Sect. 4 and E[V k j (l)] can be approxmated by (7). Fnally, n case a queue k s vsted before queue j n the ntervst perod of queue, V k and S j are obvously uncorrelated. In case queue j s vsted frst, we assume ndependence between setup tmes and vst perods as well,.e., Cov[S j,v k ] 0, and, thus, all terms n (8) have been specfed. The numercal results n Sect. 6 show that ths assumpton s vald as long as the setup tmes are not too varable. By defnton, the coeffcent of varaton c I of an uncondtonal ntervst perod s, subsequently, gven by Var[I ] c I =, =, 2,...,N. E[I ] We approxmate the varance of the condtonal ntervst perods I ( ) by assumng equalty of the coeffcents of varaton of all perods,.e., Var[I (l)] c 2 I E[I (l)] 2, l =, 2,...,k,=, 2,...,N, (9) where an approxmaton of E[I ( )] sgvenby(6). We add that we have also expermented wth other approxmatons for the varance of condtonal vst perod such as assumng equalty of the coeffcents of varaton of all condtonal cycle lengths. Approxmaton (9), however, turned out to be the most accurate one. Fnally, notce that (9) s ncreasng n l. 4 Vst perod The present secton ams to compute the dstrbuton of a vst perod for queue gven the frst two moments of the condtonal ntervst perods as computed va (6) and (9) n the precedng secton. By means of matrx-analytc technques, we analyse a sngle-staton vacaton model wth k-lmted servce, n whch the vacaton length depends on the length of the precedng vst perod. The authors are aware of only one other study n whch ths specfc dependency s studed under the restrctve assumpton of Posson nput [27]. Comprehensve surveys on vacaton models can be found n [0,, 40]. Snce the present secton s focussng on one sngle queue n a specfc teraton step n, the subscrpt and superscrpt n are dropped from all random varables. Throughout the present secton, the dstrbuton functons of the arrval and the servce tmes are needed. However, the only nformaton avalable for these random varables are the frst two moments. A common way to obtan an approxmate dstrbuton s to ft a phase-type dstrbuton on the frst two moments as elucdated n Appendx (cf., e.g., [43]). In the remander of the present secton, we assume that the ftted dstrbutons are used as substtute for the arrval and servce dstrbutons and that the number of phases needed equal n A and n B,respectvely. In the precedng subsecton, we have computed the frst two moments of the condtonal ntervst perods I( ) condtoned on the exact number of customers served n the precedng vst perod. To keep the sze of the state space for the k-lmted vacaton model manageable, some of these ntervst perods are aggregated. That s, we draw a dstncton between ntervst perods I(0), and n whch there have been zero, the maxmum number or any other number of customers served n the precedng vst perod, respectvely. In case the servce lmt at a queue equals one, only I(0) and I() have to be dstngushed. The perod s, thus, defned as, k := π(l)i(l), l= wth frst two moments, k E[]:= π(l)e[i(l)], l= k E[ 2 ]:= π(l)e[i(l) 2 ], l= and where π(l) follow from the prevous teraton step. We have to remark that we have tested ths aggregaton of ntervst perods for a wde varety of cases, from whch we concluded that t has only neglgble (negatve) mpact on the results, whch s outweghted by the gan n effcency. In sum, the system under consderaton s a sngle-server k-lmted vacaton model wth three dfferent knds of ntervst perods dependent on the number of customers served n the precedng vst perod. In order to construct these ntervst perods n an effcent way, we ntroduce the auxlary mutually ndependent random varables and, whch are ndependent of I(0) as well. These random varables satsfy = + I(0), and = +,

7 Queueng Syst (2007) 55: whch s always possble snce the varances of the condtonal ntervst perods are ncreasng n l as shown n (9). Thereupon, phase-type dstrbutons are ftted on I(0), and (see Appendx for further detals) n such a way that the frst two moments of and are correct. If we assume that the number of phases needed for the descrpton of I(0), and equal n I(0), n and n, respectvely, the total number n I of phases for the ntervst process s gven by n I = n I(0) + n + n. The k-lmted vacaton model can be descrbed by a contnuous-tme Markov process wth states (,j,m).the state varable = 0,,... denotes the total number of customers n the specfc queue under consderaton, whereas the state varable j =, 2,...,n A ndcates the phase of the arrval process A. Fnally, m =, 2,...,n D ndcates the phase of the departure process D, whch s the combnaton of the servce process and