Quasi-stationary analysis for queues with temporary overload

Motivated by the high variation in transmission rates for document transfer in the Internet and file down loads from web servers, we study the buffer content in a queue with a fluctuating service rate. The fluctuations are assumed to be driven by an independent stochastic process. We allow the queue to be overloaded in some of the server states. In all but a few special cases, either exact analysis is not tractable, or the dependence of system performance in terms of input parameters (such as the traffic load) is hidden in complex or implicit characterizations. Various asymptotic regimes have been considered to develop insightful approximations. In particular, the so-called quasistationary approximation has proven extremely useful under the assumption of uniform stability. We refine the quasi-stationary analysis to allow for temporary instability, by studying the “effective system load” which captures the effect of accumulated work during periods in which the queue is unstable

Pointwise Stationary Fluid Models for Stochastic Processing Networks

Generalizing earlier work on staffing and routing in telephone call centers, we consider a processing network model with large server pools and doubly stochastic input flows. In this model the processing of a job may involve several distinct operations. Alternative processing modes are also allowed. Given a finite planning horizon, attention is focused on the two-level problem of capacity choice and dynamic system control. A pointwise stationary fluid model (PSFM) is used to approximate system dynamics, which allows development of practical policies with a manageable computational burden. Earlier work in more restrictive settings suggests that our method is asymptotically optimal in a parameter regime of practical interest, but this paper contains no formal limit theory. Rather, it develops a PSFM calculus that is broadly accessible, with an emphasis on modeling and practical computation.admission control, dynamic routing, doubly stochastic arrivals, approximation, pointwise stationary, fluid models, abandonments, stochastic networks