Towards a unifying theory on branching-type polling systems in heavy traffic

Transcription

1 Queueng Syst : DOI 0007/s Towards a unfyng theory on branchng-type pollng systems n heavy traffc RD van der Me Receved: 2 August 2006 / Revsed: 2 May 2007 / Publshed onlne: 24 October 2007 Sprnger Scence+Busness Meda, LLC 2007 Abstract For a broad class of pollng models the evoluton of the system at specfc embedded pollng nstants s known to consttute a mult-type branchng process MTBP wth mmgraton In ths paper t s shown that for ths class of pollng models the vector that descrbes the state of the system at these pollng nstants, say X = X,,X M,satsfes the followng heavy-traffc behavor under mld assumptons: ρx d γ Ɣα,μ ρ, where γ s a known M-dmensonal vector, Ɣα,μ has a gamma-dstrbuton wth known parameters α and μ, and where ρ s the load of the system Ths general and powerful result s shown to lead to exact and n many cases even closed-form expressons for the Laplace-Steltes Transform LST of the complete asymptotc queue-length and watng-tme dstrbutons for a broad class of branchngtype pollng models that ncludes many well-studed pollng models polces as specal cases The results generalze and unfy many known results on the watng tmes n pollng systems n heavy traffc, and moreover, lead to new exact results for classcal pollng models that have not been Part of ths research has been funded by the Dutch BSIK/BRICKS proect RD van der Me Centre for Mathematcs and Computer Scence, Amsterdam, Netherlands e-mal: me@cwnl RD van der Me Mathematcs and Computer Scence, Vre Unverstet, Amsterdam, Netherlands e-mal: me@fewvunl observed before To demonstrate the usefulness of the results, we derve closed-form expressons for the LST of the watng-tme dstrbutons for models wth cyclc globallygated pollng regmes, and for cyclc pollng models wth general branchng-type servce polces As a by-product, our results lead to a number of asymptotc nsenstvty propertes, provdng new fundamental nsghts n the behavor of pollng models Keywords Pollng systems Mult-type branchng processes Heavy traffc Watng-tme dstrbuton Queue-length dstrbuton Gamma-dstrbuton Unfyng theory Mathematcs Subect Classfcaton B22 90B8 68M0 60B2 60J80 60J85 Introducton Pollng systems are mult-queue systems n whch a sngle server vsts the queues n some order to serve the customers watng at the queues, typcally ncurrng some amount of swtch-over tme to proceed from one queue to the next Pollng models fnd a wde varety of applcatons n whch processng power eg, CPU, bandwdth, manpower s shared among dfferent types of users Typcal applcaton areas of pollng models are computer-communcaton systems, logstcs, flexble manufacturng systems, producton systems and mantenance systems; the reader s referred to [2, 37] for extensve overvews of the applcablty of pollng models Over the past few decades the performance analyss of pollng models has receved much attenton n the lterature We refer to the classcal surveys [36, 38], and to a recent survey paper by Vshnevsk and Semenova [47]

2 30 Queueng Syst : for overvews of the avalable results on pollng models One of the most remarkable results s that there appears to be a strkng dfference n complexty between pollng models Resng [32] observed that for a large class of pollng models, ncludng for example cyclc pollng models wth Posson arrvals and exhaustve and gated servce at all queues, the evoluton of the system at successve pollng nstants at a fxed queue can be descrbed as a mult-type branchng process MTBP wth mmgraton Models that satsfy ths MTBP-structure allow for an exact analyss, whereas models that volate the MTBP-structure are often more ntrcate In ths paper we study the heavy-traffc behavor for the class of pollng models that have an MTBP-structure, n a general parameter settng Intated by the poneerng work of Coffman et al [2, 3], the analyss of the heavy-traffc behavor of pollng models has ganed a lot of nterest over the past decade Ths has led to the dervaton of asymptotc expressons for key performance metrcs, such as the moments and dstrbutons of the watng tmes and the queue lengths, for a varety of model varants, ncludng for example models wth mxtures of exhaustve and gated servce polces wth cyclc server routng [39], perodc server routng [26, 27], smultaneous batch arrvals [42], contnuous pollng [8], amongst others In ths context, a remarkable observaton s that n the heavy-traffc behavor of pollng models a central role s played by the gamma-dstrbuton, whch occurs n the analyss of these dfferent model varants as the lmtng dstrbuton of the scaled cycle tmes and the margnal queue-lengths at pollng nstants Ths observaton has motvated us to develop a unfyng theory on the heavy-traffc behavor of pollng models that ncludes all these model nstances as specal cases, such that everythng falls nto place We beleve that the results presented n ths paper are a sgnfcant step towards such a general unfyng theory The motvaton for studyng heavy-traffc asymptotcs n pollng models s twofold Frst, a partcularly attractve feature of heavy-traffc asymptotcs e, when the load tends to for MTBP-type models s that n many cases they lead to strkngly smple expressons for queuelength and watng-tme dstrbutons, especally when compared to ther counterparts for arbtrary values of the load, whch usually leads to very cumbersome expressons, even for the frst few moments cf, eg, [9] The remarkable smplcty of the heavy-traffc asymptotcs provdes fundamental nsght n the mpact of the system parameters on the performance of the system, and n many cases attractve nsenstvty propertes have been observed see also Sects 3 and 32 A second motvaton for consderng heavy-traffc asymptotcs s that the computaton tme needed to calculate the relevant performance metrcs usually becomes prohbtvely long when the system s close to saturaton, both for branchng-type [0] and non-branchngtype pollng models [3, 4], whch rases the need for smple and fast approxmatons Heavy-traffc asymptotcs form an excellent bass for developng such approxmatons see also Sect 33, and n fact, have been found to be remarkably accurate n several cases, even for moderate load cf, eg, [27, 39, 4] Recently, pollng models n heavy traffc have receved attenton n the lterature, and sgnfcant progress has been made n ths area For a two-queue model wth exhaustve servce and ndependent renewal arrval processes, Coffman et al [2, 3] use the theory of dffuson processes to derve expressons for the ont workload dstrbuton and the watng-tme dstrbutons under heavy traffc assumptons For models wth ndependent Posson arrvals, Kudoh et al [9] gve explct expressons for the second moment of the watng tme n fully symmetrc systems wth gated or exhaustve servce at each queue for models wth two, three and four queues, by explorng the classcal bufferoccupancy approach [36], whch s based on the relaton between the ont queue-length dstrbutons at successve pollng nstants They also gve conectures for the heavytraffc lmts of the frst two moments of the watng tmes for systems wth an arbtrary number of queues In a seres of papers, van der Me and co-authors explore the use of the Descendant Set Approach DSA [7] to derve exact expressons the watng-tme dstrbutons n models wth mxtures of exhaustve and gated servce and cyclc [39] or perodc [26] server routng Followng a smlar approach, van der Me also derves the exact asymptotcs watng-tme dstrbuton n cyclc queueng models wth smultaneous batch arrvals [42] Kroese [8] studes contnuous pollng systems n heavy traffc wth unt Posson arrvals on a rng and shows that the steady-state number of customers at each queue has approxmately a gamma-dstrbuton Vatutn and Dyakonova [46] use the theory of MTBPs to obtan the lmtng dstrbutons for several two-queue pollng models wth zero swtch-over tmes In addton to the evaluaton of the performance of heavly loaded pollng systems, the results can also be used to address stochastc schedulng problems [23, 24, 30, 3] To develop a unfyng theory on the heavy-traffc behavor of branchng-type pollng models, t s nterestng to observe that the theory of MTBPs, whch was developed largely developed n the early 970s, s well-matured and powerful [2, 5, 6, 29] Nonetheless, the theory of MTBPs has receved remarkably lttle attenton n the lterature on pollng models In fact, throughout ths paper we wll show that the followng result on MTBPs can be used as the bass for the development of a unfyng theory on branchngtype pollng models under heavy-traffc assumptons: the ont probablty dstrbuton of the M-dmensonal branchng process {Z n,n = 0,,} wth mmgraton n each

