Weining Kang

Fluid limits of many-server queues with reneging

This work considers a many-server queueing system in which im- patient customers with i.i.d., generally distributed service times and i.i.d., generally distributed patience times enter service in the order of arrival and abandon the queue if the time before possible entry into service exceeds the patience time. The dynamics of the system is represented in terms of a pair of measure-valued processes, one that keeps track of the waiting times of the customers in queue and the other that keeps track of the amounts of time each customer being served has been in service. Under mild assumptions, essen- tially only requiring that the service and reneging distributions have densities, as the number of servers goes to innity, a law of large numbers (or uid) limit is established for this pair of processes. The limit is shown to be the unique solution of a coupled pair of deterministic integral equations that admits an explicit representation. In addition, a uid limit for the virtual waiting time process is also established. This paper extends previous work by Kaspi and Ramanan, which analyzed the model in the absence of reneging. A strong motivation for understanding performance in the presence of reneging arises from models of call centers.

Fluid and Brownian approximations for an Internet congestion control model

We consider a stochastic model of Internet congestion control that represents the randomly varying number of flows present in a network where bandwidth is shared fairly amongst elastic document transfers. We focus on the heavy traffic regime in which the average load placed on each resource is approximately equal to its capacity. We first describe a fluid model (or functional law of large numbers approximation) for the stochastic model. We use the long time behavior of the solutions of this fluid model to establish a property called (multiplicative) state space collapse, which shows that in diffusion scale the flow count process can be approximately recovered as a continuous lifting of the workload process. Under proportional fair sharing of bandwidth and a mild condition, we show how state space collapse can be combined with a new invariance principle to establish a Brownian model as a diffusion approximation for the workload process and hence to yield an approximation for the flow count process. The workload diffusion behaves like Brownian motion in the interior of a polyhedral cone and is confined to the cone by reflection at the boundary, where the direction of reflection is constant on any given boundary face. We illustrate this approximation result for a simple linear network. Here the diffusion lives in a wedge that is a strict subset of the positive quadrant. This geometrically illustrates the entrainment of resources, whereby congestion at one resource may prevent another resource from working at full capacity.

Asymptotic approximations for stationary distributions of many-server queues with abandonment

A many-server queueing system is considered in which customers arrive according to a renewal process and have service and patience times that are drawn from two independent sequences of independent, identically distributed random variables. Customers enter service in the order of arrival and are assumed to abandon the queue if the waiting time in queue exceeds the patience time. The state of the system with

An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries

N

Characterization of stationary distributions of reflected diffusions

servers is represented by a four-component process that consists of the forward recurrence time of the arrival process, a pair of measure-valued processes, one that keeps track of the waiting times of customers in queue and the other that keeps track of the amounts of time customers present in the system have been in service and a real-valued process that represents the total number of customers in the system. Under general assumptions, it is shown that the state process is a Feller process, admits a stationary distribution and is ergodic. It is also shown that the associated sequence of scaled stationary distributions is tight, and that any subsequence converges to an invariant state for the fluid limit. In particular, this implies that when the associated fluid limit has a unique invariant state, then the sequence of stationary distributions converges, as

Product form stationary distributions for diffusion approximations to a flow-level model operating under a proportional fair sharing policy

N\rightarrow \infty

A Dirichlet process characterization of a class of reflected diffusions.

, to the invariant state. In addition, a simple example is given to illustrate that, both in the presence and absence of abandonments, the

On the submartingale problem for reflected diffusions in domains with piecewise smooth boundaries

N\rightarrow \infty

Fluid limits of many-server retrial queues with nonpersistent customers

and

Existence and uniqueness of a fluid model for many-server queues with abandonment

t\rightarrow \infty