Formulas on Queues in Burst Processes-I

Abstract

Queues arising in buffers due to either random interruptions of the channel or variable source rates are analyzed in the framework of a single switched system. Examples of systems to which the results of the paper may be applied are: multiplexing of speech with data in telephone channels and, in certain instances, buffering of data generated by the coding of moving images in the Picturephone® system. The switched system consists of a uniform source, buffer, switch and channel. The source feeds data to the buffer at a uniform rate. The buffer's access to the channel is controlled by the switch; if the switch is closed, the buffer empties to the extent of the channel's transmission rate. The on-off pattern of the switch is indicated by a 0 — 1 burst process {E <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</inf> } j = 0, 1, 2, − − − if E <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</inf> = 0, the switch is closed for the duration [j, j + 1). The burst phenomenon is introduced to account for two different processes responsible for the event E <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</inf> = 0. There are relatively long periods during which E <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</inf> = 0 uniformly, and the activity separated by such periods is defined to be a burst. During a burst, E <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</inf> = 0 only infrequently. The duration of a burst is an independently distributed random variable with a geometric or weighted sum of geometric distributions. The inter-burst periods are assumed to be sufficiently long for the buffer to empty at some point during these periods of inactivity. During a burst {E <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</inf> } is a Bernoulli sequence of independent random variables. Exact expressions for a variety of performance functionals related to the system described above are obtained, together with qualitative results. Recursive formulas are obtained for the following: (i) steady-state distribution of buffer content for a finite buffer of size N; (ii) mean time for first passage across a level N; (iii) the probability of overflow, for a given level N during a burst; (iv) mean time for first passage across a level N during a burst. The recursion in each case is with respect to N. The asymptotic behavior of the main recursions is determined.