Explicit Characterization of Stability Region for Stationary Multi-Queue Multi-Server Systems

Abstract

In this paper, we characterize the network stability region (capacity region) of multi-queue multiserver (MQMS) queueing systems with stationary channel distribution and stationary arrival processes.The stability region is specified by a finite set of linear inequalities. We first show that the stability region is a polytope characterized by the finite set of its facet defining hyperplanes. We explicitly determine the coefficients of the linear inequalities describing the facet defining hyperplanes of the stability region polytope. We further derive the necessary and sufficient conditions for the stability of the system for general arrival processes with finite first and second moments. For the case of stationary arrival processes, the derived conditions characterize the system stability region. Furthermore, we obtain an upper bound for the average queueing delay of Maximum Weight (MW) server allocation policy which has been shown in the literature to be a throughput optimal policy for MQMS systems. Using a similar approach, we can characterize the stability region for a fluid model MQMS system. However, the stability region of the fluid model system is described by an infinite number of linear inequalities since in this case the stability region is a convex surface. We present an example where we show that in some cases depending on the channel distribution, the stability region can be characterized by a finite set of non-linear inequalities instead of an infinite number of linear inequalities.

Sensitivity to Cumulative Perturbations for a Class of Piecewise Constant Hybrid Systems

“…1 for a simple example). This policy and its properties, e.g., stability [8], [9], [10], [11] and state space collapse [12], [13], have been studied extensively over the last three decades.…”

The Role of Information in System Stability with Partially Observable Servers

“…Performance evaluation and control of queueing systems subject to randomly varying environments is an area of research that has received much attention during the past few decades (see for example, [4], [8], [23] and references there-in). This is because numerous situations arise in practice 5 where a controller needs to decide how to best utilise resources, and these are often subject to changing conditions. Examples of such situations arise in wireless communication, supply chain logistics, health care, manufacturing and transportation.…”

“…In such a situation, it is already a non-trivial task to carry out explicit system performance analysis (see [9] as an example). Further, finding optimal or even merely stabilizing control is typically a formidable achievement (see for example [1], [25] or the more recent [5]). But in practice, the actual environment state is 20 often not a directly observed quantity, or is at best only partially observable.…”

Control and inference of structured Markov models

“…For instance, see [78,106,188] where the stability region of a fully observed system was characterised. Even in this situation, performing explicit system analysis and finding an optimal policy is not a trivial task, see for instance [23,107,188], or the more recent [78]. However, in practice, the environment states are often not directly observed.…”

“…In such a situation, it is already a non-trivial task to carry out explicit system performance analysis (see [107] as an example). Further, finding optimal or even merely stabilizing control is typically a formidable achievement (see for example [23], [188] or the more recent [78]). But in practice, the actual environment state is often not a directly observed quantity or is at best only partially observable.…”