A Geo / Geo /1 Inventory Priority Queue with Self Induced Interruption
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A Geo/Geo/1 Inventory Priority Queue with Self Induced Interruption M. P. Anilkumar1,2 · K. P. Jose2 © Springer Nature India Private Limited 2020
Abstract This paper considers discrete time (s, S) inventory Geo/Geo/1 priority queue formed by the customers’ induced interruption during service. An arriving customer enters into a high priority queue of infinite capacity. During his/her service, the service may be interrupted due to own reasons with some probability. The interrupted customer is moved to a lower priority queue of infinite capacity. An item in the inventory is supplied whenever a customer leaves from high priority queue after interrupting/completing the service. The customer in the lower priority queue is served according to preemptive priority discipline. Inter-arrival time, service time and lead time are considered to be geometrically distributed. We use the matrix analytic method to analyse the model. The necessary and sufficient condition for the stability of the system is obtained. The marginal distributions of both higher and lower priority queue lengths are studied. Numerical experiments are incorporated to draw special attention to the importance of the model. Keywords Geometric distribution · Preemptive priority · Matrix-analytic method · Stability · Marginal probability distributions · Cost function Mathematics Subject Classification 60K25 · 90B05 · 91B70
Introduction The concept of priority queues is introduced by White and Christie [24] with Poisson arrivals and exponential service rates. Priority queues are mainly classified into two; Preemptive and Non-preemptive priorities, depending on whether the service of the ongoing low priority is interrupted or not by the arrival of a priority customer during the service. The service of the interrupted customer may or may not start from the point where it was stopped.
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K. P. Jose [email protected] M. P. Anilkumar [email protected]
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T. M. Govt. College, Tirur, India
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P. G. & Research Department of Mathematics, St. Peter’s College, Kolenchery, Kerala 682 311, India 0123456789().: V,-vol
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Int. J. Appl. Comput. Math
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The resumption of service in the preemptive queue at the interrupted stage is discussed by Jaiswal [14]. Priority queues in discrete time were introduced by Alfa [2] in which the Matrixgeometric method is used for the methodology of the work. In that paper, both priority and nonpriority queues have infinite capacity and arrivals are modeled by Marked Markovian Arrival Process (MMAP). The correlation of inter-arrival time within each priority class and between two priority classes of jobs are considered. The service time of different classes follows a phase-type distribution with different parameters. The author extended the structure of the rate matrix R obtained in the continuous case by Miller [21] to a discrete case. Alfa et al. [4] generalized the preemptive case to three class priority queues and compared the probability distributions of queue length as well as waiting time of low pr
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