EURANDOM PREPRINT SERIES January 29, Analysis and optimization of vacation and polling models with retrials

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1 EURADOM PREPRIT SERIES January 29, 2015 Analyss and optmzaton of vacaton and pollng models wth retrals M. Abdn, O. Boxma, J. Resng ISS

2 Analyss and optmzaton of vacaton and pollng models wth retrals Murtuza Al Abdn, Onno Boxma and Jacques Resng EURADOM and Department of Mathematcs and Computer Scence Endhoven Unversty of Technology P.O. Box 513, 5600 MB Endhoven, The etherlands Abstract. We study a vacaton-type queueng model, and a sngleserver mult-queue pollng model, wth the specal feature of retrals. Just before the server arrves at a staton there s some determnstc glue perod. Customers both new arrvals and retrals arrvng at the staton durng ths glue perod wll be served durng the vst of the server. Customers arrvng n any other perod leave mmedately and wll retry after an exponentally dstrbuted tme. Our man focus s on queue length analyss, both at embedded tme ponts begnnngs of glue perods, vst perods and swtch- or vacaton perods and at arbtrary tme ponts. Keywords: vacaton queue, pollng model, retrals 1 Introducton Queueng systems wth retrals are characterzed by the fact that arrvng customers, who fnd the server busy, do not wat n an ordnary queue. Instead of that they go nto an orbt, retryng to obtan servce after a random amount of tme. These systems have receved consderable attenton n the lterature, see e.g. the book by Faln and Templeton [10], and the more recent book by Artaleo and Gomez-Corral [3]. Pollng systems are queueng models n whch a sngle server, alternatngly, vsts a number of queues n some prescrbed order. Pollng systems, too, have been extensvely studed n the lterature. For example, varous dfferent servce dscplnes rules whch descrbe the server s behavour whle vstng a queue and both models wth and wthout swtchover tmes have been consdered. We refer to Takag [24, 25] and Vshnevsk and Semenova [27] for some lterature revews and to Boon, van der Me and Wnands [5], Levy and Sd [16] and Takag [22] for overvews of the applcablty of pollng systems. In ths paper, motvated by questons regardng the performance modellng of optcal networks, we consder vacaton and pollng systems wth retrals. Despte Ths s an nvted, consderably extended verson of [9]. The man addtons are Subsectons 2.4 and 2.5 and Secton 3. These present respectvely an alternatve dervaton of the mean number of customers and the optmal behavour of a sngle queue system, and the performance analyss for a general number of queues.

3 the enormous amount of lterature on both types of models, there are hardly any papers havng both the features of retrals of customers and of a sngle server pollng a number of queues. In fact, the authors are only aware of a sequence of papers by Langars [12 14] on ths topc. In all these papers the author determnes the mean number of retral customers n the dfferent statons. In [12] the author studes a model n whch the server, upon pollng a staton, stays there for an exponental perod of tme and f a customer asks for servce before ths tme expres, the customer s served and a new exponental stay perod at the staton begns. In [13] the author studes a model wth two types of customers: prmary customers and secondary customers. Prmary customers are all customers present n the staton at the nstant the server polls the staton. Secondary customers are customers who arrve durng the soourn tme of the server n the staton. The server, upon pollng a staton, frst serves all the prmary customers present and after that stays an exponental perod of tme to wat for and serve secondary customers. Fnally, n [14] the author consders a model wth Markovan routng and statons that could be ether of the type of [12] or of the type of [13]. In ths paper we consder a pollng staton wth retrals and so-called glue perods. Just before the server arrves at a staton there s some determnstc glue perod. Customers both new arrvals and retrals arrvng at the staton durng ths glue perod "stck" and wll be served durng the vst of the server. Customers arrvng n any other perod leave mmedately and wll retry after an exponentally dstrbuted tme. The study of queueng systems wth retrals and glue perods s motvated by questons regardng the performance modellng and analyss of optcal networks. Performance analyss of optcal networks s a challengng topc see e.g. Maer [17] and Rogest [21]. In a telecommuncaton network, packets must be routed from source to destnaton, passng through a seres of lnks and nodes. In copper-based transmsson lnks, packets from dfferent sources are tme-multplexed. Ths s often modeled by a sngle server pollng system. Optcal fbre offers some bg advantages for communcaton w.r.t. copper cables: huge bandwdth, ultra-low losses, and an extra dmenson the wavelength of lght. However, n an optcal routng node, opposte to electroncs, t s dffcult to store photons, and hence bufferng n optcal routers can only be very lmted. Bufferng n these networks s typcally realzed by sendng optcal packets nto fbre delay loops,.e., lettng them crculate n a fbre loop and extractng them after a certan number of crculatons. Ths feature can be modelled by retral queues. Recent experments wth slow lght, where lght s slowed down by sgnfcantly ncreasng the refractve ndex n wavegudes, have up to now shown very modest bufferng tmes [11]. It should be noted that wth the very hgh speeds achevable n fbre, packet duratons are very short, so that small bufferng tmes may already allow suffcent storage of small packets. We represent the effect of slowng down lght by ntroducng a glue perod at a queue ust before the server arrves. The paper s organzed as follows. In Secton 2 we consder the case of a sngle queue wth vacatons and retrals; arrvals and retrals only "stck" durng

