Strong approximations for Markovian service networks

Transcription

1 Queueing Sytem Strong approximation for Markovian ervice network Avi Mandelbaum a, William A. Maey b and Martin I. Reiman c a Davidon Faculty of Indutrial Engineering and Management, Technion Intitute, Haifa 32, Irael avim@tx.technion.ac.il b Bell Laboratorie, Lucent Technologie, Office 2C-32, Murray Hill, NJ 7974, USA will@reearch.bell-lab.com c Bell Laboratorie, Lucent Technologie, Office 2C-35, Murray Hill, NJ 7974, USA marty@reearch.bell-lab.com Received 22 Augut 997; revied May 998 Inpired by ervice ytem uch a telephone call center, we develop limit theorem for a large cla of tochatic ervice network model. They are a pecial family of nontationary Markov procee where parameter like arrival and ervice rate, routing topologie for the network, and the number of erver at a given node are all function of time a well a the current tate of the ytem. Included in our modeling framework are network of M t/m t/n t queue with abandonment and retrial. The aymptotic limiting regime that we explore for thee network ha a natural interpretation of caling up the number of erver in repone to a imilar caling up of the arrival rate for the cutomer. The individual ervice rate, however, are not caled. We employ the theory of trong approximation to obtain functional trong law of large number and functional central limit theorem for thee network. Thi give u a tractable et of network fluid and diffuion approximation. A common theme for ervice network model with feature like many erver, prioritie, or abandonment i non-mooth tate dependence that ha not been covered ytematically by previou work. We prove our central limit theorem in the preence of thi non-moothne by uing a new notion of derivative. Keyword: trong approximation, fluid approximation, diffuion approximation, multierver queue, queue with abandonment, queue with retrial, priority queue, queueing network, Jackon network, nontationary queue Content. Introduction and ummary 5 2. The model and main reult Calculu for calable Lipchitz derivative 6 4. Claical Jackon network Queue with abandonment and retrial Jackon network with abandonment 7 J.C. Baltzer AG, Science Publiher

2 5 A. Mandelbaum et al. / Strong approximation 7. Priority queue Jackon network with tate dependent routing Proof of the trong limit theorem 78. Proof of the central limit theorem 86. Appendix: Ordinary differential equation Appendix: Scalable Lipchitz derivative 97 Acknowledgement 2 Reference 2. Introduction and ummary Motivated by the need to deign and analyze Markovian ervice network, we invetigate fluid and diffuion limit for uch ytem. The main ditinguihing feature of mot, but not all of the ytem we conider in thi paper i that ervice i provided by a large upply of erver, and there i a correponding large demand for thi ervice. It i thee large quantitie that motivate the aymptotic regime we conider. Our method allow u to conider network with time dependent parameter, tate dependent routing, abandonment, and retrial. To make the decription of our model and reult eaier to follow, we firt conider a imple example ee figure. The M t /M t /n t queue ha a timeinhomogeneou Poion arrival proce with rate λ t, a ervice rate per erver of µ t, and n t erver, for all t. We can contruct the ample path for the M t /M t /n t queue length proce a the unique et of olution to the functional equation Qt = Q A λ d A 2 µ Q n d,. where A and A 2 are given independent, tandard rate Poion procee, and for all real x and y, x y minx, y. Figure. The M t/m t/n t queue.

3 A. Mandelbaum et al. / Strong approximation 5 The aymptotic approach to the M t /M t / queue, a ued in both Maey 5,6] and Mandelbaum and Maey ], wa to create a family of aociated M t /M t / queue where the queue indexed by > ha arrival rate λ t and ervice rate µ t. We then determined the aymptotic behavior for the time evolution of thi family of queue, when. For the M t /M t /n t queue, we alo create a family of aociated procee. The key difference here i that, for the M t /M t /n t queue indexed by, we want to have both the arrival rate and number of erver grow large, i.e., caled up by, but leave the ervice rate uncaled. We are then intereted in the aymptotic behavior of the procee Q t = Q A λ d A 2 µ Q n d.2 = Q A λ d A 2 µ Q n d.3 a. Note that.3 how that the caling we want i equivalent to the imultaneou caling of λ t and µ t with multiplication by, provided that we alo divide Q by when n t, thi ditinction doe not matter ince the M t /M t / ervice rate indexed by i µ t {Q >}, where t {Q t >} i the indicator function for the event {Q t > }, which i the ame a µ t {Q t />}. Equation.3 i a pecial cae of equation 2.8, which i in turn a pecial cae of equation 2.9. Equation 2.9 define the procee of interet to u in the general ervice network etting. For ytem with infiniteimal rate that are not tate dependent, the caling ued in.3 and 2.8 i the ame a uniform acceleration, a conidered in Maey 5,6] and Mandelbaum and Maey ] for the M t /M t / queue, a well a Maey and Whitt 8] for the general cae of finite tate, timeinhomogeneou, continuou time Markov chain. In thee article a parameter ε i ued and limit are taken uch that ε. Thi parameterization can be reconciled with the notation in thi paper by etting ε = /. We refer to the caling in.3 and 2.8 a uniform acceleration alo, even though thi involve a light abue of terminology. It eem appropriate to comment here on two ditinguihing feature of the above formulation that carry over to the general reult of thi paper: many uncaled erver, and time-dependent parameter. Our original motivating example were call center, where ervice involve an interaction between either two people the cutomer and erver, or a peron and a machine the peron i the cutomer. In either cae, becaue a peron i involved, it doe not eem reaonable to cale the ervice rate with. Thu, in order to accommodate the arrival, whoe rate i proportional to, the number of erver mut be caled with. Time dependent arrival rate hould need no jutification, ince phenomena uch a ruh hour are quite common. Time dependent ervice rate can be ued to model phenomena uch a erver fatigue or change in the nature of ervice over the day. Finally, a time dependent number of erver arie with hift change and in ytem where the number of erver i varied to accommodate change in the arrival rate.

