Bounding of performance measures for a threshold-based queueing system with hysteresis

In this paper, we consider a K-server threshold-based queueing system with hysteresis in which the number of servers, employed for servicing customers, is governed by a forward threshold vector F= (F₁, F₂, …, F_K-1) (where F₁ < F₂ < … < F_K-1) and a reverse threshold vector R= (R₁, R₂, …, R_K-1) (where R₁ < R₂ < … < R_K-1). There are many applications where a threshold-based queueing system can be of great use. The main motivation for using a threshold-based approach in such applications is that they incur significant server setup, usage, and removal costs. And, as in most practical situations, an important concern is not only the system performance but rather its cost/performance ratio. The motivation for use of hysteresis is to control the cost during momentary fluctuations in workload. An important and distinguishing characteristic of our work is that in our model we consider the time to add a server to be non-negligible. This is a more accurate model, for many applications, than previously considered in other works. Our main goal in this work is to develop an efficient method for computing the steady state probabilities of a multi-server threshold queueing system with hysteresis, which will, in turn, allow computation of various performance measures.