Relative stability analysis of multiple queueing systems

Abstract

Stability is always the most fundamental issue to consider in the development of any system with finite resources, simply because an unstable system is not operable in a real-world environment. In this research, we study the stability problems for single-server systems with multiple queues in which the server services the customers arriving at the queue according to some service policy. Our contribution is to tackle the stability problems from a relative stability point of view and, as a result, to obtain a number of new systems and queue stability results. We consider two kinds of stability problems in the single-server-multiple-queue systems (SSMQSs)?bsolute stability and relative stability. The absolute stability concerns the stability status of the objects under studied. The objects could be individual queues or the entire system, and an absolute stability analysis answers questions, such as whether the objects are stable for some given system inputs. The relative stability, on the other hand, concerns the stability relations among two or more queues and answers questions, such as whether some queues are more (or less) stable than others. There are three types of absolute stability problems: queue stability, system stability, and degree of stability. The queue/system stability problems aim at obtaining the queue/system stability conditions, and the degree of stability problem measures how stable a queue is. On the other hand, the relative stability problem aims at comparing the degree of stability for two or more queues. Moreover, it involves obtaining the conditions for a given relative stability relation to hold. Knowing the relative stability also helps determine the queue stability conditions. Therefore, our focus of this work is on the relative stability analysis of the SSMQS. Obtaining the relative stability results of the SSMQSs consists of several steps. First, we provide a criterion to classify the SSMQSs. This classification allows us to identify system models in which the degree of stability of a queue can be defined through indirectly utilizing the Loynes' theorem. The relative stability among the queues can then be defined for the models. The next step is to investigate useful properties of the models, and in particular, we discover properties related to the relative stability. One property is the sufficient and necessary relative stability conditions for any two queues in the models. Another is the existence of the maximum as stable as configuration of the system. Through these properties we can solve the relative stability problems that we have introduced completely. In addition, the properties also allow us to reformulate three problems-system stability region characterization, system stabilization, and achieving maximum stable throughput-into one single optimization problem and provide clue to solving the optimization problem. The relative stability properties are not only interesting and important in themselves but also essential to solving the queue stability problems. Since queue stability is more general than system stability, the relative stability is also useful to solving the system stability problems. To show the importance of the relative stability properties, we select four practical systems from a class of SSMQS models and investigate both absolute stability and relative stability conditions of these four systems. With the relative stability properties, we can see the approach to derive the stability conditions in the single-server-multiple-queue systems is unified and straightforward, though because of the complexity of some systems, the exact absolute stability conditions for those systems may not be found. Nevertheless, through the relative stability results we can always provide necessary stability conditions for the class of systems.