aa r X i v : . [ m a t h . P R ] A ug Stochastic Systems arXiv: arXiv:0000.0000
TRANSITORY QUEUEING NETWORKS
By Harsha Honnappa , Rahul Jain
Purdue University ∗ and University of Southern California † Queueing networks are notoriously difficult to analyze sans bothMarkovian and stationarity assumptions. Much of the theoretical con-tribution towards performance analysis of time-inhomogeneous singleclass queueing networks has focused on Markovian networks, with therecent exception of work in Liu and Whitt (2011) and Mandelbaum and Ramanan(2010). In this paper, we introduce transitory queueing networks asa model of inhomogeneous queueing networks, where a large, butfinite, number of jobs arrive at queues in the network over a fixedtime horizon. The queues offer FIFO service, and we assume that theservice rate can be time-varying. The non-Markovian dynamics ofthis model complicate the analysis of network performance metrics,necessitating approximations. In this paper we develop fluid and dif-fusion approximations to the number-in-system performance metricby scaling up the number of external arrivals to each queue, followingHonnappa et al. (2014). We also discuss the implications for bottle-neck detection in tandem queueing networks.
1. Introduction.
Single class queueing networks (henceforth ‘queueingnetworks’) have been studied extensively in the literature, with much effortfocused on understanding the steady-state joint distribution of the state ofthe network (typically defined as the number of jobs in each queue). In thispaper, we consider a variation of the generalized Jackson queueing networkmodel ((Chen and Yao 2001, Chapter 7)) where a finite, but large, numberof jobs arrive at some of the nodes in the network from an extraneous source.We characterize fluid and diffusion approximations to the queue length stateprocess, as the population size scales to infinity.To motivate this model, consider a manufacturing facility that producesengines. Each part of the engine is produced and assembled in a separatemachine that requires some human supervision, with final assembly occuringat the end of the job-shop. Typically, there is a finite, but large number ofjobs that need to be completed in a shift spanning a few hours. Furthermore,jobs cannot be carried over to the next shift. It is typically the case thatthe shift horizon is not long enough for the system to reach a steady state.Furthermore, the bottleneck machine can change over the shift horizon, as
Keywords and phrases: transient analysis, fluid approximations, diffusion approxima-tions, bottlenecks HONNAPPA AND JAIN a consequence of the variation in the arrival of jobs to the job-shop, or dueto stochastic variation in the assembly time in each machine. This is alsoknown as the ‘shifting bottleneck’ phenomenon in production engineering;see Roser et al. (2002), Lawrence and Buss (1994, 1995). In the latter twopapers, Jackson networks are used to analyze the shifting bottleneck net-work. More generally, there is less reason to believe that the arrival andservice processes are stationary and ergodic, and that they are Poisson pro-cesses. We introduce transitory queueing networks as a broadly applicablemodel of such systems.Transitory queueing networks consist of a number of infinite buffer, FIFO,single server queues (a.k.a. ‘nodes’) interconnected by customer routes. Weassume that the routing matrix satisfies a so-called
Harrison-Reiman (H-R) condition that the matrix has a spectral radius of less than one. Thisimplies that, on completion of service at a particular node, a customer isrouted to another node or exits the network altogether. Jobs enter a givennode at random time epochs modeled as the ordered statistics of indepen-dent and identically distributed (i.i.d.) random variables. The arrival epochof the jobs at different nodes can be correlated (for instance, in the manu-facturing context, jobs can be submitted to multiple nodes simultaneously).We assume that the service processes at different nodes are independentwith time-inhomogeneous service rates, and modeled as a time change of aunit rate renewal counting process, generalizing the construction of a time-inhomogeneous Poisson process. The transient analysis of generalized Jack-son networks is non-trivial, as noted before. The conventional heavy-trafficdiffusion approximation that relied on long-run average rates has been usedto approximate the evolution of the state process. However, these rates donot exist in transitory queueing networks. Here, we develop a ‘populationacceleration’ approximation, by increasing the number of jobs arriving atthe network in the interval of interest to infinity, and suitably scaling (or‘accelerating’) the service process in each queue by the population size.Our results complement the existing literature on the analysis of single-class queueing networks by establishing the following results:(i) we develop a large population approximation framework for studyingsingle class queueing networks in a transitory setting, complementingand extending Markovian network analyses to non-Markovian queue-ing networks,(ii) Our diffusion approximations use the recently developed directionalderivative oblique reflection map in Mandelbaum and Ramanan (2010)to establish a diffusion scale approximation; this is substantially differ-ent from the conventional heavy-traffic approximations used to study
RANSITORY NETWORKS single-class queueing networks, and(iii) we study the evolution of the bottleneck process over the time hori-zon, identifying the bottleneck station as time progresses. This anal-ysis extends the standard bottleneck analyses, where bottlenecks areidentified in terms of the long-term average arrival and service rates.1.1. Analytical Results.
We consider a sequence of queueing networkswherein n jobs arrive at each node that receives external traffic in the n thnetwork. We first establish a functional strong law of large numbers (FSLLN)to the arrival process, as the population size scales to infinity, by generaliz-ing the Glivenko-Cantelli Theorem to multiple dimensions. We also assumethe service processes satisfy a FSLLN and functional central limit theorem(FCLT) in the population acceleration scale. The queue length fluid limitis shown to be equal to the oblique reflection of the difference of the fluidarrival and service processes (or the ‘fluid netput’ process). On the otherhand, for the FCLT we introduce the notion of a multidimensional Brow-nian bridge process, as a generalization of a the one-dimensional Brownianbridge process, and show that the diffusion scaled arrival process convergesto a multidimensional Brownian bridge in the large population limit. Thediffusion limit turns out to be complicated, and it is shown to be a re-flection of a multidimensional Gaussian bridge process - however, the re-flection is through a directional derivative of the oblique reflection of thenetput in the direction of the diffusion limit of the fluid netput process.This is a highly non-standard result. Indeed, it is only in the recent pastthat Mandelbaum and Ramanan (2010) have investigated the existence of adirectional derivative to the oblique reflection map.Leveraging the results of Mandelbaum and Ramanan (2010), we can onlyestablish a pointwise diffusion limit for an arbitrary transitory queueing net-work. This is due to the fact that the directional derivative limit can havesample paths with discontinuities that are both right- and left-discontinuous.Thus, establishing convergence in a sample path space under a suitably weaktopology such as the (‘strong’ or ‘weak’) M , for instance, is not straight-forward. Instead, we focus on the case of tandem queueing networks, withuniform and unimodal arrival time distribution functions. In this case, weshow that the discontinuities in the limit are either right or left continuous,and hence we can establish M convergence. Using these approximations, wenext address the question of bottleneck prediction. We generalize the stan-dard definition of a bottleneck in a single class network, defined as the queueswhose fluid arrival rate exceeds the fluid service capacity to the transitorysetting. HONNAPPA AND JAIN
Implications for Bottleneck Detection.
Bottleneck detection and pre-diction is likely the most important question that a system operator faces.Heavy traffic theory has been immensely successful at characterizing steadystate bottlenecks in general queueing networks, under minimal network dataassumptions. However, there are many circumstances, ranging from man-ufacturing, to healthcare, transportation and computing, where transientbottleneck detection and analysis is critical. In the purely transient, or ‘tran-sitory’ setting, it is common for the bottleneck node to change over the shifthorizon. As a consequence the plant manager moves workers around try-ing to ease bottlenecks, increasing costs and increasing the likelihood of joboverages. Another example is in healthcare where patient diagnosis relieson a number of tests that must be done with different machines. Further-more, in many time critical settings, the horizon within which tests mustbe conducted is fixed. An important question in these situations is whethertransitory bottlenecks can be accurately predicted, given network data.Note that the standard definition of a bottleneck is a ‘capacity level’ one,defined in terms of long-term averages. This definition, of course, is not sat-isfactory in the transitory setting where steady states might not be reached.Second, standard heavy-traffic analysis completely ignores non-stationaritiesin the network data, which is generally prevalent in the examples presentedabove, making it an inappropriate analytical tool to use. In particular, it ispossible that the number of bottleneck nodes in the network changes overtime, thus necessitating a more nuanced definition and analysis of bottle-necks. We introduce a natural definition of a ‘transitory bottleneck’ in termsof the diffusion approximations developed in this paper, and then use theseapproximations to analyze the evolution of the number of bottlenecks in tan-dem transitory networks. This is particularly relevant to job-shop analysisof production systems.1.3.
Related Literature.
