Analysis of multiserver retrial queueing system: A martingale approach and an algorithm of solution

Abstract

Abstract. The paper studies a multiserver retrial queueing system with m servers. Arrival process is a point process with strictly stationary and ergodic increments. A customer arriving to the system occupies one of the free servers. If upon arrival all servers are busy, then the customer goes to the secondary queue, orbit, and after some random time retries more and more to occupy a server. A service time of each customer is exponentially distributed random variable with parameter µ1. A time between retrials is exponentially distributed with parameter µ2 for each customer. Using a martingale approach the paper provides an analysis of this system. The paper establishes the stability condition and studies a behavior of the limiting queue-length distributions as µ2 increases to infinity. As µ2 → ∞, the paper also proves the convergence of appropriate queue-length distributions to those of the associated 'usual' multiserver queueing system without retrials. An algorithm for numerical solution of the equations, associated with the limiting queue-length distribution of retrial systems, is provided.

Multiserver queueing systems with retrials and losses

“…Jo Clearly, the left-hand side of relation (2.1) is the total number of customers in the system including customers in service, one in orbit (in the case of all servers being occupied) as well as all losses up to time t. Observe that the right-hand side of (2.1) is the same as the right-hand side of (2.1) in [1]. The first term of the right-hand side of (2.2) represents the total number of arrivals to the busy system where all servers are occupied.…”

“…The second term of the right-hand side of (2.2) is the number of successful retrial customers up to time t subtracted from the total number of arrivals to the busy system during the same time t. Successful retrials occur in the cases where immediately before retrial instants there is at least one busy server. Notice that relation (2.2) is structured similarly to the corresponding relation (2.2) of [1]. Relation (2.3) characterizes the number of losses.…”

“…The jump of a point process is denoted by adding A. 1 For example, AA(/) is a jump of A(t), An t (t) is a jump for n t (t) and so on.…”

“…Recently Abramov [1] studied the main multiserver retrial queueing system, where the input stream was a quite general point process (see [1] for more details). The martingale approach that has been used there can be adapted to systems with retrials and losses as well, and the present paper is devoted to analysis of these systems.…”

“…In the case of the main multiserver [3] Multiserver queueing systems with retrials and losses 299 retrial queueing system considered in [1], the point process A(t) must satisfy the condition llim --=X <nix\ = 1,…”