3 Queueng Syst : state converges n dstrbuton to vɣα,μ n the sense that cf Qune [29]: lm n π n ξ Z n d vɣα,μ ξ, 2 where ξ s the maxmum egenvalue of the so-called mean matrx, π n ξ s a scalng functon, v s a known M- dmensonal vector and Ɣα,μ s a gamma-dstrbuted random varable wth known shape and scale parameters α and μ, respectvely We emphasze that relaton 2 svald for general MTBPs under very mld moment condtons see Sect 2 for detals In ths paper, we show that ths result 2 can be transformed nto, provdng an asymptotc analyss for a very general class of MTBP-type pollng models Subsequently, we show that leads to exact asymptotc expressons for the scaled tme-average queue-length and watng-tme dstrbutons under heavy-traffc assumptons; for specfc model nstances, bascally all we have to do s calculate the parameters v, α and μ, and the dervatve of ξ as a functon of ρ at ρ =, whch s usually straghtforward In ths way, we propose a new and powerful approach to derve heavy-traffc asymptotcs for pollng models that have MTBP-structure To demonstrate the usefulness of the results we use the approach developed n ths paper to derve new and yet unknown closed-form expressons for the complete asymptotc watng-tme dstrbutons for a number of classcal pollng models To ths end, we derve closed-form expressons for the asymptotc watng-tme dstrbutons for cyclc pollng models wth the Globally-Gated GG servce polcy, and for models wth general branchng-type servce polces As a by-product, the results also lead to asymptotc nsenstvty propertes provdng new fundamental nsghts n the behavor of pollng models Moreover, the results lead to smple approxmatons for the watng-tme dstrbutons n stable pollng systems The remander of ths paper s organzed as follows In Sect 2 we gve a bref ntroducton on MTBPs and formulate the lmtng result by Qune [29] see Theorem that wll be used throughout In Sect 3 we translate ths result to the context of pollng models, and gve an approach for how to obtan heavy-traffc asymptotcs for branchng-type pollng models To llustrate the usefulness of the approach, we consder two specfc types of pollng models: cyclc models wth GG servce, and 2 cyclc models wth general branchng-type servce polces For these models, we derve a complete characterzaton of the asymptotc watng-tme dstrbutons The mplcatons of these results are dscussed extensvely Fnally, n Sect 4 we address a number of challengng topcs for further research 2 Multtype branchng processes wth mmgraton We consder a general mult-type branchng process wth M partcle types, denoted by Z ={Z n,n= 0,,}, where Z n =Z n,,z n M, where Z n s the number of type partcles n the nth generaton, for =,,M, n = 0,, The partcle producton s defned by the partcle offsprng functon fz = f z,,f M z, wthz = z,,z M, and where for z k k =,,M, =,,M, f z =,, M 0 p,, M z z M M, 3 where p,, M s the probablty that a type- partcle produces k partcles of type kk=,,mwe consder a MTBP wth an ndependent mmgraton n each state So n addton to the generaton functons f z defned n 3, representng the offsprng dstrbutons, an addtonal generatng functon gz s gven, representng the mmgraton dstrbuton: For z k k =,,M, gz =,, M 0 q,, M z z M M, 4 where q,, M s the probablty that a group of mmgrants conssts of k partcles of type kk=,,m Note that 3 and 4 provde a full characterzaton of the MTBP wth mmgraton n each state Denote g := g,,g M, where g := gz z z=, 5 and where s the M-vector where each component s equal to A key role n the analyss wll be played by the frst and second-order dervatves of fz The frst-order dervatves are denoted by the mean matrx M = m,, wth m, := f z z z=,,=,,m 6 Thus, adoptng the standard noton of chldren, for a gven type- partcle n the nth generaton, m, s the mean number of type- chldren t has n the n+st generaton Smlarly, for a type- partcle, the second-order dervatves are denoted by the matrx K, = k,k wth k,k := 2 f z z z k z=,,,k=,,m 7