vacaton processes I(0), and. These latter two processes can be modeled by one sngle varable, snce the server s ether servng customers or s on vacaton. When the server s servng customers, one has to keep track of the phase of the servce process and of the number of customers already served n the correspondng vst perod. On the other hand, when the server s on vacaton the phase of the correspondng vacaton perod s needed. Consequently, the total number of states for the departure process s n D = k n B + n I. The phases of ths departure process are grouped as follows: frst, we group all phases related to the k servce processes and, then, the phases of, and I(0). Refer by level to the set of states wth customers n the system and group the states by these levels, so that (,j,m) precedes (,j,m ) f <. Wthn each level, the states are grouped accordng to the arrval phase, so that (,j,m) precedes (, j,m ) f j<j. Lastly, the states are ordered by the departure phase, so that (,j,m) precedes (,j,m ) f m<m. Now, one may verfy that the ntroduced Markov process s a quas-brth-and-death (QBD) process where the nfntesmal generator Q has the followng blocktrdagonal structure, B 00 B B 0 A A Q = 0 A 2 A A A 2 A A Below we specfy the submatrces n Q, where we use the concept of Markovan Arrval Process (MAP) (see, e.g., []) to descrbe the arrval and departure processes. In general, a MAP s defned n terms of a contnuous-tme Markov process wth fnte state space {0,...,m } and generator G 0 + G. The element G (, j) denotes the ntensty of transtons from to j accompaned by an arrval. For j element G 0 (, j) denotes the ntensty of the remanng transtons from to j, whle the dagonal elements G 0 (, ) are strctly negatve and chosen such that the row sums of G 0 + G are zero. The arrval process can be straghtforwardly represented by such a MAP, the states of whch correspond to the phases of ths process. Its generator can be expressed as G A 0 + GA, where the transton rates n G A are the ones that correspond to an arrval of a customer to the system. The transton rates of the G A 0 and GA matrces are lsted n Appendx 2. The MAP for the departure process wth generator G D 0 + G D s a lttle more nvolved. All transtons related to the vacaton perods do not cause departures and are, thus, wthn G D 0. Completon of a servce process, obvously, leads to a departure mplyng that the correspondng rates are n G D. Transtons wthn a servce process not causng departures are, of course, part of G D 0. Further, we have to dstngush between the stuaton when there are more than two customers n the system or not. In the frst stuaton, f a departure s not the kth departure the next servce process s started and f t s the kth departure a new vacaton perod s begun. To deal wth the stuatons n whch there are only zero or one customers present, we have to ntroduce matrces G D 0 and G D, representng the transton wthn level 0 and the transtons from level to level 0, respectvely. We can recognze two dfferences between these matrces and G D 0 + GD. Frst, when a servce process s completed whch s not the kth servce, a vacaton perod s commenced nstead of the next servce. Second, when a vacaton perod s fnshed, we jump to process I(0) nstead of to the servce process of the frst customer n the vst perod. The transton rates for G D 0, GD, G D 0 and G D are summarzed n Appendx 2. Now, we are n the poston to descrbe all the submatrces n Q,.e., B 0 = G A I n D, B 00 = G A 0 I n D + I na G D 0, B 0 = I na G D, A 0 = G A I n D, A = G A 0 I n D + I na G D 0, A 2 = I na G D, where I n s the dentty matrx of sze n and f A s an n n 2 matrx and B an n 3 n 4 matrx the Kronecker product A B s an n n 3 n 2 n 4 matrx defned by A(, )B A(,n 2 )B A B =..... A(n, )B A(n,n 2 )B

8 68 Queueng Syst (2007) 55: 6 78 Fg. Algorthm of [34]for fndng the rate matrx R,where. denotes a matrx-norm and ɛ some postve number N := A L := A 0 M := A 2 W := A df := whle df >ɛ { X := N L Y := N M Z := LY df := Z W := W + Z N := N + Z + MX Z := LX L := Z Z := MY M := Z } R := A 0 W Ths completes the descrpton of the QBD. If we let q denote the equlbrum probablty vector of level, the correspondng balance equatons are gven by q n A 0 + q n A + q n+ A 2 = 0, n 2, and q 0 B 00 + q B 0 = 0, (0) q 0 B 0 + q A + qa 2 = 0. () Introducng the rate matrx R as the mnmal nonnegatve soluton of the nonlnear matrx equaton transton rates from a state where l, l =, 2,...,k, customers are served (or 0 customers when l = 0) to a vacaton, multpled by the probablty of beng n that specfc state. Further, we recall that the ndces of q ( ) wthn the brackets correspond to lexcographcally ordered states of the arrval and departure processes. So, h(0) = h(l) = n A n I(0) = j= (q (( )n D + kn B + n + n + ) B 00 (( )n D + kn B + n + n +, ( )n D + kn B + n + n + )), n A n B = j= q (( )n D + (l )n B + j)) B 0 (( )n D + (l )n B + j, ( )n D + kn B + n + ), l =,...,k, h(k) = where n A n B = j= r(( )n D + (k )n B + j) A 2 (( )n D + (k )n B + j, ( )n D + kn B + ), r = q = q R = q (I na n D R), = = whch completes the analyss of the k-lmted vacaton model. A 0 + RA + R 2 A 2 = 0, t can be proved that the equlbrum probabltes satsfy (see, e.g., [35]) q n+ = q n R, n. To determne ths matrx R we use the algorthm developed by [34] as lsted n Fg.. The vectors q 0 and q follow from the boundary condtons (0), (), and the normalzaton condton. Ths queue length dstrbuton q yelds the followng expresson for the dstrbuton of the length of a vst perod, π(l) = h(l) k=0, l = 0,,...,k, (2) h() where h(l) s the total rate of jumps to a vacaton perod after servng l customers. To calculate h(l) we have to sum all 5 Iteratve algorthm As descrbed at the end of Sect. 2, the performance characterstcs of the k-lmted pollng system are approxmated by an teratve scheme. The algorthm s as follows. Outlne of the algorthm Step 0: Choose ntal characterstcs for all queues. Step : For = ton, determne the frst two moments of the condtonal ntervst perod I ( ) for queue from (6) and (9), respectvely. Step 2: For = ton, determne the dstrbuton of the vst perod V from (2). Step 3: Repeat Step and 2 untl the characterstcs for all queues have converged. Step 4: For = ton, compute the performance measures of nterest for queue.

9 Queueng Syst (2007) 55: Intalzaton In Step 0 of the algorthm, we have to choose ntal values for π (l), l = 0,,...,k and =, 2,...,N. The assumpton s made that all of these probabltes are zero except for π (k ), =, 2,...,N. Notce that, va the approach developed n Sect. 3, the correct mean cycle lengths are obtaned as computed by (). We note that we have expermented wth a large number of ntal values, from whch we concluded that the startng values of the algorthm have no, or at least neglgble, mpact on the results. Convergence crteron After Step and 2 we check whether the teratve algorthm has converged by comparng the probabltes π ( ), =, 2,...,N,nthe(n )th and nth step. We decde to stop when the maxmum of the absolute values of the dfferences s less than ε; otherwse we repeat Step and 2. Hence, the convergence crteron s max π (n) (l) π (n ) (l) <ε, =, 2,...,N, l=0,,...,k where ε s chosen to be 0 4. Of course, we may use other stop-crtera as well, e.g., mean queue lengths or mean ntervst perods. Complexty analyss The complexty of ths method s as follows. Wthn the teratve algorthm, solvng a subsystem consumes most of the tme. In one sngle teraton step N subsystems are solved. The number of teratons needed s dffcult to predct, but n practce ths number s about 0 to 5 teratons. The tme consumng part of solvng a subsystem s the calculaton of the R-matrx. Ths can be done n O(n 3 ) tme, where n s the sze of the R matrx of subsystem. Then, the tme complexty of one teraton becomes O(N max (n 3 )). Ths means that the tme complexty s polynomal n the number of queues, the servce lmts and the number of phases for each process. 6 Numercal evaluaton The present secton reports on an extensve numercal study desgned to assess the accuracy of the approxmaton method developed. We compare the frst two moments and tal probabltes of the queue length dstrbuton wth the ones produced by dscrete event smulaton. Each smulaton run s suffcently long such that the wdths of the 95% confdence ntervals of the performance measures of nterest are smaller than %. A frst mportant remark s that the computaton tme of our algorthm s consderably less than the smulaton tme, whch can mount up to ffteen mnutes or more. Ths neffcency of smulaton technques for (k-lmted) pollng systems has been observed before by, e.g., [2]. 