4 52 A. Mandelbaum et al. / Strong approximation Our firt-order aymptotic reult take the form of a functional trong law of large number FSLLN, and yield a fluid approximation for the original proce. For the above M t /M t /n t example, the FSLLN tate that lim Q = Q implie lim Q t = Q t a..,.4 uniformly on compact et in t, where Q = {Q t t } i the unique proce that olve the integral equation Q t = Q λ µ Q n ] d,.5 for all t. The general verion of our FSLLN i theorem 2.2. The above FSLLN can be refined with a functional central limit theorem FCLT. A fundamental difficulty arie in attempting to apply prior reult to obtain the FCLT, even for the M t /M t /n t queue. The reolution of thi difficulty for general Markovian ervice network i the purpoe of thi paper. Before tating the FCLT for the M t /M t /n t queue we firt point out the eence of the difficulty. Conider a equence of real valued random variable {X n, n } that correpond to partial um of i.i.d. random variable with finite mean µ and variance σ 2. Letting Y n = X n /n and n Z n = σ Y n µ = X n nµ σ n,.6 then the trong law of large number and central limit theorem yield lim n Y n = µ a.. and lim n Z n d = Z,.7 d where lim n Z n = Z indicate convergence in ditribution, and Z i a tandard mean, variance normal random variable. Let f : R R be differentiable in a neighborhood of µ, then cf. 2] n lim fy n = fµ a.., and lim fyn fµ d = f µz..8 n n σ What happen if f i continuou but not differentiable at µ? Continuity i ufficient to enure that lim n fy n = fµ a.. If f µ lim x µ f x and f µ lim x µ f x both exit, then a more careful treatment can be ued to how that n lim fyn fµ d = f µz f µ Z, n σ where for all real x and y, x y maxx, y, x x, and x x. Going back to the M t /M t /n t example, we cannot apply previou reult, uch a Kurtz 9] to obtain an FCLT for it queue length becaue the function f t x = x n t i not differentiable with repect to x at x = n t. To circumvent thi difficulty we

5 A. Mandelbaum et al. / Strong approximation 53 introduce a new notion of derivative in the context of a multivariate function, which we call the calable Lipchitz derivative. For example, if Λf t x; y denote the calable Lipchitz derivative of f t at x for any real y, then Λf t x; y = y {x<nt} y {x=nt} = y {x<nt} y {x nt}..9 Uing thi new notion of derivative we are able to obtain an FCLT for a wider cla of tochatic model. For the M t /M t /n t queue, the FCLT that arie from uniform acceleration tate that, if {Q > } i a family of random variable ee ection 2 for the independence aumption, then Q ] lim Q d = Q. implie Q ] t lim Q d t = Q t,. where Q i defined in.5, Q = {Q t t } i the unique tochatic proce that olve the integral equation Q t = Q µ {Q <n }Q d µ {Q n }Q d B λ d B 2 µ Q n d,.2 and B, B 2 are two independent, tandard Brownian motion. Although it eem clear that the FCLT for the M t /M t /n t queue could be proved on an ad-hoc bai without the calable Lipchitz derivative, thi notion i the key to proving the FCLT for more general ytem, which i our theorem 2.3. We are alo able to obtain ordinary differential equation for the mean and covariance of the diffuion limit ariing in the FCLT. Thee are given in theorem 2.4. We actually obtain a more refined FCLT which i motivated by the work of Halfin and Whitt 4]. They identify an important aymptotic regime that correpond to parameter aymptotic of the form λ = λ l o and µ = µ m o..3 The fluid limit i unchanged. The reulting refined diffuion limit for the M t /M t /n t queue i

6 54 A. Mandelbaum et al. / Strong approximation Q t = Q µ {Q <n }Q d µ {Q n }Q d l m Q ] n d B λ d B 2 µ Q n d..4 If we et λ = λ, µ = µ, n = n, m =, and l = µβ with λ = µn, and let Q = n, we recover the M/M/n pecial cae for the diffuion limit of 4]. The SLLN and the FCLT are proved in two tep. Firt, we prove a trong approximation theorem, which in the context of the M t /M t /n t queue tate that, a t Q t = Q λ µ Q n d B λ d B 2 µ Q n d Olog a..,.5 where B and B 2 are a above and the convergence i uniform on compact t et. The general verion of thi reult i theorem 2.. The limit theorem then follow from a more detailed aymptotic analyi of thi approximation theorem. Although we leave the precie pecification of our model and aumption to ection 2, we decribe here two more example that illutrate the breath of our framework. One example ee ection 7 i a ingle node with everal cutomer clae operating under the preemptive priority dicipline figure 5, and the other ee ection 5 i a ytem with cutomer abandonment and retrial figure 3. More complicated example, uch a a Jackon network figure 2 and a network with tate dependent routing figure 6 are treated in the body of the paper ection 4 and 8, repectively. All the network example given in the paper have the feature of time-varying rate and multierver node. The priority ytem we conider ha c clae of cutomer and n t erver. Cutomer of cla i arrive a a Poion proce with rate λ i t and have ervice rate µi t. All the arrival and ervice procee are contructed from mutually independent Poion procee. Cla i i given preemptive priority over any cla j uch that i > j, i, j c. The ytem with abandonment and retrial ha a ingle ervice node with n t erver. New cutomer arrive to the ervice node in a Poion proce of rate λ t. Cutomer arriving to find an idle erver are taken into ervice with rate µ t. Cutomer that find all erver buy join the queue, from which they are taken into ervice in a FCFS manner. Each cutomer waiting in the queue abandon at rate β t. An abandoning cutomer leave the ytem with probability ψ t or join the retrial pool with probability ψ t. Each cutomer in the retrial pool leave to enter the ervice node at rate µ 2 t. Upon entry to the ervice node thee cutomer are treated the ame a new cutomer.

7 A. Mandelbaum et al. / Strong approximation 55 Sytem with an infinite number of erver, or where the number of erver grow fat enough to be effectively infinite in the limit are alo covered by our model and reult. Example of uch reult in the literature are Iglehart 6] and Whitt 23]. Although all of the example that we conider in thi paper correpond to ytem with a large number of erver, thi i not the only context in which our reult are applicable. In particular it hould be noted that our reult can be applied to ome cloed queueing network with a large number of cutomer. For dicuion of uch network, we refer the reader to the reference on finite population model found in the bibliography of Mandelbaum and Pat 4]. There ha been a great deal of work on tate dependent queue, time dependent queue, and related aymptotic. We make no attempt to urvey thi literature, focuing intead on four piece of work related cloely enough to our to merit pecific mention: 9,3,4], and 9]. The reader intereted in more reference on tate dependent queue hould conult 4] or 3]. Reference on time dependent queue are contained in Mandelbaum and Maey ], and Maey and Whitt 7]. Motivated by population, epidemic, and chemical reaction model Kurtz 9] prove a FSLLN and a FCLT for ytem with mooth parameter. Our motivation i queueing ytem that do not atify the moothne required in 9]. We alo generalize 9] in the ene that we allow time dependent rate, but thi i motly a notational iue. In Mandelbaum and Pat 4] limit theorem are proved for Markovian network with tate dependent rate. Sytem whoe limit may hit a boundary of the tate pace are allowed in 4], o that the iue of reflection mut be dealt with. The limit procee that we obtain do not have the ingular local time term typically aociated with reflection. Intuitively, thi i becaue our limit procee do not hit any boundarie. The iue of piecewie continuou derivative i treated in 3, theorem 4.3] for the one-dimenional cae and i uggeted a a ubject for future reearch in 4]. Newell 9] conider approximation for the G t /G/n queue with large n. The approximation in 9] are of fluid and diffuion type, and are motivated by the trong law of large number and the central limit theorem, but no limit theorem are tated or proved in that work. The ret of thi paper i organized a follow. The model and main reult are preented in ection 2. The ome propertie of calable Lipchitz derivative are decribed in ection 3. Some example of Markovian ervice network covered by our theorem are preented in ection 4 8. In ection 4 we conider Jackon network with many erver at each node. The ytem with abandonment and retrial dicued above i treated in ection 5. Thi i a pecial cae of a Jackon network with abandonment, which i treated in ection 6. Section 7 deal with the priority ytem decribed above, and ection 8 cover Jackon network with tate dependent routing. The proof of our main reult are contained in ection 9 and. The trong approximation theorem and the FSLLN are proved in ection 9. The FCLT i proved in ection. There are two appendice. The firt appendix ection contain reult on ordinary differential equation that we need, including a verion of Gronwall inequality and a uniquene reult for our limit procee. The econd appendix ection 2 contain proof of ome of the baic propertie of Lipchitz derivative.