4 a glue perod. We study ths case separately because t s of nterest n ts own rght, t allows us to explan the analytc approach as well as the probablstc meanng of the man components n consderable detal, t makes the analyss of the mult-queue case more accessble, and v results for the one-queue case may serve as a frst-order approxmaton for the behavour of a partcular queue n the -queue case, swtchover perods now also representng glue and vst perods at other queues. In Secton 3 the -queue case s analyzed. Secton 4 presents some conclusons and suggestons for future research. 2 Queue length analyss for the sngle-queue case 2.1 Model descrpton In ths secton we consder a sngle queue Q n solaton. Customers arrve at Q accordng to a Posson process wth rate λ. The servce tmes of successve customers are ndependent, dentcally dstrbuted..d. random varables r.v., wth dstrbuton B and Laplace-Steltes transform LST B. A generc servce tme s denoted by B. After a vst perod of the server at Q t takes a vacaton. Successve vacaton lengths are..d. r.v., wth S a generc vacaton length, wth dstrbuton S and LST S. We make all the usual ndependence assumptons about nterarrval tmes, servce tmes and vacaton lengths at the queues. After the server s vacaton, a glue perod of determnstc.e., constant length begns. Its sgnfcance stems from the followng assumpton. Customers who arrve at Q do not receve servce mmedately. When customers arrve at Q durng a glue perod G, they stck, onng the queue of Q. When they arrve n any other perod, they mmedately leave and retry after a retral nterval whch s ndependent of everythng else, and whch s exponentally dstrbuted wth rate ν. The glue perod s mmedately followed by a vst perod of the server at Q. The servce dscplne at Q s gated: Durng the vst perod at Q, the server serves all "glued" customers n that queue,.e., all customers watng at the end of the glue perod but none of those n orbt, and nether any new arrvals. We are nterested n the steady-state behavour of ths vacaton model wth retrals. We hence make the assumpton that ρ := λeb < 1; t may be verfed that ths s ndeed the condton for ths steady-state behavour to exst. Some more notaton: G n denotes the nth glue perod of Q. V n denotes the nth vst perod of Q mmedately followng the nth glue perod. S n denotes the nth vacaton of the server mmedately followng the nth vst perod. X n denotes the number of customers n the system hence n orbt at the start of G n. Y n denotes the number of customers n the system at the start of V n. otce that here we should dstngush between those who are queueng and those who are n orbt: We wrte Y n = Y q n + Y o n, where q denotes queueng and o denotes n

5 orbt. Fnally, Z n denotes the number of customers n the system hence n orbt at the start of S n. 2.2 Queue length analyss at embedded tme ponts In ths subsecton we study the steady-state dstrbutons of the numbers of customers at the begnnng of glue perods, vst perods, and vacaton perods. Denote by X a r.v. wth as dstrbuton the lmtng dstrbuton of X n. Y and Z are smlarly defned, and Y = Y q + Y o, the steady-state numbers of customers n queue and n orbt at the begnnng of a vst perod whch concdes wth the end of a glue perod. In the sequel we shall ntroduce several generatng functons, throughout assumng that ther parameter z 1. For concseness of notaton, let βz := Bλ1 z and σz := Sλ1 z. Then t s easly seen that E[z X ] = σze[z Z ], 2.1 snce X equals Z plus the new arrvals durng the vacaton; E[z Z ] = E[βz Y q z Y o ], 2.2 snce Z equals Y o plus the new arrvals durng the Y q servces; and E[z Y q q z Y o o ] = e λ1 zqg E[{1 e νg z q + e νg z o } X ]. 2.3 The last equaton follows snce Y q s the sum of new arrvals durng G and retrals who return durng G; each of the X customers whch were n orbt at the begnnng of the glue perod have a probablty 1 e νg of returnng before the end of that glue perod. Combnng Equatons , and ntroducng fz := 1 e νg βz + e νg z, 2.4 we obtan the followng functonal equaton for E[z X ]: E[z X ] = σze λ1 βzg E[fz X ]. Introducng Kz := σze λ1 βzg and Xz := E[z X ], we have: Xz = KzXfz. 2.5 Ths s a functonal equaton that naturally occurs n the study of queueng models whch have a branchng-type structure; see, e.g., [7] and [20]. Typcally, one may vew customers who newly arrve nto the system durng a servce as chldren of the served customer "branchng", and customers who newly arrve nto the system durng a vacaton or glue perod as mmgrants. Such a functonal equaton may be solved by teraton, gvng rse to an nfnte product where

6 the th term n the product typcally corresponds to customers who descend from an ancestor of generatons before. In ths partcular case we have after n teratons: n Xz = Kf zxf n+1 z, 2.6 =0 where f 0 z := z and f z := ff 1 z, = 1, 2,.... Below we show that ths product converges for n ff ρ < 1, thus provng the followng theorem: Theorem 1. If ρ < 1 then the generatng functon Xz = E[z X ] s gven by Xz = Kf z. 2.7 =0 Proof. Equaton 2.5 s an equaton for a branchng process wth mmgraton, where the number of mmgrants has generatng functon Kz and the number of chldren n the branchng process has generatng functon fz. Clearly, K 1 = λes + λρg < and f 1 = e νg + 1 e νg ρ < 1, f ρ < 1. The result of the theorem now follows drectly from the theory of branchng processes wth mmgraton see e.g., Theorem 1 on page 263 n Athreya and ey [4]. Havng obtaned an expresson for E[z X ] n Theorem 1, expressons for E[z Z ] and E[z Y q q z Y o o ] mmedately follow from 2.2 and 2.3. Moments of X may be obtaned from Theorem 1, but t s also straghtforward to obtan EX from Equatons : yeldng Hence EX = λes + EZ, 2.8 EZ = ρey q + EY o, 2.9 EY q = λg + 1 e νg EX, 2.10 EY o = e νg EX, 2.11 EX = EY q = λg + 1 e νg λes + λρg 1 ρ1 e νg λes + λρg 1 ρ1 e νg λes + λg =, ρ EY o = e νg λes + λρg 1 ρ1 e νg, 2.14 EZ = λρg + λes[ρ1 e νg + e νg ] 1 ρ1 e νg. 2.15

