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Equilibrium and optimal behavior of customers in Markovian queues with multiple working vacations

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Abstract

This paper studies the customers’ equilibrium and socially optimal joining–balking behavior in single-server Markovian queues with multiple working vacations. Different from the classical vacation policies, the server does not completely stop service but maintains a low service rate in vacation state in case there are customer arrivals. Based on different precision levels of the system information, we discuss the observable queues, the partially observable queues, and the unobservable queues, respectively. For each type of queues, we get both the customers’ equilibrium and socially optimal joining–balking strategies and make numerical comparisons between them. We numerically observe that their equilibrium strategy is unique, and especially, the customers’ equilibrium joining probability in vacation state is not necessarily smaller than that in busy state in the partially observable queues. Moreover, we also find that the customers’ individual behavior always deviates from the social expectation and makes the system more congested.

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Notes

  1. Because of the memoryless property of the exponential distribution, both V and V R follow the same distribution.

  2. For the observable queues, we can obviously find that the expected net benefit functions U e (i) (i=1,2) are decreasing in n so that our search for equilibrium is restricted in the class of threshold strategies.

  3. We directly assume that n(0)≤n(1) in the transition diagram since μ v <μ b , i.e., the opposite case cannot occur.

  4. It is easy to prove that one root \(r_{v}=\bigl(\lambda q_{0}+\theta+\mu_{v}-\sqrt{(\lambda q_{0}+ \theta+\mu_{v})^{2}-4\mu_{v}\lambda q_{0}}\bigr)/2\mu_{v}\in(0,1)\); then substituting r v into (4.7), we get (4.8).

  5. Since we focus on the customers’ decisions from an economic viewpoint, we need to replace the arrival rate λ exists in the stationary results derived by Liu et al. (2007) with λq, who studied service systems from the classical viewpoint of queueing theory.

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Acknowledgements

The authors would like to thank the support from National Natural Science Foundation of China (No. 71101124, No. 71171174), the Natural Science Foundation of Hebei Province (No. G2012203068, No. F2013203136), the Humanity and Social Science Foundation of Ministry of Education of China (No. 12YJC790101), and the China Postdoctoral Science Foundation (No. 2012M520598).

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  1. School of Economics and Management, Yanshan University, Qinhuangdao, 066004, China

    Wei Sun & Shiyong Li

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  1. Wei Sun
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  2. Shiyong Li
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Correspondence to Shiyong Li.

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Sun, W., Li, S. Equilibrium and optimal behavior of customers in Markovian queues with multiple working vacations. TOP 22, 694–715 (2014). https://doi.org/10.1007/s11750-013-0288-6

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