8 56 A. Mandelbaum et al. / Strong approximation 2. The model and main reult The primitive for our model are {A i i I}, a collection of mutually independent, tandard rate Poion procee, indexed by a et I which i at mot countably infinite; a eparable Banach pace V with norm ; a collection of jump vector {v i V i I} uch that v i < ; 2. a random initial tate vector Q in V that i alway aumed to be independent of the collection of Poion procee {A i i I}; and a collection of real-valued, non-negative Lipchitz rate function on V, { αt, i t, i I }, 2.2 that jointly atify αt, i βt γ i 2.3 for ome β t, a locally integrable function, and {γ i i I}, a ummable equence of real number; here i the Lipchitz norm for real-valued function on V, namely f fx fy up f. 2.4 x,y V, x y x y It follow that for all x and y in V, we have fx fy f x y 2.5 and o f i a Lipchitz function whenever f <. Moreover, for all x V, fx f x. 2.6 For all of the example that we conider in thi paper, V = R N for ome N < and the number of element in I i finite. Thu, although we prove the main reult of the paper for a more general etting, any reader uncomfortable with the trapping of Banach pace can replace V with R N and till follow the example we preent. In that cae i the tandard Euclidean norm on R N. In term of the primitive, we repreent our Markovian ervice network to be the V-valued tochatic proce Q {Qt t }, whoe ample path are uniquely determined by Q and the functional equation Qt = Q A i α Q, i d v i 2.7 for all t for the M t /M t /n t example, V = R, I = {, 2}, v =, v 2 =, α t x, = λ t, α t x, 2 = µ t x n t. Uniquene of the olution to 2.7 i hown in theorem 9.2. The pecial uniform acceleration that i ued for the rate function of the

9 A. Mandelbaum et al. / Strong approximation 57 M t /M t /n t queue in.3 now generalize to an aymptotic analyi of the procee {Q > } a, where Q t = Q A i α Q, i d v i. 2.8 Our goal i to characterize thi aymptotic behavior a with a functional trong law of large number and a central limit theorem, but we do thi with a more general type of aymptotic behavior for the rate function. The aymptotic analyi that we decribe above wa carried out by Kurtz 9] for the pecial cae of rate function having no explicit time dependence and tate dependence that i continuouly differentiable. In thi paper, we extend hi analyi to the following general cla of procee: Q t = Q A i α Q, i d v i, 2.9 where In our extenion, we allow the following: α t, i βt γ i. 2.. The rate function α t, i are function of time a well a tate. 2. The rate function, which are indexed by the parameter, are uch that for each i I, α t, i ha the following aymptotic expanion a : α t, i = α t, i α t, i o The rate function, a a function of the tate pace V, have a more general type of differentiability that include function on the real line that are everywhere left and right differentiable. The firt condition i a minor extenion of Kurtz but the latter two condition are ignificant new extenion. The lat condition i the mot ignificant in that a new nonmooth differential calculu mut be developed to deal with thee continuou but piecewie differentiable rate function. Thee new condition allow u to apply the limit theorem to a wider cla of Markov procee that arie in the tudy of queueing network with large number of erver. Within the framework of trong approximation, we firt approximate the amplepath repreentation 2.9 of the family {Q > } by the following theorem, which i proved in ection 9. Theorem 2. Strong approximation. If 2. and 2. hold, then a we have

10 58 A. Mandelbaum et al. / Strong approximation Q t = Q α Q d B i α Q, i d v i Olog a where the convergence i uniform on compact et in t. From thi trong approximation, we deduce a FSLLN theorem 2.2, followed by a FCLT theorem 2.3. The limit theorem enable ample-path 2.7 and ditributional approximation 2.3, which upport computation and confidence interval. The proof of the following functional trong law of large number i preented in ection 9. Theorem 2.2 FSLLN. Aume that 2. and 2. hold. Moreover, aume that lim α, i α, i d =, 2.3 for all t. If {Q > } i any family of random initial tate vector in V, then Q lim = Q Q t a.. implie lim = Q t a.., 2.4 where the convergence i uniform on compact et in t, and Q i the unique determinitic proce {Q t t } that olve the integral equation Here α t, given by Q t = Q α Q d, t. 2.5 α t x = α t x, iv i, x V, 2.6 i a Lipchitz mapping of V into itelf and it Lipchitz norm α t i a locally integrable function of t. We call Q the fluid approximation aociated with the family {Q t t }. It give rie to firt-order macrocopic fluid approximation of the form Q t, ω = Q t o a.., t. 2.7 In the development of a functional central limit theorem for our tochatic network, which refine the above fluid approximation, it i neceary to differentiate α t over the Banach pace V. There are pecific example of queueing ytem that we analyze, like the M t /M t /n t queue, where the correponding α t i piecewie differentiable but not everywhere differentiable. Thi poe a problem that i not eaily

11 A. Mandelbaum et al. / Strong approximation 59 ignored ince thee derivative are evaluated at value for the fluid model Q t. So even if α t ha no derivative at only a finite number of point, the fluid proce could pend all of it time at thee point. We reolve thi iue by introducing a new type of differentiability. If f i a mapping from V into V 2, we extend the Banach pace norm and 2 on V and V 2, repectively, to define the following norm on f: f fx fy 2 up f 2, 2.8 x,y V, x y x y and ay that f i Lipchitz on V whenever f <. If O i an open ubet of V and x O, we ay that f i locally Lipchitz at x if f O fy fz 2 up f 2 <. 2.9 y,z O, y z y z Now we define f to have a calable Lipchitz derivative at x V if there exit another mapping from V into V 2, denoted Λfx;, uch that fx y fx Λfx; y 2 lim =, 2.2 y y where the function Λfx; i Lipchitz on V o that Λfx; <, 2.2 and for all real calar with λ, λλfx; y = Λfx; λy Since all bounded linear mapping between V and V 2 poe thee lat two propertie, we ee that differentiability i a pecial cae of calable Lipchitz differentiability. Nonmooth differentiation in the context of generalizing directional derivative ha been defined before, ee Clarke ] and Rockafellar 22] for detail. Our definition 2.2 can be viewed a the analogue to the multivariate definition of differentiability or contructing the Jacobian. We ometime write thi new derivative a Λf x y Λfx; y 2.23 to emphaize that we hould fix x and view the derivative a a function of y. We can now tate the functional central limit theorem, whoe proof i potponed to ection. Theorem 2.3 FCLT. Aume that 2. and 2. hold. Moreover, aume that lim α, i α, i] d < 2.24