7 otce that the denomnators of the above expressons equal 1 f 1. Also notce that t makes sense that the denomnators contan both the factor 1 ρ and the probablty 1 e νg that a retral returns durng a glue perod. In a smlar way as the frst moments of X, Y q, Y o and Z have been obtaned, we can also obtan ther second moment. For further use we here calculate E[XX 1]: E[XX 1] = K 1 1 ρ1 e νg 1 + ρ1 e νg + e νg + K 1[2K 1ρ1 e νg + e νg + 1 e νg λ 2 EB 2 ] 1 ρ 2 1 e νg ρ1 e νg + e νg, 2.16 where K 1 = λes +λρg and K 1 = λ 2 ES 2 +2ρλ 2 GES +λ 3 GEB 2 +λgρ 2. Remark 1. Specal cases of the above analyss are, e.g.: Vacatons of length zero. Smply take σz 1 and ES = 0 n the above formulas. ν =. Retrals now always return durng a glue perod. We then have fz = βz, whch leads to mnor smplfcatons. Remark 2. It seems dffcult to handle the case of non-constant glue perods, as t seems to lead to a process wth complcated dependences. If G takes a few dstnct values G 1,..., G wth dfferent probabltes, then one mght stll be able to obtan a knd of multnomal generalzaton of the nfnte product featurng n Theorem 1. One would then have several functons f z := 1 e νg βz + e νg z, and all possble combnatons of teratons f f h f k... z arsng n functons K z := σze λ1 βzg, = 1, 2,...,. By way of approxmaton, one mght stop the teratons after a certan number of terms, the number dependng on the speed of convergence hence on 1 ρ and on 1 e νg. 2.3 Queue length analyss at arbtrary tme ponts Havng found the generatng functons of the number of customers at the begnnng of glue perods E[z X ], vst perods E[z Y q q z Y o o ], and vacaton perods E[z Z ], we can also obtan the generatng functon of the number of customers at arbtrary tme ponts. Theorem 2. If ρ < 1, we have the followng results: a The ont generatng functon, R va z q, z o, of the number of customers n the queue and n the orbt at an arbtrary tme pont n a vacaton perod s gven by R va z q, z o = E[z Z o ] 1 Sλ1 z o λ1 z o ES. 2.17

8 b The ont generatng functon, R gl z q, z o, of the number of customers n the queue and n the orbt at an arbtrary tme pont n a glue perod s gven by R gl z q, z o = G t=0 e λ1 zqt E[{1 e νt z q + e νt z o } X ] dt G c The ont generatng functon, R v z q, z o, of the number of customers n the queue and n the orbt at an arbtrary tme pont n a vst perod s gven by [ z q E[z Y q q z Y o o ] E[ Bλ1 ] z o Y q z Y o o ] 1 R v z q, z o = E[Y q ] z q Bλ1 Bλ1 z o z o λ1 z o EB d The ont generatng functon, Rz q, z o, of the number of customers n the queue and n the orbt at an arbtrary tme pont s gven by G Rz q, z o = ρr v z q, z o + 1 ρ G+ES R glz q, z o + 1 ρ ES G+ES R vaz q, z o Proof. a Follows from the fact that durng vacaton perods the number of customers n the queue s 0 and the fact that the number of customers at an arbtrary tme pont n the orbt s the sum of two ndependent terms: The number of customers at the begnnng of the vacaton perod and the number that arrved durng the past part of the vacaton perod. The generatng functon of the latter s gven by 1 Sλ1 z o λ1 z o ES. b Follows from the fact that f the past part of the glue perod s equal to t, the generatng functon of the number of new arrvals n the queue durng ths perod s equal to e λ1 zqt and each customer present n the orbt at the begnnng of the glue perod s, ndependent of the others, stll n orbt wth probablty e νt and has moved to the queue wth probablty 1 e νt. c Durng an arbtrary pont n tme n a vst perod the number of customers n the system conssts of two parts: the number of customers n the system at the begnnng of the servce tme of the customer currently n servce, leadng to the term see Remark 3 below: z q E[z Y q q z Y o o E[Y q ] ] E[ Bλ1 z o Y q z Y o o ] z q Bλ1 z o ; 2.21 the number of customers that arrved durng the past part of the servce of the customer currently n servce, leadng to the term 1 Bλ1 z o λ1 z o EB.

9 d Follows from the fact that the fracton of tme the server s vstng Q s equal to ρ, and f the server s not vstng Q, wth probablty ES/G + ES the server s on vacaton and wth probablty G/G + ES the system s n a glue phase. Remark 3. A straghtforward way to prove 2.21 s to condton on the number of customers, say,, n queue at the end of a glue perod, and to average the number of customers n queue and n orbt over the begnnngs of the servces. A more elegant proof of Formula 2.21 uses the observaton that typcally n vacaton and pollng systems each tme a vst begnnng or a servce completon occurs, ths concdes wth a servce begnnng or a vst completon. Ths observaton yelds see, e.g., Boxma, Kella and Kosnsk [8] γv b z q, z o + S c z q, z o = S b z q, z o + γv c z q, z o. Here, V b z q, z o and V c z q, z o are the ont generatng functons of the number of customers n the queue and n the orbt at vst begnnngs and vst completons, respectvely. Smlarly, S b z q, z o and S c z q, z o are the ont generatng functons of the number of customers n the queue and n the orbt at servce begnnngs and servce completons, respectvely. Fnally, γ s the recprocal of the mean number of customers served per vst. Clearly, γ = and 1 E[Y q ], V bz q, z o = E[z Y q q z Y o o ], V c z q, z o = E[ Bλ1 z o Y q z Y o o ], S c z q, z o = S bz q, z o z q Bλ1 zo, whch yelds that S b z q, z o s gven by From Theorem 2, we now can obtan the steady-state mean number of customers n the system at arbtrary tme ponts n vacaton perods E[R va ], n glue perods E[R gl ], n vst perods E[R v ] and n arbtrary perods E[R]. These are gven by E[R va ] = E[Z] + λ E[S2 ] 2E[S], E[R gl ] = E[X] + λ G 2, E[R v ] = 1 + λ E[B2 ] 2E[B] + E[Y q Y o ] + 1+ρE[Y q Y q 1], E[Y q ] 2E[Y q ] G E[R] = ρe[r v ] + 1 ρ G+ES E[R gl] + 1 ρ ES G+ES E[R va] Remark that the quanttes E[Y q Y o ] and E[Y q Y q 1] can be obtaned usng 2.3: E[Y q Y o ] = λge νg E[X] + 1 e νg e νg E[XX 1], E[Y q Y q 1] = λg e νg 2 E[XX 1] + 2λG 1 e νg E[X].

