EURANDOM PREPRINT SERIES September 30, Networks of /G/ Queues with Shot-Noise-Driven Arrival Intensities

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1 EURANDOM PREPRINT SERIES Sepember 3, 216 Neworks of /G/ Queues wh Sho-Nose-Drven Arrval Inenses D. Koops, O. Boxma, M. Mandjes ISSN

2 Neworks of /G/ Queues wh Sho-Nose-Drven Arrval Inenses D.T. Koops 1, O.J. Boxma 2, and M.R.H. Mandjes 1 1 Koreweg-de Vres Insue, Unversy of Amserdam 2 Eurandom and Deparmen of Mahemacs and Compuer Scence, Endhoven Unversy of Technology Sepember 3, 216 Absrac We sudy nfne-server queues n whch he arrval process s a Cox process (or doubly sochasc Posson process, of whch he arrval rae s gven by sho nose. A sho-nose rae emerges naurally n cases where he arrval rae ends o exhb sudden ncreases (or: shos a random epochs, afer whch he rae s nclned o rever o lower values. Exponenal decay of he sho nose s assumed, so ha he queueng sysems are amenable o analyss. In parcular, we perform ransen analyss on he number of cusomers n he queue jonly wh he value of he drvng sho-nose process. Addonally, we derve heavy-raffc asympocs for he number of cusomers n he sysem by usng a lnear scalng of he sho nensy. Frs we focus on a one dmensonal seng n whch here s a sngle nfne-server queue, whch we hen exend o a nework seng. 1 Inroducon In he queueng leraure one has radonally suded queues wh Posson npu. The Posson assumpon ypcally faclaes explc analyss, bu does no always algn well wh acual daa, see e.g. [11] and references heren. More specfcally, sascal sudes show ha n many praccal suaons, Posson processes underesmae he varably of he queue s npu sream. Ths observaon has movaed research on queues fed by arrval processes ha beer capure he bursness observed n pracce. The exen o whch bursness akes place, can be measured by he dsperson ndex,.e. he rao of he varance of he number of arrvals n a gven nerval, and he correspondng expeced value. In arrval sreams ha dsplay bursness, he dsperson ndex s 1

3 larger han uny (as opposed o Posson processes, for whch s equal o uny, a phenomenon ha s usually referred o as overdsperson. I s desrable ha he arrval process of he queueng model can ake he observed overdsperson no accoun. One way o acheve hs, s o make use of Cox processes, whch are Posson processes, condonal on he sochasc me-dependen nensy. I s an mmedae consequence of he law of oal varance, ha Cox processes do have a dsperson ndex larger han uny. Therefore, hs class of processes makes for a good canddae o model overdspersed npu processes. In hs paper we conrbue o he developmen of queueng models fed by npu sreams ha exhb overdsperson. We analyze nfne-server queues drven by a parcular Cox process, n whch he rae s a (sochasc sho-nose process. The sho-nose process ha we use, s one n whch here are only upward jumps (or: shos, ha arrve accordng o a homogeneous Posson process. Furhermore, we employ an exponenal response or decay funcon, n whch s encoded how quckly he process wll declne afer a jump. In hs case, he sho-nose process s a Markov process, see [16, p. 393]. There are several varaons on sho-nose processes; see e.g. [1] for a comprehensve overvew. I s no a novel dea o use a sho-nose process as sochasc nensy. For nsance, n nsurance mahemacs, he auhors of [5] use a sho-nose-drven Cox process o model he clam coun. They assume ha dsasers happen accordng o a Posson process, and each dsaser can nduce a cluser of arrvng clams. The dsaser corresponds o a sho upwards n he clam nensy. As me passes, he clam nensy process decreases, as more and more clams are seled. In our queueng model, here could be smlar underlyng dynamcs causng a sho-nose arrval rae. For nsance, could be ha many people go o a webse afer a ceran commercal has been on elevson. Anoher example of sho-nose arrval processes s found n he famous paper [13], where s used o model he occurences of earhquakes. The arrval process consdered n hs paper has one crucal dfference wh he one used n he presen paper: makes use of Hawkes processes [9], whch do have a sho-nose srucure, bu have he specal feaure ha hey are self-excng. More specfcally, n Hawkes processes, an arrval can nduce a sho n he arrval rae, whereas n our sho-nose-drven Cox model hese shos are merely exogenous. The Hawkes process s less racable han he sho-nose-drven Cox process. A very recen effor o analyze /G/ queues ha are drven by a Hawkes process has been made n [8], where a funconal cenral lm heorem s derved for he number of cusomers n he sysem. In hs model, s much harder (or mpossble o oban explc resuls (n a non-asympoc seng, as we are able o do n he sho-nose-drven Cox varan. In order o successfully mplemen a heorecal model, s crucal o have mehods o esmae s parameers from daa. The sho-nose-drven Cox process s aracve snce 2

