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Abstract
This paper is concerned with a discrete-time $ Geo/G/1$ retrial queueing system with non-preemptive priority, working vacations and vacation interruption where the service times and retrial times are arbitrarily distributed. If an arriving customer finds the server free, his service commences immediately. Otherwise, he either joins the priority queue with probability $ α$, or leaves the service area and enters the retrial group (orbit) with probability $ \mathit{\bar{\alpha }}\left( = 1-\alpha \right)$. Customers in the priority queue have non-preemptive priority over those in the orbit. Whenever the system becomes empty, the server takes working vacation during which the server can serve customers at a lower service rate. If there are customers in the system at the epoch of a service completion, the server resumes the normal working level whether the working vacation ends or not (i.e., working vacation interruption occurs). Otherwise, the server proceeds with the vacation. Employing supplementary variable method and generating function technique, we analyze the underlying Markov chain of the considered queueing model, and obtain the stationary distribution of the Markov chain, the generating functions for the number of customers in the priority queue, in the orbit and in the system, as well as some crucial performance measures in steady state. Furthermore, the relation between our discrete-time queue and its continuous-time counterpart is investigated. Finally, some numerical examples are provided to explore the effect of various system parameters on the queueing characteristics.
Highlights
IntroductionIn the study of the classical queueing theory, it is always assumed that once an arriving customer finds the server temporarily unavailable (e.g., the server is busy, on vacation or under repair), he either joins the waiting line to2010 Mathematics Subject Classification
In the study of the classical queueing theory, it is always assumed that once an arriving customer finds the server temporarily unavailable, he either joins the waiting line to2010 Mathematics Subject Classification
In day-to-day queueing activities, we often meet the scenarios where the customers who cannot receive service immediately leave the service zone and rejoin the system to access the server after some random length of time
Summary
In the study of the classical queueing theory, it is always assumed that once an arriving customer finds the server temporarily unavailable (e.g., the server is busy, on vacation or under repair), he either joins the waiting line to2010 Mathematics Subject Classification. Discrete-time retrial queue, working vacations, vacation interruption, non-preemptive priority, Markov chain. ∗ Corresponding author: Shaojun Lan. SHAOJUN LAN AND YINGHUI TANG get service or leaves the service system forever. In day-to-day queueing activities, we often meet the scenarios where the customers who cannot receive service immediately leave the service zone and rejoin the system to access the server after some random length of time. In literature such queueing phenomena are referred to as retrial queues. For comprehensive discussions on retrial queues, we refer the reader to the survey papers by Yang and Templeton [48], Falin [17], Artalejo [2, 3] and Phung-Duc [33], and the monographs by Falin and Templeton [18], and Artalejo and Gomez-Corral [4]
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