Algorithmic Discussions Of Distributions Of Numbers Of Busy Channels For GI/Geom/m/m Queues

Abstract

The high speed multi-access communication channels such as BISDN and ATM operate on a discrete-time basis rather than on a continuous-time basis. The usual Erlang loss formula developed for continuous-time is not valid for discrete-time queues, though models such as these can be derived from the corresponding discrete-time models. In this paper, we consider the discrete-time GI/Geom/m/m queue. Two variants of this model viz. late-arrival system with delayed access and early arrival system have been discussed and analyzed. Distributions of numbers of busy channels at various epochs have been obtained using the supplementary variable technique. The performance measures such as probability of loss, and average number of busy channels have been presented for interarrival distributions: geometric, deterministic, arbitrary, etc. It is hoped that the results obtained in this paper may provide useful information to designers of telecommunication systems, practitioners, and others.

Résumé

Les canaux de communication de multi-accès à haute vitesse tels que BISDN et ATM opèrent sur une base de temps discret plutôt que sur une base de temps continue. La formule classique d’Erlang, développée pour des temps continus n’est pas valide pour des files d’attente à temps discret. Cependant, dans quelques cas simples, le premier cas peut être dérivé du deuxième. Dans cet article, on considère la file d’attente discrète GI/Geom/m/m. Deux variantes de ce modèle, à savoir, des systèmes d’arrivée avancée sont discutées et analysées. Les distributions de nombre de canaux occupées à différentes époques ont été obtenues en utilisant une technique de variable supplémentaire. Les mesures de performance telles qu’une perte de probabilité, et un nombre moyen de canaux occupés ont été présentés pour des distributions entre-arrivées: géométrique, déterministe, arbitraire, etc. Il est espéré que les résultats obtenus dans cet article peuvent fournir une information utile aux modélistes de systèmes de télécommunication, utilisateurs, et à d’autres.

Key Words:

• ATM
• discrete-time
• Erlang loss probabilities
• recursive equations
• steady-state distribution
• supplementary variable

Mots-clés:

• ATM
• temps discret
• formule de pertes d’Erlang
• récursivité
• distribution à l’état d’équilibre
• variable supplémentaire

Additional information

Notes on contributors

M.L. Chaudhry

Mohan Chaudhry is a Distinguished Professor at the Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, Ontario. He is also a Distinguished Alumnus of Kurukshetra University. He has served and continues to serve in various professional societies in different capacities. He has won several awards and grants and has held appointments at several universities in India, Canada and USA. Currently, he is an adjunct professor at DalTech, University of Manitoba and University of Toronto. Having published over 6 dozen articles in numerous international Operations Research and Statistics journals, he has coauthored two books: (i) A First Course in Bulk Queues, 1983, Wiley, New York and (ii) An Introduction to Queueing Theory, 1988, A & A Publications, 395 Carrie Crescent, Kingston, Ontario, K7M 5X7. In addition, he has edited or coedited special issues of several journals and has prepared software packages for several queueing models.

U.C. Gupta

U.C. Gupta is currently Associate Professor in the Department of Mathematics, Indian Institute of Technology, Kharagur, India. He received the Ph.D degree from Indian Institute of Technology, Delhi in 1982. His research interests include analysis of continuous/discrete time bulk and non-bulk queues. He has published several research papers in international journals such Journal of Applied Probability, Performance Evaluation, Stochastic Processes and Their Applications, Queueing Systems, Operations Research Letters, INFOR, EJOR, OR Spektrum.