Sojourn time distributions in modulated G-queues with batch processing

Abstract

Quantiles on response times, given by probability distribution functions, are a critical metric for quality of service in computer networks as well as many other logistical systems. We derive explicit expressions in the time domain for the sojourn (or response) time probability distribution in a modulated, batched G-queue. More precisely, this queue is Markovian with arrival streams of both positive (normal) and negative customers. Arrivals occur in batches of geometric size and service completions also release batches of geometric size, truncated at the current queue length. All the queue's parameters are modulated by an independent, stationary, continuous time Markov chain. This highly complex queue is able to model many characteristics observed in modern distributed computer systems and telecommunications traffic, such as burstiness, autocorrelation and failures. However, previously, sojourn time distributions have not been obtained even for an MMPP/M/1 queue. We simplify a previous result for their Laplace transform which we then show takes a rational form and can be inverted to give a mixture of exponential and Erlang distributions, possibly modified with sine-factors. An algorithm is described which generates these Junctions from any given model parameterization and is applied to a range of problems to illustrate graphically the potentially diverse density functions that ensue.