4 32 Queueng Syst : Denote by v = v,,v M and w = w,,w M the left and rght egenvectors correspondng to the largest realvalued, postve egenvalue ξ of M, commonly referred to as the maxmum egenvalue cf, eg, [2], normalzed such that v = v w = 8 The followng condtons are necessary and suffcent condtons for the ergodcty of the process Z cf [32]: ξ< and q,, M log + + M < M >0 Throughout the followng defntons are convenent For any varable x that depends on ξ we use the hat-notaton ˆx to ndcate that x s evaluated at ξ = Moreover, for ξ>0 let π 0 ξ := 0, and π n ξ := n ξ r 2, n=, 2, 0 r= A non-negatve contnuous random varable Ɣα,μ s sad to have a gamma-dstrbuton wth shape parameter α>0 and scale parameter μ>0 f t has the probablty densty functon f Ɣ x = μα Ɣα xα e μx x > 0 wth Ɣα := t=0 t α e t dt, and Laplace-Steltes Transform LST μ α Ɣ s = Res > 0 2 μ + s Note that n the defnton of the gamma-dstrbuton μ s a scalng parameter, and that Ɣα,μ has the same dstrbuton as μ Ɣα, Usng these defntons, the followng result holds Theorem Assume that all dervatves of fz through order two exst at z = and that 0 <g < =,,M Then lm n π n ξ Z ṇ Z M n d A ˆv ˆv M Ɣα, ξ, 3 where ˆv = ˆv,, ˆv M s the normalzed left egenvector of ˆM, and where Ɣα, s a gamma-dstrbuted random varable wth scale parameter and shape parameter α := Aĝ ŵ = A wth A := M ĝ ŵ, = M ˆv ŵ ˆK ŵ>0 4 = Proof See [29] Theorem 4 In the next secton we wll show how ths result, whch was derved n the context of generc MTBPs, can be transformed nto results for a general class of pollng models 3 Heavy-traffc asymptotcs for pollng models In ths secton we show how Theorem can be transformed to derve new closed-form expressons for the LST of the queue-length and watng-tme dstrbutons for a broad class of pollng models, under heavy-traffc scalngs To ths end, we consder two classcal models that have been wdely studed n the lterature In Sect 3 we derve the LST of the asymptotc watng-tme dstrbuton for cyclc pollng models wth globally-gated GG servce In Sect 32 we derve asymptotc expressons for cyclc pollng models wth general branchng-type servce polces In Sect 33 we dscuss the mplcatons, the generalty and the lmtatons of the results To avod duplcaton, the followng model assumptons and notaton are ntroduced for both types of models Consder an asymmetrc cyclc pollng model that conssts of N 2 queues, Q,,Q N, and a sngle server that vsts the queues n cyclc order Customers arrve at Q accordng to a Posson process wth rate λ, and are referred to as type- customers The total arrval rate s := N = λ The servce tme of a type- customer s a random varable B, wth LST B and kth moment bk, whch s assumed to be fnte for k =, 2 The kth moment of the servce tme of an arbtrary customer s b k := N = λ b k / k =, 2 The total load of the system s ρ := N = ρ We defne a pollng nstant at Q to be the moment at whch the server arrves at Q, and a departure epoch at Q a moment at whch the server departs from Q The vst tme at Q s defned as the tme elapsed between a pollng nstant and ts successve departure epoch at Q Moreover, an -cycle s the tme between two successve pollng nstants at Q Upon departng from Q the server mmedately proceeds to Q +,ncurrng a swtch-over tme R wth LST R and frst two moments r k k =, 2, whch are assumed to be fnte De- note by r>0 and r 2 > 0 the frst two moments of the total swtch-over tme per -cycle of the server along the queues The nterarrval tmes, servce tmes and swtch-over tmes

5 Queueng Syst : are assumed to be mutually ndependent and ndependent of the state of the system Throughout, we focus on the behavor of the model when the load ρ approaches For ease of the dscusson we assume that as ρ changes the total arrval rate changes whle the servce-tme dstrbutons and ratos between the arrval rates are kept fxed; note that n ths way, the lmt for ρ, whch wll be used frequently throughout ths paper, s unquely defned Smlar to the hat-notaton for the MTBPs defned n Sect 2, for each varable x that s a functon of ρ we use the hat-notaton ˆx to ndcate ts value at ρ = For both models to be dscussed below, ont queuelength vector at successve moments when the server arrves at a fxed queue say Q k consttutes an MTBP wth mmgraton n each state To ths end, the followng notaton s useful Let X k,n be the number of type- customers n the system at the nth pollng nstant at Q k,for, k =,,N and n = 0,,, and let X k n = X k,n,,xk N,n be the ont queue-length vector at the nth pollng nstant at Q k Moreover, X k ={X k n,n= 0,,} s the MTBP descrbng the evoluton of the state of the system at successve pollng nstants at Q k Forρ<, we have X k n d X k for n, where X k denotes the steady state ont queuelength vector at an arbtrary pollng nstant at Q k 3 Globally-Gated servce The Globally-Gated GG servce dscplne works as follows [6] At the begnnng of a -cycle, marked by a pollng nstant at Q see above, all customers present at Q,,Q N are marked Durng the comng -cycle e, the vst of queues Q,,Q N, the server serves all and only the marked customers Customers that meanwhle arrve at the queues wll have to wat untl beng marked at the next cycle-begnnng, and wll be served durng the next -cycle Snce at each cycle the server serves all the work that arrved durng the prevous cycle, the stablty condton s ρ<, whch s both necessary and suffcent [6, 4] Throughout ths paper, ths model wll be referred to as the GG-model In Sect 3 we show how Theorem can be used to derve expressons for the LST of the asymptotc scaled watng-tme dstrbutons at each of the queues In Sect 32 we dscuss several nterestng mplcatons that follow from these expressons 3 Analyss To analyze the heavy-traffc behavor of the GG-model, we establsh the relaton wth the general MTBP-model descrbed n Sect 2 To ths end, recall that for the model consdered here the ont queue-length process at embedded pollng nstants at Q k for any k can be descrbed as an N-dmensonal MTBP wth mmgraton n each state For notatonal ease of the dscusson that wll follow, we proceed along two steps Frst we focus on the heavy-traffc asymptotcs for the ont queue-length vector at the successve moments at whch the server arrves at Q Theorem 2 Second, we wll transform these results to the ont queuelength dstrbuton at pollng nstants at Q k,k=,,n Theorem 3 To start, we consder the MTBP X := {X n, n = 0,,} descrbng the evoluton of the ont queuelength vector at successve pollng nstants of the server at Q Then the process X s characterzed by the offsprng generatng functons, for =,,N, z k k =,,N, N f z,,z N = B λ z = and the mmgraton functon gz,,z N = N = R 5 N λ z 6 = Note that t follows drectly from 6 that, for =,,N, g = r λ = rλ 7 = To derve the lmtng dstrbuton of the ont queue-length vector at pollng nstants at Q, we need to specfy the followng parameters: the mean matrx M and ts correspondng left and rght egenvectors ˆv and ŵ at ρ = normalzed accordng to 8, and 2 the parameters A and ĝ These parameters are obtaned n the followng two lemmas Lemma For the GG-model, the mean matrx M s gven by the followng expresson: b λ b λ 2 b λ N b 2 M = λ b 2 λ N b N λ b N λ N b ˆλ b ˆλ 2 b ˆλ N b ˆM ˆλ 2 b = 2 ˆλ N b N ˆλ b N ˆλ N and hence, 8

