The Phenomenon of Secondary Flow Explosion in Retrial Priority Queueing System with Randomized Push-Out Mechanism

Abstract

We consider a single-server queueing system with finite buffer size, Poisson arrivals and exponentially distributed service time. If the arriving customer finds the completely filled queue of the system, the customer joins a special retrial waiting group (called the orbit) and after a random period of time that has an exponential distribution tries to come to the system again. Primary customers take priority over secondary customers. We also introduce the so-called randomized push-out buffer management mechanism. It allows primary customers to push secondary ones out of the system to free up space. Such a queueing system can be reduced to a similar model without retrials, which had been studied by the authors earlier. Using generating functions approach, we obtain loss probabilities for both types of customers. Theoretical results allow to investigate the dependence of the loss probabilities on the main parameters of the model (such as the push-out and retrial probabilities). We considered in details the cases of preemptive and non-preemptive priorities and discovered an interesting phenomenon. When the intensity of the primary flow increases smoothly after it reaches a certain critical value, an avalanche-like increase in the intensity of the secondary flow occurs (up to tens of thousands of times). In other words, there is a kind of “explosion” of the flow of secondary customers. This article is a strictly quantitative study of this phenomenon, which is of great interest in the calculation of telematic devices.