6. Parameter settng We use a broad set of parameters for the tests. The number of queues n the system s vared between 2, 5 and 0, whereas the servce lmts are ether, 5 or 0. The total load on the system vares between 0.45, 0.60 and 0.75; as mentoned n Sect. 2 ths load does not nclude the setup tmes. Hence, especally for small values of the servce lmts k the effectve load on the system s consderably hgher. For ths reason, some cases are unstable, meanng that (4) does not hold, and are thus removed from the test bed. The squared coeffcents of varaton of the nterarrval, servce and setup tmes for each queue are dentcal and are vared between 0.25 and 2 and between 0.25 and, respectvely. We have to remark that we envson producton systems as the man applcaton for the present paper (see also Sect. ). Snce the varatons n the setup and servce tmes tend to be small n such systems n contrast to telecommuncaton systems where heavy-taled random varables are common we only consder cases n whch these varatons are ndeed relatvely small. Furthermore, we test cases for whch the setup tmes are 0 tmes smaller than the servce tmes and cases for whch setup and servce tmes are equal. Furthermore, both balanced and mbalanced pollng systems are consdered. In the balanced cases we set the arrval rates of all queues equal to. We test mbalance n the average nterarrval tmes by makng the load of the most heavly loaded queue 0 tmes hgher then that of the least heavly loaded queue, and by lettng the arrval rates of the other queues change lnearly such that the overall mean arrval rate s mantaned at. For example, n case of 5 queues we get arrval rates (0.82, 0.59,.000,.409,.88). Testng mbalance n the servce tmes proceeds along the same lnes. Ths leads to a total of = 2592 test cases, whch are summarzed n Table. After removng the unstable Table Test bed Test bed Parameter Notaton Value Low Medum Hgh Number of queues N Load ρ Servce lmt k 5 0 SCV nterarrval tmes A SCV servce tmes B 0.25 SCV setup tme S 0.25 Imbalance nterarrval tmes I A : :0 Imbalance servce tme I B : :0 Rato servce and setup tmes I B /S : 0: Number of nstances 2592

10 70 Queueng Syst (2007) 55: 6 78 cases, we end up wth a total of 2088 cases. For further reference, we have classfed the values for each parameter n the categores low, medum and hgh. The performance measures under consderaton n the present numercal study are the mean, standard devaton, 0.90-quantle and 0.95-quantle of the margnal queue length dstrbutons, where the α-quantle of the dstrbuton of a random varable X can be defned as the smallest value x such that P[X x] α. The mportance of the quantles of the queue length dstrbutons les n the fact that the optmal base-stock levels n the producton applcaton descrbed n Sect. precsely equal these quantles. 6.2 Results Table 2 summarzes the performance of the approach developed n the present paper showng the average errors and for four error-ranges the percentage of the cases whch fall n that range. Overall, we can say that for all performance measures the average error s around 7%, whle the errors are for the majorty of the cases less than 0%. We beleve that these errors are n general satsfactory n vew of the complexty of the system under consderaton: we study a k-lmted servce dscplne contanng the exhaustve polcy as specal case under the assumpton of general arrval processes, whlst the fact that our nterest s n the complete queue length dstrbuton consttutes an addtonal complcatng factor. To gve ths statement a more scentfc bass, we compare the performance of our approach to the standard de- Table 2 Overall results approach of present paper Errors approach of present paper Aver. (%) 0 0% 0 20% 20 30% >30% Mean queue lengths SD queue lengths quantle quantle Table 3 Overall results standard approach Errors standard approach Aver. (%) 0 0% 0 20% 20 30% >30% Mean queue lengths SD queue lengths quantle quantle composton approach. In such a standard decomposton approach the dependences among the ndvdual queues are completely gnored. That s, the ntervst perods are assumed to be ndependent of the length of the precedng vst perod, thus the need for condtonal cycles and condtonal (nter)vst perods cancels, and the correlatons among the ndvdual vst perods are set equal to zero. Remark that the applcaton of ths standard approach to k-lmted pollng systems has not been publshed n the open lterature. The results for the latter approach are lsted n Table 3. Comparng ths table to Table 2, we can conclude that our approach not only halves the mean errors for all performance measures, but also that the standard approach, n contrast to our approach, qute often results n more than 30% error. Ths observaton clearly underpns the statement made n the ntroducton that t s extremely mportant to capture the correlatons among the dfferent queues, snce these correlatons have a sgnfcant mpact on the performance measures. In partcular, the performance of the standard approach sgnfcantly degrades as the total load ncreases as shown n Table 5, whch s n agreement wth the result of [32] that the correlaton between successve vst tmes converges to one as the total load tends to one for the cases of exhaustve and