Tail asymptotics for the $$M_1,M_2/G_1,G_2/1$$ retrial queue with non-preemptive priority

Abstract

Stochastic networks with complex structures are key modelling tools for many important applications. In this paper, we consider a specific type of network: retrial queueing systems with priority. This type of queueing system is important in various applications, including telecommunication and computer management networks with big data. The system considered here receives two types of customers, of which Type-1 customers (in a queue) have non-pre-emptive priority to receive service over Type-2 customers (in an orbit). For this type of system, we propose an exhaustive version of the stochastic decomposition approach, which is one of the main contributions made in this paper, for the purpose of studying asymptotic behaviour of the tail probability of the number of customers in the steady state for this retrial queue with two types of customers. Under the assumption that the service times of Type-1 customers have a regularly varying tail and the service times of Type-2 customers have a tail lighter than Type-1 customers, we obtain tail asymptotic properties for the numbers of customers in the queue and in the orbit, respectively, conditioning on the server’s status, in terms of the exhaustive stochastic decomposition results. These tail asymptotic results are new, which is another main contribution made in this paper. Tail asymptotic properties are very important, not only on their own merits but also often as key tools for approximating performance metrics and constructing numerical algorithms.