10 By combnng these relatons wth 2.22, 2.8, 2.12 and 2.16, we obtan after tedous calculatons the followng relatvely smple expresson for the mean number of customers E[R]: E[R] = ρ+ λ2 E[B 2 ] 21 ρ +λe[g + S2 ] λρe[g + S] + +λρg+e[s] 2E[G + S] 1 ρ 1 ρ1 e νg, 2.23 whch we rewrte for later purposes as E[R] = ρ + λ2 E[B 2 ] 21 ρ + λ E[S] G + E[S] + λρg + E[S] 1 ρ e νg E[S 2 ] 2E[S] + λ E[S] G + E[S] G + λ G G + E[S] e νg + λρg + E[S] 1 ρ1 e νg Remark 4.. It should be notced that the frst two terms n the rghthand sde of 2.23 together represent the mean number of customers n the ordnary M/G/1 queue, wthout vacatons and glue perods. The thrd term represents the mean number of arrvals durng the resdual part of a vacaton plus glue perod. The fourth term can be nterpreted as the mean number of arrvals durng a vst perod of the server snce the mean length of one cycle,.e., vst plus vacaton plus glue perod, s va a balance argument seen to equal E[C] = G+E[S] 1 ρ, whle a mean vst perod equals ρe[c]. The ffth term s the only term nvolvng the retral rate ν. In partcular, that term dsappears when ν. In the latter case, our model reduces to an M/G/1 queue wth gated vacatons, wth vacaton lengths G + S. The resultng expresson for the mean number of customers concdes wth formula 5.23 of [23] see also formula of [26].. A more nterestng lmtng operaton s to smultaneously let ν and G 0, such that νg remans constant. The resultng model s an M/G/1 queue wth vacatons and bnomally gated servce; see, e.g., Levy [15]. Here, each customer who s present at the end of a vacaton, wll be served n the next vst perod wth probablty p = 1 e νg. In ths case, the mean number of customers n the system s gven by E[R] = ρ + λ2 E[B 2 ] 21 ρ + λe[s2 ] + λρe[s] 2E[S] 1 ρ G 2 λe[s]1 p p1 ρ Ths formula concdes wth the results obtaned n [15] see e.g., formula 7.1 wth = 1 n [15] for the mean soourn tme of customers n ths model. Observe that our formula, after applcaton of Lttle s formula, does not fully agree wth the mean delay expresson 5.50b n [23] and wth a smlar formula on p. 90 of [26]. Those mean delay expressons for the bnomal gated model seem to refer to the case where customers who are not served durng a vst n those mean delay expressons.. Formula 2.24 mmedately shows how the mean number of customers behaves for very small and for very large values of the glue perod length G: w.p. 1 p are lost; hence factors lke 1 1 pρ E[R] λe[s], G 0, 2.26 Gν1 ρ

11 and E[R] λ1 + ρ G, G ρ In subsecton 2.5 we explore the effect of G on E[R] more deeply. 2.4 An alternatve dervaton of the mean number of customers Alternatvely, equaton 2.24 can be obtaned usng mean value analyss. Let T denote the steady-state soourn tme of customers n the system. The mean value relatons for the system are gven by Lttle s law and the arrval relaton E[T ] = E[R] ρ E[B] + ρ E[B2 ] 2E[B] + E[B] + ρ E[S] + G 1 e νg + 1 ρ G E[S] + 1 ρ G + E[S] E[R] = λe[t ] 2.28 G 2 G + E[S] E[S 2 ] G + E[S] + G + e νg 2E[S] 1 e νg Combnng 2.28 and 2.29, one readly arrves at The man deas to obtan 2.29 are the followng: 1. The steady-state soourn tme can be wrtten as the sum of the busy tme of the server durng the soourn tme and the dle tme of the server durng the soourn tme. 2. The mean busy tme of the server durng the soourn tme s gven by E[R] ρ E[B] + ρ E[B2 ] 2E[B] + E[B]. Ths follows by changng the order of servce n the system nto the Frst Come Frst Serve order remark that changng the order of servce does not change the mean busy tme durng a soourn tme. 3. If a customer arrves durng a servce perod, the mean dle of the server durng ts soourn tme s gven by a geometrc number of vacaton plus glue perods. Ths gves the ρ E[S]+G n e 4. If a customer arrves durng a νg glue perod, the mean dle of the server durng G G G+E[S] 2 ts soourn tme s gven by G/2. Ths gves the 1 ρ n If a customer arrves durng a vacaton perod, the mean dle of the server durng ts soourn tme s gven by a resdual vacaton perod plus a glue perod plus, wth probablty e νg, a geometrc number of vacaton plus glue perods. Ths gves the term n E[S] 1 ρ G + E[S] E[S 2 ] G + E[S] + G + e νg 2E[S] 1 e νg

12 2.5 Optmzng the length of the glue perod The results of the prevous subsectons can, e.g., be used to determne the value of G whch mnmzes the mean number of customers n the system at any arbtrary tme pont. The mean soourn tme of an arbtrary customer follows from Lttle s formula. Therefore we can fnd the value of G whch mnmzes the mean soourn tme of an arbtrary customer. Let us frst present a sample of numercal results that we obtaned for E[R] as a functon of G. We consder four cases: the servce tme dstrbuton and vacaton tme dstrbuton are exponental, the servce tme and vacaton tme are constant, the servce tme dstrbuton s exponental and the vacaton tme s constant and v the servce tme s constant and the vacaton tme dstrbuton s exponental. For all these cases we assume that the expected servce tme EB = 1, expected vacaton tme ES = 10 and retral rate ν = 0.5. We plot G vs E[R] for λ = 0.5, 0.9. a λ = 0.5 b λ = 0.9 Fg. 1: Exponental servce and vacaton tme dstrbutons From the examples n Fg 1 we observe the followng results: The mean number of customers at any arbtrary pont seems to be convex w.r.t. glue perod length,.e., there exsts a glue perod G mn where the system has mnmum mean number of customers E[R mn ] and hence mnmum mean soourn tme. For a very small G, E[R] decreases exponentally wth G as confrmed by For a large G, E[R] ncreases lnearly wth G as confrmed by Indeed, f G s very small, the number of customers onng the queue n each glue perod s very small and thereby the effcency of the system s decreased.