12 6 A. Mandelbaum et al. / Strong approximation and lim α, i ] α, i α It follow that α t, given by 2.6, and α t, given by, i d = α t x = α t x, iv i, x V, 2.26 are both Lipchitz mapping of V into itelf, and their Lipchitz norm are locally integrable function of t, Moreover, if we aume that α t ha a calable Lipchitz derivative Λα t Q t; and we have a family of random initial tate vector {Q > } in V, then for all random vector Q and Q in V, it follow that Q ] lim Q d = Q 2.27 implie Q ] t lim Q d t = Q t, 2.28 the convergence being weak-convergence in D V,, the pace of V-valued function that are right-continuou with left-limit, equipped with the Skorohod J topology. Finally, the limit Q {Q t t } i the unique tochatic proce that olve the tochatic integral equation Q t = Q B i Λα Q ; Q α Q ] d α Q, i d v i, t, 2.29 where the {B i i I} are a family of mutually independent, tandard Brownian motion. We call Q the diffuion approximation aociated with the family {Q t t }. It quantifie deviation from the fluid approximation, and it give rie to econd-order meocopic diffuion approximation of the form Q t = d Q t Q t o 2.3 a for all t, with the approximation being in ditribution. Although we tate and prove theorem for the etting of 2.9, all but one of the example are preented in the more retrictive context of 2.8. Thi i done mainly to reduce the notational burden. The full generality of 2.9 i employed for

13 A. Mandelbaum et al. / Strong approximation 6 the ytem with abandonment and retrial in ection 5. It hould be clear from thee reult how to extend the other example to the etting of 2.9. Now conider the cae of V being either a finite dimenional vector pace or a Banach pace that can be embedded into it own dual pace like a Hilbert pace, o that we can define the notion of a tranpoe, denoted by a upercript T for V = R N, thi correpond to the tandard tranpoe of a matrix. One conequence of the diffuion limit i an aociated et of differential equation that become ueful in the computation of it mean and covariance matrix. The proof i given at the end of ection. Theorem 2.4. If condition 2., 2., 2.24, and 2.25 all hold, then the mean vector and covariance matrix for Q t olve the following et of differential equation: and d dt E Q t ] = E Λα t Q t; Q t ] α t Q t 2.3 d dt Cov Q t, Q t ] = { Cov Q t, Λα t Q t; Q t ]} for almot all t, where α t Q t, i v T i v i 2.32 Cov Q t, Q t ] E Q t T Q t ] E Q t ] T E Q t ] 2.33 and for all operator A on V, {A} A A T Moreover, if Λα t Q t; i a linear operator for almot all t, then EQ t] i the unique olution for 2.3 and CovQ t, Q t] i the unique olution for Finally, for all < t, CovQ, Q t] olve the ame et of differential equation in t a doe EQ t], but with a different et of initial condition. 3. Calculu for calable Lipchitz derivative Certain baic propertie of the calable Lipchitz derivative are ueful in doing calculation for the diffuion limit of our ervice network procee. All of the theorem in thi ection are proved in ection 2. The firt theorem tate general propertie for thee function. Theorem 3.. Scalable Lipchitz differentiability ha the following propertie:. If the function f : V V 2 i calable Lipchitz differentiable at x, then the reulting Lipchitz derivative function Λf x : V V 2 i unique.

14 62 A. Mandelbaum et al. / Strong approximation 2. If f : V V 2 and g : V 2 V 3 are both calable Lipchitz differentiable at fx, then g f : V V 3 i calable Lipchitz differentiable at x, with Λg f x y = Λg fx Λf x y If f : V V 2 i locally Lipchitz, a defined in ection 2, in an open neighborhood O V of x V and ha a calable Lipchitz derivative at x, then Λfx f O. 3.2 The next theorem i ueful in the identification of calable Lipchitz differentiable function that act on finite dimenional vector pace. Theorem 3.2. The following reult hold:. If f : R n R m i differentiable at x R n with Jacobian matrix Dfx, then it i calable Lipchitz differentiable there and it calable Lipchitz derivative i matrix multiplication by the Jacobian matrix o that for all y R m, viewing y a a row vector. Λf x y = y Dfx If f : R n R m i locally Lipchitz at x R n and ha all it radial derivative at x, then f ha a calable Lipchitz derivative at x. One imple conequence of the econd tatement of thi theorem i that if f : R R ha left and right derivative everywhere, then it i everywhere calable Lipchitz differentiable and Λf x y = f xy f x y, 3.4 for all real x and y. For all x and y in R m, let x y be the R m -vector whoe ith component equal x i y i and define x y in a imilar fahion. We can then define x x and x x. Now let f, g : V R m and define I {fx>gx} to be the projection operator on R m uch that for any unit bai vector e i for i =,..., m we have { ei if f i x > g i x, e i I {fx>gx} 3.5 if f i x g i x. where fx = f x,..., f m x and gx = g x,..., g m x. The projection operator I {fx<gx} and I {fx=gx} are defined imilarly. We ue thee operator in the following theorem which give u a non-mooth calculu for computing thee calable Lipchitz derivative. Theorem 3.3. The following operation preerve calable Lipchitz differentiability:

15 A. Mandelbaum et al. / Strong approximation 63. If f : V V 2 and g : V V 2 are both calable Lipchitz differentiable at x, then f g i calable Lipchitz differentiable at x, where Λf g x y = Λf x y Λg x y If f : V R and g : V R are both calable Lipchitz differentiable at x, then f g i calable Lipchitz differentiable at x, where Λfg x y = fxλg x y gxλf x y If f : V R m and g : V R m are both calable Lipchitz differentiable at x, then f g and f g are both calable Lipchitz differentiable at x, where and Λf g x y = Λf x yi {fx>gx} Λg x yi {fx<gx} Λf x y Λg x y I {fx=gx} 3.8 Λf g x y = Λf x yi {fx<gx} Λg x yi {fx>gx} Λf x y Λg x y I {fx=gx}. 3.9 Note that if f : R n R, then we have fx = fx fx, and o Λ f x y = Λf x y {fx>} Λfx y {fx=} Λf x y {fx<} Claical Jackon network We now conider the claical Jackon network but with the additional feature of time varying rate and number of erver ee figure 2. We extend Kendall notation and call it the M t /M t /n t N network, where N denote the number of node. We contruct the M t /M t /n t N network by firt defining the following et of parameter: Figure 2. The Jackon network.