There has been significant interest in the anal-ysis of single class queueing networks. Under the assumption of Poissonarrival and service processes, Jackson (1957) showed that the steady statedistribution of the state of the network (the number of jobs waiting in eachnode) is equal to the product of the distribution of the state of each node inthe network. This desirable property implies that, in steady state, the net-work exhibits a nice independence property. This property does not extendto networks with general arrival and service processes; these are also knownas generalized Jackson networks.Reiman first established the heavy-traffic diffusion approximation to opengeneralized Jackson networks in Reiman (1984). In particular, the diffu-
RANSITORY NETWORKS sion approximation is shown to be a multi-dimensional reflected Brownianmotion in the non-negative orthant, reflected through the oblique reflec-tion mapping. Such reflection maps have come to be called as Harrison-Reiman maps following the seminal work in Harrison and Reiman (1981).Chen and Mandelbaum (1991b,a) characterize a homogeneous fluid network,and establish fluid and diffusion approximations. The analysis of non-stationaryand time inhomogeneous queueing systems is non-trivial in general. For sin-gle server queues, see Keller (1982), Massey (1985), Mandelbaum and Massey(1995) among others. In Honnappa et al. (2014, 2017) we develop fluid anddiffusion approximations to the ∆ ( i ) /GI/ ( I ) /GI/ G t /M t /s t + GI t ) m /M t network with m nodes, time-varying arrivals, staffing and abandonments, and inhomoge-neous Poisson service and routing, and characterize the performance of thenetwork as a direct extension of the single-server queue case. However, theirscaling regime and the limit processes are completely different from the ourresults.The rest of the paper is organized as follows. We collect relevant nota-tion in Section 2. We start our analysis with a description of the transitorygeneralized Jackson network model in Section 3, and we develop fluid anddiffusion approximations to the network primitives. In Section 4, we developfunctional strong law of large numbers approximations to the queueing equa-tions, and identify the fluid model corresponding to the transitory network.We identify the diffusion network model in Section 5, and establish a weakconvergence result for a tandem network with unimodal arrival time distri-bution. We end with conclusions and future research directions in Section7
2. Notation.
Following standard notation, C K represents the space ofcontinuous R K -valued functions, and D K the space of functions that are HONNAPPA AND JAIN right continuous with left limits (RCLL) and are R K -valued. The space D Kl,r consists of R K -valued functions that are either right- or left-continuous ateach point in time, while D K lim is the space of R K -valued functions thathave right and left limits at all points in time. D Kusc is the space of RCLLfunctions that are upper semi-continuous as well. We represent the space of L × L matrices by M L . The space and mode of convergence of a sequence ofstochastic elements is represented by ( X, Y ), where X is the space in whichthe stochastic elements take values and Y the mode of convergence. In thispaper our results will be proved under the uniform mode of convergenceand occasionally in the “strong” M ( SM ) topology (see (Whitt 2001a,Chapter 11)). Weak convergence of measures will be represented by ⇒ .Finally, diag( x , . . . , x K ) represents a K × K diagonal matrix with entries x , . . . , x K .
3. Transitory Queueing Network.
Let (Ω , F , P ) be an appropriateprobability space on which we define the requisite random elements. Let K := { , . . . , K } be the set of nodes in the network, and E ⊂ K the set ofnodes where exogenous traffic enters the network. Each node in E receives n jobs that arrive exogenously to the node. We assume a very general modelof the traffic: let T m := ( T e ,m , . . . , T e J ,m ) , m ≤ n, represent the tuple ofarrival epoch random variables where T e j ,m is the arrival epoch of the m thjob to node e j ∈ E (here J := |E| ). By assumption T e j ,m ∈ [0 , T ] for all e j ∈ E and 1 ≤ m ≤ n . We also assume that { T m ; m = 1 , . . . , n } forms a sequenceof independent random vectors. Let F e j be the distribution function of thearrival epochs to node e j ∈ E ; that is E [ { T ej,m ≤ t } ] = F e j ( t ) with support[0 , T ]. Users sample a time epoch to arrive at the node and enter the queue inorder of the sampled arrival epochs; thus the arrival process to each node isa function of the ordered statistics of the arrival epoch random variables. Inmany situations, it is plausible that there is correlation between the arrivalprocesses to the nodes in E . To model such phenomena, we assume thatthe joint distribution of the arrival epochs are fully specified. To be precise,we assume that P ( T e ,m ≤ t, . . . , T e J ,m ≤ t ) for all m ∈ { , . . . , n } is welldefined. Let a m ( t ) := ( { T e ,m ≤ t } , . . . , { T eJ ,m ≤ t } ) ∈ D J [0 , ∞ ) and(1) A n,e j := n X m =1 { T ej,m ≤ t } for 1 ≤ j ≤ J, then A n ( t ) := P nm =1 a m ( t ) = ( A n,e ( t ) , . . . , A n,e J ( t )) ∈ D J [0 , ∞ ) is thevector of cumulative arrival processes to the nodes in E . Then, E [ A n ( t )] = n F ( t ) := n ( F e ( t ) , . . . , F e J ( t )) RANSITORY NETWORKS and E [ A n ( t ) A n ( t ) T ] = [ nF e i ,e j ( t ) + n ( n − F e i ( t ) F e j ( t )], where F e i ,e j ( t ) := P ( T e i ,m ≤ t, T e j ,m ≤ t ). This ‘multivariate empirical process’ representationfor the traffic affords a very natural model of correlated traffic in networks,and stands in contrast with generalized Jackson networks where externaltraffic to each node in E is independent.Recall from Donsker’s Theorem (for empirical sums) that n − / ( A n,e i − nF e i ) ⇒ W e i ◦ F e i , where W e i is a standard Brownian bridge process. TheBrownian bridge process W is also well defined as a ‘tied-down’ Brownianmotion process equal in distribution to ( W ( t ) − tW (1) , t ∈ [0 , t ∈ [0 , W is a standard Brownian motion process. Definition 1
Let W = ( W , . . . , W J ) be a J -dimensional standard Brown-ian motion process with identity covariance matrix. If R is a J × J positive-definite matrix with lower-triangular Cholesky factor L , then W R = L W is a J -dimensional Brownian motion with covariance matrix R . By directlyextending the definition of a one-dimensional Brownian bridge process, (cid:0) W ( t ) = W R ( t ) − t W R (1) , t ∈ [0 , (cid:1) is a J -dimensional Brownian bridge process with covariance matrix R . It is straightforward to see that E [ W ( t )] = 0 for all t ∈ [0 ,
1] and E [ W ( t ) W ( s ) T ] = t (1 − s ) R ≡ [ t (1 − s ) r i,j ] when t ≤ s . More gener-ally, we define a Brownian bridge process with time-dependent covariancematrix as follows. Recall that a stochastic process is defined as a Gaussianprocess provided its finite dimensional distributions are jointly Gaussian. Definition 2
Let R : [0 , T ] × [0 , T ] → M J be a right continuous, symmet-ric function such that R ( t, s ) is positive-definite for each t, s ∈ [0 , T ] , withthe restriction that R (0 ,
0) = 0 and r i,j ( T, T ) = r i ( T ) r j ( T ) , where r i,j is the ( i, j ) th entry of R and r i is the i th diagonal element of R . The Gaussian pro-cess W is a J -dimensional Brownian bridge process if it has mean zero andcovariance function E [ W ( t ) W ( s ) T ] = R ( t, s ) − diag ( R ( t, t )) diag ( R ( s, s )) ≡ [ r i,j ( t, s ) − r i ( t ) r j ( s )] , where diag : M J → R J is a function that maps a ma-trix to a vector of the diagonal elements. Note that this definition is a natural generalization of the bridge pro-cess in Definition 1. The terminal condition r i,j ( T, T ) = r i ( T ) r j ( T ) ensuresthat W ( T ) = 0 a.s. While we do not argue the existence of this objectrigorously, the right continuity of the covariance matrix and the assumedGaussianity of the marginals imply that it can be inferred from the Daniell-Kolmogorov theorem; see Rogers and Williams (2000). Now, observe thatthe covariance function of the pre-limit traffic process satisfies R n ( t, s ) = HONNAPPA AND JAIN E [( A n ( t ) − E [ A n ( t )])( A n ( s ) − E [ A n ( s )]) T ] = n [ F e i ,e j ( t, s ) − F e i ( t ) F e j ( s )] ∈C J × J , where F e i ,e j ( t, s ) := P ( T e i ,m ≤ t, T e j ,m ≤ s ) and t ≤ s . Now, followingDefinition 2 let W ◦ F represent a multidimensional Brownian bridge withcovariance function R ( t, s ) = [ F e i ,e j ( t, s ) − F e i ( t ) F e j ( s )] , (2)when t ≤ s . Note that we are “overloading” the composition operator ◦ inthis notation, but the usage should be clear from the context. Theorem 1 be-low establishes multivariate generalizations of the classical Glivenko-Cantelliand Donsker’s theorems. Theorem 1
We have,(i) n − A n → F in ( C J , U ) a.s. as n → ∞ , and(ii) ˆ A n := n − / ( P nm =1 a m − n F ) ⇒ W ◦ F in ( C J , U ) as n → ∞ , where W ◦ F ∈ C J [0 , ∞ ) is the J -dimensional Brownian bridge process with co-variance matrix defined in (2) . The proof of the theorem is available in the appendix. We refer to the k thcomponent process by W k ◦ F k .Next, we consider a sequence of service processes indexed by the popu-lation size n ≥ S n,k : Ω × [0 , ∞ ) → N for k ∈ K . We assume that foreach k ∈ K the function µ n,k : [0 , ∞ ) → [0 , ∞ ) is Lebesgue-integrable andthat M n,k ( t ) := R t µ n,k ( s ) ds satisfies M n,k → M k in ( C , U ) as n → ∞ , where M k : [0 , ∞ ) → [0 , ∞ ) is non-decreasing and continuous. We also assume that(3) M n := ( M ,n , . . . , M n,k ) → M := ( M , . . . , M K ) in ( C K , U ) as n → ∞ , where K = |K| . Let S n := ( S ,n , . . . , S n,k ) represent the ‘network’ serviceprocess, where the component service processes are independent of eachother. We assume that S n satisfies the following fluid and diffusion limits. Assumption 1
The service processes { S n , n ≥ } satisfies(i) (cid:2) n − S n − M n (cid:3) → in ( C K , U ) a.s. as n → ∞ , and(ii) ˆ S n ( t ) := n − / ( S n − n M ) ⇒ W ◦ M in ( C K , U ) as n → ∞ , where W := ( W , . . . , W K ) is a K − dimensional Brownian motion process withidentity covariance matrix. Note that the covariance function of the process W ◦ M is the diago-nal matrix with entries ( M , . . . , M K ). This service process is analogous tothe time-dependent ‘general’ traffic process G t proposed in Liu and Whitt(2014). It’s possible to anticipate a proof of this result when the centeredservice process S n − M n is a martingale. This would be the case when RANSITORY NETWORKS S n is a K -dimensional stochastic process where the marginal processes arenonhomogeneous Poisson processes and M n,k = E [ S n,k ]. Here, we leave thedevelopment of a general result to a separate paper and, instead, assumethat such a sequence of service processes exist.On completion of service at node i , a job will join node j with prob-ability p i,j ≥ , i, j ∈ { , . . . , K } , or exit the network with probability1 − P j p i,j . Thus, the routing matrix P := [ p i,j ] is sub-stochastic. Notethat, we also allow feedback of jobs to the same node; i.e., p i,i ≥
0. Let φ il : Ω → { , . . . , K } , ∀ i ∈ K and ∀ l ∈ N , be a measurable function suchthat φ il = j implies that the l th job at node i will be routed to node j and E [ { φ il = j } ] = p i,j . Define the random vector R l ( m ) := P mi =1 e φ li , where e i is the i th K -dimensional unit vector and the k th component of R l ( m ),denoted R kl ( m ), represents the number of departures from node l to node k out of m departures from that node. Then, R ( m ) := ( R ( m ) , . . . , R K ( m ))is a K × K matrix whose columns are the routing vectors from the nodes inthe network. Assumption 2
The stationary routing process { R ( m ) , m ≥ } satisfiesthe following functional limits:(i) n − R ( ne ) → P e in ( C K × K , U ) a.s. as n → ∞ , where e : [0 , ∞ ) → [0 , ∞ ) is the identity function, and(ii) ˆ R n := n − / ( R ( ne ) − n P e ) ⇒ ˆ R , in ( C K × K , U ) as n → ∞ , where ˆ R = [ W i,j ] and W i,j are independent Brownian motion processes with meanzero and diffusion coefficient p i,j (1 − p i,j ) . As a direct consequence of Assumption 2 we have the following corollary,which will prove useful in our analysis of the network state process in thenext section.