4 has hs propery. Sascal mehods ha fler he unobservable nensy process, based on Markov Chan Mone Carlo (MCMC echnques, have been developed; see [3] and references heren. By flerng, hey refer o he esmaon of he nensy process n a gven me nerval, gven a realzed arrval process. Subsequenly, gven a sample pah of he nensy process, he parameers of he sho-nose process can be esmaed by a Mone Carlo verson of he expecaon maxmzaon (EM mehod. Furhermore, he sho-nosedrven Cox process can also be easly smulaed; see e.g. he hnnng procedure descrbed n [12]. In hs paper we sudy neworks of nfne-server queues wh sho-nose-drven Cox npu. We assume ha he servce mes a a gven node are..d. samples from a general dsrbuon. The oupu of a queue s roued o a nex queue, or leaves he nework. Infne-server queues have he nheren advanage ha cusomers do no nerfere wh one anoher, whch consderably smplfes he analyss. Furhermore, nfne-server sysems are frequenly used o produce approxmaons for correspondng fne-server sysems. In he nework seng, we can model queueng sysems ha are drven by correlaed sho-nose arrval processes. Wh regards o applcaons, such a sysem could represen he call ceners of a fre deparmen and polce deparmen n he same own. The conrbuons and organzaon of hs paper are as follows. In hs paper we derve exac and asympoc resuls. The man resul of he exac analyss s Thm. 4.6, where we fnd he jon Laplace ransform of he numbers of cusomers n he queues of a feedforward nework, jonly wh he sho-nose-drven arrval raes. We buld up owards hs resul along he followng roue. In Secon 2 we nroduce noaon and we sae he mporan Lemma 2.1 ha we repeaedly rely on. Then we commence he analyss by dervng exac resuls for he sngle nfne-server queue, wh a sho-nose arrval rae, n Secon 3.1. Subsequenly, n Secon 3.2, we show ha afer an approprae scalng he number of cusomers n he sysem sasfes a funconal cenral lm heorem (Thm. 3.4; he lmng process s an Ornsen-Uhlenbeck (OU process drven by a superposon of a Brownan moon and an negraed OU process. We hen exend he heory o a nework seng n Secon 4. Before we consder full-blown neworks, we frs consder a andem sysem conssng of an arbrary number of nfne-server queues n Secon 4.1. Then s argued n Secon 4.2 ha a feedforward nework can be seen as a number of andem queues n parallel. We analyze wo dfferen ways n whch dependency can ener he sysem hrough he arrval process. Frsly, n Model (M1, parallel servce facles are drven by a muldmensonal sho-nose process n whch he shos are smulaneous (whch ncludes he possbly ha all sho-nose processes are equal. Secondly, n Model (M2, we assume ha here s one sho-nose arrval nensy ha generaes smulaneous arrvals n all andems. In Secon 5 we fnsh wh some concludng remarks. 3

5 2 Noaon and prelmnares Le (Ω, F, {F }, P be a probably space, n whch he flraon {F } s such ha Λ( s adaped o. A sho-nose process s a process ha has random jumps a Posson epochs, and a deermnsc response or decay funcon, whch governs he behavor of he process. See [16, Secon 8.7] for a bref accoun of sho-nose processes. The sho nose ha we use n hs paper has he followng represenaon: Y ( Λ( = Λ(e r + B e r(, (1 where he B are..d. shos from a general dsrbuon, he decay funcon s exponenal wh rae r >, Y s a homogeneous Posson process wh rae ν, and he epochs of he shos, ha arrved before me, are labelled 1, 2,..., Y (. As explaned n he nroducon, he sho-nose process serves as a sochasc arrval rae o a queueng sysem. I s sraghforward o smulae a sho-nose process; for an llusraon of a sample pah, consder Fg. 1. Usng he hnnng mehod for nonhomogeneous Posson processes [12], and usng he sample pah of Fg. 1 as he arrval rae, one can generae a correspondng sample pah for he arrval process, as s dsplayed n Fg. 2. Typcally, mos arrvals occur shorly afer peaks n he sho-nose process n Fg. 1, as expeced. We wre Λ (.e., whou argumen for a random varable wh dsrbuon equal o lm Λ(. We now presen well-known ransen and saonary momens of he shonose process, see Appendx B and e.g. [16]: wh B dsrbued as B 1, E Λ( = Λ(e r + ν E B r =1 (1 e r, E Λ = ν E B, r Var Λ( = ν E B2 (1 e 2r, Var Λ = ν E B2, (2 2r 2r Cov(Λ(, Λ( + δ = e rδ Var Λ(. We remark ha, for convenence, we hroughou assume Λ( =. The resuls can be exended o Λ( > n a sraghforward fashon, a he cos of somewha more cumbersome noaon. In he one-dmensonal case, we denoe β(s = E e sb, and n he muldmensonal case, where s = (s 1, s 2,..., s d, for some neger d 2, now denoes a vecor, we wre β(s = E e s B. In he remander of he paper we ofen use he followng lemma. 4

6 Λ( Fgure 1: Sample pah of sho-nose process number of arrvals Fgure 2: A realzaon of arrval process correspondng o he sample pah of he arrval rae process n Fg. 1 Lemma 2.1. Le Λ( be a sho-nose process. Le f : R R d R be a funcon whch s pecewse connuous n s frs argumen, wh a mos a counable number of dsconnues. Then holds ha ( E exp f(u, zλ(u du sλ( = exp ν Proof. See appendx A. ( β se r( v e rv f(u, ze ru du 1 dv. v 3 A sngle nfne-server queue In hs secon we sudy he M S /G/ queue. Ths s a sngle nfne-server queue, of whch he arrval process s a Cox process drven by he sho-nose process Λ(, as defned n Secon 2. Frs we derve exac resuls n Secon 3.1, where we fnd he jon ransform 5

