Stability of a priority queueing system with customer transfers
Introduction
The queueing model of interest has types of customers: type 1, type 2, …, and type customers. The priority increases from type 1 customers to type customers, with type 1 customers having the lowest priority and type customers the highest priority. This paper finds simple conditions for the stability/instability of the queueing model.
The queueing model of interest can find applications in the design of emergency departments in healthcare systems and custom inspection systems. For example, in a hospital emergency department, patients are categorized into critical and non-critical groups. A patient in critical condition will be attended by a doctor, if one is available, as soon as the patient arrives. The status of a patient in non-critical condition may deteriorate and become critical. For a custom inspection system, perishable products such as food require immediate attention. Other products may be expedited while waiting and need to be inspected at the earliest available time. In such systems, items have different service priorities. The service priority of an item may increase while waiting.
Queueing systems with customer priorities and queueing systems with customer transfers have wide applications in manufacturing, computer networks, telecommunication systems, and vehicle traffic control. The study of such queueing systems is extensive. Existing works address issues related to system stability, optimal scheduling, routing, and performance analysis [1], [9], [10], [12], [14], [15], [16], [17]. For example, some of the existing works focus on system stability conditions, some on the stationary analysis of the queue length(s) and waiting times, and some on customer transfer strategies. The queueing model of interest is also related to, but not included in, stochastic transfer networks [2]. In addition, the literatures focus on a product-form solution, rather than stability conditions.
In the priority queueing system of interest, a customer transfer scheme is given. The model is different from those in the existing literature. As a consequence, the stability/instability conditions of the system are different from those of the existing models. An interesting result is that the stability/instability conditions depend only on the ratio of the sum of the arrival rates and the service rate of customers of the highest priority, i.e., the stability/instability conditions are independent of the service rates of lower priority customers and the transfer rates. That result can be useful in the design of such queueing systems.
The mean-drift method is the main mathematical tools used in this paper [5], [6], [7], [8], [11]. This method has been used in the past in the study of classification of Markov processes and queueing models. One of the keys in using this method is the construction of the Lyapunov (test) functions. In this paper, several Lyapunov functions are introduced and they lead to the findings of simple conditions for system stability and instability.
The remainder of the paper is organized as follows. The queueing model of interest is introduced in Section 2. The main results–stability and instability conditions–are presented. Section 3 gives proofs of the main results.
Section snippets
Queueing model and main results
The queueing model of interest consists of identical servers serving types of customers: type 1, type 2, …, and type customers. Type , and customers form queue , and , respectively. Type customers have the highest service priority, type the second highest service priority, …, and type 1 the lowest service priority. When a server is available, it chooses a customer from the non-empty queue of the highest priority and begins to serve it. If some servers are serving type
Proof of Theorem 1
In this proof of Theorem 1, the mean-drift method [6] is utilized. Theorem 1.18 in Chen [3] for ergodicity of Markov chains, Theorem 1 in Choi and Kim [4] for non-ergodicity, and Part (ii) of Theorem 2.2 in Tweedie [13] for recurrence are applied.
Proof of (1.1) The proof of (1.1) includes three steps: (i) Construction of a Lyapunov function (test function); (ii) Calculation of the mean drift; and (iii) Application of Theorem 1.18 in [3]. We begin with the selection of the parameters in the Lyapunov function.
Acknowledgements
The authors would like to thank an anonymous referee for valuable comments and suggestions. The research was partially supported by NSF of China under grant 70325004.
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