13 On the other hand, a large G means the system stays n the glue perod for a long tme and ths decreases the effcency of the system. Hence t s logcal to have a G mn whch optmzes the system. We wll now prove that E[R] s ndeed convex n G. Dfferentatng the expresson for E[R] n Equaton 2.23 w.r.t. G gves d dg E[R] = λ λe[g + S2 ] 2E[G + S] 2 + λρ 1 ρ1 e νg λρg+e[s] νe νg 1 ρ1 e νg Dfferentatng 2.30 agan w.r.t. G gves [ ] [ ] d 2 dg 2 E[R] = λ E[G + S 2 ] E[G + S] E[G + S] 2 1 λνe νg + 1 ρ1 e νg 2 νρg+e[s] 1 + e νg 1 e νg 2ρ The frst term n the rghthand sde of 2.31 s nonnegatve, because E[G + S 2 ] E[G + S] 2. Let QG := νρg + E[S] 1 + e νg 1 e νg 2ρ. We can see QG as G 0 or G. Let QG attan ts mnmum at G = g. Hence at G = g, d QG = 0, dg whch mples Therefore ρg + E[S] = ρ 2νe νg 1 e 2νg. Qg = ρ 2 [eνg 2 + e νg ] 0. We observe that the mnmum value of QG s always nonnegatve. Snce both d terms of 2.31 are nonnegatve, 2 dg E[R] 0. 2 Hence E[R], the mean number of customers at an arbtrary pont of tme n the system, s convex n G. So the system can mprove the qualty of servce by settng an optmal value G for the fxed glue perod. In Table 1 we analyze the behavour of G mn and E[R mn ] as we ncrease E[S] for an exponental dstrbuton B wth EB = 1, arrval rate λ = 0.5 and retral rate ν = 0.5. Table 1 suggests that, n the case under consderaton, G mn and E[R mn ] ncrease when E[S] ncreases. G mn and E[R mn ] ncrease when the varance of S becomes larger. When ES approaches 0, G mn also approaches 0. When there s no customer n the queue, the system wll then have a seres of very short glue perods, and when a customer arrves or returns from orbt, t can almost nstantaneously be taken nto servce. In ths case, the system reduces to an ordnary M/G/1 retral queue; ndeed, Formula 2.23 reduces to E[R] = ρ + λ2 E[b 2 ] 21 ρ + λρ ν1 ρ whch s n agreement wth Formula 1.37 of [10].

14 Table 1: Exponental servce tme dstrbuton, EB = 1, λ = 0.5 and ν = 0.5 ES G mn for exponental S G mn for constant S E[R mn ] for exponental S E[R mn ] for constant S 0 ɛ, ɛ 0 ɛ, ɛ Queue length analyss for the -queue case 3.1 Model descrpton In ths secton we consder a one-server pollng model wth multple queues, Q, = 1,,. Customers arrve at Q accordng to a Posson process wth rate λ ; they are called type- customers, = 1,,. The servce tmes at Q are..d. r.v., wth B denotng a generc servce tme, wth dstrbuton B and LST B, = 1,,. The server follows cyclc pollng, thus after a vst of Q, t swtches to Q +1. Successve swtchover tmes from Q to Q +1 are..d. r.v., wth S a generc swtchover tme, wth dstrbuton S and LST S, = 1,,. We make all the usual ndependence assumptons about nterarrval tmes, servce tmes and swtchover tmes at the queues. After a swtch of the server to Q, there frst s a determnstc.e., constant glue perod G, before the vst of the server at Q begns, = 1,,. As n the one-queue case, the sgnfcance of the glue perod stems from the followng assumpton. Customers who arrve at Q do not receve servce mmedately. When customers arrve at Q durng a glue perod G, they stck, onng the queue of Q. When they arrve n any other perod, they mmedately leave and retry after a retral nterval whch s ndependent of everythng else, and whch s exponentally dstrbuted wth rate ν, = 1,,. The servce dscplne at all queues s gated: Durng the vst perod at Q, the server serves all "glued" customers n that queue,.e., all type- customers watng at the end of the glue perod but none of those n orbt, and nether any new arrvals. We are nterested n the steady-state behavour of ths pollng model wth retrals. We hence assume that the stablty condton =1 ρ < 1 holds, where ρ := λ EB. Some more notaton: G n denotes the nth glue perod of Q. V n denotes the nth vst perod of Q. S n denotes the nth swtch perod out of Q, = 1,,.

15 X n1, X n2,, X n denotes the vector of numbers of customers of type 1 to type n the system hence n orbt at the start of G n, = 1,,. Y n1, Y n2,, Y n denotes the vector of numbers of customers of type 1 to type n the system at the start of V n, = 1,,. We dstngush between those who are queueng n Q and those who are n orbt for Q : We wrte Y n = Y q n + Y o n, = 1,,, where q denotes queueng and o denotes n orbt. Fnally, Z n1, Z n2,, Z n denotes the vector of numbers of customers of type 1 to type n the system hence n orbt at the start of S n, = 1,,. 3.2 Queue length analyss In ths secton we study the steady-state ont dstrbuton of the numbers of customers n the system at begnnngs of glue perods. Ths wll also mmedately yeld the steady-state ont dstrbutons of the numbers of customers n the system at the begnnngs of vst perods and of swtch perods. We follow a smlar generatng functon approach as n the one-queue case, throughout makng the followng assumpton regardng the parameters of the generatng functons: z 1, z q 1, z o 1. Observe that the generatng functon of the vector of numbers of arrvals at Q 1 to Q durng a type servce tme B s β z 1, z 2,, z := B =1 λ 1 z. Smlarly, the generatng functon of the vector of numbers of arrvals at Q 1 to Q durng a type- swtchover tme S s σ z 1, z 2,, z := S =1 λ 1 z. Snce the server serves Q +1 after Q we can successvely express n terms of generatng functons X +1 n1, X +1 n2,, X +1 n nto Z n1, Z n2,, Z n, Z n1, Z n2,, Z n nto Y n1, Y n2,, Y q n, Y o n,, Y n, and Y n1, Y n2,, Y q n, Y o n,, Y n nto X n1, X n2,, X n ; etc. Denote by X 1, X 2,, X the vector wth as dstrbuton the lmtng dstrbuton of X n1, X n2,, X n, = 1,,, and smlarly ntroduce Z 1, Z 2,, Z and Y 1, Y 2,, Y, wth Y = Y q + Y o, for = 1,,. We have: E[z X z X z X+1 ] = σ z 1, z 2,, z E[z Z 1 1 z Z 2 2 z Z ]. 3.1 E[z Z 1 = 1 z Z 2 yeldng z h E[z Z 1 1 z Z 2 Y 2 z Z z ho [β z 1, z 2,, z ] hq, 2 z Z 1 = h 1, Y 2 = h 2,, Y q ] = E[[β z 1, z 2,, z ] Y q = h q, Y o z Y 1 1 z Y 2 2 z Y o = h o,, Y = h ] z Y ]. 3.2

