The performance measures and randomized optimization for an unreliable server M [ x] /G/1 vacation system

This paper examines an M [ x] /G/1 queueing system with a randomized vacation policy and at most J vacations. Whenever the system is empty, the server immediately takes a vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be immedia...

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Bibliographic Details
Published in:Applied mathematics and computation Vol. 217; no. 21; pp. 8277 - 8290
Main Authors: Ke, Jau-Chuan, Huang, Kai-Bin, Pearn, Wen Lea
Format: Journal Article
Language:English
Published: Amsterdam Elsevier Inc 01.07.2011
Elsevier
Subjects:
Batch arrival queue
Breakdown
Calculus of variations and optimal control
Customer satisfaction
Exact sciences and technology
Mathematical analysis
Mathematical models
Mathematics
Minimum cost
Numerical analysis
Numerical analysis. Scientific computation
Numerical methods in mathematical programming, optimization and calculus of variations
Numerical methods in optimization and calculus of variations
Optimization
Queues
Randomized control
Reliability
Sciences and techniques of general use
Server breakdown
Servers
Size distribution
Supplementary variable technique
Vacations
Supplementary variable technique
Randomized control
Server breakdown
Vacations
Reliability
Batch arrival queue
Optimization method
Probability distribution
Poisson process
Numerical analysis
Queueing system
Applied mathematics
Distribution function
Variational calculus
Queue
Mathematical programming
ISSN:0096-3003, 1873-5649
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Summary:This paper examines an M [ x] /G/1 queueing system with a randomized vacation policy and at most J vacations. Whenever the system is empty, the server immediately takes a vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be immediately activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with probability p or leaves for another vacation with probability 1 − p. This pattern continues until the number of vacations taken reaches J. If the system is empty by the end of the Jth vacation, the server becomes idle in the system. Whenever one or more customers arrive at server idle state, the server immediately starts providing service for the arrivals. Assume that the server may meet an unpredictable breakdown according to a Poisson process and the repair time has a general distribution. For such a system, we derive the distributions of important system characteristics, such as system size distribution at a random epoch and at a departure epoch, system size distribution at busy period initiation epoch, the distributions of idle period, busy period, etc. Finally, a cost model is developed to determine the joint suitable parameters ( p ∗, J ∗) at a minimum cost, and some numerical examples are presented for illustrative purpose.
Bibliography:ObjectType-Article-2
SourceType-Scholarly Journals-1
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ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2011.03.008