Corollary 1
The routing process R also satisfies the following fCLT: (4) ˆ R Tn ⇒ ˆ R T = ˜ W in ( C K , U ) as n → ∞ , where = (1 , . . . , is a K -dimensional vector of one’s and ˜ W = K X k =1 W ,k , . . . , K X k =1 W K,k ! is a K -dimensional Brownian motion with mean zero and diagonal covari-ance matrix with entries K X k =1 p ,k (1 − p ,k ) , . . . , K X k =1 p K,k (1 − p K,k ) ! . HONNAPPA AND JAIN
Finally, we claim the following joint convergence result that summarizesand generalizes the convergence results in the afore-mentioned theorems.
Proposition 1
Assume that for each n ≥ , A n , S n and R ( n ) are mutuallyindependent. Then,(i) n − ( A n , S n , R ( ne )) → ( F , M , P ′ e ) in ( C J × C K × C K × K , U ) a.s. as n →∞ , and(ii) (cid:16) ˆ A n , ˆ S n , ˆ R n (cid:17) ⇒ (cid:16) W ◦ F , W ◦ M , ˆ R (cid:17) in ( C J × C K × C K × K , U ) as n → ∞ . The joint convergence follows from the assumed independence of the pre-limit random variables, and is straightforward to establish under the uniformconvergence criterion.
4. Functional Strong Law of Large Numbers.
Let Q n,k ( t ) = E n,k ( t ) − D n,k ( t ) be the queue length sample path at node k , where E n,k ( t ) is the totalnumber of jobs arriving at node k in the interval [0 , t ] and D n,k is the cumu-lative departure process. We assume that the server is non-idling implyingthat D n,k ( t ) = S n,k ( B n,k ( t )), where B n,k ( t ) := R t { Q n,k ( s ) > } ds is the totalbusy time of the server. Therefore, the queue length process is Q n,k ( t ) := A n,k ( t ) + K X l =1 R kl ( S n,l ( B n,l ( t ))) − S n,k ( B n,k ( t )) . (5)The K -dimensional multivariate stochastic process Q n := ( Q n, , . . . , Q n,K ) ∈D K represents the network state. Our first result establishes a fluid limit ap-proximation to a rescaled version of Q n by establishing a functional stronglaw of large number result as the exogenous arrival population size n scales toinfinity. Consider the queue length in the k th node, Q k . Centering each termon the right hand side by the corresponding fluid limits (and subtracting RANSITORY NETWORKS those terms), and introducing the term R t µ n,k ( s ) ds , we obtain n − Q n,k ( t )= (cid:18) n A n,k ( t ) − F k ( t ) (cid:19) + n K X l =1 (cid:20) R kl ( S n,l ( B n,l ( t ))) − p l,k S n,l ( B n,l ( t )) (cid:21)! − S n,k ( B n,k ( t )) n − Z B n,k ( t )0 µ n,k ( s ) ds ! + F k ( t ) − Z B n,k ( t )0 µ n,k ( s ) ds + 1 n K X l =1 p l,k S n,l ( B n,l ( t )) ! = (cid:18) n A n,k ( t ) − F k ( t ) (cid:19) + n K X l =1 (cid:20) R kl ( S n,l ( B n,l ( t ))) − p l,k S n,l ( B n,l ( t )) (cid:21)! − S n,k ( B n,k ( t )) n − Z B n,k ( t )0 µ n,k ( s ) ds ! + (cid:18) F k ( t ) − Z t µ n,k ( s ) ds (cid:19) + (1 − p k,k ) Z tB n,k ( t ) µ n,k ( s ) ds + n K X l =1 p l,k (cid:20) S n,l ( B n,l ( t )) − n Z B n,l ( t )0 µ n,l ( s ) ds (cid:21)! + K X l =1 p l,k (cid:18)Z t µ n,l ( s ) ds (cid:19) − X l = k p l,k Z tB n,l ( t ) µ n,l ( s ) ds. (6)Note that we used the fact that B n,k ( t ) ≤ t so that Z t µ n,k ( s ) ds = Z B n,k ( t )0 µ n,k ( s ) ds + Z tB n,k ( t ) µ n,k ( s ) ds. Recall too that I n,k ( t ) := t − B n,k ( t ) = R t { Q n,k ( s )=0 } ds is the idle timeprocess, which measures the amount of time in [0 , t ] that the node is notserving jobs (i.e., the queue is empty). Now, n − Q n,k can be decomposed as HONNAPPA AND JAIN the sum of two processes, ¯ X n,k and ¯ Y n,k , where¯ X n,k ( t ) = (cid:18) n A n,k ( t ) − F k ( t ) (cid:19) + n K X l =1 (cid:20) R kl ( S n,l ( B n,l ( t ))) − p l,k S n,l ( B n,l ( t )) (cid:21)! − S n,k ( B n,k ( t )) n − Z B n,k ( t )0 µ n,k ( s ) ds ! + (cid:18) F k ( t ) − (cid:18)Z t µ n,k ( s ) ds (cid:19)(cid:19) + K X l =1 p l,k (cid:18)Z t µ n,l ( s ) ds (cid:19) + n K X l =1 p l,k (cid:20) S n,l ( B n,l ( t )) − n Z B n,l ( t )0 µ n,l ( s ) ds (cid:21)! , and(7) ¯ Y n,k ( t ) = (1 − p k,k ) Z tB n,k ( t ) µ n,k ( s ) ds − X l = k p l,k Z tB n,l ( t ) µ n,l ( s ) ds. (8)While this expression appears formidable, the analysis is simplified signifi-cantly by the fact that ¯ Q n := n − ( Q n, , . . . , Q n,k ) and ¯ Y n := ( ¯ Y n, , . . . , ¯ Y n,k )are solutions to the K -dimensional Skorokhod/oblique reflection problem.First, we recall the definition of the oblique reflection problem. Theorem 2 [Oblique Reflection Problem] Let V be a K × K M -matrix ,also known as the reflection matrix. Then, for every x ∈ D K := { x ∈D K : x (0) ≥ } , there exists a unique tuple of functions ( y, z ) in D K × D K satisfying z = x + V y ≥ ,dy ≥ and y (0) = 0 , (9) z j dy j = 0 , j = 1 , . . . , K. The process ( z, y ) := (Φ( x ) , Ψ( x )) is the so-called oblique reflection map,where Φ( x ) = x + V Ψ( x ) . Note that, in general, if G is a nonnegative M -matrix then so is V = I − G T (Lemma 7.1 of Chen and Yao (2001)). The following lemma shows that thequeue length satisfies the Oblique Reflection Mapping. Lemma 1
Consider ¯X n ( t ) = ( ¯ X n, ( t ) , . . . , ¯ X n,k ( t )) ∈ D K , where ¯ X n,k ( t ) k ∈{ , . . . , K } is defined in (7) , ¯Q n ∈ D K and ¯Y n ∈ D K . Then, ( ¯Q n , ¯Y n ) =(Φ( ¯X n ) , Ψ( ¯X n )) , with reflection matrix V = I − P T . An M -matrix is a square matrix with spectral radius less than one.RANSITORY NETWORKS Next, we establish a functional strong law of large numbers result for(7), which will subsequently be used in Theorem 3 for the queue lengthapproximation.
