Abstract
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.
Highlights
The present article is motivated by mathematical challenges that arise in the study of queueing systems that involve two types of customers that compete for the same server
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
While the calculations for the general case are cumbersome, in some fortunate cases, usually with a simple arrival process, there is little to no numerical work needed; this is for example possible in the discrete-time 2 × 2 switch model [6,7]
Summary
The present article is motivated by mathematical challenges that arise in the study of queueing systems that involve two types of customers that compete for the same server. We consider the heavy-traffic limit of the joint system content distribution of both models In our case, this means that we let the mean arrival rate of the models go to its critical value of instability such that the queues are nearly saturated. The first reason is that by considering the heavy-traffic limit, the boundary value method can be applied in a simpler manner compared to the non-heavy-traffic case, such that the numerical work is limited (model II) or even not necessary (model I) Since it is typical for heavy-traffic results to be rather insensitive to the exact form of the arrival (and service) process [20], we can assume a general batch arrival process for our queueing models.
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