Mean value and tail distribution of the message delay in statistical multiplexers with correlated train arrivals

In this paper, we study a statistical multiplexer which is modelled as a discrete-time single-server infinite-capacity queueing system. This multiplexer is fed by messages generated by an unbounded population of users. Each message consists of a generally distributed number of fixed-length packets. We assume the packet arrival process to exhibit simultaneously the following two types of correlation. First, the messages arrive to the multiplexer at the rate of one packet per slot, which results in what we call a primary correlation in the packet arrival process. Also, on a higher level, the arrival process contains an additional secondary correlation , resulting from the fact that the behaviour of the user population is governed by a two-state Markovian environment. Specifically, the state of this user environment in a particular slot determines the distribution of the number of newly generated messages in that slot. In previous work on this model, we provided analytical results for the moments and the tail distribution of the system contents. Using these results, we now concentrate on the message delay performance of this system, under the important assumption of a first-come-first-served queueing discipline for packets, whereby packets that arrive during the same slot are stored in random order. Closed-form expressions are derived for the mean value of both the total delay and the transmission time of an arbitrary message. Additionally, we provide a reasonably tight upper and lower bound for the tail probabilities of the message delay. By means of some numerical examples, we discuss the influence of the environment parameters on the delay performance.

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