7 of he number of cusomers n he sysem and he sho-nose rae, and derve expressons for he expeced value and varance. Subsequenly, n Secon 3.2, we derve a funconal cenral lm heorem for hs model. 3.1 Exac analyss We le J denoe he servce requremen of he -h cusomer, where J 1, J 2,... are assumed o be..d.; n he sequel J denoes a random varable ha s equal n dsrbuon o J 1. Our frs objecve s o fnd he dsrbuon of he number of cusomers n he sysem a me, n he sequel denoed by N(. Ths can be found n several ways; because of he appealng underlyng nuon, we here provde an argumen n whch we approxmae he arrval rae on nervals of lengh by a consan, and hen le. Ths procedure works as follows. We le Λ( = Λ(ω, be an arbrary sample pah of he drvng sho-nose process. Gven Λ(, he number of cusomers ha arrved n he nerval [k, (k + 1 and are sll n he sysem a me, has a Posson dsrbuon wh parameer P(J > (k + U k Λ(k + o(, where U 1, U 2,... are..d. sandard unform random varables. Summng over k yelds ha he number of cusomers n he sysem a me has a Posson dsrbuon wh parameer / P(J > (k + U k Λ(k + o(, k=1 whch converges, as, o P(J > uλ(u du. (3 The argumen above s no new: a smlar observaon was menoned n e.g. [6], for deermnsc rae funcons. Snce Λ( s acually a sochasc process, we conclude ha he number of cusomers has a mxed Posson dsrbuon, wh he expresson n Eqn. (3 as random parameer. As a consequence, we fnd by condonng on F, ( ξ(, z, s := E z N( e sλ( = E e sλ( E (z N( F ( = E exp (z 1P(J > Λ(u du sλ(. (4 We have found he followng resul. Theorem 3.1. Le Λ( be a sho-nose process. Then ( log ξ(, z, s = ν β (1 ze rv P(J > ue ru du + se r( v 1 dv. (5 v 6

8 Proof. The resul follows drecly from Lemma 2.1 and Eqn. (4. In Thm. 3.1 we found ha N( has a Posson dsrbuon wh he random parameer gven n Eqn. (3. Ths leads o he followng expresson for he expeced value E N( = E Λ(u P(J > u du. (6 In addon, by he law of oal varance we fnd ( ( Var N( = Var Λ(u P(J > u du + E Λ(u P(J > u du. (7 The laer expresson we can furher evaluae, usng an approxmaon argumen ha resembles he one we used above. Usng a Remann sum approxmaon, we fnd ( / Var Λ(u P(J > u du = lm Λ( P(J > / = 2 lm = 2 / = j> v Var = Cov(Λ( P(J >, Λ(j P(J > j Cov(Λ(u, Λ(v P(J > u P(J > v du dv. Assumng ha u v, we know ha Cov(Λ(u, Λ(v = e r(u v Var Λ(v (cf. Lemma B.1 n he appendx for a proof. We hus have ha (7 equals 2 v e r(u v Var Λ(v P(J > u P(J > v du dv + We can make hs more explc usng he correspondng formulas n (2. E Λ(u P(J > u du. Example 3.2 (Exponenal case. Consder he case n whch J s exponenally dsrbued wh mean 1/µ and Λ( =. Then we can calculae he mean and varance explcly. For µ r, E N( = E Λ µ h r,µ(, where he funcon h r,µ ( s defned by µ(1 e r r(1 e µ f µ r, µ r 1 e r re r f µ = r. For he varance, we hus fnd for µ r Var N( = ν E B2 2r r 2 (1 e 2µ + µ 2 (1 e 2r + µr(4e (µ+r e 2µ e 2r 2 µ(µ r 2 (µ + r + ν E B h r,µ (, r 7

9 and for µ = r Var N( = ν E B2 ( 4r 3 1 e 2r 2r(1 + re 2r + ν E B h r,r (. r 3.2 Asympoc analyss Ths subsecon focuses on dervng a funconal cenral lm heorem (FCLT for he model under sudy, afer havng appropraely scaled he sho rae of he sho-nose process. In he followng we assume ha he servce requremens are exponenally dsrbued wh rae µ, and we pon ou how can be generalzed o a general dsrbuon n Remark 3.6 below. We follow he sandard approach o derve he FCLT for nfneserver queueng sysems; we mmc he argumenaon used n e.g. [1, 14]. As he proof has a relavely large number of sandard elemens, we resrc ourselves o he mos mporan seps. We apply a lnear scalng o he sho rae of he sho-nose process,.e. ν nν. I s readly checked ha under hs scalng, he seady-sae level of he sho-nose process, as well as he seady-sae number of cusomers n he queue, blow up by a facor n, oo. I s our objecve o prove ha, afer approprae cenerng and normalzaon, he process recordng he number of cusomers n he sysem converges o a Gaussan process. In he n-h scaled model, he number of cusomers n he sysem a me, denoed by N n (, has he followng (obvous represenaon: wh A n ( denong he number of arrvals n [, ], and D n ( he number of deparures, N n ( = N n ( + A n ( D n (. (8 Here, A n ( corresponds o a Cox process wh a sho-nose-drven rae, and herefore we have, wh Λ n (s he sho-nose n he scaled model a me s and S A ( a un-rae Posson process, ( A n ( = S A Λ n (udu ; n lne wh our prevous assumpons, we pu Λ n ( =. For our nfne-server model he deparures D n ( can be wren as, wh S D ( a un-rae Posson process (ndependen of S A (, ( D n ( = S D µn n (u du. We sar by denfyng he average behavor of he process N n (. Followng he reasonng of [1], assumng ha N n (/n ρ(, N n (/n converges almos surely o he soluon of ρ( = ρ( + E Λ(u du µρ(u du. (9 8