16 64 A. Mandelbaum et al. / Strong approximation λ i t = external arrival rate to node i at time t, µ i t = ervice rate for node i at time t, φ ij φ i t t = ervice routing probability to node i from node j at time t, = ervice departure probability from node i at time t, n i t = number of erver for node i at time t. All thee rate function are aumed to be locally integrable function of t and we require that φ i t N j= for all t and i =,..., N. We then et V = R N and define N N Qt = Q A a i i= j= A c ij λ i d A b i φ ij t = 4. Qi n i µ i φ ij d e j e i Qi n i µ i φ i d e i ], 4.2 where A a i, Ab i, and Ac ij for i, j =,..., N are mutually independent tandard Poion procee. For all x R N, we define x to be the diagonal N N matrix where the ith diagonal entry i the ith component of the vector x. We define λ t, µ t, and n t to be row vector where their ith component equal λ i t, µi t, and ni t, repectively. We alo define Φ t to be the N N matrix whoe i, j entry i φ ij t. Theorem 4.. Defining Q by uniform acceleration a in 2.8, the fluid limit for the M t /M t /n t N network i the olution to the integral equation Q t = Q λ Q n µ Φ I ] d. 4.3 Moreover, the diffuion limit for the M t /M t /n t N network i the unique olution to the integral equation Q t = Q N N i= j= B c ij B a i Q I {Q <n } Q I {Q n } µ Φ I d Q i n i µ i φ ij e d j e i λ i d Bi b Q i n i µ i φ i d e i ],

17 A. Mandelbaum et al. / Strong approximation 65 where Bi a, Bb i, and Bc ij for i, j =,..., N are mutually independent tandard Brownian motion. Proof. From 4.2, it follow that N α t x = λ i t e i ] N x i n i t µ i t φ i t e i xj n j t µ j t φji t e i e j i= i= j= j= ] N N = λ i t xj n j t µ j t φji t x i n i tµ i t = λ t x n t µ t Φ t x n t µ t = λ t x n t µ t Φ t I. 4.4 e i The fluid limit now follow from applying theorem 2.2. If fx x and gx = n t for all x R m, then by theorem 3.2 Applying theorem 3.3 to f gx = x n t give Λfx; y = y and Λgx; y =. 4.5 Λf gx; y = yi {x<nt} y I {x=nt} = y y I {x<nt} y I {x=nt} = y I {x<nt} y I {x nt}. 4.6 Uing 4.4 and 4.6, the calable Lipchitz derivative of α t i Λα t x; y = y I {x<nt} y I {x nt} µt Φ t I 4.7 and the diffuion limit follow from applying theorem 2.3. The following reult, which follow immediately from theorem 2.4, provide ordinary differential equation for the mean vector and covariance matrix of Q. Theorem 4.2. The mean vector for the diffuion limit olve the differential equation d dt E Q t ] = E Q t ] I {Q t<n t} E Q t ] I {Q t n t} µ t Φ t I and the covariance matrix for the diffuion limit olve the differential equation d dt Cov Q t, Q t ] { ] } = Cov Q t, Q t I {Q t<n t} Q t I {Q t n t} µ t Φ t I λ t Q t n t µt Φ t I { Q t n t µt Φ t }. Proof. Given 4.4 and 4.6, the proof i imply an application of theorem 2.4.

18 66 A. Mandelbaum et al. / Strong approximation 5. Queue with abandonment and retrial We contruct the multierver queue with abandonment and retrial ee figure 3 by firt defining the following et of parameter: λ t = external arrival rate to the ervice node at time t, β t = abandonment rate from the ervice node at time t, µ t = ervice rate for the ervice node at time t, µ 2 t = ervice rate for the retry pool at time t, ψ t = probability that a cutomer abandoning at time t doe not retry, n t = number of erver in ervice node at time t. We then et V = R 2 and define Qt = Q t, Q 2 t, where Q t = Q A a λ d A c 2 A c t Q n µ d A b A b β 2 Q n ψ d Q 2 µ 2 d β Q n ψ d Figure 3. The abandonment queue with retrial.

19 and Q 2 t = Q 2 A b 2 A. Mandelbaum et al. / Strong approximation 67 β Q n ψ d A c 2 Q2 µ 2 d. Here we have a network with two node where the firt one correpond to the ervice node. The econd node i the retrial pool and ha an infinite number of erver to model retrial delay. Moreover, the act of abandoning the ervice queue due to impatience i modeled a abandonment routing where the cutomer enter the retrial pool with ome probability or leave the network entirely. Service routing intruct cutomer to leave the entire network after ervice completion at the firt node and to enter the ervice queue after ervice completion at the retrial pool. Theorem 5.. Defining Q by uniform acceleration a in 2.8, the fluid limit for the multierver queue with abandonment and retrial i the unique olution to the differential equation d dt Q t = λ t µ 2 t Q 2 t µ t Q t n t βt Q, t n t 5. d dt Q 2 t = β t ψ t Q t n t µ 2 t Q 2 t. 5.2 Moreover, the diffuion limit for the multierver queue with abandonment and retrial i the unique olution to the integral equation Q and t = Q µ {Q n} β {Q >n} Q µ {Q <n} β {Q n} Q µ 2 Q 2 ]d B2 b t Q n β ψ d B2 c B a λ d B b Q n β ψ d B c Q n µ d Q 2 t = Q 2 Bc 2 B b 2 Q 2 µ 2 d Q n β ψ d Q β ψ µ 2 Q 2 ] d, Q 2 µ 2 d

20 68 A. Mandelbaum et al. / Strong approximation where Q t = Q t {Q Q t nt} t {Q t>nt}. 5.3 Proof. Thee reult follow from theorem 2.2 and 2.3. Theorem 5.2. The mean vector for the diffuion limit olve the et of differential equation d dt E Q t] = µ t {Q t nt} β t {Q t>nt} E Q t ] µ t {Q t<nt} β t {Q t nt} E Q t] µ 2 t E Q 2 t] 5.4 and d dt E Q 2 t] = β t ψ t E Q t] {Q t nt} E Q t ] {Q t>nt} µ 2 t E Q 2 t] 5.5 and the covariance matrix for the diffuion limit olve the differential equation d dt Var Q t] = 2 β t {Q t>nt} µ t {Q t nt} Cov Q t, Q t ] 2 β t {Q t nt} µ t {Q t<nt} Cov Q t, Q t] and d dt Var Q 2 t] λ t β t Q t n t µ t Q t n t µ 2 t Q 2 t, 5.6 = 2µ 2 t Var Q 2 t] β t ψ t Q t n t µ 2 t Q 2 t 2β t ψ t Cov Q 2 t, Q t] {Q t nt} Cov Q 2 t, Q t ] {Q t>nt} d dt Cov Q t, Q 2 t] = β t {Q t>nt} µ t {Q t nt} Cov Q 2 t, Q t ] β t {Q t nt} µ t {Q t<nt} Cov Q 2 t, Q t] µ 2 t Var Q 2 t] Cov Q t, Q 2 t] β t ψ t Q t n t µ 2 t Q 2 t. 5.7

