Abstract
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.
Highlights
In this paper, we continue our investigation of sample-path conditions for rate stability in general input-output processes
This paper provides a generalization of stability results given by Stidham and E1-Taha [17]
We provide a counterexample to confirm an assertion that the workload process exhibits a stronger form of rate stability that the queue length process
Summary
We continue our investigation of sample-path conditions for rate stability in general input-output processes. We establish conditions for rate stability that can be verified from information on input (primary) processes in a deterministic framework that makes it possible to characterize the sample-path behavior of non-stationary stochastic processes. E1-Taha and Stidham [3, 4, 5] provide a sample-path characterization of (rate) stability and establish connections between rate stability and other measures of interest, such as the finiteness of the limiting average number of customers in a queueing system. Stidham and EI-Taha [17] consider an input-output process with a single output stream and establish rate stability conditions using only sample-path information available from primary processes.
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