10 Ths equaon s solved by ρ( = E N(, wh E N( provded n Example 3.2. Now we move o he dervaon of he FCLT. Followng he approach used n [1], we proceed by sudyng an FCLT for he npu rae process. To hs end, we frs defne ˆΛ n ( := ( 1 n n Λ n( E Λ( ; ˆKn ( := ˆΛ n (u du. The followng lemma saes ha ˆK n ( converges o an negraed Ornsen-Uhlenbeck (OU process, correspondng o an OU process ˆΛ( wh a speed of mean reverson equal o r, long-run equlbrum level, and varance σ 2 Λ := ν E B2 /(2r. We le denoe weak convergence. Lemma 3.3. Assume ha for a sho B holds ha E B, E B 2 <. Then ˆK n ( ˆK( as n, where ˆK( = n whch ˆΛ sasfes, wh W 1 ( a sandard Brownan moon, ˆΛ( = σ Λ W 1 ( r ˆΛ(u du, (1 ˆΛ(u du. (11 Proof. Ths proof s sandard; for nsance from [2, Prop. 3], by pung he λ d n ha paper o zero, follows ha ˆΛ n ( ˆΛ(. Ths mples ˆK n ( ˆK(, usng (1 ogeher wh he connuous mappng heorem. Ineresngly, he above resul enals ha he rae process dsplays mean-reverng behavor. Ths also holds for nfne-server queues n general. In oher words, he number of cusomer process, n he queueng sysem we are sudyng, can be consdered as he composon of wo mean-reverng processes. We make hs more precse n he followng. From now on we consder he followng cenered and normalzed verson of he number of cusomers n he sysem: ˆN n ( := ( 1 n n N n( ρ(. We assume ha ˆNn ( ˆN( as n. To prove he FCLT, we rewre ˆN n ( n a convenen form. Mmckng he seps performed n [1] or [14], wh S A ( := S A (, S D ( := S D (, ( ( R n ( := S A Λ n (udu S D µ N n (udu, and usng he relaon (9, we evenually oban ˆN n ( = ˆN n ( + R n( n + ˆK n ( µ ˆN n (udu. 9

11 Our nex goal s o apply he marngale FCLT o he marngales R n (/ n; see for background on he marngale FCLT for nsance [7] and [17]. The quadrac varaon equals [ ] ( ( Rn = 1 (udu S A Λ n (udu + S D µ N n, n n whch converges o E Λ(udu + µ ρ(udu. Appealng o he marngale FCLT, he followng FCLT s obaned. Theorem 3.4. The cenered and normalzed verson of he number of cusomers n he queue sasfes an FCLT: ˆNn ( ˆN( as n, where ˆN( solves he sochasc negral equaon ˆN( = ˆN( + E Λ(u + µρ(u dw2 (u + ˆK( µ ˆN(udu, wh W 2 ( a sandard Brownan moon ha s ndependen of he Brownan moon W 1 ( we nroduced n he defnon of ˆK(. Remark 3.5. In passng, we have proven ha he arrval process as such obeys an FCLT. Wh  n ( := ( 1 n n A n( E Λ(u du, we fnd ha Ân( Â( as n, where Â( := E Λ(u + µρ(u dw2 (u + ˆK( = 2µρ(u + ρ (u dw 2 (u + ˆK(; he las equaly follows from he fac ha ρ( sasfes (9. Remark 3.6. The FCLT can be exended o non-exponenal servce requremens, by makng use of [15, Thm. 3.2]. Ther approach reles on wo assumpons: The arrval process should sasfy an FCLT; The servce mes are..d. non-negave random varables wh a general c.d.f. ndependen of he arrval process. As noed n Remark 3.5, he frs assumpon s sasfed for he model n hs paper. The second assumpon holds as well. In he non-exponenal case he resuls are less clean; n general, he lmng process can be expressed n erms of a Kefer process, cf. e.g. [4]. 4 Neworks Now ha he reader s famlar wh he one-dmensonal seng, we exend hs o neworks. In hs secon, we manly sudy feedforward neworks n whch each node corresponds o an nfne-server queue. Feedforward neworks are defned as follows. 1

12 Fgure 3: Snce he cusomers are no nerferng wh each oher, he nework on he lef s equvalen o he graph on he rgh. Node 3 s a copy of node 3: works a he same speed and nduces he same servce requremens. Defnon 4.1 (feedforward nework. Le G = (V, E be a dreced graph wh nodes V and edges E. The nodes represen nfne-server queues and he dreced edges beween he facles demonsrae how cusomers move hrough he sysem. We suppose ha here are no cycles n G,.e. here s no sequence of nodes, sarng and endng a he same node, wh each wo consecuve nodes adjacen o each oher n he graph, conssen wh he orenaon of he edges. We focus on feedforward neworks o keep he noaon manageable. In Thm. 4.6, we derve he ransform of he numbers of cusomers n all nodes, jonly wh he sho-nose process(es for feedforward neworks. Noneheless, we provde Example 4.5, o show ha analyss s n fac possble f here s a loop, bu a he expense of more nvolved calculaons. Snce all nodes represen nfne-server queues, one can see ha whenever a node has mulple npu sreams, s equvalen o mulple nfne-server queues ha work ndependenly from each oher, bu have he same servce speed and nduce he same servce requremen for arrvng jobs. Consder Fg. 3 for an llusraon. The reason why hs holds s ha dfferen cusomer sreams move ndependenly hrough he sysem, whou creang wang mes for ohers. Therefore, mergng sreams do no ncrease he complexy of our nework. The same holds for spls n cusomer sreams. By hs we mean ha afer cusomers fnshed her servce n a server, hey move o server wh probably q (wh q = 1. Then, one can smply sample he enre pah ha he cusomer wll ake hrough he sysem, a he arrval nsance a s frs server. If one recognzes he above, hen all feedforward neworks reduce o parallel andem sysems n whch he frs node n each andem sysem s fed by exernal npu. The procedure o decompose a nework no parallel andems, s o fnd all pahs n he nework ha sar from a node n whch here s exernal npu and end n a node of whch he deparures leave he sysem. Each of hese pahs wll subsequenly be consdered as a andem queue, whch are hen se n parallel. 11