16 Furthermore, = yeldng E[z Y 1 E[z Y 1 1 z Y 2 1 z Y 2 2 z Y q q z Y o o z Y X 1 = a 1, X 2 = a 2,, X = a ] z a e λ1 zg e λ1 zqg [1 e νg z q + e νg z o ] a, 2 z Y q q z Y o o z Y ] = e e λ1 zg λ1 zqg It follows from 3.1, 3.2 and 3.3, wth that E[z X z X+1 2 and E[ z X [1 e νg z q + e νg z o ] X ]. f z 1, z 2,, z := 1 e νg β z 1, z 2,, z + e νg z, z X+1 ] = σ z 1, z 2,, z e e λ1 zg λ1 βz1,z2,,z G E[ z X Let z denote vector z 1, z 2,, z ; further [f z 1, z 2,, z ] X ]. 3.4 h z := f z 1,, z, h +1 z,, h z 3.5 h z := f z 1,, z. 3.6 Snce the server moves to Q 1 after Q, substtutng = n 3.4, we have E[z X1 1 1 z X1 2 2 z X1 ] = σ z 1, z 2,, z e λ1 zg From 3.4 we have E[ z X E[ e λ 1 β z 1,z 2,,z G z X [h z] X ]. 3.7 [h z] X ] = σ 1 z 1, z 2,, z 1, h z 2 =1 e λ1 zg 1 e λ 11 β 1 z 1,z 2,,z 1,h zg 1 e λ 1 h zg 1 2 E[ [h 1 z] X 1 1 [h z] X 1 ]. =1 z X 1 3.8

17 From 3.7 and 3.8 we have E[z X1 1 1 z X1 2 2 z X1 ] = σ z 1, z 2,, z σ 1 z 1,, z 1, h z e λ 1 h zg 1 E[ [ = = 1 1 E[ =1 2 e λ1 zg e λ 1 β z 1,z 2,,z G 2 =1 2 =1 z X 1 σ z 1,, z, h +1 z,, h z e λ1 zg 1 e λ 11 β 1 z 1,,z 1,h zg [h 1 z] X 1 1 [h z] X 1 ]. e λ1 zg e λ1 βz1,,z,h+1z,,h zg =1 z X 1 [h 1 z] X 1 1 [h z] X 1 ]. =+1 e λ1 hzg ] By recursvely substtutng as above we get E[z X1 1 1 z X1 2 where 2 z X1 ] = σ z 1,, z, h +1 z,, h z =1 =1 =1 e GDz E [ ] [h 1 z] X1 1, [h 2 z] X1 2,, [h z] X1, 1 D z = λ 1 z +λ 1 β z 1,, z, h +1 z,, h z + = λ 1 h z. Equaton 3.9 can be dvded nto three factors, representng the swtchover perod, glue perod and vst perod respectvely. The frst factor, for a partcular value of, represents the arrvals durng the swtchover tme after the vst of Q. The second factor represents the arrvals durng the glue perod before a vst of Q. It s further dvded nto three generatng functons. Frst are the arrvals of type < ; these don t have any further effect on the system. Then the arrvals of type, these are served durng the followng vst and produce new chldren.e., arrvals durng ther servce of each type. Fnally those of type > whch may or may not be served n future vsts and f served produce new chldren of each type. These two factors are taken for all = 1,,. The thrd factor represents the descendants arrvals durng servces, arrvals durng servces of customers who arrved durng servces, etc. of X 1 1,, X1.

18 Consder Xz = E[ =1 z X1 ], wth an obvous defnton of Kz, we can rewrte 3.9 nto where Xz = KzXhz 3.10 hz := h 1 z,, h z and h z are as defned n 3.5 and 3.6. Defne h 0 z = z, h n z = h h n 1 1 z, h n 1 2 z,, h n 1 z, = 1,,. Theorem 3. If ρ < 1, then the generatng functon Xz s gven by Xz = Kh m 1 z, h m 2 z,, h m z m=0 Proof. Equaton 3.11 follows from 3.10 by teraton. We stll need to prove that the nfnte product converges f ρ < 1. Equaton 3.10 s an equaton for a mult-type branchng process wth mmgraton, where the number of mmgrants of dfferent types has generatng functon Kz and the number of chldren of dfferent types of a type ndvdual n the branchng process has generatng functon h z, = 1,,. An mportant role n the analyss of such a process s played by the mean matrx M of the branchng process, M = m 11 m m 1 m, 3.12 where m represents the mean number of chldren of type of a type ndvdual. The elements of the matrx M are the same as gven n Secton 5 of Resng [20], whch s m = f 1[ ] + f k m k, 3.13 k=+1 where m = h z 1, 1,, 1 and f = f z 1, 1,, 1. We observe that the equaton for m s the sum of two terms. Frst the chldren of type, who do not affect the system n the future. ext the chldren of type produced by the chldren of type k > n the subsequent vsts. The theory of mult-type branchng processes wth mmgraton see Qune [19] and Resng [20] now states that f the expected total number of mmgrants n a generaton s fnte and the maxmal egenvalue λ max of the mean

