Abstract
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.
Highlights
Priority models with multiple servers constitute an important class of queueing systems, having applications in areas as diverse as manufacturing, wireless communication and the service industry
Our method first makes use of a slight tweak of the clearing analysis on phases (CAP) method featured in Doroudi et al [10], in that we show how CAP can be modified to study Laplace transforms of transition functions
While the focus of our work is on Laplace transforms of transition functions, analogous results can be derived for the stationary distribution of the M/M/c 2-class preemptive priority model as well
Summary
Priority models with multiple servers constitute an important class of queueing systems, having applications in areas as diverse as manufacturing, wireless communication and the service industry. While the focus of our work is on Laplace transforms of transition functions, analogous results can be derived for the stationary distribution of the M/M/c 2-class preemptive priority model as well. The Laplace transforms we derive can be numerically inverted to retrieve the transition functions with the help of the algorithms of Abate and Whitt [3,4] or den Iseger [9] These transition functions can be used to study—as a function of time—key performance measures such as the mean number of customers of each priority class in the system; the mean total number of customers in the system; or the probability that an arriving customer has to wait in the queue. Having explicit expressions for the Laplace transforms of the transition functions greatly simplifies the computation of some performance measures These transforms yield explicit expressions for the Laplace transforms of the distribution of the number of low-priority customers in the system at time t. The appendices provide supporting results on combinatorial identities and single-server queues used in deriving the expressions for the Laplace transforms
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