gated pollng systems wth Posson arrvals. Table 4 shows that the accuracy of our approach decreases n heavy traffc as well; the decrease n accuracy s, however, not so severe as for the standard decomposton approach (see, also, Sect. 6.3). It would also be nterestng to compare the performance of our approach to the one of the alternatve approach developed n [24]. In ths study, t s proposed to take a weghted Table 4 Relatve errors for approach of present paper as functon of total utlzaton ρ Errors approach of present paper as functon of ρ (%) Low Medum Hgh Mean queue lengths SD queue lengths quantle quantle Table 5 Relatve errors for standard approach as functon of total utlzaton ρ Errors standard approach as functon of ρ (% ) Low Medum Hgh Mean queue lengths SD queue lengths quantle quantle

11 Queueng Syst (2007) 55: sum of a completely uncorrelated and a perfectly correlated system n order to capture the correlatons among the queues. A good choce of the desred mxng probablty s an nterestng problem n tself and the probablty used n [24] has not been developed for the k-lmted pollng system covered n the present paper, rather for a modfcaton of ths system,.e., ncluson of a reservaton mechansm. Drectly applyng the same mxng probablty to our settng would certanly wrong the approach of [24] leadng to an unfar comparson. Essentally, ths observaton reveals a weakness of the procedure of [24]: the qualty of ths procedure strongly depends on the choce of the mxng probablty. Takng the above nto account, we confne ourselves to a more qualtatve comparson between the two approaches. That s, when comparng the errors reported n [24] tothe ones lsted n Table 2, one can conclude that they are of the same order of magntude. The approxmaton method of [24] has, however, only been tested n a system wth smaller nherent dependences for the specal case of Posson arrvals. We have to remark that Tables 6, 7, 8 and 9 show that the nterarrval dstrbuton has no or at least neglgble effect on the accuracy of our approach. Table 6 Relatve errors for mean queue lengths Errors mean queue lengths (%) Parameter Low Medum Hgh N ρ k A B S I A I B I B /S More specfcally, Tables 6 through 9 show the detaled results for our approach, when fxng one parameter at a certan level. When a row s partally empty, t means that ths parameter s only tested on two levels. Our approxmaton method seems to be farly nsenstve to dfferent parameter settngs. In ths respect, the parameter havng the largest mpact on the performance s the total utlzaton ρ as earler llustrated n Table 5. Moreover, we observe that mbalance n the servce tmes and an ncrease n the setup tmes have negatve mpact on the accuracy, whereas the accuracy of our approach ncreases as the servce lmts become larger. Ths latter observaton tempts one to use the approach of the present paper as approxmaton for the exhaustve polcy as well as touched upon n Sect. 7. In the next subsecton, we present results for varous asymptotc regmes n order to study the effect of the ndvdual parameters even further. Remark 6. In the past, so-called pseudo-conservaton laws, ntensty-weghted sums of mean delays, have been appled qute often to develop accurate and elegant approxmatons for mean delays n pollng systems (and, thus, mean queue lengths as well). Throughout the present paper, we have delberately left ths approach asde, because Table 8 Relatve errors for 0.90-quantle Errors 0.90-quantle (%) Parameter Low Medum Hgh N ρ k A B S I A I B I B /S Table 7 Relatve errors for SD queue lengths Errors SD queue lengths (%) Parameter Low Medum Hgh N ρ k A B S I A I B I B /S Table 9 Relatve errors for 0.95-quantle Errors 0.95-quantle (%) Parameter Low Medum Hgh N ρ k A B S I A I B I B /S

12 72 Queueng Syst (2007) 55: 6 78 our approach does not use ths technque and because ths technque only gves approxmatons for mean performance measures for the specal case of Posson arrvals (for more nformaton see, e.g., [4] and the references theren). An addtonal complexty that shows up when applyng pseudoconservatons laws to pollng systems wth k-lmted servce s that n such systems these laws stll contan some unknown terms that have to be approxmated as ndependently shown by [5] and [2]. Note that the most accurate algorthm [6] based on such a pseudo-conservaton law can stll gve up to 20% errors for the mean delays n k-lmted pollng systems. Table Relatve errors for hghly varable setup and/or servce tmes (c 2 S = ) Errors for hghly varable setup and/or servce tmes (c 2 S = ) c 2 B = c 2 B = 4 c 2 B = 6 P S P S P S Mean queue lengths SD queue