Journal of Applied Mathematics and Stochastic Analysis
Volume 9 (1996), Issue 2, Pages 185-204

Queueing system with passive servers

Alexander N. Dudin and Valentina I. Klimenok

Byelorussian State University, Department of Applied Mathematics, Minsk, Belarus

Received 1 January 1995; Revised 1 January 1996

Copyright © 1996 Alexander N. Dudin and Valentina I. Klimenok. 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.


In this paper the authors introduce systems in which customers are served by one active server and a group of passive servers. The calculation of response time for such systems is rendered by analyzing a special kind of queueing system in a synchronized random environment. For an embedded Markov chain, sufficient conditions for the existence of a stationary distribution are proved. A formula for the corresponding vector generating function is obtained. It is a matrix analog of the Pollaczek-Khinchin formula and is simultaneously a matrix functional equation. A method for solving this equation is proposed.