Journal of Applied Mathematics and Stochastic Analysis
Volume 3 (1990), Issue 4, Pages 227-244
On some queue length controlled stochastic processes
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 May 1990; Revised 1 October 1990
Copyright © 1990 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.
The authors study the input, output and queueing processes in a general controlled single-server bulk queueing system. It is supposed that inter-arrival time, service time, batch size of arriving units and the capacity of the server depend on the queue length.
The authors establish an ergodicity criterion for both the queueing process with continuous time parameter and the embedded process, study their
transient and steady state behavior and prove ergodic theorems for some functionals of the input, output and queueing processes. The following results are
obtained: Invariant probability measure of the embedded process, stationary
distribution of the process with continuous time parameter, expected value of
a busy period, rates of input and output processes and the relative speed of convergence of the expected queue length. Various examples (including an optimization problem) illustrate methods developed in the paper.