13 To buld up o he man resul, we frs sudy andem sysems n Secon 4.1. Subsequenly, we pu he andem sysems n parallel n Secon 4.2 and fnally we presen he man heorem and some mplcaons n Secon Tandem sysems As announced, we proceed by sudyng andem sysems. In Secon 4.2 below, we sudy d parallel andem sysems, where = 1,..., d. In hs subsecon we consder he -h of hese andem sysems. Suppose ha andem has S servce facles and he npu process a he frs node s Posson, wh a sho-nose arrval rae Λ (. We assume ha cusomers ener node 1. When hey fnsh servce, hey ener node 2, ec., unl hey ener node S afer whch hey leave he sysem. We use as a subscrp referrng o node j n andem sysem and we refer o he node as node. Hence N ( and J denoe he number of cusomers n node a me, and a copy of a servce requremen, respecvely, where j = 1,..., S. Fx some me >. Agan we derve resuls by splng me no nervals of lengh. Denoe M (k, for he number of cusomers presen n node a me ha have enered node 1 beween me k and (k + 1 ; as we keep fxed we suppress n our noaon. Because cusomers are no nerferng wh each oher n he nfne-server realm, we can decompose he ransform of neres: S E z N ( = lm / k= S E z M (k,. (12 Supposng ha he arrval rae s a deermnsc funcon of me λ (, by condonng on he number of arrvals n he k-h nerval, S E z M (k, = e λ (k (λ (k m ( f (k, z m m! m= ( = exp λ (k (f (k, z 1, where S f (u, z := p (u + z p (u, (13 n whch p (u (p (u, respecvely denoes he probably ha he job ha enered andem a me u has already lef he andem (s n node j, respecvely a me. Noe ha S p (u = P J l < u, l=1 j 1 j p (u = P J l < u, J l > u. (14 l=1 l=1 12

14 Recognzng a Remann sum and leng, we conclude ha Eqn. (12 akes he followng form: ( S E = exp λ (u(f (u, z 1 du. z N ( In case of a sochasc rae process Λ (, we oban S ( E z N ( Λ ( = exp Λ (u(f (u, z 1 du. Therefore holds ha S E z N ( and we consequenly fnd S E z N ( e sλ ( e sλ ( 4.2 Parallel (andem sysems S = E E z N ( S = E e sλ( E e sλ ( z N ( Λ ( Λ (, ( = E exp Λ (u(f (u, z 1 du sλ (. (15 Now ha he andem case has been analyzed, he nex sep s o pu he andem sysems as descrbed n Secon 4.1 n parallel. We assume ha here are d parallel andems. There are dfferen ways n whch dependence beween he parallel sysems can be creaed. Two relevan models are lsed below, and llusraed n Fg. 4. (M1 Le Λ Λ( be a d-dmensonal sho-nose process (Λ 1,..., Λ d where he shos n all Λ occur smulaneously (he sho dsrbuons and decay raes may be dfferen. The process Λ, for = 1,..., d, corresponds o he arrval rae of andem sysem. Each andem sysem has an arrval process, n whch he Cox processes are ndependen gven her sho-nose arrval raes. (M2 Le Λ Λ( be he sho-nose rae of a Cox process. The correspondng Posson process generaes smulaneous arrvals n all andems. 13

15 Λ 1 Λ 2 Tandem 1 Tandem 2 Λ Tandem 1 Tandem 2 Fgure 4: Model (M1 s llusraed on he lef, and Model (M2 s llusraed on he rgh. The recangles represen andem sysems, whch conss of an arbrary number of nodes n seres. Remark 4.2. The model n whch here s essenally one sho-nose process ha generaes arrvals for all queues ndependenly, s a specal case of Model (M1. Ths can be seen by seng all componens of Λ = (Λ 1,..., Λ d equal, by leng he shos and decay rae be dencal. In Model (M1, correlaon beween he sho-nose arrval raes nduces correlaon beween he numbers of cusomers n he dfferen queues. In Model (M2, correlaon clearly appears because all andem sysems have he same npu process. Of course, he andem sysems wll no behave dencally because he cusomers may have dfferen servce requremens. In shor, correlaon across dfferen andems n Model (M1 s due o lnked arrval raes, and correlaon n Model (M2 s due o smulaneous arrval epochs. We feel ha boh versons are relevan, dependng on he applcaon, and hence we analyze boh. Analyss of (M1 Suppose ha he dependency s of he ype as n (M1. Ths means ha Λ = (Λ 1,..., Λ d for whch he dsurbances n Λ 1,..., Λ d arrve smulaneously. Recall he defnon of f as saed n Eqn. (13. I holds ha d S E e s Λ ( z N ( d S = E E z N ( e s Λ ( Λ ( =1 =1 d = E S e s Λ ( E =1 where he las equaly holds due o (15. z N ( Λ ( ( d ( = E exp Λ (u(f (u, z 1 du s Λ, (16 =1 Analyss of (M2 Now suppose ha he dependency n hs model s of ype (M2,.e., here s one sho-nose process ha generaes smulaneous arrvals n he parallel andem sysems. 14