Lemma 2
The fluid-scaled netput process ¯ X n converges to a deterministiclimit as n → ∞ : ¯X n ( t ) → ¯ X ( t ) := ( ¯ X ( t ) , . . . , ¯ X K ( t )) in ( C K , U ) a.s. , where, ¯ X k ( t ) = F k ( t ) − Z t µ k ( s ) ds + K X l =1 p l,k Z t µ l ( s ) ds. (10)We can now establish the functional strong law of large numbers limit forthe queue length process. The proof essentially follows from the continuityof the oblique reflection map (Φ( · ) , Ψ( · )). Theorem 3
Let ¯X n ( t ) and ¯X ( t ) be as defined in (7) and (10) respectively.Then, ( ¯Q n ( t ) , ¯ Y n ( t )) satisfy Theorem 2 and, as n → ∞ , ( ¯ Q n ( t ) , ¯ Y n ( t )) → (Φ( ¯ X ( t )) , Ψ( ¯ X ( t ))) in ( C K × C K ) a.s. ∀ t ∈ [0 , ∞ ) . Proof:
It follows by Lemma 1 that ( ¯ Q n ( t ) , ¯ Y n ( t )) satisfy the oblique reflec-tion mapping theorem. Therefore, ( ¯ Q n ( t ) , ¯ Y n ( t )) ≡ (Φ( ¯ X n ( t )) , Ψ( ¯ X n ( t ))) . Now, the reflection regulator map, Ψ( · ), is Lipschitz continuous under theuniform metric (Theorem 7.2, Chen and Yao (2001)). By the ContinuousMapping Theorem and Lemma 2 it follows that,(Φ( ¯ X n ( t )) , Ψ( ¯ X n ( t ))) → (Φ( ¯ X ( t )) , Ψ( ¯ X ( t ))) u.o.c. a.s. as n → ∞ , ∀ t ∈ [0 , ∞ ) . Proof: [Proof of Lemma 1] First, by definition we have ¯Q n = ¯X n + ( I − P T ) ¯Y n . Note that P is a non-negative (sub-stochastic) matrix with spectralradius less than unity and, therefore, an M -matrix, implying that I − P T isalso an M -matrix. Once again by definition Q n,k and Y n,k satisfy the con-ditions in (9) for all k ∈ K . Thus, Theorem 2 is satisfied and the lemma isproved. Proof: [Proof of Lemma 2] The result follows by an application of part (i)of Proposition 1 to (7). Noting that B n,k ( t ) ≤ t , the random time change HONNAPPA AND JAIN theorem (Theorem 5.5, Chen and Yao (2001)) and Assumption 1(i) togetherimply that,1 n S n,k ( B n,k ( t )) − Z B n,k ( t )0 µ n,k ( s ) ds → C , U ) a.s. as n → ∞ ∀ t ∈ [0 , ∞ ) . Similarly, applying the random time change theorem along with Assump-tion 2 (i) and Assumption 1(i) we obtain1 n (cid:16) R kl ( S n,k ( B n,k ( t ))) − p l,k S n,k ( B n,k ( t )) (cid:17) → C , U )a.s. as n → ∞ ∀ t ∈ [0 , ∞ ) . Applying these results to (7) it follows that¯ X n,k ( t ) → ¯ X k ( t ) in ( C , U ) a.s. as n → ∞ . The joint convergence follows au-tomatically from these results and Proposition 1.Note that neither Theorem 2 nor Theorem 3 provide an explicit functionalform for the reflection regulator Ψ( · ). It can be shown (see (Chen and Yao2001, Chapter 7)) that the regulator map is the unique fixed point, y ∗ ∈ D K ,of the map π ( x, y )( t ) := sup ≤ s ≤ t [ − x ( s ) + G y ( s )] + ∀ t ∈ [0 , ∞ ), where G isan M -matrix. Extracting a closed form expression for y ∗ is not straightfor-ward, barring a few special cases. The following corollary shows that thereflection map and fluid limit of the queue length process for a parallel nodequeueing network is particularly simple and an obvious generalization ofthat of a single queue. Corollary 2
Consider a K -node parallel queueing network. The fluid limitto the queue length and cumulative idleness processes are ( ¯ Q , ¯ Y ) = (Φ( ¯ X , Ψ( ¯ X ))) ∈ C K × C K , where ¯ X = ( X , . . . , X K ) , Ψ( ¯ X ( t )) = sup ≤ s ≤ t [ − ¯ X ( s )] + and Φ( ¯ X ) = ¯ X +Ψ( ¯ X ) .Proof: Note that for a parallel queueing network P = 0. Therefore, thefixed point of the map π ( · , · ) is simply sup ≤ s ≤ t [ − x ( s )] + . It follows thatthe regulator map of the fluid scaled queue length process is Ψ( ¯ X n ( t )) =sup ≤ s ≤ t [ − ¯ X n ( s )] + . It follows by Theorem 3 thatΨ( ¯ X n ( t )) → sup ≤ s ≤ t [ − ¯ X ( s )] + and Φ( ¯ X n ( t )) → ¯ X ( t ) + Ψ( ¯ X ( t )) in ( C K , U ) a.s. RANSITORY NETWORKS as n → ∞ .A slightly more complicated example would be a series queueing network.Corollary 3 establishes the fluid limit to the network state of a two queuetandem network, when a large, but finite, number n of users arrive at queue1 over a finite time horizon [ − T , T ]. This result can be rather straightfor-wardly extended to a network of more than two queues. Let P = (cid:18) (cid:19) , be the matrix of Markov routing probabilities. Corollary 3
Consider a tandem queueing network and recall that V = I − P T . Let F = F be the arrival epoch distribution with support [ − T , T ] where T , T > , and assume that µ and µ are the fixed service rates. Then, the(joint) fluid limit to the queue length and cumulative idleness processes is ( ¯ Q , ¯ Y ) = (Φ( ¯X ) , Ψ( ¯X )) ∈ C K × C K , where ¯X := ( X , X ) = (( F − µ e ) , ( µ − µ ) e ) , and Ψ( ¯ X ) = ( Y , Y ) with Y ( t ) = sup ≤ s ≤ t ( − X ( s )) + and Y ( t ) = sup ≤ s ≤ t ( − X ( s ) + Y ( s )) + =sup ≤ s ≤ t [ − X ( s ) + sup ≤ r ≤ s ( − X ( r )) + ] + , and Φ( ¯X )( t ) = ¯ X + V Ψ( ¯ X ) =( X + Y , X + Y − Y ) . The proof is straightforward by substitution and we omit it. Note that thequeue length fluid limit to the downstream queue appears quite complicated:¯ Q = X + Y − Y where Y ( t ) = sup ≤ s ≤ t ( − X ( s )+ Y ( s )) + . By substitutingin the expression for X we have¯ Q = ( µ − µ ) e + F − F − Y + Y = ( F − ¯ Q − µ e ) + Y . Note that F − ¯ Q is just the cumulative fluid departure function from theupstream queue, which is precisely the input to the downstream queue.Next, suppose the service process is stationary such that µ k ( t ) = µ k forall t ≥ k ∈ K . Then, the busy time process satisfies the followingtheorem. Theorem 4
Let ¯ B n ( t ) = ( B n, ( t ) , . . . , B n,k ( t )) . Then, as n → ∞ , ¯ B n → e − M Ψ( ¯ X ) in ( C K , U ) a.s. , (11) where, M = diag (1 /µ , . . . , /µ K ) . HONNAPPA AND JAIN
Proof:
By definition ¯ B n ( t ) = t − ¯ I n ( t ) , where ¯ I n ( t ) = ( I n, ( t ) , . . . , I n,k ( t )) T .Recalling the definition of the process ¯Y n ( t ) it is straightforward to see that ¯I n ( t ) = ( I − P T ) − ¯Y n ( t ) for all t ≥
0. Therefore, ¯ B n ( t ) = t − ( I − P T ) − ¯ Y n ( t ) . Theorem 3 implies that, as n → ∞ , ¯ B n ( t ) → t − Ψ( ¯ X ( t )) in ( C K , U ) a.s. ∀ t ∈ [0 , ∞ ) . The following corollary establishes the fluid busy time process for theparallel queue case. The proof follows that of Corollary 2 and we omit it.
Corollary 4
Consider a K -node parallel queueing network. Then, ¯ B n → e − ( I − P T ) − sup ≤ s ≤· [ − ¯ X ( s )] + in ( C K , U ) a.s. n → ∞ . In the stationary case we considered here, the busyness time-scale is effec-tively fixed by the service rate through the matrix M . On the other hand, ifthe service processes are non-stationary this time-scale itself is time-varying.Thus, computing the busy time (or equivalently the idle time) process whenthe service process is non-stationary is complicated. Note that the function¯ Y represents the number of “blanks” or the amount of unused capacity inthe network at each point in time, providing an indication of whether aparticular queue in the network is busy or not.Note that the population acceleration scale we use in the current analysisensures that (in the limit) the amount of time each user spends in serviceis infinitesimally small. The ‘behavior’ of the queue state under populationacceleration scaling is akin to the conventional heavy-traffic scaling intro-duced in Reiman (1984) for stationary single class queueing networks. Thecorresponding diffusion heavy-traffic scaling identifies the critical time-scaleof the stationary queueing network. The population acceleration scaling dif-fers from the conventional heavy-traffic scaling by the fact that the fluidlimit process (in general) is non-linear in nature. This implies that queuesin the network can enter idle and busy periods, and arriving jobs will onlyface delays in the latter time intervals. We should anticipate that the crit-ical time-scale of the queue state in the diffusion scale should itself changedepending on whether the queue is busy or idle, leading to a non-stationarydiffusion approximation. Indeed, this is precisely what is implied by theresults in the next section.
