Interdeparture time distributions in ΣiMi/Gi/1 priority queues

This paper reviews existing results for the stationary interdeparture time distribution in the M/G/1 nonpreemptive and preemptive resume queues, and introduces a unified approach which exploits for the first time the common structure for the interdeparture time process that is present in all classical preemptive priority service disciplines. This approach confirms previously known results for the preemptive resume discipline, and presents new results for several variants of the preemptive repeat model. Exact expressions for the squared coefficient of variation of the interdeparture time distribution are also provided. Several numerical examples are given and discussed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Exact tail asymptotics in a priority queue—characterizations of the non-preemptive model

This is a companion paper to Li and Zhao (Queueing Syst. 63:355–381, 2009) recently published in Queueing Systems, in which the classical preemptive priority queueing system was considered. In the current paper we consider the classical non-preemptive priority queueing system with two classes of independent Poisson customers and a single exponential server serving the two classes of customers at possibly different rates. A complete characterization of the regions of system parameters for exact tail asymptotics is obtained through an analysis of generating functions. This is done for the joint stationary distribution of the queue length of the two classes of customers, for the two marginal distributions and also for the distribution of the total number of customers in the system, respectively. This complete characterization is supplemental to the existing literature, which would be useful to researchers.     

Exact tail asymptotics in a priority queue—characterizations of the preemptive model

In this paper, we consider the classical preemptive priority queueing system with two classes of independent Poisson customers and a single exponential server serving the two classes of customers at possibly different rates. For this system, we carry out a detailed analysis on exact tail asymptotics for the joint stationary distribution of the queue length of the two classes of customers, for the two marginal distributions and for the distribution of the total number of customers in the system, respectively. A complete characterization of the regions of system parameters for exact tail asymptotics is obtained through analysis of generating functions. This characterization has never before been completed. It is interesting to note that the exact tail asymptotics along the high-priority queue direction is of a new form that does not fall within the three types of exact tail asymptotics characterized by various methods for this type of two-dimensional system reported in the literature. We expect that the method employed in this paper can also be applied to the exact tail asymptotic analysis for the non-preemptive priority queueing model, among other possibilities.     

An optimal repair–replacement policy for a cold standby system with use priority

In this paper, the maintenance problem for a cold standby system consisting of two dissimilar components and one repairman is studied. Assume that both component 1 and component 2 after repair follow geometric process repair and component 1 is given priority in use when both components are workable. Under these assumptions, using geometric process repair model, we consider a replacement policy N under which the system is replaced when the number of failures of component 1 reaches N. Our purpose is to determine an optimal replacement policy N1 such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimized. The explicit expression for the average cost rate of the system is derived and the corresponding optimal replacement policy N1 can be determined analytically or numerically. Finally, a numerical example is given to illustrate some theoretical results and the model applicability.     

On the stationary distribution of queue lengths in a multi-class priority queueing system with customer transfers

This paper deals with a multi-class priority queueing system with customer transfers that occur only from lower priority queues to higher priority queues. Conditions for the queueing system to be stable/unstable are obtained. An auxiliary queueing system is introduced, for which an explicit product-form solution is found for the stationary distribution of queue lengths. Sample path relationships between the queue lengths in the original queueing system and the auxiliary queueing system are obtained, which lead to bounds on the stationary distribution of the queue lengths in the original queueing system. Using matrix-analytic methods, it is shown that the tail asymptotics of the stationary distribution is exact geometric, if the queue with the highest priority is overloaded.      

Time-dependent analysis of an <Emphasis Type="Italic">M</Emphasis> / <Emphasis Type="Italic">M</Emphasis> / <Emphasis Type="Italic">c</Emphasis> preemptive priority system with two priority classes

We analyze the time-dependent behavior of an M / M / c priority queue having two customer classes, class-dependent service rates, and preemptive priority between classes. More particularly, we develop a method that determines the Laplace transforms of the transition functions when the system is initially empty. The Laplace transforms corresponding to states with at least c high-priority customers are expressed explicitly in terms of the Laplace transforms corresponding to states with at most \(c - 1\) high-priority customers. We then show how to compute the remaining Laplace transforms recursively, by making use of a variant of Ramaswami’s formula from the theory of M / G / 1-type Markov processes. While the primary focus of our work is on deriving Laplace transforms of transition functions, analogous results can be derived for the stationary distribution; these results seem to yield the most explicit expressions known to date.