lengths quantle quantle Asymptotc regmes The foregong subsecton showed the accuracy of the developed approxmaton for a wde range of cases. The test bed s, undoubtedly, not only representatve for practcal nstances of the producton applcaton motvatng the present research but also for most applcatons n communcaton systems. In the present subsecton we, however, want to test the applcablty of the approxmaton beyond all lmts and test the accuracy of the approxmaton n the followng asymptotc regmes:. Hghly varable setup and/or servce tmes 2. Heavy traffc,.e., ρ 3. Large setup tmes,.e., E[S] 4. Large number of queues,.e., N Before we dscuss these regmes n detal, t s mportant to stress that for none of these regmes any, qualtatve or quanttatve, results are known for the k-lmted polcy. However, there are (partal) asymptotc results known for the less ntrcate exhaustve and gated polces (and, sometmes, for branchng-type polces). We menton these results n the present subsecton. Frst of all, we want to gve the reader a feelng for what mght happen for the k-lmted dscplne Table 2 Relatve errors for hghly varable setup and/or servce tmes (c 2 S = 4) Errors for hghly varable setup and/or servce tmes (c 2 S = 4) c 2 B = c 2 B = 4 c 2 B = 6 P S P S P S Mean queue lengths SD queue lengths quantle quantle Table 3 Relatve errors for hghly varable setup and/or servce tmes (c 2 S = 6) Errors for hghly varable setup and/or servce tmes (c 2 S = 6) c 2 B = c 2 B = 4 c 2 B = 6 P S P S P S Mean queue lengths SD queue lengths quantle quantle Table 0 Test bed for hghly varable setup and/or servce tmes Table 4 Test bed for heavy traffc Test bed Parameter Value(s) Test bed Parameter Value(s) N 5 ρ 0.6 k 5 ca 2 cb 2 46 cs 2 46 I A : I B : I B /S : Number of nstances 9 N 5 ρ k 35 ca 2 cb 2 cs 2 I A : I B : I B /S 0: Number of nstances 0

13 Queueng Syst (2007) 55: Table 5 Relatve errors for heavy traffc (k = 3) Errors for heavy traffc (k = 3) ρ = 0.75 ρ = 0.8 ρ = 0.85 ρ = 0.90 ρ = 0.95 P S P S P S P S P S Mean queue lengths SD queue lengths quantle quantle Table 6 Relatve errors for heavy traffc (k = 5) Errors for heavy traffc (k = 5) ρ = 0.75 ρ = 0.8 ρ = 0.85 ρ = 0.90 ρ = 0.95 P S P S P S P S P S Mean queue lengths SD queue lengths quantle quantle n the correspondng regmes. Secondly, these results for the exhaustve and gated polces clearly show that pollng systems dsplay aberrant behavor n these asymptotc cases mplyng that one cannot expect to be able to develop one sngle algorthm whch s accurate both n standard traffc settngs and for all possble asymptotc regmes Hghly varable setup and/or servce tmes The frst case, as summarzed n Table 0, nvestgates the mpact of the squared coeffcent of varaton of both the setup and servce tmes. Thereto, these quanttes are vared between, 4 and 6. Tables, 2 and 3 summarzes the results for ths case. In these tables, and all other tables throughout ths subsecton, the values n the column P refer to the relatve errors of the approach of the present paper, whereas column S shows the relatve errors of the standard approach. As observed from these tables, the accuracy of our approach s somewhat dsappontng. The reason for ths observaton s perdu n the way of condtonng ntroduced n Sect. 3. That s, we condton on the number of customers served wthout takng the length of each servce perod nto account. In case of hghly varable servce tmes ths can cause dffcultes. Condtonng on the length sgnfcantly complcates our analyss, snce the length of a vst perod s contnuous, has an nfnte support and s more dffcult to be montored. For the mpact of the varance of the setup tmes smlar observatons apply, where the assumpton of ndependence between setup and (subsequent) vst perods s the man reason for the decrease n accuracy. Fnally, we should stress that the standard approach agan clearly tastes Table 7 Test bed for large setup tmes Test bed Parameter Value(s) N 5 ρ 0.6 k 00 ca 2 cb 2 cs 2 I A : I B : I B /S : :2 :4 :8 :6 :32 Number of nstances 6 defeat. Also, the alternatve approach of [24] brngs no relef, snce ths approach has specfcally been developed for determnstc dstrbutons Heavy traffc Next, we analyze the case as shown n Table 4, where we ncrease the total load as follows: ρ = 0.75, 0.8, 0.85, 0.9, 0.95 to study the effect of heavy traffc. We have to stress that also n the extensve test bed examned n the prevous subsecton we studed (many) heavy-traffc cases. However, n the present paragraph the system reaches saturaton due to an ncrease n the traffc ntensty, whereas n the prevous subsecton the system got saturated manly due to the