6 34 Queueng Syst : Moreover, the rght and left egenvectors of ˆM e, M at ρ = are b ŵ = b b 2, and ˆλ ˆλ 2 ˆv = b ˆλ N wth b N, respectvely, b := b,,b N, and b := = 9 b 20 Proof The frst equaton of 8 follows drectly from 5 by dfferentaton: For, =,,N, m, := f z z z= = N B λ z = b z λ, 2 = z= and the second equaton n 8 then follows drectly by evaluatng the frst equaton at ρ = To prove that ŵ s a rght egenvector of ˆM, note that t follows drectly from 8 that, for =,,N, = b ˆλ b = b ˆρ = b, 22 so that ˆMb = b, and hence, ˆMŵ =ŵ Smlarly, to show that ˆv s a left egenvector of ˆM, note that for =,,N, = ˆλ b ˆλ =ˆρ ˆλ = ˆλ, 23 whch mples ˆM ˆv =ˆv The proof of Lemma s then completed by properly normalzng the egenvectors accordng to 8 Lemma 2 For the GG-model, we have ĝ ŵ = b r, 24 and A = b b2 25 b Proof To start, 24 follows drectly the followng sequence of equaltes: ĝ ŵ := ĝ ŵ = r b ˆλ b = = = ˆρ b r = b r, 26 whch follows drectly from 7 and 9, and usng the fact that ˆρ = by defnton To prove 25, we frst observe that by dfferentatng 5 two tmes we have, for =,,N, λ 2 λ λ 2 λ λ N K = b 2 ˆK = b 2 λ 2 λ λ 2 2 λ 2 λ N, λ N λ λ 2 N ˆλ 2 ˆλ ˆλ 2 ˆλ ˆλ N ˆλ 2 ˆλ ˆλ 2 2 ˆλ 2 ˆλ N ˆλ N ˆλ ˆλ 2 N and so Consequently, usng 9wehavefor =,,N, ŵ ˆK ŵ = b 2 b 2 = k= b ˆλ ˆλ k b k = b 2 b 2, and hence, combnng 9 and 28wehave A := ˆv ŵ ˆK ŵ = b = = b ˆ b 2 = b b2 N = ˆλ b , 29 b where the last equalty follows from the fact that ˆ = /b Ths completes the proof of Lemma 2 Let us consder the heavy-traffc behavor of the maxmum egenvalue ξ of M Note that n general, ξ s a nonnegatve real-valued functon of ρ cf [2], say ξ = ξρ, 30 for ρ>0 Then the followng result descrbes the behavor of ξ n the neghbourhood of ρ = Lemma 3 For the GG-model, the maxmum egenvalue ξ = ξρ has the followng propertes: ξ< f and only f 0 <ρ<, ξ = f and only f ρ =, and ξ> f and only f ρ>; 2 ξ = ξρ s a contnuous functon of ρ;

7 Queueng Syst : lm ρ ξρ = ξ = ; 4 The dervatve of ξ at ρ = s gven by ξ ξρ := lm = 3 ρ ρ Proof See Appendx We are now ready to transform Theorem to the model under consderaton Theorem 2 For the GG-model, the steady-state ont queuelength dstrbuton at pollng nstants at Q satsfes the followng lmtng behavor: ρ where X X N d b 2 b ˆλ ˆλ N Ɣα, ρ, 32 α = r b 33 b 2 Proof Frst, t s readly verfed that the ont-queue-length process X := {X n = X,n,,X N,n, n = 0,,} at embedded pollng nstants at Q consttutes an N- dmensonal MTBP wth offsprng functon f z and mmgraton functon gz defned n 5 and 6 and wth mean matrx M defned n 8 Moreover, s t easy to verfy that the assumptons of Theorem are satsfed wth M = N Then usng Lemmas to 3 and Theorem t follows that X n, ˆλ lm n π n ξρ b 2 d b Ɣα, ˆλ M X n,n ρ, 34 where α s defned n 33 Consequently, relaton 32 follows from the followng sequence of equatons: X lm ρ ρ X N = lm lm ρ ρ n X n, X n,n = lm ρ lm n ρπ nξρ π n ξρ X n, X n,n = lm ρ lm n ρπ nξρ lm ρ lm n = b2 b π n ξρ ˆλ ˆλ N X n, X n,n Ɣα,, 35 where the last equalty n 35 follows from Theorem and the fact that 0 mples lm ρ lm n ρπ nξρ ρ = lm ρ ξρ lm ξρ n = =, 36 n ξρ by usng the propertes formulated n Lemma 3 The next result generalzes Theorem 2, whch gves the asymptotc scaled queue-length dstrbuton at an arbtrary pollng nstant at Q, to the asymptotc queue-length dstrbuton at an arbtrary pollng nstant at Q k k =,,N Theorem 3 For the GG-model, the steady-state ont queuelength dstrbuton at pollng nstants at Q k k =,,N satsfes the followng lmtng behavor: X k ρ d b 2 b where X k N ˆρ + + ˆρ k ˆλ ˆλ k ˆλ k ˆλ N ˆλ k ˆλ N Ɣα, ρ, 37 α = r b 38 b 2 Proof For k =,,N, denote by Xk z,,z N the PGF of X k,,xk N, the ont queue length at an arbtrary pollng nstant at Q k Then t s readly verfed that, for z, =,,N, k =,,N,

8 36 Queueng Syst : Xk z,,z N N λ z k = = R X = B B k N λ z,, = N λ z,z k,z k+,,z N = 39 To ths end, consder the customer populaton at a pollng nstant Pk at Q k k > ; note that for k = the result was shown n Theorem 2 and s therefore not consdered here agan Then ths populaton conssts of three ndependent parts: the customers that were present at Q = k,k +,,N at the last precedng pollng nstant at Q, 2 the customers who arrved durng the servce tmes of the customers that were present at Q =, 2,,k at the precedng pollng nstant at Q, and 3 the customers who arrved durng the past swtch-over tmes R =, 2,,k Then 39 follows drectly by usng standard generatng functon manpulatons Theorem 3 then follows drectly from Theorem 2 by usng 32 and takng the proper lmts We are now ready to obtan the man result for the GGmodel Theorem 4 For the GG-model, the watng-tme dstrbuton satsfes the followng lmtng behavor: For =,,N, ρw d W ρ 40 where the LST of W s gven by, for Res > 0, { W s = μ α ˆρ rs μ + s ˆρ + + ˆρ } μ α, 4 μ + s +ˆρ + + ˆρ where α = r b b, and μ= 42 b 2 b 2 Proof Denote by X and Y the number of customers at Q at the begnnng and at the end of a vst perod to Q,respectvely, and denote by N the number of customers at Q at an arbtrary customer departure epoch from Q Denote the correspondng PGFs by X, Y and N Then the followng result was obtaned by Borst and Boxma [8]: For z, =,,N, N z = ρ zb λ z B λ z z Y z X z zλ ρ r/ ρ 43 Then from Theorem 3, takng the th component only, we have that n the lmtng case ρ, ρx b 2 d b ˆλ +ˆρ + + ˆρ Ɣα, 44 Then, to determne the number of type- customers Y at the end of a vst of the server to Q, note that we can wrte, for =,,N, X + = d Y + A, 45 where A stands for the number of arrvals at Q durng a swtch-over tme from Q to the next queue, wth Y and A ndependent Then, t s readly seen that ρa 0 almost surely as ρ Moreover, t follows from Theorem 3 by lettng k = + and takng the th component n 37 that, for =,,N, ρx + d b 2 b ˆλ ˆρ + + ˆρ Ɣα, 46 Then combnng 46 and 45 mmedately mples that ρy d b 2 b ˆλ ˆρ + + ˆρ Ɣα, 47 Combnng 43, 44 and 47, usng the dstrbutonal form of Lttle s formula and the observaton that a departng customer sees the tme average [35] s then easly seen to lead to 4, whch completes the proof of Theorem 4 32 Implcatons Theorem 4 leads to a number of nterestng mplcatons that wll be dscussed below Corollary Insenstvty propertes For =,,N, the asymptotc watng-tme dstrbuton W, Is ndependent of the vst order assumng the order s cyclc,