19 matrx M satsfes λ max < 1, then the generatng functon of the steady state dstrbuton of the process s gven by To complete the proof of Theorem 3, we shall now verfy and. Ad : The expected total number of mmgrants n a generaton s λ 1 G 1 m 1 + ES + G =1 =2 +λ 2 G 1 + S 1 1 e ν2g2 + G λ = =1 m 2 + G 1 + S 1 e ν2g2 + ES + G =2 =3 1 m + G + S e ν G + ES G + S 1 e ν G + G =1 =1 1 1 λ G + S 1 e νg + G m + G + S e νg + ES + Snce the above equaton s a fnte sum/product of fnte terms t s ndeed fnte. Here, the term λ 1 G 1 m 1 corresponds to the type 1 customers arrvng durng the glue perod of Q 1 and ther subsequent chldren of all types. The term λ 1 =1 ES + =2 G corresponds to the type 1 customers arrvng durng the glue perods of Q, = 2,,, and swtchover perods after Q, = 1,,. These customers arrve after the vst of Q 1 and hence do not get served or produce chldren. The term λ 2 G 1 +S 1 1 e ν2g2 +G 2 m 2+ G 1 + S 1 e ν2g2 corresponds to the type 2 customers arrvng durng the glue perod of Q 1, Q 2, the swtchover perod after Q 1 and ther subsequent chldren. The term λ 2 =2 ES + =3 G corresponds to the type 2 customers arrvng durng the glue perods of Q, = 3,,, and swtchover perods after Q, = 2,,. These customers do not produce any chldren. Smlarly the term λ 1 =1 G + S 1 e ν G + G m + 1 =1 G + S e ν G corresponds to the type customers arrvng durng the glue perod of Q 1,, Q, the swtchover perods after Q 1,, Q 1 and ther subsequent chldren. The term λ ES corresponds to the type customers arrvng durng the swtchover perod after Q, whch do not produce any chldren. Ad : Defne the matrx =1 = =+1 e ν1g1 + 1 e ν1g1 ρ1 1 e ν 1G 1 λ2 EB 1 1 e ν 1G 1 1 λ EB 1 e ν 2G 2 λ1 EB 2 e ν2g2 + 1 e ν2g2 ρ2 1 e ν 2G 2 λ EB 2 H =... 1 e ν G λ1 EB 1 e ν G λ2 EB e ν G +, 1 e ν G ρ 3.15 G. 3.14

20 where the elements h of the matrx H represent the mean number of type customers that replace a type customer durng a vst perod of Q ether new arrvals f the customer s served, or the customer tself f t s not served. We have that [e EB ν1g1 + ] 1 e ν1g1 1 ρ EB 1 EB 2 H. = [e ν2g2 + ] EB 1 1 e ν2g2 ρ EB 2 EB 2 <. EB n [e ν G e ν G ] ρ EB EB n f and only f ρ < 1. Usng ths result and followng the same lne of proof as n Secton 5 of Resng [20], we can show that the stablty condton ρ < 1 mples that also the maxmal egenvalue λ max of the mean matrx M satsfes λ max < 1. Ths concludes the proof. We can now obtan the moments, EX +1, ether from 3.11 or n a smlar way as n Secton 2.2, n terms of EX When, else Further EX +1 EZ EZ EY EY q EY o and EX : = λ ES + EZ. = λ EB EY q = λ EB EY q = λ G + EX, + EY, + EY o. = λ G + 1 e νg EX, = e νg EX. From the above equatons we get, when : and EX +1 EX +1 = λ ES + λ 1 + ρ G + λ EB 1 e νg EX + EX, = λ ES + λ ρ G + ρ 1 e νg + e νg EX. Usng flow balance arguments mean number of customers of type served per cycle equals mean number of type customers arrvng per cycle and the obvous fact that the mean cycle tme equals EC := ES + G /1 ρ, we obtan EY q = λ ES + G ρ

21 We can also use a smlar argument for mean number of type customers leavng the orbt, 1 e νg EX, to equal the mean number of type customers enterng t, λ EC G, per cycle, yeldng EX λ = 1 e [ ES + G G ] νg 1 ρ We can observe that 3.17 and 3.18 satsfy the above relaton of EY q EX. Further for each, cyclcally substtutng we get all EX EY and EZ. and and therefore The second moments of X 1 and the varous terms E[X X k ] can be obtaned by solvng a set of equatons whch s derved by twce dfferentatng 3.10 w.r.t. z and z k,, k = 1,,, and calculatng the value at z = 1, 1,, 1. Snce the system s cyclc, once we obtan EX 1 2, = 1,,, we can smlarly obtan EX 2, = 1,,, by changng ndces. It s not dffcult to develop an effcent procedure for determnng hgher moments n pollng systems wth a branchng dscplne, cf. [20]. 3.3 Queue length analyss at arbtrary tme ponts In the prevous secton we have gven the procedure for fndng the dstrbuton of the number of customers at the begnnng of glue perods E[ zx ], vst perods E[ zy z Y q q z Y o o ], and swtchover perods E[ zz ], for = 1,,. Smlar to the sngle queue case, we now obtan the generatng functon of the number of customers at arbtrary tme ponts. Theorem 4. If ρ < 1 and z q, z o := z 1q, z 1o,, z q, z o, we have the followng results: a The ont generatng functon, R swz q, z o, of the number of customers n the queue and n the orbt at an arbtrary tme pont n a swtchover perod after Q s gven by R swz q, z o = E[ z Z o ]1 S λ 1 z o λ 1 z o ES b The ont generatng functon, R gl z q, z o, of the number of customers n the queue and n the orbt at an arbtrary tme pont n a glue perod of Q s gven by R gl z q, z o = G E[z X o ] t=0 e λ1 zot e λ1 zqt E[{1 e νt z q +e νt z o } X ] dt. G 3.20

