Many useful descriptions of stochastic models can be obtained from functional limit theorems (invariance principles or weak convergence theorems for probability meastires on function spaces). These descriptions typically come from standard functional limit theorems via the o^ntinuous mapping theorem. This paper facilitates applications of the continuous mapping theorem by determining when several important ftmctions and sequences of functions preserve convergence. The functions considered are composition, addition, composition plus addition, multiplication, supremtun, reflecting barrier, first passage time and time reversal. These functions provide means for proving new functional limit theorems from previous ones. These functions are useful, for example, to establish the stability or continuity of queues and other stochastic models.

We consider a series of n single-server queues, each with unlimited waiting space and the first-in first-out service discipline. Initially, the system is empty; then k customers are placed in the first queue. The service times of all the customers at all the queues are i.i.d. with a general distribution. We are interested in the time D(k, n) required for all k customers to complete service from all n queues. In particular, we investigate the limiting behavior of D(k, n) as n and/or k . There is a duality implying that D(k, n) is distributed the same as D(n , k) so that results for large n are equivalent to results for large k. A previous heavy-traffic limit theorem implies that D(k, n) satisfies an invariance principle as n , converging after normalization to a functional of k-dimensional Brownian motion. We use the subadditive ergodic theorem and a strong approximation to describe the limiting behavior of D(k n , n) where k n as n . The case of k n = xn corresponds to a hydrodyna...

this paper we take a decision-theoretic viewpoint towards understanding the abandonment phenomena: the abandonment time for each customer is based on an individual utility optimization, which balances perceived waiting costs against the benets of service, and from which the patience distribution emerges as an equilibrium point

In polling systems, M 2 queues are visited by a single server in cyclic order. These systems model such diverse applications as token-ring communication networks and cyclic production systems. We study polling systems with exhaustive service and zero switchover (walk) times. Under standard heavy-traffic assumptions and scalings, the total unfinished work converges to a one-dimensional reflected Brownian motion, whereas the workloads of individual queues change at a rate that becomes infinite in the limit. Although it is impossible to obtain a multidimensional limit process in the usual sense, we obtain an `averaging principle' for the individual workloads. To illustrate the use of this principle, we calculate a heavy-traffic estimate of waiting times. Keywords: Polling systems, cyclic servers, diffusion approximations, heavy-traffic limits. June 21, 1995 The research of this author was supported in part under contract with AT&T Bell Laboratories, and in part under a grant from the I...

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Consider a single-server queue with a Poisson arrival process and exponential processing times in which each customer independently reneges after an exponentially distributed amount of time. We establish that this system can be approximated by either a reflected Ornstein-Uhlenbeck process or a reflected affine diffusion when the arrival rate exceeds or is close to the processing rate and the reneging rate is close to 0. We further compare the quality of the steady-state distribution approximations suggested by each diffusion.

Using a slight modification of the framework in Bramson [7] and Williams [52], we prove heavy traffic limit theorems for six families of multiclass queueing networks. The first three families are single station systems operating under first-in first-out (FIFO), generalized head-of-the-line proportional processor sharing (GHLPPS) and static buffer priority (SBP) service disciplines. The next two families are re-entrant lines operating under first-buffer-first-serve (FBFS) and last-buffer-first-serve (LBFS) service disciplines; the last family consists of certain 2-station, 5-class networks operating under an SBP service discipline. Some of these heavy traffic limits have appeared earlier in the literature; our new proofs demonstrate the significant simplifications that can be achieved in the present setting.

diffusion processes, fluid limit, heavy traffic. We consider an infinite-capacity s-server queue in a finite-state random environment, where the traffic intensity exceeds 1 in some environment states and the environment states change slowly relative to arrivals and service completions. Queues grow in unstable environment states, so that it is useful to look at the system in the time scale of mean environment-state sojourn times. As the mean environment-state sojourn times grow, the queue-length and workload processes grow. However, with appropriate normalizations, these processes converge to fluid processes and diffusion processes. The diffusion process in a random environment is a refinement of the fluid process in a random environment. We show how the scaling in these limits can help explain numerical results for queues in slowly changing random environments. For that purpose, we apply recently developed numerical-transform-inversion algorithms for the MAP/G/1 queue and the piecewise-stationary Mt/Gt/1 queue. 1.

We establish functional large deviation principles (FLDPs) for waiting and departure processes in single-server queues with unlimited waiting space and the first-in first-out service discipline. We apply the extended contraction principle to show that these processes obey FLDPs in the function space D with one of the non-uniform Skorohod topologies whenever the arrival and service processes obey FLDPs and the rate function is finite for appropriate discontinuous functions. We apply our previous FLDPs for inverse processes to obtain an FLDP for the waiting times in a queue with a superposition arrival process. We obtain FLDPs for queues within acyclic networks by showing that FLDPs are inherited by processes arising from the network operations of departure, superposition and random splitting. For this purpose, we also obtain FLDPs for split point processes. For the special cases of deterministic arrival processes and deterministic service processes, we obtain convenient explicit express...

Consider a single server queue with renewal arrivals and i.i.d. service times in which the server operates under a processor sharing service discipline. To describe the evolution of this system, we use a measure valued process that keeps track of the residual service times of all jobs in the system at any given time. From this measure valued process, one can recover the traditional performance processes, including queue length and workload. We show that under mild assumptions, including standard heavy traffic assumptions, the (suitably rescaled) measure valued processes corresponding to a sequence of processor sharing queues converge in distribution to a measure valued diffusion process. The limiting process is characterized as the image under an appropriate lifting map, of a one-dimensional reflected Brownian motion. As an immediate consequence, one obtains a diffusion approximation for the queue length process of a processor sharing queue. 1. Introduction. Consider