Heavy-Traffic Comparison of a Discrete-Time Generalized Processor Sharing Queue and a Pure Randomly Alternating Service Queue

This paper compares two discrete-time single-server queueing models with two queues. In both models, the server is available to a queue with probability 1/2 at each service opportunity. Since obtaining easy-to-evaluate expressions for the joint moments is not feasible, we rely on a heavy-traffic limit approach. The correlation coefficient of the queue-contents is computed via the solution of a two-dimensional functional equation obtained by reducing it to a boundary value problem on a hyperbola. In most server-sharing models, it is assumed that the system is work-conserving in the sense that if one of the queues is empty, a customer of the other queue is served with probability 1. In our second model, we omit this work-conserving rule such that the server can be idle in case of a non-empty queue. Contrary to what we would expect, the resulting heavy-traffic approximations reveal that both models remain different for critically loaded queues.

Heavy-traffic approx­imations for a layered network with limited resources

HEAVY-TRAFFIC APPROXIMATIONS FOR A LAYERED NETWORK WITH LIMITED RESOURCESMotivated by a web-server model, we present a queueing network consisting of two layers. The first layer incorporates the arrival of customers at a network of two single-server nodes. We assume that the interarrival and the service times have general distributions. Customers are served according to their arrival order at each node and after finishing their service they can re-enter at nodes several times for another service. At the second layer, active servers act as jobs that are served by a single server working at speed one in a processor-sharing fashion. We further assume that the degree of resource sharing is limited by choice, leading to a limited processor-sharing discipline. Our main result is a diffusion approximation for the process describing the number of customers in the system. Assuming a single bottleneck node and studying the system as it approaches heavy traffic, we prove a state-space collapse property.

A unified approach for large queue asymptotics in a heterogeneous multiserver queue

We are interested in a large queue in a GI/G/k queue with heterogeneous servers. For this, we consider tail asymptotics and weak limit approximations for the stationary distribution of its queue length process in continuous time under a stability condition. Here, two weak limit approximations are considered. One is when the variances of the interarrival and/or service times are bounded, and the other is when they become large. Both require a heavy-traffic condition. Tail asymptotics and heavy-traffic approximations have been separately studied in the literature. We develop a unified approach based on a martingale produced by a good test function for a Markov process to answer both problems.