9 Queueng Syst : Depends on the varablty of the servce-tme dstrbutons only through b 2, and 3 Depends on the swtch-over tme dstrbutons only through r Note that smlar nsenstvty propertes are generally not vald for stable systems e, ρ<, n whch case the watng-tme dstrbutons do depend on the vst order, the complete servce-tme dstrbutons and each of the ndvdual swtch-over tme dstrbutons Apparently, these dependences are of lower order, and hence ther effect on the watng-tme dstrbutons becomes neglgble, n heavy traffc Corollary 2 Zero swtch-over tmes For the case of zero swtch-over tmes, the LST of W for the GG-model s gven by the followng expresson: For =,,N,Res 0, lm W s = r 0 b ˆρ s b 2 μ + s +ˆρ + + ˆρ log μ + s ˆρ + + ˆρ, 48 where α and μ are defned n 42, and where log s an nverse functon of the complex functon lz := expz Corollary 3 Expected asymptotc delay For the GGmodel, the asymptotc expected delay at Q s gven by the followng expresson: For =,,N, E[ W ]= b ˆρ +ˆρ 2 b + r 49 = Remark Pseudo-conservaton law The pseudo-conservaton law PCL for the present model s as follows cf [6]: For ρ<, λ b 2 ρ E[W ]=ρ 2 ρ + ρ r2 2r = = + ρ 2 r N ρ + =2 ρ = r 50 By takng heavy-traffc lmts, t follows drectly that = ρ E[ W ]= b2 + r 5 2b Then t s easy to verfy that 49 ndeed satsfes 5, whch supports the valdty of Theorem 4 32 Cyclc pollng models wth general branchng-type servce polces In ths secton we consder the cyclc pollng model ntroduced at the begnnng of Sect 3, wth general branchngtype servce polces that satsfy the followng property cf [32]: Branchng property If the server arrves at Q to fnd k customers there, then durng the course of the server s vst, each of these k customers wll effectvely be replaced n an d manner by a populaton of customers havng ont probablty generatng functon PGF h z = h z,,z N, whch can be any N-dmensonal PGF We assume that the servce dscplnes are work conservng, n the sense that the server always works durng a vst to a queue From the branchng property, a vst perod of the server startng wth k orgnal customers, say C,,C k, consst of k mutually ndependent sub-busy perod, each of whch s characterzed by the ont PGF-LST: For =,,N,Reu > 0, v, ψ u, v := E[e ut v L ], 52 where T s the duraton of a sub-busy perod, and L s the so-called sub-busy perod resdue, e, the number of type- chldren of the orgnal customer that generates ths sub-busy perod Ths class of servce polces contans a varety of classcal servce polces, ncludng the exhaustve, gated, bnomal-gated [20] and bnomal-exhaustve [32] polces, amongst others For gated and exhaustve servce at Q,we have for z k k =,,N, N h z = B λ z and = 53 h z = λ z, respectvely, where denotes the LST of a busy perod n an M/G/ queue wth arrval rate λ and servce tme dstrbuton B Smlarly, for the case of bnomalgated servce wth parameter 0 <p and bnomalexhaustve servce wth parameter 0 <q we have for z k k =,,N, N h z = p z + p B λ z = h z = q z + q λ z, and 54

10 38 Queueng Syst : respectvely Adoptng the termnology ntroduced n [4], we defne the exhaustveness of the servce polcy at Q =,,Nby f := E[L ], 55 where L s the sub-busy perod resdue, defned n 52 The exhaustveness f has the followng smple nterpretaton: each customer present at Q at the begnnng of a vst of the server to Q s effectvely replaced by a number of customers at Q whose mean value s f In other words, f can be seen as f := E[number of customers at Q at the end of a vst to Q ] E[number of customers at Q at the begnnng of that vst to Q ] 56 It s readly verfed from that for the case of exhaustve and gated servce we have f = and f = ρ, respectvely see also Remark 2 below Notce also that the work conservng property mples the followng relaton between the sub-busy perod duraton T and the sub-busy perod resdue L :For =,,N, E[T ]= E[L ] b b = f 57 ρ ρ 32 Analyss To establsh the relaton wth the general MTBP-model descrbed n Sect 2, we observe that for the model consdered here the ont queue-length process at embedded pollng nstants at Q can be descrbed as an N-dmensonal MTBP wth mmgraton n each state [32] Ths process s characterzed by the offsprng generatng functons, for z k k =,,N, =,,N, f z,,z N wth = h z,z 2,,z,f + z,,z N,, f N z,,z N, 58 h z,,z N := ψ λ z, z, 59 where ψ, s defned n 52, and the mmgraton functon, for z k k =,,N, gz,,z N = N = + R λ k z k k=+ k= λ k f k z,,z N 60 To derve the lmtng dstrbuton of the ont queue-length vector at pollng nstants at Q, we need to specfy the followng parameters: the mean matrx ˆM and ts correspondng normalzed left and rght egenvectors ˆv and ŵ, and 2 the parameters ĝ and A These parameters are obtaned n the followng two lemmas Lemma 4 For the cyclc branchng-type pollng model, the mean matrx M s gven by the followng expresson: M = M M N, 6 where for =,,N, M = λ f ϕ λ 2 f ϕ λ f ϕ f λ + f ϕ λ N f ϕ,