21 A. Mandelbaum et al. / Strong approximation 69 Proof. Thee reult follow from theorem 2.4. A we how in the next ection, they are alo a pecial cae of theorem 6.2 where the matrice that are pecified by the given parameter rate are then ] ] βt µ t β t =, µ t =, µ 2 t ] ] 5.8 ψt Ψ t =, and Φ t =. Reulting product of thee matrice are ] βt ψ t β t Ψ t = Cov Q 2 and µ t Φ t = t, Q t Q t ] Var Q 2 t] ] µ t The pecial matrice that are functional of the diffuion proce are Cov Q t, Q t ] Cov Q t, Q = t ] ] Cov Q 2 t, Q t ] 5. and Cov Q t, Q t Q t ] Cov Q t, Q t Q = t ] Cov Q t, Q 2 t] ], 5. where and Q t = Q t I {Q t n t} Q t I {Q t>n t} 5.2 Q t = Q t {Q The vector formula of theorem 6.2 reduce to Q t nt} t {Q t>nt}. 5.3 λ Q βt t n t Ψ t I Q t n t µt Φ t I = λ t ] Q t n ] ] β t ] ψ t t ] µ ] Q t n t Q 2 t] t µ 2 t = λ t β t Q t n t µ t Q t n t µ 2 t Q 2 t e β t ψ t Q t n t µ 2 t Q 2 t e Combining all thee identitie give u our anwer.

22 7 A. Mandelbaum et al. / Strong approximation Finally, we explore the aymptotic regime uggeted in Halfin and Whitt 4] by applying the full power of theorem 2.2 and 2.3 to thi multierver queue with abandonment and retrial. Firt, we modify our rate function o that λ t λ t l t, 5.5 µ i t µ i t m i t for i =, 2, 5.6 β t β t b t, 5.7 ψ t ψ t p t, 5.8 n t n t, 5.9 where like λ t, µ i t, β t and φ t, the function l t, m i t, b t, and p t are locally integrable, but unlike them, not necearily non-negative. By theorem 2.2 and 2.3 we ee that thee additional term of order or / have no effect on the fluid approximation of Q. However, the diffuion approximation i now the unique olution to the integral equation Q t = Q and µ {Q n} β {Q >n} Q µ {Q <n} β {Q n} Q µ 2 Q 2 ]d l m 2 Q 2 m Q n b Q n ] d B2 b Q n β ψ d B2 c B a λ d B b Q n β ψ d B c Q n µ d Q 2 t = Q 2 Bb 2 B c 2 Q n β ψ d Q 2 µ 2 d Q {Q n} Q 2 µ 2 d ]

23 A. Mandelbaum et al. / Strong approximation 7 Q {Q >n} β ψ µ 2 Q 2 ]d b ψ β p Q n m 2 Q 2 ] d. The differential equation for the covariance matrix of Q are unchanged but the equation for the mean vector are now and d dt E Q t] = d dt E Q 2 t] µ t {Q t nt} β t {Q t>nt} µ t {Q t<nt} β t {Q l t m 2 t Q 2 t m t = β t ψ t t nt} Q t n t E Q t ] E Q t] {Q t nt} E Q t ] {Q µ 2 t E Q 2 t] b t ψ t β t p t ] Q t n t E Q t] µ 2 t E Q 2 t] bt Q t n t 5.2 t>nt} m 2 t Q t Jackon network with abandonment The multierver queue with abandonment and retrial i a pecial cae of a more general network that we dicu in thi ection. Here, we contruct a time varying analogue of the Jackon network that ha the added feature of ervice abandonment ee figure 4. Extending Kendall notation, we call it the M t /M t \M t /n t N network for hort. We contruct it by firt defining the following et of parameter: λ i t = external arrival rate to node i at time t, βt i = abandonment rate for node i at time t, = ervice rate for node i at time t, µ i t ψ ij φ ij ψt i t = abandonment routing probability from node i to node j at time t, t = ervice routing probability from node i to node j at time t, = abandonment departure probability from node i at time t, φ i t = ervice departure probability from node i at time t. n i t = number of erver for node i at time t, where we require that ψ i t N j= ψ ij t = and φ i t N j= φ ij t = 6.

24 72 A. Mandelbaum et al. / Strong approximation Figure 4. The Jackon network with abandonment. for all t and i =,..., N. We then et V = R N and define Qt = Q N N i= j= N i= N i= N i= j= N A a i A c i A c ij A b ij Qi n i β i ψ ij d e j e i Qi n i µ i φ ij d e j e i λ i d A b i Qi n i β i ψd i e i Qi n i µ i φ i d e i. 6.2 Theorem 2.2 and 2.3 yield the following limiting reult for thee network. Theorem 6.. Defining Q by uniform acceleration a in 2.8, the fluid limit for the M t /M t \M t /n t N network i the unique olution to the integral equation Q t = Q λ Q n β Ψ I ] d Q n µ Φ I d. 6.3

25 A. Mandelbaum et al. / Strong approximation 73 Moreover, the diffuion limit for the M t /M t \M t /n t N network i the unique olution to the integral equation Q t Proof. = Q N i= Q I {Q n } Q I {Q >n } β Ψ I d Q I {Q <n } Q I {Q n } µ Φ I d N j= B b ij Q i n i β i ψ ij d Bij c Q i n i µ i φ ij e d j e i Bi a λ i d Bi b Q i n i β i ψ i d ] t Bi c Q i n i µ i φ i d e i. 6.4 From 6.2, it follow that α t x = λ t x n t β t Ψ t I x n t µ t Φ t I. 6.5 The fluid limit now follow from applying theorem 2.2. The calable Lipchitz derivative of α t i Λα t xy = y I {x nt} y I {x>nt} βt Ψ t I y I {x<nt} y I {x nt} µt Φ t I. 6.6 The diffuion limit now follow from applying theorem 2.3. Theorem 6.2. The mean vector for the diffuion limit olve the differential equation d dt E Q t ] ] = E Q t I {Q t n t} Q t I {Q t>n t} β t Ψ t I ] E Q t I {Q t<n t} Q t I {Q t n t} µ t Φ t I and the covariance matrix for the diffuion limit olve the differential equation d ] dt Cov Q t, Q t { ] } = Cov Q t, Q t I {Q t n t} Q t I {Q t>n t} β t Ψ t I, { ] } Cov Q t, Q t I {Q t<n t} Q t I {Q t n t} µ t Φ t I

