Journal of Applied Mathematics and Stochastic Analysis
Volume 7 (1994), Issue 3, Pages 437-456

Sample-path stability conditions for multiserver input-output processes

Muhammad El-Taha1 and Shaler Stidham Jr.2

1University of Southern Maine, Department of Mathematics and Statistics, 96 Falmouth Street, Portland 04103, ME, USA
2University of North Carolina, Department of Operations Research, CB 3180, Smith Building, Chapel Hill 27599-3180, NC, USA

Received 1 April 1993; Revised 1 March 1994

Copyright © 1994 Muhammad El-Taha and Shaler Stidham. 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.


We extend our studies of sample-path stability to multiserver input-output processes with conditional output rates that may depend on the state of the system and other auxiliary processes. Our results include processes with countable as well as uncountable state spaces. We establish rate stability conditions for busy period durations as well as the input during busy periods. In addition, stability conditions for multiserver queues with possibly heterogeneous servers are given for the workload, attained service, and queue length processes. The stability conditions can be checked from parameters of primary processes, and thus can be verified a priori. Under the rate stability conditions, we provide stable versions of Little's formula for single server as well as multiserver queues. Our approach leads to extensions of previously known results. Since our results are valid pathwise, non-stationary as well as stationary processes are covered.