11 Queueng Syst : wth ϕ := b / ρ Moreover, f we defne for =,,N, u := λ ρ f f + λ =+ ρ, 63 then the normalzed rght and left egenvectors of ˆM are gven by ŵ = ˆv = b δ wth ŵ ŵ N = b û û N δ := û ŵ =, b b N, and ˆρ ˆρ fˆ +ˆρ ˆ = and where b and b are defned n 20 f =+ 64 ˆρ, 65 Proof To prove 6 and 62, consder a tagged type- customer, say C, present at Q at the begnnng of a servce perod at Q Followng the branchng property, C generates a sub-busy perod wth ont PGF-LST ψ,, defned n 52 Durng ths sub-busy perod, the average number of chldren C has at Q s λ E[T ]=λ f ϕ,byusng 57 Moreover, t s readly seen that the number of type- chldren of C s exactly the resdue of the sub-busy perod generated by C, and ts mean value equals E[L ]= f Based on these observatons, equatons 6 and 62 are easly seen to hold, for =,,N To show that ŵ s a rght egenvector at ˆM, note that t follows drectly from 62 that, for =,,N, ˆλ fˆ ˆϕ b + b fˆ fˆ ˆϕ + b fˆ fˆ b + b ˆλ b ˆ f = b, 66 so that ˆM ŵ = ŵ =,,N, and hence, ˆMŵ = ŵ, whch shows that ŵ s ndeed a rght egenvector of ˆMSmlar arguments can be used to show that ˆv s a left egenvector of ˆM along the lnes dscussed n the Appendx of [40] The detals are omtted for compactness of the presentaton, and are left as an exercse to the reader Ths completes the proof of Lemma 4 Lemma 5 For the cyclc branchng-type pollng model, ĝ ŵ = b r, 67 and A = b δ b b Proof Assume ρ = To show 67 we frst observe that t follows from 60 that the mean number of type- customers that mmgrate durng a cycle s gven by ĝ = = r ˆλ I { } + k=+ ˆλ k ˆm k,, 69 where I E stands for the ndcator functon on the event E Ths mples ĝ ŵ := ĝ ŵ = b = = b + N = k=+ = b r r ˆλ k N = N = ĝ b ˆλ b I { } ˆm k, b ˆρ = b r, 70 = by usng 64, 69, and the fact that N = ˆm k, b = b k, whch s an mmedate consequence of the second part of Lemma,see9 Fnally, the proof of 68 can be obtaned along smlar lnes as the proof of 25 n26 29, but wth notatonally cumbersome dervatons, the detals of whch are omtted for compactness of the presentaton See also Remark 5 for an alternatve proof of 68 Lemma 6 For the cyclc branchng model, the maxmum egenvalue ξ = ξρ has the followng propertes: ξ< f and only f ρ<, ξ = f and only f ρ = and ξ> f and only f ρ>; 2 ξρ s a contnuous functon of ρ; 3 lm ρ ξρ = ξ = ; 4 The dervatve of ξρ at ρ = s gven by ξ = lm ρ ξρ ρ = δ, 7

12 40 Queueng Syst : where δ s defned n 65 X n, lm lm Proof See Appendx 2 ρ n π n ξρ We are now ready to present the man result for the model under consderaton Theorem 5 For the cyclc branchng-type pollng model, the ont queue-length vector at pollng nstants at Q has the followng asymptotc behavor: ρ X X N d b 2 2b δ û û N Ɣα, ρ, 72 ˆv = δ A ˆv N = δ b 2 2b Ɣα, û û N X n,n Ɣα, 75 Here, the second equalty n 75 follows from 74 and the fact that where α = 2rδ b, 73 b 2 and where δ and û =,,N are defned n 65 and 63, respectvely Proof To start, note that the ont-queue-length process X := {X n = X,n,,X N,n, n = 0,,} at embedded pollng nstants at Q consttutes an N-dmensonal MTBP wth offsprng functon f z and mmgraton functon gz defned n 58 and 60, and wth mean matrx M defned n 6, 62 Moreover, s t easy to verfy that the assumptons of Theorem are satsfed wth M = N Then usng Lemmas 4 to 6 and Theorem t follows that lm n π n ξρ X n, X n,n d A ˆv ˆv N Ɣα, ρ, 74 where α, ˆv and A are gven n 73, 64 and 68, respectvely Hence, smlar to the dervaton of Theorem 2, relaton 72 follows from the followng sequence of equatons: lm ρ ρ X X N = lm lm ρ ρ n X n, X n,n = lm ρ lm n ρπ nξρ lm ρ lm n ρπ nξρ ρ = lm ρ ξρ lm ξρ n = δ = δ, 76 n ξρ whch follows drectly by usng 0 and the propertes lsted n Lemma 6 Fnally, equaton 75 follows from 64 and 68 Ths completes the proof of Theorem 5 Theorem 6 For the cyclc branchng-type pollng model, the watng-tme dstrbuton satsfes the followng lmtng behavor: For =,,N, ρw d W ρ 77 where the LST of W s gven by { W s = ˆρ rs μ μ + s ˆ f α α } μ μ + s Res > 0, 78 where α = 2rδ b b 2, and where δ s gven n 65 μ = 2δ b fˆ b 2, 79 ˆρ Proof Wthout loss of generalty, we focus on the watng tme dstrbuton at Q Adoptng the notaton used n the proof of Theorem 4, relaton 43 s also applcable to the cyclc branchng-type model under consderaton and hence also for the specal case =, so t remans to determne the lmtng behavor for X and Y, e the number of type- customers present at the begnnng and the end of a vst

13 Queueng Syst : perod to Q, respectvely To ths end, note that Theorem 4 mples that n the lmtng case ρ, ρx d b 2 2b δ u Ɣα, 80 Then, usng the branchng structure of the servce polcy at Q t s then readly seen that, for ρ, ρy d fˆ b 2 2b δ u Ɣα, 8 To see the latter, note that at the end of the vst perod V at Q, each type- customer that was present at the begnnng of V has been replaced by a populaton of customers whose PGF s gven by ψ,, defned n 52 Focusng on type- customers only, each type- customer present n Q at the begnnng of V s replaced by, on average, f type- customers at the end of V Then, combnng 80, 8, usng the dstrbutonal form of Lttle s formula and the observaton that a departng customer sees the tme average [35] s easly seen to lead to 77, 78, recallng that we assumed = wthout loss of generalty The results presented n Theorem 6 are new and have not been observed before n the general context of the model consdered We emphasze that the results are vald n the general parameter settng of the model defned above Remarkably, the results can be obtaned n closed form, and moreover, are strkngly smple, and explctly show the mpact of the system parameters on the asymptotc delay at each of the queues 322 Implcatons Theorem 6 leads to a number of nterestng mplcatons that wll be addressed below Corollary 4 Insenstvty propertes For =,,N, the asymptotc watng-tme dstrbuton of W, Depends on the servce polces only through the exhaustveness factors f,,f N, 2 Is ndependent of the vst order assumng the order s cyclc, 3 Depends on the varablty of the servce-tme dstrbutons only through b 2, and 4 Depends on the swtch-over tme dstrbutons only through r Recall from Corollary 4 that n general these nsenstvty propertes do not hold for stable systems, n whch case the watng-tme dstrbutons depend on the complete dstrbuton of the sub-busy perods defned n 52, the vst order, the complete servce-tme dstrbutons and each of the ndvdual swtch-over tme dstrbutons Apparently, these dependences are of lower order, and hence ther effect on the watng-tme dstrbutons becomes neglgble, n heavy traffc Corollary 5 Zero swtch-over tmes For the specal case of zero swtch-over tmes, we have: For =,,N, Res > 0, lm W s = r 0 2δ ˆρ s b log b 2 μ + s μ + s ˆ f, 82 where α, μ and δ are defned n 79 and 65, respectvely, and where log s an nverse functon of the complex functon lz := expz Corollary 6 Mean asymptotc delay For the cyclc branchng model, the asymptotc expected delay at Q s gven by the followng expresson: For =,,N, E[ W ]= ˆρ 2 fˆ N= ˆρ ˆρ 2 b 2 2b fˆ r ˆρ 83 fˆ Note that ths result was also shown n [42], where we obtaned the result va the Descendant Set Approach [7] We end ths subsecton wth a number of remarks Remark 2 Generalzaton of known results Theorem 6 generalzes and unfes known results that have been shown before Van der Me [39] derved the result for the specal case of mxtures of gated and exhaustve servce at each queue More precsely, f E denotes the set of queues that receve exhaustve servce and G denotes the set of queues that receve gated servce, then t follows from 53 that f = for E, and f = ρ for G Moreover, t s easly verfed that n that case δ = E ˆρ2 + G ˆρ2 /2 Smlarly, 54 mples that for the case of bnomal-gated servce wth parameter p 0 <p we have f = p ρ, whle for the fractonal exhaustve polcy wth parameter q 0 <q we have f = q Remark 3 Pseudo-conservaton law The pseudo-conservaton law PCL for the present model s as follows cf [44]: For ρ<, λ b 2 ˆρ E[W ]=ρ 2 ρ + ρ r2 2r = =