5. Functional Central Limit Theorems.
We now consider the sec-ond order refinement to the fluid limit by establishing a functional centrallimit theorem (FCLT) satisfied by the queue length state process. We show,in particular, that the FCLT is a reflected diffusion, where the diffusion
RANSITORY NETWORKS process ˆ X is a function of the multi-dimensional Brownian bridge processas in Definition 1. Unlike the heavy traffic limits for generalized Jacksonnetworks (see (Chen and Yao 2001, Chapter 7) Reiman (1984)), the diffu-sion is not reflected through the oblique reflection map (see (Chen and Yao2001, Definition 7.1)). The non-homogeneous traffic and non-stationary ser-vice processes induce a time-varying critical time-scale under the popula-tion acceleration scaling. we show that this time-varying critical time-scalemanifests as a time-varying reflection boundary in transitory queueing net-works. To be precise, the reflection regulator for the queue length diffusionis the directional derivative of the oblique reflection of ¯ X (from Lemma 2)in the direction of the diffusion limit ˆ X to the netput process. A similarresult was observed in the case of a single ∆ ( i ) /GI/ V is a K × K M -matrix and P T = I − V . Let x ∈ C then,under the hypothesis of Theorem 2, there exists a unique oblique reflectionmap ( z, y ) := (Φ( x ) , Ψ( x )) ∈ C × C such that z = x + V y , y j is non-decreasing and y j grows only when z j is zero (for all j = 1 , . . . , K ). Thedirectional derivative of the oblique reflection of x in the direction of theprocess χ ∈ C is defined as follows (see Mandelbaum and Ramanan (2010)as well): Definition 3 (Directional Derivative Reflection Map)
Given ( x, χ ) ∈C K × C K and M -matrix V , the directional derivative of the oblique reflectionmap Φ( x ) = x + V Ψ( x ) in the direction of χ is the pointwise limit of ∆ nχ ( x ) := √ n (cid:18) Φ (cid:18) χ √ n + x (cid:19) − Φ( x ) (cid:19) ∈ C n ≥ as n → ∞ . Theorem 1.1 (ii) of Mandelbaum and Ramanan (2010) identifies the limitprocess, which we state as a lemma for completeness. Here,
Lemma 3 If ( x, χ ) ∈ C K × C K then the directional derivative limit ∆ χ ( x ) exists and convergence in Definition 3 is uniformly on compact subsets ofcontinuity points of the limit ∆ χ ( x ) . Further, if ( z, y ) solve the oblique re-flection problem for x then ∆ χ ( x ) = χ + V γ ( x, χ ) , where V = I − P T γ := γ ( x, χ ) lies in D Kusc and is the unique solution tothe system of equations γ i ( t ) = ( sup s ∈∇ it [ − χ i ( s ) + [ P γ ] i ( s )] + t ∈ [0 , t iu ] , sup s ∈∇ it [ − χ i ( s ) + [ P γ ] i ( s )] t > t iu , HONNAPPA AND JAIN for i = 1 , . . . , K , where ∇ it := { s ∈ [0 , t ] | z i ( s ) = 0 and y i ( s ) = y i ( t ) } , and t iu := inf { t ≥ y i ( t ) > } . Consider the second order refinement to the netput process,ˆ X n := n − / (cid:0) X n − n ¯ X (cid:1) ∈ D K . Using Proposition 1, and the fact that the limit processes have sample pathsin C K , the following Lemma is straightforward to establish. Lemma 4
The diffusion-scaled netput process satisfies, ˆ X n ⇒ ˆ X in ( C K , U ) as n → ∞ , where ˆ X k := W k ◦ F k − W k ◦ R t µ k ( s ) ds + T ( ˆ R k ◦ M ) , ˆ R k is the k th row ofthe matrix valued process ˆ R defined in part (ii) of Assumption 2, and M isdefined in (3) . The proof of the lemma follows from an application of part (ii) of Propo-sition 1 and using the fact that the addition operator is a continuous mapunder the uniform metric. We omit it for brevity.Now, The diffusion scale queue length process isˆ Q n := n − / (cid:0) Q n − n ¯Q (cid:1) ∈ D K . Recall, from Lemma 1, that ¯Q n = ¯X n + V Ψ( ¯X n ) and, from Theorem 3, that ¯Q = ¯X + V Ψ( ¯X ). It follows that ˆQ n = √ n (cid:0) ¯X n + V Ψ( ¯X n ) − ¯X − V Ψ( ¯X ) (cid:1) = ˆ X n + V √ n Ψ ˆ X n √ n + ¯ X ! − Ψ (cid:0) ¯ X (cid:1)! + V √ n Ψ( ¯ X n ) − Ψ ˆ X n √ n + ¯ X !! = ∆ n ˆ X n (cid:0) ¯ X (cid:1) + V √ n Ψ( ¯ X n ) − Ψ ˆ X n √ n + ¯ X !! . Our next result shows that ∆ n ˆX n ( ¯X ) is asymptotically equal to ∆ n ˆX ( ¯X ). Lemma 5
Let ∆ n ˆX ( ¯X ) and ∆ n ˆX n ( ¯X ) be defined as in Definition 3. Then, ∆ n ˆX n ( ¯X ) − ∆ n ˆX ( ¯X ) → in ( C K , U ) a.s. as n → ∞ . RANSITORY NETWORKS Lemma 5 implies it suffices to consider the process(12) ˆ Q n ≡ ∆ n ˆ X (cid:0) ¯ X (cid:1) + V √ n Ψ( ¯ X n ) − Ψ ˆ X √ n + ¯ X !! (where, by an abuse of notation, we call this process ˆ Q n as well). Now, ifwe show that √ n Ψ( ¯ X n ) − Ψ ˆ X √ n + ¯ X !! → C K , U )a.s. as n → ∞ , then Lemma 3 implies that ˆ Q n converges to the process ∆ ˆX ( ¯X ) pointwise in the large population limit. The following lemma estab-lishes the required result under general conditions. Lemma 6
Let x n , x ∈ D K be stochastic processes that satisfy k√ n ( x n − x ) k → χ a.s. as n → ∞ . Then, (13) (cid:13)(cid:13)(cid:13)(cid:13) √ n (cid:18) Ψ( x n ) − Ψ (cid:18) χ √ n + x (cid:19)(cid:19)(cid:13)(cid:13)(cid:13)(cid:13) → a.s. as n → ∞ , where χ ∈ C K . The main result of this section follows as a consequence of these lemma’s.
Theorem 5
For any fixed t ∈ [0 , ∞ ) , as n → ∞ (14) ˆ Q n ( t ) ⇒ ˆ Q ( t ) = ∆ ˆ X ( ¯ X )( t ) , where ∆ ˆ X ( ¯ X )( t ) = ˆ X ( t ) + V γ ( ¯ X , ˆ X )( t ) .Proof: First, using the Skorokhod representation theorem Billingsley (1968),it follows from Lemma 4 that there exist versions of the stochastic processes n ˆ X n o and ˆ X , referred to using the same notation, such thatˆ X n → ˆ X in ( C K , U ) a.s. as n → ∞ . It follows that ¯ X n = ¯ X + ( √ n ) − ˆ X + o (1) a.s. Lemma 6 implies that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) √ n Ψ( ¯ X n ) − Ψ ˆ X √ n + ¯ X !!(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) → n → ∞ . Next, using Lemma 5 and Lemma 3, it follows that ˆ Q n ( t ) → ∆ ˆ X ( ¯ X )( t ) a.s. as n → ∞ for any fixed t ∈ [0 , ∞ ), which in turn implies weak HONNAPPA AND JAIN convergence of the stochastic processes thus proving the desired result.
