Journal of Applied Mathematics and Stochastic Analysis
Volume 13 (2000), Issue 4, Pages 415-427

Complex-analytic and matrix-analytic solutions for a queueing system with group service controlled by arrivals

Lev Abolnikov1 and Alexander Dukhovny2

1Loyola Marymount University, Department of Mathematics, Los Angeles 91316, CA, USA
2San Francisco State University, Department of Mathematics, San Francisco 94132, CA, USA

Received 1 February 2000; Revised 1 October 2000

Copyright © 2000 Lev Abolnikov and Alexander Dukhovny. 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.


A bulk M/G/1 system is considered that responds to large increases (decreases) of the queue during the service act by alternating between two service modes. The switching rule is based on two “up” and “down” thresholds for total arrivals over the service act. A necessary and sufficient condition for the ergodicity of a Markov chain embedded into the main queueing process is found. Both complex-analytic and matrix-analytic solutions are obtained for the steady-state distribution. Under the assumption of the same service time distribution in both modes, a combined complex-matrix-analytic method is introduced. The technique of “matrix unfolding” is used, which reduces the problem to a matrix iteration process with the block size much smaller than in the direct application of the matrix-analytic method.