14 42 Queueng Syst : [ ] r + ρ 2 ρ 2 Note that 89 s ndeed n lne wth 88 More generally, + E[M ], 2 ρ from smple balancng arguments t follows drectly that, for = = ρ<,, k =,,N, 84 where the mean amount of at Q at a server departure nstant at Q s, for ρ<, =,,N, E[X k ]= λ r ρ f + λ r f ρ ρ k =+ ρ 9 E[M ]= rρ ρ f 85 f ρ By takng heavy-traffc lmts, t follows drectly that = ρ E[ W ]= b2 2b + r 2 = 2 ˆρ ˆρ 86 fˆ Then t s easy to verfy that 83 ndeed satsfes 86, whch supports the valdty of Theorem 6 Remark 4 Drect calculaton of mean values The mean values of X k, k =,,Ncan also be obtaned drectly va smple balancng arguments To ths end, note frst that for = k smple balancng arguments lead to the followng equatons: For ρ<, =,,N, E[X ]=λ r + λ E[X ]E[T ]+E[X ]E[L ], 87 whch s readly seen to lead to the followng expresson cf also [44]: For ρ<, =,,N, E[X ]= r ρ ρ E[T ] = λ r ρ f ρ 88 Notce that for the specal case = t follows from Theorem 5 that lm ρe[x ρ ]= b2 2b δ û α = b2 2b δ ˆλ ˆρ ˆ f 2rδ b b 2 = r ˆλ ˆρ, 89 fˆ where the second equalty follows from the fact that û = ˆλ ˆρ fˆ + ˆλ ˆρ = ˆλ ˆρ 90 fˆ fˆ Then t s readly verfed from Theorem 5 that for the case k = wthout loss of generalty, for =,,N, lm ρe[x ρ ] = δ b 2 2b û α = rû = ˆλ r ˆρ ˆ f ˆ f whch s n lne wth Theorem 5 + ˆλ r =+ ˆρ, 92 Remark 5 Alternatve proof of 68 The results n provde an alternatve proof for relaton 68 n Lemma 5 To ths end, note that from 75 and 92, by takng =, t s readly seen that ˆλ ˆρ r ˆ f = δ A ˆv α 93 Equaton 68 then follows drectly by combnng 93 wth the defntons of δ, ˆv and α n 65, 64 and 79, respectvely, usng standard algebrac manpulatons 33 Dscusson and further remarks Model extensons: The results presented n Sect 3 and 32 can be readly extended to a broader set of models The requrements for the dervaton of heavy-traffc lmts smlar to Theorems 2 to 6 are that the evoluton of the system at specfc moments can be descrbed as a multdmensonal branchng process wth mmgraton, and 2 that the system s work conservng In addton to the models addressed above, ths class of models ncludes as specal cases for example models wth gated/exhaustve servce and non-cyclc perodc server routng [26], models wth smultaneous batch arrvals [22, 42], contnuous pollng models [8], models wth customer routng [34], globally-gated models wth elevator-type routng [], models wth local prortes [33], amongst many other model varants Bascally, all that needs to be done for each of these model varants s to determne the parameters α, û and the dervatve of ξ = ξρ at ρ =, whch s usually straghtforward

