Formulas on Queues in Burst Processes--I

A convenient framework for an unified analysis of a variety of digital communication systems involving buffering-some are discussed later-is provided by the system in Fig. 1. The source emits data uniformly at the rate of one symbol per unit time. The transmission rate of the channel is (k + 1) symbols per unit time where k is some positive integer. The buffer has access to the channel only when the switch is closed. The switch is controlled by a burst process {Ej} j = 0, 1, 2, · · ·. Ej, for every j, is either 0 or 1. If Ej = 0, the switch is closed for the duration [ J , j + 1); otherwise the switch is open. The burst process is introduced to account for cases where two basically different types of phenomena are responsible for the event Ej = 0. There are relatively long periods during which Ej = 0 uniformly; the activity separated by such periods is defined to be a burst. On the other hand, during a burst, Ej = 0 only infrequently. The duration or length of a burst is a random variable. It is assumed that the burst length is independently distributed with a geometric or a weighted sum of geometric distributions. The interburst periods are assumed to be sufficiently long for the buffer to empty during these periods. The statistical assumption made in the paper about the controlling sequence [Ej] within a burst is that it is a Bernoulli sequence of independent random variables and P r { E j = 1} = IT where 0