Stochastic Bounds for Queueing Systems with Multiple On–Off Sources

Consider a queueing system where the input traffic consists of background traffic, modeled by a Markov Arrival Process, and foreground traffic modeled by N ≥ 1 homogeneous on–off sources. The queueing system has an increasing and concave service rate, which includes as a particular case multiserver queueing systems. Both the infinite-capacity and the finite-capacity buffer cases are analyzed. We show that the queue length in the infinite-capacity buffer system (respectively, the number of losses in the finite-capacity buffer system) is larger in the increasing convex order sense (respectively, the strong stochastic order sense) than the queue length (respectively, the number of losses) of the queueing system with the same background traffic and M N homogeneous on–off sources of the same total intensity as the foreground traffic, where M is an arbitrary integer. As a consequence, the queue length and the loss with a foreground traffic of multiple homogeneous on–off sources is upper bounded by that with a single on–off source and lower bounded by a Poisson source, where the bounds are obtained in the increasing convex order (respectively, the strong stochastic order). We also compare N ≥ 1 homogeneous arbitrary two-state Markov Modulated Poisson Process sources. We prove the monotonicity of the queue length in the transition rates and its convexity in the arrival rates. Standard techniques could not be used due to the different state spaces that we compare. We propose a new approach for the stochastic comparison of queues using dynamic programming which involves initially stationary arrival processes.