International Journal of Mathematics and Mathematical Sciences
Volume 17 (1994), Issue 3, Pages 571-586

First passage processes in Queuing system MX/Gr/1 with service delay discipline

Lev Abolnikov,1 Jewgeni H. Dshalalow,2 and Alexander M. Dukhovny3

1Department of Mathematics, Loyola Marymount University, Los Angeles 90045, CA, USA
2Department of Applied Mathematics, Florida Institute of Technology, Melbourne 32901, FL, USA
3Department of Mathematics, San Francisco State University, San Francisco 94132, CA, USA

Received 1 February 1992

Copyright © 1994 Lev Abolnikov et al. 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.


This article deals with a general single-server bulk queueing system with a server waiting until the queue will reach level r before it starts processing customers. If at least r customers are available the server takes a batch of the fixed size r of units for service. The input stream is assumed to be a compound Poisson process modulated by a semi-Markov process and with a multilevel control of service time.

The authors evaluate the steady state probabilities of the queueing processes with discrete and continuous time parameter preliminarily establishing necessary and sufficient conditions for the ergodicity of the processes. The authors use the recent results on the first excess level processes to explicitly find all characteristics of the named processes. Some characteristics of the input process, service cycle, intensity of the system, and both idle and busy periods are also found. The results obtained in the article are illustrated by numerous examples.