Journal of Applied Mathematics and Stochastic Analysis
Volume 5 (1992), Issue 3, Pages 237-260

On a multilevel controlled bulk queueing system MX/Gr,R/1

Lev Abolnikov1 and Jewgeni H. Dshalalow2

1Department of Mathematics, Loyola Marymount University, Los Angeles 90045, CA, USA
2Department of Applied Mathematics, Florida Institute of Technology, Melbourne 32901, FL, USA

Received 1 December 1991; Revised 1 July 1992

Copyright © 1992 Lev Abolnikov and Jewgeni H. Dshalalow. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The authors introduce and study a class of bulk queueing systems with a compound Poisson input modulated by a semi-Markov process, multilevel control service time and a queue length dependent service delay discipline. According to this discipline, the server immediately starts the next service act if the queue length is not less than r; in this case all available units, or R (capacity of the server) of them, whichever is less, are taken for service. Otherwise, the server delays the service act until the number of units in the queue reaches or exceeds level r.

The authors establish a necessary and sufficient criterion for the ergodicity of the embedded queueing process in terms of generating functions of the entries of the corresponding transition probability matrix and of the roots of a certain associated functions in the unit disc of the complex plane. The stationary distribution of this process is found by means of the results of a preliminary analysis of some auxiliary random processes which arise in the “first passage problem” of the queueing process over level r. The stationary distribution of the queueing process with continuous time parameter is obtained by using semi-regenerative techniques. The results enable the authors to introduce and analyze some functionals of the input and output processes via ergodic theorems. A number of different examples (including an optimization problem) illustrate the general methods developed in the article.