Queues with Correlated Service and Inter-Arrival Times and Their Application to Optical Buffers

Abstract

This paper presents an algorithmic procedure to calculate the queue length and delay distribution of customers in a discrete time D-MAP/PH/1 queue, where the service time distribution of a customer depends on the inter-arrival time between himself and his predecessor. Setting up a Markov chain that keeps track of the contents of such a queue will result in a state space explosion as the inter-arrival times of all customers present in the system must be remembered. We avoid these difficulties by making use of the age process, a process that keeps track of the “age” of the customer in the service facility. From this process, which we solve by means of matrix analytic methods, we compute the queue length and sojourn time distribution by means of a simple formula and obtain an expression for the stability of the system. We also demonstrate that the D-MAP arrival process can be easily replaced by the more general semi-Markovian arrival process, without any additional computational costs. Queueing systems of this type arise in the domain of synchronous optical buffers. Based on the numerical analysis of such a queueing system, some guidelines for the design of optical buffers are presented. We also show the impact on the numerical results when the cross-correlation that exists between the service and inter-arrival times is neglected.