A Single-Server Bulk-Service Queue With Varying Capacity And Erlang Input

Abstract

This paper deals with a bulk-service queuing model denoted by It is assumed that the interarrival times are i.i,d.r,v,’s each having the A-phase Erlangian distribution, the service is in bulk, and there is a single exponential server of capacity b. The random variable X denotes the number of units already present with the server at a service epoch, and if X = m (≤ b), then the server takes (b - m) more units or the whole queue length, whichever is smaller. Under the assumption that at a service epoch the server does not wait for the units to arrive if he finds the queue empty, we derive limiting distributions of queue sizes at random, post-departure, and pre-arrival epochs. The first two moments of the distributions are listed. The results are specialized to suit the case of deterministic input. One of the main features of this study is that the results are expressed in terms of a unique real root of the characteristic equation and are thus easy to compute numerically. A sample listing of the root is given. The model under discussion is potentially useful in transportation systems.

Résumé

Cet article traite d’un modèle de files d’attente à services en groupe dénoté par. On suppose que les interarrivées sont des variables aléatoires indépendantes et équidistribuées. Chacune d’elles obeit à une distribution d’Erlang avec A-phases, Le service est en groupe et il y a un serveur unique et exporteritiel de capacité b. La variable aléatoire X dénote le nombre d’unités déjà presént avec le serveur à l’époque d’un service. Si X=m (≤ b), alors le serveur prénd le minimum entre (b - m) et le reste de la file d’attente d’unités suppémentaires, Utilisant l’hypothèse qu’a l’époque d’un service, le serveur n’attend pas que les unités arrivent s’il trouve la file d’attente vide, on dérive la distribution en régime permanent de la grandeur de la file d’attente à trois époques: aléatoire, post-départ et pré-arrivées, Les deux premiers moments des distributions sont données, Les résultats sont ensuite specialisés au cas d’arrivées déterministiques. Une des caractéristiques importantes de cette étude est que les résultats sont exprimés en termes de la racine unique de l’équation caractéristique et ainsi peuvent être aisément calculeés numériquement. Une liste des racines pour différentes valeurs des paramètres est donnée, Le modèle discuté ici peut s’avérer utile pour l’étude des systémes de transports.

Additional information

Notes on contributors

N.S. Kambo

N.S. KAMBO received MA (Mathematics) from Delhi University, India, in 1961 and MASC AND PHD degrees in Industrial Engineering from University of Toronto in 1965 and 1968, respectively. He has held visiting appointments in Canada and Iraq. He is a member of the Operations Research Society of India. Currently, he is a professor of mathematics at the Indian Institute of Technology, New Delhi. His research interests include information theory, numerical analysis, and applications of optimization and statistical techniques.

M.L. Chaudhry

M.L. CHAUDHRY is a Professor in the Department of Mathematics and Computer Science at the Royal Military College of Canada, Kingston, Ontario. He was a senior post-doctoral Fellow at the University of Toronto. Since receiving his PHD in Operations Research from Kurnkshetra University, India, he has held visiting appointments at several institutes of learning. He has contributed articles to numerous international journals and is a member of several professional societies. His most recent contribution (co-authored with J.G.C. Templeton) is a book titled, A First Course in Bulk Queues, published by Wiley. Included among Professor Chaudhry’s current interests are applied probability and stochastic processes.