Proof: [Proof of Lemma 5] First, recall that ∆ n ˆX n ( ¯X ) = ˆX n + V √ n Ψ ˆX n √ n + ¯X ! − Ψ( ¯X ) ! . By Lemma 4 and the Skorokhod representation theorem Billingsley (1968),it follows that(15) ˆX n → ˆX in ( C K , U ) a.s.as n → ∞ . The lemma is proved once we show that √ n Ψ ˆX n √ n + ¯X ! − Ψ ˆX √ n + ¯X !! → C K , U ) a.s. as n → ∞ . Chen and Whitt Chen and Whitt (1993) show that the oblique reflectionmap and the reflection regulator are Lipschitz continuous with respect tothe uniform metric topology. Recall that k · k represents the uniform metricon C K over the interval [0 , T ]. Then, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) √ n Ψ ˆX n √ n + ¯X ! − Ψ ˆX √ n + ¯X !!(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ K (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) √ n ˆX n √ n + ¯X − ˆX √ n − ¯X (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , = K (cid:13)(cid:13)(cid:13) ˆX n − ˆX (cid:13)(cid:13)(cid:13) , where K is the Lipshitz constant associated with the oblique reflection map.The conclusion follows as a consequence of (15). Proof: [Proof of Lemma 6] The condition on x n , x implies that x n a.s. = x +( √ n ) − χ + o ( √ n ). Therefore, it follows that (cid:13)(cid:13)(cid:13)(cid:13) √ n (cid:18) Ψ( x n ) − Ψ (cid:18) χ √ n + x (cid:19)(cid:19)(cid:13)(cid:13)(cid:13)(cid:13) a.s. = k√ n (cid:18) Ψ (cid:18) χ √ n + x + o (1) (cid:19) − Ψ (cid:18) χ √ n + x (cid:19)(cid:19) k≤ K √ n k o (1) k , where the last inequality follows from the Lipshitz continuity of the obliquereflection map. The final conclusion follows from the fact that the indeter-minate form on the right hand side converges to 0 as n → ∞ . RANSITORY NETWORKS Remarks:
We include a short summary of the relevant results in Mandelbaum and Ramanan(2010) that imply that process-level convergence might be near impossi-ble to prove (in general) in a transitory queueing network. Lemma 2 inHonnappa et al. (2014) (an extension of Theorem 3.2 in Mandelbaum and Massey(1995)) proves the process-level diffusion limit result in the M topology fora single queue. The fact that the limit process has right- or left-discontinuitypoints that are ‘unmatched’ by the pre-limit process necessitates that con-vergence be proved in the M topology as opposed to the more natural J topology. On the other hand, Mandelbaum and Ramanan (2010) show thatit is not possible to prove a process-level convergence result even in the W M topology (‘weak’ M topology (see Whitt (2001a)), due to the fact that themultidimensional limit process can have discontinuity points that are bothright- and left-discontinuous . For completeness, we state the relevant portionofTheorem 1.2 of Mandelbaum and Ramanan (2010) that encapsulates thevarious necessary conditions for discontinuities in the sample paths of the di-rectional derivative limit process, ∆ ˆX ( ¯X ). First, given ( z, y ) as the solutionto the oblique reflection problem for x ∈ C define, for each t ∈ [0 , ∞ ), O ( t ) := { i ∈ { , . . . , K } : z i ( t ) > } , U ( t ) := { i ∈ { , . . . , K } : z i ( t ) = 0 , ∆ y i ( t +) = 0 , ∆ y i ( t − ) = 0 } , C ( t ) := { , . . . , K }\ [ O ( t ) ∪ U ( t )] , EO ( t ) := { i ∈ C ( t ) : ∃ δ > z i ( s ) > ∀ s ∈ ( t − δ, t ) } , SU ( t ) := { i ∈ C ( t ) : ∆ z i ( t − ) = 0 , ∆ z i ( t +) = 0 . } When x = ¯X , O ( t ) is the set of nodes in the network that are overloaded attime t , U ( t ) is the set of underloaded nodes, C ( t ) the set of critically loadednodes, EO ( t ) is the set of critically loaded queues that are at the end of over-loading and SU ( t ) is the set of critically loaded nodes that are at the startof under-loading. Note that the definitions of overloading, under-loadingand critical loading conform to the standard notions for G/G/
Definition 4 (Def. 1.5 Mandelbaum and Ramanan (2010))
Given a K × K routing matrix P and the oblique reflection map Ψ and x ∈ C K sothat y = Ψ( x ) . Then a sequence j , j , . . . , j m with j k ∈ { , . . . , K } for k =0 , , . . . , m that satisfies P j k − ,j k > for k = 0 , , . . . , m is said to be a chain.The chain is said to be a cycle if there exist distinct k , k ∈ { , . . . , m } suchthat j k = j k , the chain is said to precede i if j = i and is said to be empty HONNAPPA AND JAIN at t if y j k ( t ) = 0 for every k = 1 , . . . , m . For i = 1 , . . . , K and t ∈ [0 , ∞ ) ,we consider the following two types of chains:1. An empty chain preceding i is said to be critical at time t if it is eithercyclic or j m is at the end of overloading at t .2. An empty chain preceding i is said to be sub-critical at time t if it iseither cyclic or j m is at the start of overloading at t . Theorem 1.2 of Mandelbaum and Ramanan (2010) gives necessary con-ditions so that, in general, the sample paths of the directional derivativecan have both a right and left discontinuity at t ∈ [0 , ∞ ). Simply put, thestructure of the routing matrix P determines whether we see such a point. Proposition 2 (Thm. 1.2 Mandelbaum and Ramanan (2010))
Underthe conditions of Definition 4 and given a process χ ∈ C k , if the directionalderivative ∆ χ ( x ) has both a right and a left discontinuity at t ∈ [0 , ∞ ) thenone of the following conditions must hold at time t :a) i is at the end of overloading, and a sub-critical chain precedes i , inwhich case ∆ χ ( x ) i ( t − ) < ∆ χ ( x ) i ( t ) i = 0 < ∆ χ ( x ) i ( t +) , b) i is at the start of under-loading and a critical chain precedes i , inwhich case ∆ χ ( x ) i ( t − ) > ∆ χ ( x ) i ( t ) > ∆ χ ( x ) i ( t +) = 0 , c) i is not underloaded and there exist both critical and sub-critical chainspreceding i ; if, in addition, i is overloaded then the discontinuity is aseparated discontinuity of the form ∆ χ ( x ) i ( t ) < min { ∆ χ ( x ) i ( t − ) , ∆ χ ( x ) i ( t +) } . Note that the sample paths of ∆ ˆX ( ¯X ) lie in D K lim and establishing M con-vergence in this space is non-trivial. Recall that the standard descriptionof M convergence is through the graphs of the functions - which can bedescribed via linear interpolations in D and D Kl,r . However, in D K lim no suchsimple description exists (see Chapter 12 of Whitt (2001a) and Chapter 6,8 of Whitt (2001b) for further details on these issues).Given the inherent difficulty in establishing a general process-level result,we first focus on a two queue tandem network, where the arrival time distri-bution is uniform on the interval [ − T , T ] and T , T > RANSITORY NETWORKS Theorem 6
Consider a tandem queueing network with P = (cid:18) (cid:19) , and R = I − P T = (cid:18) − (cid:19) . Assume that F = F is uniform over [ − T , T ] , andservice rate at node 1 is µ and at node 2 µ . Then, ˆQ n ⇒ ˆQ := ∆ ˆX ( ¯X ) in ( D l,r , SM ) as n → ∞ , where ˆX = ( ˆ X , ˆ X ) with ˆ X = W ◦ F − W ◦ M k , ˆ X = W ◦ M k − W ◦ M and M k ( · ) = R · µ k ( s ) ds for k ∈ { , } , ¯X =(( F − µ e ) , ( µ − µ ) e ) T and e : R → R is the identity map.Proof: Recall that F ( t ) = t + T T + T for all t ∈ [ − T , T ]. We consider three sub-cases and establish the weak convergence result for each of them separately.(i) Let µ < µ . Then,(16) ¯ Q ( t ) = ( ( F ( t ) − µ t { t ≥ } ) ∀ t ∈ [ − T , τ ) , ∀ t ∈ [ τ , ∞ ) , and ¯ Q ( t ) = 0 ∀ t ≥ , where τ := inf { t > | F ( t ) = µ t } . These follow as aconsequence of Corollary 3, and noting that ¯X = ( F ( t ) − µ e, ( µ − µ ) e ).Thus, we have ∇ t := {− T } ∀ t ∈ [0 , τ ) , {− T , τ } t = τ , { t } ∀ t > τ , and(17) ∇ t := { t } ∀ t ∈ [0 , ∞ ) . (18)Thus, node 1 is in O ( t ) for all t ∈ [ − T , τ ), C ( t ) for t = τ and in U ( t ) for t > τ , and node 2 is in U ( t ) for all t .The limit process ˆQ has a discontinuity only in the first component atˆ Q ( τ ) = ˆ X ( τ ) + max { , − ˆ X ( τ ) } . Note that ˆ Q ( τ − ) = ˆ X ( τ ) andˆ Q ( τ +) = 0, implying that ˆ Q has either a right or left discontinuity at τ . Ifˆ X ( τ ) ≥ Q ( τ ) = ˆ X ( τ ) = ˆ Q ( τ − ) > ˆ Q ( τ +) = 0 and has a rightdiscontinuity. Else, if ˆ X ( τ ) < Q ( τ ) = 0 = ˆ Q ( τ +) > ˆ Q ( τ − )and has a left discontinuity. Thus, the limit process ˆQ has sample paths in D l,r . The proof of convergence for ˆQ n = ( ˆ Q n, , ˆ Q n, ) in this case is simple.First, Theorem 2 of Honnappa et al. (2014) shows that ˆ Q n, ⇒ ˆ Q := ˆ X +sup s ∈∇ · ( − ˆ X ( s )) in ( D l,r , M ) as n → ∞ , and ˆ Q n, ⇒ D l,r , M ). Recallthat Disc ( ˆ Q ) and Disc ( ˆ Q ) are the (respective) sets of discontinuity point,and it is obvious that Disc ( ˆ Q ) ∩ Disc ( ˆ Q ) = φ . Therefore, by (Whitt 2001b,Corollary 6.7.), ˆ Q n, + ˆ Q n, ⇒ ˆ Q in ( D l,r ( R ) , M ) as n → ∞ . Consequentto (Whitt 2001b, Theorem 6.7.2), it follows that ˆQ n ⇒ ˆQ := ( ˆ Q , T in( D l,r , SM ) as n → ∞ . HONNAPPA AND JAIN (ii) Let µ > µ . Then, ¯ Q and ∇ t follow (16) and (17) (resp.). ¯ Q on theother hand, is more complex now:¯ Q ( t ) = ( µ − m ) t ∀ t ∈ [0 , τ ] , ( F ( t ) − µ t ) ∀ t ∈ [ τ , τ ] , ∀ t > τ , where τ := inf { t > τ : F ( t ) = µ t } (note that τ > τ since µ > µ ). Itfollows that ∇ t = { } ∀ t ∈ [0 , τ ) , { , τ } t = τ , { t } ∀ t > τ . It follows that node 2 is in O ( t ) for all t ∈ [0 , τ ), C ( t ) at t = τ and U ( t ) for t > τ .The diffusion limit ˆQ := ( ˆ Q , ˆ Q ) has discontinuities in both compo-nents. For node 1, if ˆ X ( τ ) ≥ Q ( τ ) has a right discontinuity, whileˆ X ( τ ) < Q ( τ ) has a left discontinuity. Similarly, if ˆ X ( τ ) ≥ Q ( τ ) has a right discontinuity, and if ˆ X ( τ ) < ˆQ has sample paths in D l,r . Furthermore, it is clear that Disc ( ˆ Q ) ∩ Disc ( ˆ Q ) = φ . Therefore, the weak convergence result followsby the same reasoning as in part (i).(iii) Assume µ = µ . Once again, ˆ Q and ∇ t follow (16) and (17) (resp.).On the other hand, for node 2 ˆ Q = 0, but unlike case (i), the queue is emptybut the server operates at full capacity till τ , and then enters underload.Thus, ∇ t = ( [0 , t ] ∀ t ∈ [0 , τ ] , { t } ∀ t > τ . It is clear that node 2 switches from C ( t ) in [0 , τ ] to U ( t ) for t > τ . Fur-thermore, at τ itself, the node is in SU ( t ) (the regulator is flat to the leftof τ and increasing to the right).The diffusion limit, once again, has discontinuities in both components.However, it is clear that Disc ( ˆ Q ) = Disc ( ˆ Q ) = { τ } . For any T > − T , itis straightforward to see that ( ˆ Q ( t ) − ˆ Q ( t − ))( ˆ Q ( t ) − ˆ Q ( t − )) ≥ − T ≤ t ≤ T : clearly, for any t < τ , ˆ Q i , i = 1 , τ , ˆ Q ( τ ) ≥ ˆ Q ( τ − ) and ˆ Q ( τ ) = ˆ Q ( τ − ). Finally, for any t > τ , ˆ Q ( τ ) = ˆ Q ( τ − ) and ˆ Q ( τ ) = ˆ Q ( τ − ). Now, by Theorem 6.7.3of Whitt (2001b), it follows that ˆ Q n, + ˆ Q n, ⇒ ˆ Q + ˆ Q in ( D l,r ( R ) , M ) as n → ∞ . Then, by Theorem 6.7.2 of Whitt (2001b), ˆQ n ⇒ ˆQ in ( D l,r , SM ) RANSITORY NETWORKS as n → ∞ . This concludes the proof.Theorem 6 shows that in the case of a tandem network, with uniformarrival time distribution, the weak convergence result can be established inthe space D l,r and in the SM topology. In fact this result is true, if F is unimodal such that node 1 is overloaded in the initial phase (i.e., in theinterval [ − T , τ ), with T ≥ T = 0. Corollary 5
Let F be a unimodal distribution function with finite support [0 , T ] , and consider a tandem queue as defined in Theorem 6. Then, ˆQ n ⇒ ˆQ := ∆ ˆX ( ¯X ) in ( D l,r , SM ) as n → ∞ , where ˆX := (cid:16) W ◦ F − σ µ / W , ( σ µ / W − σ µ / W ) (cid:17) T , ¯X = ( F − µ e, ( µ − µ ) e ) T and e : R → R is the identity map. The proof follows that of Theorem 6 and is omitted. Note that the com-pact support assumption is required, due to the fact that we prove weak con-vergence over compact intervals of time (see Section 7.2 of Honnappa et al.(2014) for a discussion on this point).
6. High-intensity Analysis of Tandem Networks.
We illustratethe utility of the afore-developed approximations in bottleneck analysis oftransitory tandem networks. Almost all of the analysis in the literature hasfocused on the characterization and detection of bottlenecks in stationaryqueueing networks. Of particular relevance to our results in this paper is theheavy-traffic bottleneck phenomenon identified in Suresh and Whitt (1990),Whitt (2001a). To recall, the heavy-traffic bottleneck phenomenon corre-sponds to the state space collapse that is observed when the traffic intensityat a single queue approaches 1, while the traffic intensity at other queuesremains below 1. In this case, the well known heavy-traffic approxima-tions in Iglehart and Whitt (1970), Reiman (1984), Chen and Mandelbaum(1991a) imply that the network workload process will collapse to a singledimensional process determined by the bottleneck node. In other words, thenon-bottleneck nodes behave like ‘switches’ where the service time is ef-fectively zero. In general, exact bottleneck analysis is very difficult (if notimpossible), and several approximations have been proposed in the liter-ature, particularly the parametric-decomposition approach Whitt (1983),Buzacott and Shanthikumar (1992), the stationary-interval method Whitt(1984), and Reiman’s individual (IBD) and sequential bottleneck decompo-sition (SBD) algorithms Reiman (1990). Nonetheless, the natural metric to HONNAPPA AND JAIN use to study bottlenecks would be the waiting time at each node. The fluidand diffusion workload approximations can be established as a corollary toTheorem 3 and Theorem 5, assuming that the service process is stationary:
Corollary 6 (Workload Approximation)
Recall that M is a diagonalmatrix defined as M := diag (1 /µ , . . . , /µ , /µ K ) . Then the fluid workload process ¯Z = M ¯Q , and for each t ∈ [0 , ∞ ) thediffusion workload process is ˆZ ( t ) = M ˆQ ( t ) . The proof of this corollary follows by analogous arguments to (Honnappa et al.2014, Proposition 4), and we omit it.Bottleneck analysis, however, has largely been ignored in transitory net-works in particular. The key difference (and difficulty) in the transitorysetting is that, for general arrival epoch distributions F it is possible thatthe number of bottleneck queues can change with time. The situation is con-siderably simpler when F is uniform, however, and we focus on this case firstto illustrate the main ideas. We commence with a definition of a bottleneckqueue in a transitory network, in the large population limit. Definition 5 (Transitory Bottleneck Queue)
Queue k ∈ K in the tran-sitory queueing network is a bottleneck at time t if and only if the diffusionworkload process satisfies | ˆ Z k ( t ) | > . Note that we choose to use a sample path definition of the bottleneck nodeowing to the fact that the temporal stochastic variations can produce differ-ing numbers of bottlenecks, even compared with the average/fluid variation.This definition is natural to consider in job-shop type production systemsand complements the definitions in (Lawrence and Buss 1994, P. 23) thatclassifies bottlenecks in terms of short, intermediate and long time horizons.
Example 1 (Tandem network with uniform traffic)
Consider a se-ries network of K queues. Let the service rate at queues through K − be µ and µ K at queue K . Without loss of generality we assume that µ K < ≤ µ .Assume that the traffic arrival epochs are randomly scattered per a uni-form distribution function, over the interval [0 , . Then, in the fluid pop-ulation acceleration limit as observed in Theorem 3, it can be observedthat each of the queues , . . . , K − behave like instantaneous switches and O ( n ) fluid accumulates at the final queue. Extending the analysis in Corol-lary 3 to a K -node tandem network it is straightforward to compute that ¯ X = ( ¯ X , . . . , ¯ X K − , ¯ X K ) , where ¯ X ( t ) = F ( t ) − µ t ≤ and ¯ X k ( t ) = 0 for RANSITORY NETWORKS all k = 2 , . . . , K − , and ¯ X K ( t ) = ( µ − µ K ) t > . Since the routing matrixis P = . . . . . . . . . ... . . . a simple (if tedious) calculation shows that ¯ Q ( t ) = ( (0 , . . . , , ( µ − µ K ) t ) t ∈ [0 , /µ K ] , (0 , . . . , t > /µ K . Now, It follows that, in the case of the tandem queueing network underconsideration ¯ Z ( t ) = ( (0 , . . . , , ( µ µ − K − t ) t ∈ [0 , /µ K ] , (0 , . . . , t > /µ K . Thus, in the fluid limit, we find that the tandem queueing network “col-lapses” to a single queue in the fluid limit (this is an example of a statespace collapse as defined in Reiman (1984)), and the sojourn time throughthe network, in the fluid scale and large population limit, is determined en-tirely by the delay at node K .On the other hand, as the diffusion limit in Theorem 6 shows, there isnon-zero variability in the queue length at each node in the network. Indeed,Theorem 6 and Corollary 6 imply that the diffusion limit of the workloadvector in a tandem network is ˆZ = M∆ ˆ X ( ¯ X ) , where ˆ X ( t ) = (cid:18) ( W ( t ) − σµ / W ( t )) , ( σ µ / W ( t ) − σ µ / W ( t ) , . . . ,σ µ / W K − ( t ) − σ K µ / K W K ( t )) (cid:19) . Now, if µ > , then ˆ Z k D = 0 for k = 1 , . . . , K − and ˆ Z K ( t ) D = µ − K ( ˆ X K ( t ) +sup ≤ s ≤ t ( − ˆ X K ( s ))) with ˆ X K = σ µ / W K − − σ K µ / K W K . That is, in thepopulation acceleration scaling, the distribution of the sojourn time throughthe network is asymptotically equal to the delay distribution of the last queue.On the other hand, if µ = 1 , then ˆ Z = µ − ( ˆ X ( t ) + sup ≤ s ≤ t ( − ˆ X ( s ))) HONNAPPA AND JAIN with ˆ X = W − σµ / W , Z k D = 0 for k = 2 , . . . , K − and ˆ Z K = µ − K ( σ µ / W K − − σ K µ / K W K ) ∀ t ∈ [0 , µ − K ( − σ K µ / K W K ) ∀ t ∈ (1 , /µ K ]0 ∀ t > /µ K . This indicates that there are two bottlenecks at queues and K . Thus,there is a state space collapse to a two-dimensional vector ˆZ = ( Z , Z K ) , andthe sojourn time through the network is asymptotically equal in distributionto the sum of the delays in these two queues. Example 2 (Tandem network with unimodally traffic)
Now, suppose F is not uniform, but unimodal with support on [0 , . For simplicity, we as-sume that the distribution function is symmetric around τ := argmax { F ′ ( t ) : t ∈ [0 , } , where F ′ represent the first derivative of the arrival epoch distri-bution (assuming it is well defined), and that the service rates are the sameas in Example 1. The uni-modality of the arrival epoch distribution impliesthat up to time τ the distribution function is convex increasing, while after τ it is concave decreasing. A simple example of such a distribution functionwould be, F ′ ( t ) = ( t t ∈ [0 , / − t ) t ∈ (1 / , . In this case, τ = 1 / and F ′ ( τ ) = 2 .We first focus on the case where the service rates satisfy µ K < µ < F ′ ( τ ) .Observe that ¯ X ′ ( t ) = F ′ ( t ) − µ ≤ ∀ t ∈ [0 , τ ) > ∀ t ∈ [ τ , τ ) , ≤ ∀ t ∈ [ τ , , where τ := inf { t > F ′ ( t ) = µ } and τ := inf { t > τ : F ′ ( t ) = µ } ; thatis, these are the two points in time where the derivative of the arrival epochdistribution equals the service rate in queue 1. We also have ¯ X k ( t ) = 0 forall k = 2 , . . . , K − and ¯ X K ( t ) = ( µ − µ K ) t > , for all t ∈ [0 , . Considerthe fluid queue length at node 1 ¯ Q ( t ) = ¯ X ( t ) + sup ≤ s ≤ t [ − ¯ X ( s )] + , andobserve that ¯ Q ( t ) = t ∈ [0 , τ )¯ X ( t ) − ¯ X ( τ ) t ∈ [ τ , τ )0 t ≥ τ . (19) RANSITORY NETWORKS Following the arguments