16 Frs we assume a deermnsc arrval rae funcon λ(. Le M (k, be he number of jobs presen n andem sysem a node j a me ha have arrved n he sysem beween k and (k + 1. Noe ha d S E z N ( =1 = lm / k=1 d S E =1 z M (k, To furher evaluae he rgh hand sde of he prevous dsplay, we observe ha we can wre d S E z M (k, λ(k λ(k m = e (f(k, z m = e λ(k (f(k,z 1, m! where =1 m= f(u, z := d S j+1 p l1,...,l d l j =1 =1. d z l ; (17 n hs defnon p l1,...,l d p l1,...,l d (u equals he probably ha a job ha arrved a me u n andem s n node l a me (cf. Eqn. (14. The suaon ha l = S + 1 means ha he job lef he andem sysem; we defne z,s +1 = 1. In a smlar fashon as before, we conclude ha ( d S E e sλ( = E exp Λ(u(f(u, z 1 du sλ(, (18 =1 z N ( wh f defned n Eqn. (17. Example 4.3 (Two-node parallel sysem. In case we would be analysng a parallel sysem of smply wo nfne-server queues, hen f(u, z smplfes consderably o f(u, z 11, z 21 = 2 l 1 =1 l 2 =1 2 z 1l1 z 2l2 p l1,l 2 = z 11 z 21 p 11 + z 21 p 21 + z 11 p 12 + p 22. Remark 4.4 (Roung. Consder a feedforward nework wh roung. As argued n he begnnng of hs secon, he nework can be decomposed as a parallel andem sysem. In case here s splng a some pon, hen one decomposes he nework as a parallel sysem, n whch each andem receves he cusomer wh probably q, such ha q = 1. Ths can be ncorporaed smply by adjusng he probables conaned n f n Eqn. (16, whch are gven n Eqn. (14, so ha hey nclude he even ha he cusomer joned he andem under consderaon. For nsance, he expresson for p (u n he lef equaon n (14 would become ( P S Q =, l=1 15 J l < u,

17 where Q s a random varable wh a generalzed Bernoull (also called caegorcal dsrbuon, where P(cusomer s assgned o andem = P(Q = = q, for = 1,... d, wh q = 1; he rgh equaon n (14 s adjused smlarly. Oher han ha, he analyss s he same for he case of spls. Remark 4.5 (Neworks wh loops. So far we only consdered feedforward neworks. Neworks wh loops can be analyzed as well, bu he noaon becomes que cumbersome. To show he mehod n whch neworks wh loops and roung can be analyzed, we consder a specfc example. Suppose ha arrvals ener node one, afer whch hey ener node wo. Afer hey have been served n node wo, hey go back o node one wh probably η, or leave he sysem wh probably 1 η. Noe ha hs nework has a loop. In hs case, wh smlar echnques as before, we can fnd ( E z N 1( 1 z N 2( 2 = exp Λ(u(f(u, z 1, z 2 1 du, wh f(u, z 1, z 2 = P( job(u lef sysem + 2 z P( job(u s n node, n whch job(u s he job ha arrved a me u and we are examnng he sysem a me. Now, f we denoe servce mes n he j-h node by J (j, hen, a a specfc me, k+1 P( job(u lef sysem = P (J (1 + J (2 u η k (1 η. k= =1 Analogously, P( job(u s n node 1 equals, by condonng on he job havng aken k loops, k k η k P J (1 k+1 + (J (1 + J (2 > u, (J (1 + J (2 u ; k= =1 =1 =1 lkewse, P( job(u s n node 2 equals k+1 k η k P (J (1 + J (2 > u, J (1 k+1 + k= =1 =1 + J (2 u. (J (1 For example, n case all J (j are ndependen and exponenally dsrbued wh mean 1/µ, we can calculae hose probables explcly. Indeed, f we denoe Y for a Posson 16

18 process wh rae µ, hen e.g., k+1 k P (J (1 + J (2 > u, J (1 k+1 + =1 and hus =1 P( job(u s n node 2 = + J (2 u = P(Y ( u = 2k + 1 (J (1 η m µ( u (µ( u2m+1 e. (2m + 1! m= µ( u (µ( u2k+1 = e (2k + 1! A smlar calculaon can be done for he probably ha he job s n node one. Recallng ha a sum of exponenals has a Gamma dsrbuon, we can wre f(u, z 1, z 2 = z 1 η m µ( u (µ( u2m e + z 2 η m µ( u (µ( u2m+1 e (2m! (2m + 1! m= m= + η m (1 ηf Γ(2m+2,µ ( u m= = z 1 e µ( u cosh ( µ η( u + z 2 e µ( u η snh ( µ η( u +(1 η η m F Γ(2m+2,µ ( u, m= where F Γ(2m+2,µ denoes he dsrbuon funcon of a Γ-dsrbued random varable wh rae µ and shape parameer 2m Man resul In hs subsecon we summarze and conclude wh he followng man resul. Recall Defnon 4.1 of a feedforward nework. In he begnnng of Secon 4 we argued ha we can decompose a feedforward nework no parallel andems. In Secon 4.2 we suded exacly hose sysems, leadng up o he followng resuls. Theorem 4.6. Suppose we have a feedforward nework of nfne-server queues, where he npu process s a Posson process wh sho-nose arrval rae. Then he nework can be decomposed no parallel andem sysems. Under assumpon (M1, holds ha ( d S d ( E e sλ ( = E exp Λ (u(f (u, z 1 du s Λ =1 z N ( = exp ( ν =1 ( β(g(s, v 1 dv, 17