26 74 A. Mandelbaum et al. / Strong approximation λ t Q t n t βt Ψ t I Q t n t µt Φ t I { Q t n t βt Ψ t Q t n t µt Φ t }. 7. Priority queue A multierver queue with preemptive prioritie ee figure 5 i defined uing the following parameter: λ i t = arrival rate for cla i cutomer at time t, µ i t = ervice rate for cla i cutomer at time t, n t = number of erver at time t, c = number of cutomer clae. We then et V = R c and define c t i ] Qt = Q λ i d A b i µ i Q i n t Q j d e i. i= A a i j= 7. Theorem 7.. Defining Q by uniform acceleration a in 2.8, the fluid limit for the priority queueing model i the olution to the integral equation Q t = Q where Θ = {θ ij i, j c} i the c c matrix { for i < j, θ ij = for i j. λ Q n Q Θ µ ] d, 7.2 Moreover, the diffuion limit i the olution to the integral equation Q t = Q Q I {Q <n t Q Θ } µ d Q Θ I{Q >n t Q Θ, n t Q Θ} µ d Q Θ I{Q >n t Q Θ, n t<q Θ} µ d 7.3 Q Q Θ I {Q =n t Q Θ } µ d c ] t λ i d Bi b µ i Q i i n t Q j d e i, i= B a i j=

27 A. Mandelbaum et al. / Strong approximation 75 Figure 5. The preemptive priority queue. where Q tθ = Q tθ I{nt<QtΘ} Q tθ I{nt QtΘ}. 7.4 Proof. From 7., it follow that α t x = λ t x n t xθ µ t. 7.5 The fluid limit now follow from applying theorem 2.2. Applying 3.8 and 3.9, the calable Lipchitz derivative of α t i Λα t x; y = yθ I {x>nt xθ, n t xθ} yθ I {x>nt xθ, n t<xθ} µt y yθ I {nt<xθ} yθ I {nt xθ} I{x=nt xθ } µ t yi {x<nt xθ } µ t. The diffuion limit now follow from applying theorem 2.3. Theorem 7.2. The mean vector for the diffuion limit olve the differential equation d dt E Q t ] = E Q t ] I {Q t<n t xθ } E Q tθ ] I {Q t>n t Q tθ, n t Q tθ} E Q tθ ] I{Q t>n t Q tθ, n t<q tθ} E Q t Q tθ ] I {Q t=n t Q tθ } and the covariance matrix for the diffuion limit olve the differential equation d dt Cov Q t, Q t ] { = Cov Q t, Q t ] } I {Q t<n t xθ } { Cov Q t, Q tθ } ] I{Q t>n t Q tθ, n t Q tθ}

28 76 A. Mandelbaum et al. / Strong approximation { Cov Q t, Q tθ ] } I {Q t>n t Q tθ, n t<q tθ} { Cov Q t, Q t Q tθ ] } I {Q t=n t Q tθ } λ t Q t n t Q tθ µt, where Q tθ i given by Jackon network with tate dependent routing We now conider another generalization of the claical Jackon network where the arrival rate, ervice rate, and routing probabilitie are all function of the tate of the joint queue length vector Qt ee figure 6. We extend Kendall notation and call it the M t Q/M t Q/n t N network. The example conidered in ection 4 and 7 are pecial cae of thi network. We contruct the M t Q/M t Q/n t N network by firt defining the following et of parameter: λ i tqt = external arrival rate to node i at time t, µ i t Qt = ervice rate for node i at time t, φ ij t Qt = ervice routing probability to node i from node j at time t, φ i t Qt = ervice departure probability from node i at time t, n i t = number of erver for node i at time t, where λ i t, µi t, φij t, and φi t are all Lipchitz function with calable Lipchitz derivative and we require that ψ i t x N j= ψ ij t x = and φi t x N for all t, all x V = R N, and i =,..., N. j= φ ij t x = 8. Figure 6. The Jackon network with tate dependent routing.

29 We then define Qt = Q N i= N A. Mandelbaum et al. / Strong approximation 77 N i= A b i N i= j= A a i A c ij λ i Q d e i Qi n i µ i Q φ i Q d e i Qi n i µ i Q φ ij Q d e j e i. 8.2 Theorem 8.. Defining Q by uniform acceleration a in 2.8, the fluid limit for the M t Q/M t Q/n t N network i the unique olution to the integral equation Q t = Q λ Q d Q n µ Q Φ Q I d. 8.3 Moreover, the diffuion limit for the M t Q/M t Q/n t N network i the unique olution to the integral equation Q t = Q where N Λλ t Q ; Q d Q µ Q Φ Q I d Q n Λµt Q ; Q Φ t Q I d Q n µt Q ΛΦ t Q ; Q d N i= j= N i= N i= B c i B a i B c ij Q i n i µ i Q φ ij Q i n i µ i Q φ i Q d e i λ i Q d e i, Q d e j e i Q t = Q t I {Q t<n t} Q t I {Q t n t}. 8.4

30 78 A. Mandelbaum et al. / Strong approximation Proof. From 8.2, it follow that α t x = λ t x x n t µ t x Φ t x I. 8.5 The fluid limit now follow from applying theorem 2.2. The calable Lipchitz derivative of α t i Λα t x; y = Λλ t x; y y I {x<nt} y I {x nt} µt x Φ t x I x n t Λµ t x; y Φ t x I µ t x ΛΦ t x; y. 8.6 The diffuion limit now follow from applying theorem 2.3. Theorem 8.2. The mean vector for the diffuion limit olve the differential equation d dt E Q t ] = E Λλ t Q t; Q t ] E Q t ] I {Q t<n t} µ t Q t Φ t Q t I E Q t ] I {Q t n t} µ t Q t Φ t Q t I Q t n t E Λµt Q t; Q t ] Φ t Q t I Q t n t µt Q t E ΛΦ t Q t; Q t ] 8.7 and the covariance matrix for the diffuion limit olve the differential equation d dt Cov Q t, Q t ] = { Cov Q t, Λλ t Q t; Q t ]} {Cov Q t, Q t ] I {Q t<n t} µ t Q t Φ t Q t I } {Cov Q t, Q t ] I {Q t n t} µ t Q t Φ t Q t I } { Cov Q t, Q t n t Λµt Q t; Q t Φ t Q t I ]} { Cov Q t, Q t n t µt Q t ΛΦ t Q t; Q t ]} λ t Q t n t µt Q t Φ t Q t I { Q t n t µt Q t Φ t Q t } Proof of the trong limit theorem In thi ection, we prove the trong approximation and trong law of large number theorem tated in ection 2. A preparation, we firt how exitence and uniquene for the proce Q = {Qt t }. In defining thi proce, we alo contruct a proce Z = {Zt t } that we ue a a bound on it growth. In a fahion imilar