in Example 1 it can be shown that the fluid queuelength in the downstream nodes satisfies ¯ Q k ( t ) = 0 for all k = 2 , . . . , K − for all t > . Similarly, in the terminal node ¯ Q K ( t ) = t ∈ [0 , τ ′ )( F ( t ) − F ( τ ′ )) − µ K ( t − τ ′ ) t ∈ [ τ ′ , τ )( F ( τ ) − F ( τ ′ )) − µ ( τ − τ ′ ) + ( µ − µ K )( t − τ ′ ) t ∈ [ τ , τ )( F ( t ) − F ( τ ′ )) − µ K ( t − τ ′ ) − ( F ( τ ) − µτ ) t ∈ [ τ , τ ′ )0 t ≥ τ ′ , where τ ′ := inf { t > F ′ ( t ) ≥ µ K } and τ ′ := sup { t > τ ′ : F ′ ( t ) ≥ µ K } .This follows from the fact that F ′ ( t ) − µ K ≤ ∀ t ∈ [0 , τ ′ ) > ∀ t ∈ [ τ ′ , τ ′ ) , ≤ ∀ t ∈ [ τ ′ , , In contrast to Example 1 the state-space collapse is not straightforward here.The fluid tandem queueing network switches between collapsing to a singlequeue network in time intervals [ τ ′ , τ ) and [ τ , τ ′ ) and a two queue networkin the interval [ τ , τ ) . Thus, the state-space collapse itself exhibits non-stationary behavior.Next, considering the diffusion limit, extending Corollary 5 to a K -nodenetwork we have ˆ X = (cid:18) ( W ◦ F − σµ / W ) σ µ / ( W − W ) ,. . . , ( σ µ / W K − − σ K µ / K W K ) (cid:19) , and from Corollary 6 the workload diffusion limit process is ˆ Z = M∆ ˆ X ( ¯ X ) .Note that ˆ X k := σ µ / ( W k − − W k ) D = σ µ / W ∗ k for k = 2 , . . . , K − where W ∗ k are independent but identically distributed Brownian motion processes.Following the fluid limit discussion above, the diffusion limit workloadprocess at queue 1 satisfies ˆ Z ( t ) = t ∈ [0 , τ ) µ − (cid:16) ˆ X ( t ) − ˆ X ( τ ) (cid:17) t ∈ [ τ , τ )0 t ≥ τ . HONNAPPA AND JAIN
On the other hand, following Corollary 6 and using the description of thedirectional derivative regulator map in Lemma 3, the processes ˆ Z k for k =2 , . . . , K − can be shown to satisfy ˆ Z k ( t ) = t ∈ [0 , τ ) µ − (cid:16) ˆ X k ( t ) + sup τ ≤ s ≤ t [ − ˆ X ( s )] + (cid:17) t ∈ [ τ , τ )0 t ≥ t ≥ τ . Observe that jobs flowing through queues , . . . , K − will experience non-zero delays in the interval [ τ , τ ) , determined by the reflected Brownianmotion process. The reason why this happens is manifest: the departure ratefrom queue 1 reaches its maximum value ( µ ) in this interval, so that thedownstream queues , . . . , K − become critically loaded in this interval.Note that the jobs experience (effectively) zero service delay in the latterqueues, and they are instantaneously switched through to downstream nodes.Thus, the “surge” period [ τ , τ ) is the same in all of these nodes. Finally,in queue K the diffusion limit workload process satisfies ˆ Z K ( t ) = t ∈ [0 , τ ′ ) µ − K ˆ X K ( t ) t ∈ [ τ ′ , τ ′ )0 t ≥ τ ′ . (20) Unlike Example 1, jobs experience delays in the last queue only in the interval [ τ ′ , τ ′ ) . Note that this interval is includes the interval [ τ , τ ] , due to theassumption that F is unimodal.Now, consider the alternative case where the serve rates satisfy µ K
7. Concluding Statements.
In this paper we developed asymptotic‘population acceleration’ approximations of the queue length and (implic-itly) the workload processes in a network of transitory queues. These resultscomplement and add to the body of research studying single class gener-alized Jackson networks. In particular, our fluid limit results accomodaterather general traffic and service models. On the other hand, we can onlyestablish point-wise diffusion approximations in the most general case, ow-ing to the difficulties in the existence of the so-called directional derivativeoblique reflection map. Nonetheless, we establish functional central limittheorems in the special case of a tandem network and we also present directconsequences of these developments on bottleneck analysis.There are several directions in which this research will be expanded in thefuture. The extension of these results to general polling queueing networkswill be interesting, exploiting some recently observed connections betweenacceleration scalings and polling networks in Rawal et al. (2014). Second,the arrival counts in non-overlapping intervals under the ∆ ( i ) traffic modelhave strong negative association. How soon will this correlation be ‘forgot-ten’ as traffic passes through multiple stages of service? This requires astudy of the possible sample paths of the workload process. We believe thisquestion has deep connections with directed percolation models; this is nota novel observation: Glynn and Whitt (1991) identify this connection whenthere are no traffic dynamics. In on-going work we are working towards ex-tending their analysis to transitory networks. A further interesting questionis how the last passage percolation time scales with the population size ina non-stationary setting (as opposed to the classical setting where the per-colation model is only studied in the stationary setting). The connectionbetween percolation time and the sojourn time through the network affordsyet another bottleneck/performance analysis measure in networks of tran-sitory queues that will be highly relevant in the context of manufacturinglines. We will consider these questions in future papers.7.1. Appendix subsection.
Proof of Theorem 1
We prove lemma’s foreach of the claims in the theorem. The first lemma establishes the FSLLN.
Lemma 7 (FSLLN)
The multivariate traffic process A n = ( A , . . . , A J ) := HONNAPPA AND JAIN P nm =1 a m satisfies n − A n → F in ( C J , U ) a.s.as n → ∞ , where F = ( F , . . . , F J ) and F j ( t ) = E [ { T j ≤ t } ] for all t ∈ [0 , T ] .Proof: First, for each j ∈ E , the classical Glivenko-Cantelli theorem impliesthat(21) n − A j → F j in ( C , U ) a.s.as n → ∞ . By the multivariate strong law of large numbers it follows thatfor a fixed t ∈ [0 , T ] A n ( t ) → F ( t ) a.s. as n → ∞ . The functional limitfollows as a consequence of (21).This proves part ( i ) of Theorem 1. The next lemma establishes part ( ii ). Lemma 8
The multivariate traffic process A n satisfies a functional centrallimit theorem where √ n (cid:0) n − A n − F (cid:1) ⇒ W ◦ F in ( C J , U ) , where W ◦ F is a J -dimensional Brownian bridge process as defined in Defi-nition 2, with covariance function ( R ( t ) , t ≥
0) = ([ F i,j ( t ) − F i ( t ) F j ( t )] , t ≥ .Proof: Once again, Donsker’s theorem for empirical processes implies that(22) ˆ A j := √ n (cid:0) n − A j − F j (cid:1) ⇒ W j ◦ F j in ( C , U )as n → ∞ for every j ∈ K . This implies that the marginal arrival processesare tight. (Whitt 2001a, Theorem 11.6.7) implies that the multivariate pro-cess A n is also tight.The multivariate central limit theorem (Whitt 2001a,Theorem 4.3.4) implies that the scaled process ˆ A n ( t ) = ( ˆ A ( t ) , . . . , ˆ A J ( t ))(for fixed t ∈ [0 , T ]) satisfiesˆ A n ( t ) = √ n (cid:18) A n ( t ) n − F ( t ) (cid:19) ⇒ N (0 , R ( t )) , where N (0 , R ( t )) is a mean zero J -dimensional Gaussian random vector withcovariance matrix R ( t ) = [ F i,j ( t ) − F i ( t ) F j ( t )]. The Cram´er-Wold device to-gether with this result implies that the finite-dimensional distributions of A n converge weakly to a tuple of Gaussian random vectors. The tightness of theprocesses { A n } , the continuity of the limit process and Prokhorov’s theoremimplies that ˆ A n converges weakly to the multivariate Gaussian stochasticprocess W ◦ F with mean zero and covariance function ( R ( t ) , t ≥
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School of Industrial Engineering,Purdue University,West Lafayette IN 47906.E-mail: [email protected]