19 wh f (, as defned n Eqn. (13 and where g(s, v s a vecor-valued funcon n whch componen s gven by s e r ( v e r v v (f (u, z 1e r u du. Furhermore, under assumpon (M2, ( d S E e sλ( = E exp Λ(u(f(u, z 1 du sλ( =1 = exp ν z N ( ( β se r( v e rv (f(u, z 1e ru du 1 dv, v wh f(, as defned n Eqn. (17. Proof. These are Eqns. (16 and (18 o whch we appled Lemma 2.1. Nex we calculae covarances beween nodes n andem and beween nodes n parallel. Covarance n Tandem Sysem Suppose ha we have a andem sysem conssng of wo nodes and we wan o analyze he covarance beween he numbers of cusomers n boh nodes. Droppng he ndex of he andem sysem, we denoe N 1 ( and N 2 ( for he numbers of cusomers n node 1 and 2, respecvely. From Eqn. (16, E N 2 ( = P(J 1 < u, J 1 + J 2 > u E Λ(u du and ( E N 1 (N 2 ( = E P(J 1 < u, J 1 + J 2 > uλ(u du so ha Cov(N 1 (, N 2 ( ( = Cov P(J 1 < u, J 1 + J 2 > uλ(u du, = 2 = 2 v v P(J 1 > vλ(v dv P(J 1 > vλ(v dv P(J 1 < u, J 1 + J 2 > u P(J 1 > v Cov(Λ(u, Λ(v du dv P(J 1 < u, J 1 + J 2 > u P(J 1 > ve r(u v Var Λ(u du dv, cf. Lemma B.1 n Appendx B for he las equaly. 18

20 Covarance parallel (M1 Suppose ha we have a parallel sysem conssng of wo nodes only. By makng use of Eqn. (16, we fnd ( E N 1 (N 2 ( = E Λ 1 (u P(J 1 > u du Λ 2 (v P(J 2 > v dv. Ths mples ( Cov(N 1 (, N 2 ( = Cov Λ 1 (u P(J 1 > u du, = 2 = 2 v v Cov(Λ 1 (u, Λ 2 (v P(J 1 > u P(J 2 > v du dv Λ 2 (v P(J 2 > v dv ν E B 11 B ( 12 1 e (r 1+r 2 u e r2(u v P(J 1 > u P(J 2 > v du dv r 1 + r 2 where we made use of he fac ha, for u v, Cov(Λ 1 (u, Λ 2 (v = Cov(Λ 1 (u, Λ 2 (ue r2(u v = ν E B ( 11B 12 1 e (r 1+r 2 u e r2(u v, r 1 + r 2 cf. Appendx B. Covarance parallel (M2 From Eqn. (18 we derve ( E N 1 (N 2 ( = E Λ(u P(J 1 > u, J 2 > u du ( + E Λ(u P(J 1 > u du Ths mples ( Cov(N 1 (, N 2 ( = Cov Λ(u P(J 1 > u du, + Λ(u P(J 2 > u du E Λ(u P(J 1 > u, J 2 > u du. Λ(u P(J 2 > u du The followng proposon compares he correlaons presen under assumpons (M1 and (M2. In he proposon we refer o he number of cusomers n queue j n model a me as N ( j (. We fnd he ancpaed resul, ha he correlaon under assumpon (M2 s sronger han under assumpon (M1. Proposon 4.7. Le Λ( be he sho-nose process ha generaes smulaneous arrvals n boh queues and le Λ 1 (, Λ 2 ( be processes ha have smulaneous jumps and generae arrvals n boh queues ndependenly. Suppose ha Λ 1 ( d = Λ 2 ( d = Λ(, for. Then, for any, Corr(N (1 1 (, N (1 2 (2 (2 ( Corr(N 1 (, N 2 (. 19.

21 Proof. Because of he assumpon Λ 1 ( = d Λ 2 ( = d Λ(, we have ha, for all combnaons, j {1, 2}, he N (j ( are equal n dsrbuon. Therefore s suffcen o show ha Cov(N (1 1 (, N (1 2 (2 (2 ( Cov(N 1 (, N 2 (. The expressons for he covarances, whch are derved earler n hs secon, mply ha Cov(N (2 1 (, N (2 2 (1 (1 ( Cov(N 1 (, N whch s non-negave, as desred. 2 ( = E Λ(u P(J 1 > u, J 2 > u du, 5 Concludng remarks We have consdered neworks of nfne-server queues wh sho-nose-drven Coxan npu processes. For he sngle queue, we found explc expressons for he Laplace ransform of he jon dsrbuon of he number of cusomers and he drvng sho-nose arrval rae, as well as a funconal cenral lm heorem of he number of cusomers n he sysem under a parcular scalng. The resuls were hen exended o a nework conex: we derved an expresson for he jon ransform of he numbers of cusomers n he ndvdual queues, jonly wh he values of he drvng sho-nose processes. We ncluded he funconal cenral lm heorem for he sngle queue, bu s ancpaed ha a smlar seup carres over o he nework conex, albe a he expense of consderably more nvolved noaon. Our fuure research wll nclude he sudy of he deparure process of a sngle queue; he oupu sream should reman Coxan, bu of anoher ype han he npu process. Acknowledgemens The auhors hank Marn Jansen for nsghful dscussons abou he funconal cenral lm heorem n hs paper. The research for hs paper s parly funded by he NWO Gravaon Projec NETWORKS, Gran Number The research of Onno Boxma was also parly funded by he Belgan Governmen, va he IAP Bescom Projec. Appendces A Proof of Lemma 2.1 There are varous ways o prove hs resul; we here nclude a procedure ha nensvely reles on he probablsc properes of he sho-nose process nvolved. Observe ha, 2

