Comparisons of Exhaustive and Nonexhaustive M/M/1/N Queues with Working Vacation and Threshold Policy
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This paper compares the performance of exhaustive and nonexhaustive M/M/1/N queues with working vacation and threshold policy. In an exhaustive queue, the server slows down its service rate only when no customers exist in the system, and turns to normal service until the number of customers achieves a threshold. However, in a nonexhaustive queue, the server switches service rate between a low and a high value depending on system congestion. To get equilibrium arrival rate of customers and social welfare for the two types of queues, we first derive queue length distributions and expected busy circle. Then, by making sensitivity analysis of busy circle, system cost, arrival rate and optimal social welfare, we find that customers tend to join exhaustive queues instead of nonexhaustive queues, and the optimal threshold in an exhaustive queue is probably inconsistent with the one in a nonexhaustive queue. Moreover, in general, whether to consider system cost or not in social welfare will obviously affect the tendencies of optimal arrival rate and optimal social welfare with the threshold and system capacity for the two types of queues, especially for the nonexhaustive queues, and then affect the final decisions of social planner or system manager.
KeywordsMarkovian queue service discipline limited capacity working vacations threshold policy equilibrium arrival rate busy circle social welfare system cost
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The authors would like to thank the anonymous reviewers for the useful comments on this work, and the support from the National Natural Science Foundation of China under Grant 71671159, the Humanity and Social Science Foundation of Ministry of Education of China under Grant 16YJC630106, the Natural Science Foundation of Hebei Province under Grants G2016203236 and G2018203302, and the project Funded by Hebei Education Department under Grants BJ2016063 and BJ2017029, and Hebei Talents Program under Grant A2017002108.
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