31 A. Mandelbaum et al. / Strong approximation 79 to reult found in Kurtz 9], the Z proce play the key role in a tochatic analogue to Gronwall inequality. Recall, from ection 2, that for all the reult in thi ection we make the following et of aumption:. The family of Lipchitz rate function {α t, i i I} ha the growth condition αt, i βt γ i, 9. for all i I, where β t i a poitive, locally integrable function and {γ i i I} i an abolutely ummable equence. Similarly, the family of Lipchitz rate function {α t, i i I, > } ha the property that α t, i βt γ i The family of tranition vector {v i i I} ha the property that v i <. 9.3 It hould be noted that the lat condition i not a limiting a it eem. If V i the Banach pace l of abolutely ummable equence and {v i i I} i the et of unit bai vector, merely redefine the norm to give each bai vector a weight where all of the weight are ummable. Lemma 9.. There exit a poitive, increaing proce Z {Zt t } that i the unique olution to the equation Zt X β Z d, 9.4 for all t, where the proce X {Xt t } i defined by a random variable X > that i independent of the collection of Poion procee {A i i I} and Xt X A i γ i t v i, 9.5 i an increaing pure jump proce with no exploion. Moreover, the proce M {Mt t } defined by Mt Zt exp γ i v i β d, 9.6 i a martingale. Proof. Since γi <, it follow that the proce A {At t }, where At = A i γ i t 9.7

32 8 A. Mandelbaum et al. / Strong approximation i Poion with mean rate γi. Given that v i < alo, we then have for all t > E Xt ] = E X ] γ i v i t < E X ] γ i v i t <. 9.8 Hence Xt < a.. for all t and it jump time are given by the Poion proce A. Now let bt β d. If β i a trictly poitive function then b, the invere function for b, i well defined. For all t, define the random proce {τt t } uch that τ t = b d. 9.9 X Thi i well defined ince Xt X > for all t. Hence, we can define the proce {Zt t } to be Zt τ t β t. 9. Since τt β Z d, we have uniquene. Now conider the proce M {M t t }, where M t Zt γ i v i β Z d. 9. Thi proce i a martingale, ince {Xt γi v i t t } i one, and by 9.9 we ee that τt = β Z d i a topping time with repect to the filtration generated by the proce X for all t. Finally, M i a martingale ince M i one of bounded variation and Mt = Zt exp γ i v i β d = Z exp γ i v i β r dr dm, 9.2 which complete the proof. Theorem 9.2. Given the rate function {α t, i t, i I} and the initial tate vector Q i independent of the collection of Poion procee {A i i I}, we can contruct a unique tochatic proce Q {Qt t } uch that Qt = Q A i α Q, i d v i. 9.3

33 A. Mandelbaum et al. / Strong approximation 8 Moreover, we have for all t, up Q Zt, 9.4 t where the proce Z {Zt t } i uniquely defined by 9.4 and 9.5 with X = Q. Proof. Define the following equence Q n {Q n t t }, where Q t Q for all t and for all poitive integer n we have Q n t Q Tn A i α Qn, i d v i, 9.5 where { T n inf t A i We are done once we prove the following two tatement:. Q n t = Q n t for all t < T n. 2. lim n T n = a.. α Qn, i } d = n. 9.6 We can then contruct the deired proce Q {Qt t } by defining for all n, Qt = Q n t for all t < T n. 9.7 Uniquene follow by uing induction on n. Uing 9.5 how that the uniquene of Q n implie the uniquene of Q n. The firt tatement i proved by uing induction on n. The reult hold for n =, ince t < T implie that Q t = Q t = Q, ince t < T mean that A i α Q, i d =, 9.8 and for all i I, we have A i α Q, i d =. 9.9 If we aume that Q n t = Q n t for all t < T n, then it follow that α Qn, i d = for all i I and t T n. We then mut have α Qn, i d 9.2 Q n t = Q n t for all t < T n. 9.2

34 82 A. Mandelbaum et al. / Strong approximation Now conider the cae of T n t < T n. By definition of the Q n we have Q n t = Q n T n 9.22 and Q n t = Q A i α Qn, i d v i However, by the definition of T n and T n, we have n A i α Qn, i d < n Thi follow from the fact that the A i are increaing procee. Combining thi with the fact that the procee are alo integer valued not only how that the um in 9.24 equal n, but that for all i I, A i α Qn, i Tn d = A i α Qn, i d = A i Tn α Qn, i d, where the lat equality follow from 9.2. Finally, thi give u which mean that Q n t = Q n t = Q n T n for all T n t < T n, 9.25 Q n t = Q n t for all t < T n, 9.26 completing the induction argument. To prove the econd tatement, we oberve that Qn t Q A i α, i Qn d v i 9.27 Q A i γ i It follow by induction that for all n and t we have β Q n d v i Qn t Zt, 9.29 where Zt i the proce defined in 9.4, with X = Q. Conequently, A i α Qn, i d A i γ i β Z d. 9.3

35 A. Mandelbaum et al. / Strong approximation 83 If we et γ γi, then A iγ i t i a Poion proce with rate γ. Let T γ n be the time of the n-th jump for thi Poion proce. It now follow from 9.6 and 9.3 that for all n, where T γ n = S γ n T n, 9.3 S γ n β Z d Since lim n Tn γ = a.., then lim n Sn γ = a.., and o we have lim n T n = a.. Finally, 9.4 follow from the fact that Qt Zt for all t and Z i a non-decreaing proce. Now conider the family of uniformly accelerated procee {Q > } a defined in 2.9. We will alway aume that every element of {Q > } i a random vector in V that i independent of the collection of Poion procee {A i i I}. Lemma 9.3. If {Q > } i a family of random vector independent of the Poion procee {A i i I}, then which implie Proof. and lim Q < a.. implie lim Z < a.., 9.33 lim up t Q Let {Z t t } be the unique proce uch that Z t = X β Z d < a X t Q A i γ i t v i A imple modification of the proof for theorem 9.2 give u up Q Z t t A imilar modification of the proof for lemma 9. how that { Z t exp } γ i v i β d t 9.38

36 84 A. Mandelbaum et al. / Strong approximation i a martingale, and o for all t. E Z t ] = Q exp γ β d Uing Chebyhev inequality, we ee that for all t >, the et {Z t > } i a tight family of random variable and o lim Z t < a which combined with 9.37, complete the proof. Our fundamental reult, tated in ection 2, are proved within the framework of trong approximation. The framework i baed on a pathwie approximation of the Poion proce, articulated in the following lemma. Lemma 9.4 Kurtz 9, lemma 3.]. A tandard rate Poion proce {At t } can be realized on the ame probability pace a a tandard Brownian motion {Bt t } in uch a way that the poitive random variable X, given by At t Bt X up <, 9.4 t log2 t ha a finite moment generating function in a neighborhood of the origin. In particular it ha a finite mean. Uing lemma 9.4, we aociate with our tochatic primitive {A i i I}, namely the family of mutually independent tandard Poion procee, another family {B i i I} of mutually independent tandard Brownian motion, uch that A i t t B i t X i up < a t log2 t Moreover, the random variable X i can be taken i.i.d. with a finite mean. Note that both familie, a well a the X i, mut be realized on a common probability pace. In the equel, we write ω Ω for elementary outcome in thi common probability pace. Now we give the proof for our trong approximation theorem. Proof of theorem 2.. Uing lemma 9.4, we have up Q Q α r t Q r dr B i α r Q r, i dr v i X i log 2 α Q, i d v i 9.43