Hybrid queueing systems with hysteretic bilevel control policies

This article analyzes a stochastic hybrid system with compound Poisson input, general batch service, and two vacation modes that operate in accordance with a so-called "bilevel hysteretic control." In the context of queues with vacations, most stochastic systems function either under multiple or single vacation regimes and they are initiated whenever the queue length becomes zero. In this model we combine the two modes in one hybrid system that switches to a multiple or single vacation modes (which consists of multiple or single segments), respectively, whenever the queue drops below some r₁ or falls into [r₁, r₂) upon the end of a service. The servicing facility then assumes some other activities. The main service is restored when the queue length reaches or exceeds some N (≥ r₂) upon the completion of a vacation segment. We use fluctuation analysis and semi-regenerative techniques to arrive at closed form functionals for the steady state probabilities of queueing processes both for discrete and for continuous time parameter processes and discuss special cases to demonstrate the applicability of the results. The latter can potentially be used to optimize the system with respect to parameters r₁, r₂ and N.