Marginal queue length approximations for a two-layered network with correlated queues

TL;DR

This paper develops an approximation method for analyzing queue lengths in a layered network with correlated machine downtimes, extending classical models to account for dependence and providing highly accurate results.

Contribution

It introduces a novel approximation approach for marginal queue lengths in layered networks with correlated downtimes, combining explicit dependence assumptions with numerical validation.

Findings

The approximation accurately predicts queue length distributions.

Correlation in downtimes significantly impacts queue performance.

Numerical experiments confirm the method's high accuracy.

Abstract

We consider an extension of the classical machine-repair model, where we assume that the machines, apart from receiving service from the repairman, also serve queues of products. The extended model can be viewed as a layered queueing network, where the first layer consists of the queues of products and the second layer is the ordinary machine-repair model. Since the repair time of one machine may affect the time the other machine is not able to process products, the downtimes of the machines are correlated. This correlation leads to dependence between the queues of products in the first layer. Analysis of these queue length distributions is hard, since the exact dependence structure for the downtimes, or the queue lengths, is not known. Therefore, we obtain an approximation for the complete marginal queue length distribution of any queue in the first layer, by viewing such a queue as a single server queue with correlated server downtimes. Under an explicit assumption on the form of the downtime dependence, we obtain exact results for the queue length distribution for that single server queue. We use these exact results to approximate the machine-repair model. We do so by computing the downtime correlation for the latter model and by subsequently using this information to fine-tune the parameters we introduced to the single server queue. As a result, we immediately obtain an approximation for the queue length distributions of products in the machine-repair model, which we show to be highly accurate by extensive numerical experiments.