22 c The ont generatng functon, R v z q, z o, of the number of customers n the queue and n the orbt at an arbtrary tme pont n a vst perod of Q s gven by z q R v z q, z o = E[z Y q q zy o z Y o o E[Y q ] ] E[ B λ 1 z o Y q z q B λ 1 z o zy o z Y o o ] 1 B λ 1 z o λ 1 z o EB d The ont generatng functon, Rz q, z o, of the number of customers n the queue and n the orbt at an arbtrary tme pont s gven by Rz q, z o = ρ R v z q, z o +1 ρ G G+ESR gl z ES q, z o +1 ρ 3.22 G+ESR swz q, z o. Proof. The proof follows the same lnes as the proof of Theorem 2, n partcular for parts a and d. We restrct ourselves here to an outlne of the proof of parts b and c. b Follows from the fact that f the past part of the glue perod s equal to t, the generatng functon of the number of new arrvals of type n the queue durng ths perod s equal to e λ1 zqt and each type customer present n the orbt at the begnnng of the glue perod s, ndependent of the others, stll n orbt wth probablty e νt and has moved to the queue wth probablty 1 e νt. Further the generatng functon of the number of new arrvals of any type n the queue durng ths perod s equal to e λ1 zot. c Durng an arbtrary pont n tme n a vst perod the number of customers n the system conssts of two parts: the number of customers n the system at the begnnng of the servce tme of the customer currently n servce, leadng to the term z q E[z Y q q zy o z Y o o ] E[ B λ 1 z o Y q zy o E[Y q ] z q B λ 1 z o z Y o o ] ; see Remark 3. the number of customers that arrved durng the past part of the servce of the customer currently n servce, leadng to the term 1 B λ 1 z o λ 1 z o EB.

23 From Theorem 4, we now can obtan the steady-state mean number of customers n the system at arbtrary tme ponts n swtchover perods E[R sw] after Q, n glue perods E[R gl ] and n vst perods E[R v ] of Q, for = 1,,, and at any arbtrary tme pont E[R]. These are gven by E[R sw] = E[R gl ] = E[R v ] = 1 + E[R] = E[Z ] + λ E[S 2 ] 2E[S ], E[X G ] + λ 2, λ E[B2 ] 2E[B ] ρ E[R v ] + 1 ρ q E[Y Y + o ] + E[Y q ] G G+ESE[R E[Y q Y ] E[Y q ] gl ] + 1 ρ + 1+EB λe[y q 2E[Y q ] ES G+ESE[R sw] The mean number of type k customers n the system at arbtrary tme ponts n a swtchover perod after Q and a glue perod before Q are gven by the values of E[R sw] and E[R gl ] at = k. The mean number of type customers n the system at arbtrary tme ponts n a vst perod of Q s gven by 1 + λ E[B 2 ] 2E[B ] ] and E[Y q Y q 1] can be ob- The quanttes E[Y q taned usng 3.3. q E[Y Y + o ] + E[Y q ] Y o ], E[Y q Y 1+ρE[Y q 2E[Y q ] Y q 1]. Y q 1]., 4 Conclusons and suggestons for future research In ths paper we have studed vacaton queues and -queue pollng models wth the gated servce dscplne and wth retrals. Motvated by optcal communcatons, we have ntroduced a glue perod ust before a server vst; durng such a glue perod, new customers and retrals "stck" nstead of mmedately gong nto orbt. For both the vacaton queue and the -queue pollng model, we have derved steady-state queue length dstrbutons at an arbtrary epoch and at varous specfc epochs. Ths was accomplshed by establshng a relaton to branchng processes. In future studes, we would lke to consder other servce dscplnes. Furthermore, the followng model varants seem to fall wthn our framework: customers may not retry wth a certan probablty; the arrval rates may be dfferent for vst, swtchover and glue perods; one mght allow that new arrvals durng a glue perod are already served durng that glue perod. We also wsh to take non-constant glue perods nto account. We beleve that a workload

24 decomposton and pseudoconservaton law, as dscussed n [6], can be derved for these varants and generalzatons, and they may be exploted for analyss and optmzaton purposes. We shall then also try to explore the followng observaton: One may vew our -queue model as a pollng model wth a new varant of bnomal gated, wth adaptve probablty p of servng a customer at a vst of Q ; p = 1 when the customer arrved n the precedng glue perod, and p = 1 e νg otherwse. We would also lke to explore the possblty to study the heavy traffc behavor of these models va the relaton to branchng processes, cf. [18]. Fnally, we would lke to pont out an mportant advantage of optcal fbre: the wavelength of lght. A fbre-based network node may thus route ncomng packets not only by swtchng n the tme-doman, but also by wavelength dvson multplexng. In queueng terms, ths gves rse to multserver pollng models, each server representng a wavelength. We refer to [1] for the stablty analyss of multserver pollng models, and to [2] for a mean feld approxmaton of large passve optcal networks. Therefore we would lke to study multserver pollng models wth the addtonal features of retrals and glue perods. Acknowledgment The authors gratefully acknowledge frutful dscussons wth Kevn ten Braak and Tuan Phung-Duc about retral queues and wth Ton Koonen about optcal networks. The research s supported by the IAP program BESTCOM, funded by the Belgan government, and by the Gravty program ETWORKS, funded by the Dutch government. References 1.. Antunes, Chr. Frcker and J. Roberts Stablty of mult-server pollng system wth server lmts. Queueng Systems, 68, Antunes, Chr. Frcker, Ph. Robert and J. Roberts Traffc capacty of large WDM Passve Optcal etworks. Proceedngs 22nd Internatonal Teletraffc Congress ITC 22, Amsterdam, September J.R. Artaleo and A. Gomez-Corral Retral Queueng Systems: A Computatonal Approach Sprnger-Verlag, Berln. 4. K.B. Athreya and P.E. ey Branchng Processes Sprnger-Verlag, Berln. 5. M.A.A. Boon, R.D. van der Me, and E.M.M. Wnands Applcatons of pollng systems. SORMS, 16, O.J. Boxma Workloads and watng tmes n sngle-server systems wth multple customer classes. Queueng Systems, 5, O.J. Boxma and J.W. Cohen The M/G/1 queue wth permanent customers. IEEE J. Sel. Areas n Commun., 9, O.J. Boxma, O. Kella and K.M. Kosnsk Queue lengths and workloads n pollng systems. Operatons Research Letters, 39, O.J. Boxma and J.A.C. Resng Vacaton and pollng models wth retrals. 11th European Workshop on Performance Engneerng EPEW 11, Florence, September 2014.

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