Performance Evaluation 14 (1992) 103-121 103 North-Holland A unified stochastic stream from periodic model for the packet sources Guy Latouche Uniuersitd Libre de Brt~elles-CP 212, Sdminaire de Thdorie des Probabilit~s, Boulet,ard du Triomphe, 1050 Bruxelles, Belgium V. Ramaswami Bell Communications Research Inc., Red Bank, NJ 07731, USA Received June 1989 Revised January 1991 Abstract Latouche, G. and V. Ramaswami, A unified stochastic model for the packet stream from periodic sources, Performance Evaluation 14 (1992) 103-121. Motivated by applications to performance analysis of packet communication systems with synchronous slotted cells, a unified stochastic model is developed to describe the packet stream generated by periodic sources. The model permits the incorporation of the statistical characteristics of the call setup instants as well as those of the holding time and packet generation scenarios of individual calls. The description of an individual call takes into account discretization due to packetization and can also accommodate very general scenarios. The stationary version of the aggregate packet stream from a collection of periodic sources is characterized, and algorithmically implementable formulae are derived for its first two moments. Through examples, we demonstrate that certain characteristics of the aggregate stream are significantly affected by the packet generation of individual calls. Keywords: packet communication, ATM, source models, multiplexing, periodicity, correlations, phase type models. I. Introduction The present work is motivated by packet communications systems over synchronous slotted channels, of which packetized speech over DS-1 [4,11,13] and broadband systems based on asynchronous transfer mode (ATM) [1,5] are examples. In such settings, while the communication medium operates syn- chronously using slotted cells (of fixed length), the input lines are asynchronous. Further, under fixed-rate sampling of the incoming analog signal, the successive packets in a talk spurt are separated by a constant interval equal to the time it takes to gather enough bits to form a packet. We develop a general stochastic model which incorporates both the asynchronous nature of call set-up instants as well as the discretization introduced by packetization of individual calls. Through an example, we demon- strate that the model can also incorporate variable rate coding schemes. Unlike in the circuit-switched setting where contentions for output channels arise only at call setup instants, under a statistical multiplexing scheme where the packets of many calls contend for cells through a common queue, contentions could occur at the packet level and can result in packet losses due to buffer overflow or in undesirable delays. To address the resulting performance issues, a careful understanding of the statistical properties of the aggregate packet stream is an essential first step. It is well recognized that the lockstep nature of packet generation within individual talk spurts can result in significant correlations in the aggregate stream; these in turn can significantly affect perfor- mance of the communication system. We present here a traffic process which is quite general and realistic. Our motivation for this process arises from the desire to develop a unified and versatile model, 0166-5316/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved