Elsevier

Operations Research Letters

Volume 36, Issue 6, November 2008, Pages 705-709
Operations Research Letters

Stability of a priority queueing system with customer transfers

https://doi.org/10.1016/j.orl.2008.06.007Get rights and content

Abstract

This paper is concerned with the stability of a preemptive priority queueing system with customer transfers. Conditions for the queueing system to be stable/unstable are found. An interesting result is that the stability/instability conditions are independent of the service rates of lower priority customers and the transfer rates.

Introduction

The queueing model of interest has N types of customers: type 1, type 2, …, and type N customers. The priority increases from type 1 customers to type N customers, with type 1 customers having the lowest priority and type N 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 s identical servers serving N types of customers: type 1, type 2, …, and type N customers. Type 1,2,, and N customers form queue 1,2,, and N, respectively. Type N customers have the highest service priority, type N1 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 j

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.

References (17)

  • B.D. Choi et al.

    Non-ergodicity criteria for denumerable continuous time Markov processes

    Operations Research Letters

    (2004)
  • I.J.B.F. Adan et al.

    Analysis of the asymmetric shortest queue problem with threshold jockeying

    Stochastic Models

    (1991)
  • X. Chao et al.

    Queueing Networks, Customers, Signals and Production Form Solutions

    (1999)
  • M.F. Chen

    On three classical problems for Markov chains with continuous time parameters

    Journal of Applied Probability

    (1991)
  • K.L. Chung

    Markov Chains with Stationary Transition Probabilities

    (1967)
  • J.W. Cohen

    The Single Server Queue

    (1982)
  • G. Fayolle et al.

    Topics in the Constructive Theory of Countable Markov Chains

    (1995)
  • F.G. Foster

    On stochastic matrices associated with certain queueing processes

    Annals of Mathematical Statistics

    (1953)
There are more references available in the full text version of this article.

Cited by (17)

  • Tail asymptotics for service systems with transfers of customers in an alternating environment

    2019, Operations Research Letters
    Citation Excerpt :

    Tang and Zhao [20] studied a tandem queue with feedback, wherein all customers in queue 2 could be transferred to queue 1 with a definite probability. Xie et al. [21–23] obtained the stationary distribution of queue lengths in a multi-class priority queueing system with customer transfers and examined the performance of service systems with priority upgrades. In this study, the specific service system being developed is well-motivated by applications in various fields.

  • Multi-server Queueing Model with Many Types of Priority Customers

    2023, Industrial Engineering and Management Systems
  • A QUEUEING SYSTEM WITH A BATCH MARKOVIAN ARRIVAL PROCESS AND VARYING PRIORITIES

    2022, Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika
View all citing articles on Scopus
View full text