Proportional fairness is a popular service allocation mechanism to describe and analyze the performance of data networks at flow level. Recently, several authors have shown that the invariant distribution of such networks admits a product form distri
bution under critical loading. Assuming exponential job size distributions, they leave the case of general job size distributions as an open question. In this paper we show the conjecture holds for a dense class of distributions. This yields a key example of a stochastic network in which the heavy traffic limit has an invariant distribution that does not depend on second moments. Our analysis relies on a uniform convergence result for a fluid model which may be of independent interest.
Fluid models have become an important tool for the study of many-server queues with general service and patience time distributions. The equilibrium state of a fluid model has been revealed by Whitt (2006) and shown to yield reasonable approximations
to the steady state of the original stochastic systems. However, it remains an open question whether the solution to a fluid model converges to the equilibrium state and under what condition. We show in this paper that the convergence holds under a mild condition. Our method builds on the framework of measure-valued processes developed in Zhang (2013), which keeps track of the remaining patience and service times.
We propose a unified approach to establishing diffusion approximations for queues with impatient customers within a general framework of scaling customer patience time. The approach consists of two steps. The first step is to show that the diffusion-
scaled abandonment process is asymptotically close to a function of the diffusion-scaled queue length process under appropriate conditions. The second step is to construct a continuous mapping not only to characterize the system dynamics using the system primitives, but also to help verify the conditions needed in the first step. The diffusion approximations can then be obtained by applying the continuous mapping theorem. The approach has two advantages: (i) it provides a unified procedure to establish the diffusion approximations regardless of the structure of the queueing model or the type of patience-time scaling; (ii) and it makes the diffusion analysis of queues with customer abandonment essentially the same as the diffusion analysis of queues without customer abandonment. We demonstrate the application of this approach via the single server system with Markov-modulated service speeds in the traditional heavy-traffic regime and the many-server system in the Halfin-Whitt regime and the non-degenerate slowdown regime.
We consider a processor sharing queue where the number of jobs served at any time is limited to $K$, with the excess jobs waiting in a buffer. We use random counting measures on the positive axis to model this system. The limit of this measure-valued
process is obtained under diffusion scaling and heavy traffic conditions. As a consequence, the limit of the system size process is proved to be a piece-wise reflected Brownian motion.
We study many-server queues with abandonment in which customers have general service and patience time distributions. The dynamics of the system are modeled using measure- valued processes, to keep track of the residual service and patience times of
each customer. Deterministic fluid models are established to provide first-order approximation for this model. The fluid model solution, which is proved to uniquely exists, serves as the fluid limit of the many-server queue, as the number of servers becomes large. Based on the fluid model solution, first-order approximations for various performance quantities are proposed.