22 recognzng a Remann sum, / f(u, zλ(u du = lm f(k, zλ(k. (19 Wh P B ( a Posson process wh rae ν and he U..d. samples from a unform dsrbuon on [, 1], holds ha Λ(k = k P B (l l=1 =P B ((l 1 +1 k=1 B e r U e r(k l. We hus oban ha he expresson n (19 equals (where he equaly follows by nerchangng he order of he summaons whch behaves as / lm f(k, z k=1 / = lm / lm l=1 k l=1 =P B ((l 1 +1 e rl l P B (l l=1 =P B ((l 1 +1 P B (l f(z, ue ru du Furhermore, we have he represenaon / Λ( = lm B e r U e r(k l / B e r U f(k, ze r(k l, P B (l l=1 =P ((l 1 +1 We conclude ha ξ(, z, s (cf. Eqn. (4 equals / P lm E exp B ( (l B e r U e rl l=1 =P B ((l 1 +1 k=l P (l =P B ((l 1 +1 l B e r( l. B e r U. f(u, ze ru r( l du se. (2 21

23 Condonng on he values of P B (l P B ((l 1, for l = 1,..., /, and usng ha he B are..d., we fnd ha he expresson n Eqn. (2 equals / ( = lm e ν (ν k l E exp B 1 e r U e rl f(u, ze ru r( l du se k l! l=1 l=1 k l = ( / = lm e ν exp ν E exp B 1 e r U e rl f(u, ze ru du se = lm exp ( / ν E exp B 1 e r U e rl l=1 exp l=1 l l l r( l f(u, ze ru r( l du se ν whch can be wren as ( / lm ν β se r( l e r U e rl f(u, ze ru du 1. By connuy of he exponen and he defnon of he Remann negral, hs equals ( exp ν β se r( v e rv f(u, ze ru du 1 dv. v l k l, B (Proof of Lemma B.1 Lemma B.1. Le Λ 1 (, Λ 2 ( be sho-nose processes of whch he jumps occur smulaneously accordng o a Posson arrval process P B ( wh rae ν. Le he decay be exponenal wh rae r 1, r 2, respecvely. Then holds ha, for δ >, whch, n case Λ 1 = Λ 2, reduces o correspondng o [16, p. 394]. Cov(Λ 1 (, Λ 2 ( + δ = e r 2δ Cov(Λ 1 (, Λ 2 (, Cov(Λ (, Λ ( + δ = e r δ Var Λ (, for = 1, 2, Proof. Le E,δ (n be he even ha P B ( + δ P B ( = n. By condonng on he number of arrvals n he nerval (, + δ], we fnd E Λ 1 (Λ 2 ( + δ = E(Λ 1 (Λ 2 ( + δ E,δ (n P(E,δ (n = n= n= E(Λ 1 (Λ 2 ( + δ E,δ (n (δνn e νδ. n! 22

24 We proceed by rewrng he condonal expecaon as E(Λ 1 (Λ 2 ( + δ E,δ (n = 1 δ n +δ... +δ E(Λ 1 (Λ 2 ( + δ F 1,..., n,δ(n d 1... d n, denong by F 1,..., n,δ(n he even E,δ (n and he arrval epochs are 1,..., n. Noe ha we have, condonal on F 1,..., n,δ(n, he dsrbuonal equaly and consequenly Λ 2 ( + δ = Λ 2 (e r 2δ + E ( Λ 1 (Λ 2 ( + δ F 1,..., n,δ(n = E Λ 1 (Λ 2 (e r 2δ + E Λ 1 ( Noe ha for all = 1,..., n we have +δ... +δ +δ n B 2 e r 2(+δ, (21 =1 n E B 2 e r 2(+δ. (22 e r 2(+δ d 1 d 2... d n = 1 r 2 (1 e r 2δ δ n 1. (23 Afer uncondonng Eqn. (22 wh respec o he arrval epochs by negrang over all from o + δ and dvdng by δ n, we hus oban E(Λ 1 (Λ 2 ( + δ E,δ (n = E Λ 1 ( E Λ 2 (e r 2δ + E Λ 1 ( 1 r 2 δ (1 e r 2δ n E B 12 and hence, denong Λ := lm Λ ( for = 1, 2, E Λ 1 (Λ 2 ( + δ = n= ( E Λ 1 (Λ 2 (e r2δ + E Λ 1 ( 1 (δν r 2 δ (1 e r 2δ n n E B 12 e νδ n! =1 = E Λ 1 (Λ 2 (e r 2δ + E Λ 1 ( ν r 2 (1 e r 2δ E B 12 = E Λ 1 ( E Λ 2 + e r 2δ ( E Λ 1 (Λ 2 ( E Λ 1 ( E Λ 2, where we made use of E Λ = ν E B 1 /r n he las equaly. I follows ha Cov(Λ 1 (, Λ 2 ( + δ = E Λ 1 (Λ 2 ( + δ E Λ 1 ( E Λ 2 ( + δ from whch he lemma follows. = E Λ 1 (Λ 2 (e r 2δ + (1 e r 2δ E Λ 1 ( E Λ 2 E Λ 1 ( ( E Λ 2 (e r 2δ + (1 e r 2δ E Λ 2, 23

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