15 Queueng Syst : Generalty of the results: The queston rases whch pollng models fall wthn the class of branchng-type models for whch the approach presented n ths paper s applcable As stated above, the key requrements are the exstence of a sutable embedded process such that the evoluton of the state of the system can be descrbed by an MTBP, and that the system s work conservng Although most pollng systems that are used n practce are ndeed work conservng, t s not nconcevable that there exst non work-conservng pollng models for whch an embedded process does satsfy an MTBP-structure In those cases, propertes smlar to those stated n Lemma s 3 and 6 are no longer vald, so that the translaton of Theorem to results for pollng models smlar to Theorems 3 and 5, whch explctly use Lemma s 3 and 6, may be more complcated Moreover, the requred MTBP-structure of a proper embedded process mples that the arrval processes should be memoryless, and hence must be Posson, or some batched varant of the Posson process For example, models wth renewal processes wth non-exponental nterarrval tmes generally volate the requred branchng structure, and hence, fall beyond the scope of the branchng-type models for whch our results hold see also the remarks about ths n Sect 4 below Choce of the embedded process: In general, the MTBP need not always be the ont queue-length vector at embedded pollng nstants at a fxed queue, wth M = N Forexample, n the case of perodc server routng wth pollng table π := π,,π L of length L N a proper choce for the MTBP s the M := L-dmensonal ont queue-length s afxedpseudo-queue [26] As another example, n the case of two-stage pollng models wth cyclc routng [28], one should most lkely consder the M := 2N-dmensonal state vector descrbng the numbers of customers at both stages of all N types at embedded pollng nstants at a fxed queue; here, the state of the system cannot be descrbed completely by an N-dmensonal state vector Assumptons on the fnteness of moments: Theorems 4 and 6 are vald under the assumpton that the second moments of the servce tmes and the frst moments of the swtch-over tmes are fnte; these assumptons are an mmedate consequence of the assumptons on the fnteness of the mean mmgraton functon g and the second-order dervatves of the offsprng functon K,k, defned n 5 and 7, respectvely It s nterestng to observe that the results obtaned n Van der Me [39] va the use of the Descendant Set Approach DSA assume the fnteness of all moments of the servce tmes and swtch-over tmes; these assumptons were requred, snce the DSA-based proofs n [39] are based on a bottom-up approach n the sense that the lmtng results for the watng-tme dstrbutons are obtaned from the asymptotc expressons for the moments of the watng tmes obtaned n [40, 4] Note that n ths way the DSAbased approach dffers fundamentally from the top-down approach taken n the present paper, where the asymptotc expressons for the moments of W can be obtaned from the expressons for the asymptotc watng-tme dstrbutons n Theorems 4 and 6 Note, however, that to prove convergence of the kth moment n the general context of the present paper requres a stronger means of convergence namely, convergence n L k than the convergence n dstrbuton shown n ths paper [], whch addresses a challengng topc for further research Local and global branchng: Although the GG-model dscussed n Sect 3 the ont queue-length vector at successve pollng nstants at a fxed queue consttutes an MTBP, the GG-model does not occur as a specal case of the branchng model dscussed n Sect 32 To ths end, note that for the GG-model the servce polcy at Q does not satsfy the local branchng property descrbed n Sect 32, for > To see ths, consder an arbtrary pollng nstant at Q >, whch marks the begnnng of a vst V to Q Then the number of customers present at that moment, say L total L total, can be wrtten as = L front + L behnd, 94 where L front, L behnd stands for the number of type- customers that n front of and behnd the global gate, respectvely Then at the end of V all L front customers that were standng n front of the gate have been served and hence have been effectvely replaced by a populaton of customers whose ont PGF s gven by B N = λ z, whereas the remanng L behnd customers have not been served, and hence, are effectvely replaced by a populaton whose PGF equals z Approxmatons: The results presented n Theorems 4 and 6 suggest the followng smple approxmatons for the watng-tme dstrbutons for stable systems: For ρ<, =,,N, Pr{W <x} Pr{ W <x ρ} 95 Extensve valdaton of ths approxmaton falls beyond the scope of ths paper We refer to [39] for a dscusson on the accuracy of the approxmaton for the specal case of exhaustve and gated servce 4 Topcs for further research The results presented n ths paper provde a sgnfcant step towards the development of a unfed theory of pollng n heavy traffc Nonetheless, the results rase a number of challengng open questons for further research Frst, n ths

16 44 Queueng Syst : paper t s assumed that the second moments of the servcetme dstrbutons are fnte, forced by the second-moment assumpton on the offsprng functon, needed for the valdty of Theorem An nterestng area for further research s to obtan heavy-traffc results for heavy-taled servcetme dstrbutons wth nfnte varance In ths context, nterestng results have been obtaned by Boxma et al [7], who study the tal behavor of the watng tmes n pollng systems wth so-called regularly varyng servce tmes and swtch-over tmes, and by Boxma and Cohen [5], who derve the heavy-traffc lmtng dstrbuton for the watng tmes n the sngle-server queue wth heavy-taled servce-tme dstrbutons Second, n order to use the theory of MTBPs the arrval processes must be Posson or batched Posson Interestngly, n specal cases smlar heavy-traffc results have been obtaned under the weaker assumpton of ndependent renewal processes, where also the gamma-dstrbuton appears to play a key role see for example [3, 27] Note, however, that the proofs of these results for N>2 are based on partal conectures Moreover, for several pollng models t was found that the heavy-traffc lmts of a Posson-type model and ts renewal counterpart only dffer by a smple scalng constant see for example [26, 27] for non-cyclc perodc pollng models Hence, based on these nsghts we may formulate strong conectures about the asymptotc behavor of pollng models n the general settng of the present paper, wth renewal arrvals Thrd, t s well known that an exact analyss of pollng models wth servce polces that volate the branchng structure eg, lmted-type polces are fundamentally more complex, and do not allow for an exact detaled analyss of the watng tmes Nonetheless, focusng on the heavy-traffc behavor t s stll an open queston whether exact or approxmate asymptotc results can be obtaned for non-branchng type pollng models Fourth, the dervaton of rgorous proofs for the momentwse convergence of the results shown n Theorem 4 and 6 are to be obtaned To ths end, note that moment-wse convergence of the results n Theorems 4 and 6 requres extenson of the results to convergence n L 2 In ths context, note that the results n [29] suggest that convergence n L 2 ndeed holds Provdng rgorous proofs of the convergence of moments s an nterestng area for further research Fnally, a related area of research s the analyss of the watng tmes n pollng systems wth multple say m> servers Multpleserver pollng models are notorously hard, and do not leave any hope for an exact analyss Interestngly, based on numercal expermentaton t was observed n [9, 25, 43] that f the servers follow the same route they tend to cluster together, partcularly when the system s heavly loaded These results suggest that n the lmtng case all servers tend to effectvely work as a sngle server that works m tmes as fast Ths, n turn, suggests that we may use our heavy-traffc results for sngle-server pollng models to develop smple approxmatons for the delay fgures at each of the queues Prelmnary expermentaton wth smulatons show promsng results, openng up an nterestng area for further research Acknowledgements The author wshes to thank Ton Deker and Lasse Leskela for ther useful suggestons Appendx : Proof of Lemma 3 Partwasshownn[32] Part 2 follows from the fact that all entres of M are contnuous functons of ρ, whch mples contnuty of ξ = ξρ wth respect to ρ see for example [2] The fact that ξ = follows drectly from the fact that ˆMb = b, whch s an mmedate consequence of the fact that the GG-model descrbed n Sect 3 s work conservng Fnally, to prove Part 4 we adopt the concept and notaton of the Descendant Set Approach DSA from [7] The DSA focuses on an arbtrary pollng nstant of the server at Q, called the reference pont, and focuses on X, the number of type- customers n the system at that moment Denotng by A,c the contrbuton to X of a type- customer that was present n the system at a pollng nstant c cycles before the reference pont, the mean values α,c := E[A,c ] can be obtaned va the followng recursve relatons cf [7] for detals: For =,,N, α, := I {=}, 96 and for c = 0,,, α,c = b λ α,c 97 = Then f we defne, for ρ<, := λ α,c, 98 = c=0 then substtuton of 96 and 97 mmedately leads to the observaton that, for ρ<, = ρ + λ = λ ρ ρ 99 Alternatvely, t s easly verfed that we can wrte α,c = e Me cf 22n[40], where e s the th unt vector =,,N Then usng Lemma 4 n [40] seealso[2], we can wrte for c = 0,,, M c = ξ c vw + S c, 00