On Various Age and Residual Life Distributions Associated with the M/G/1 Queue

Abstract

We show how a simple modification of the time-dependent Little’s law can be used to study various types of joint age and residual life distributions associated with any customer waiting in line in equilibrium in a work-conserving M/G/1 queue operating under the first-come-first-served service discipline, not necessarily the customer receiving service. This addresses an open question posed at the end of (Adan and Haviv, Stochastic Models, 2009). We also analyze the joint distribution of the number of customers in the system at time t, the remaining amount of work possessed by the customer currently in service at time t, the amount of work that has already been processed by the customer currently in service at time t, and the amount of time the current customer in service at time t spent waiting in the queue.