On Queueing with Customer Impatience Until the End of Service

Abstract

We study queueing systems where customers have strict deadlines until the end of their service. An analytic method is given for the analysis of a class of such queues, namely, M(n)/M/1 + G models with ordered service. These are single-server queues with state-dependent Poisson arrival process, exponential service times, FCFS service discipline, and general customer impatience. We derive a closed-form solution for the conditional probability density function of the offered sojourn time, given the number of customers in the system. This is a novel result that has not been seen before. Using this result, we show how the probability measure induced by the offered sojourn time is computed, and consequently how the probability of missing deadline and the probability of blocking of a customer are obtained. We also show how the probability of loss of a customer may be affected by various types of customer impatience. These are further illustrated through a numerical example.