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The mean-field computation in a supermarket model with server multiple vacations

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Abstract

While vacation processes are considered to be ordinary behavior for servers, the study of queueing networks with server vacations is limited, interesting, and challenging. In this paper, we provide a unified and effective method of functional analysis for the study of a supermarket model with server multiple vacations. Firstly, we analyze a supermarket model of N identical servers with server multiple vacations, and set up an infinite-dimensional system of differential (or mean-field) equations, which is satisfied by the expected fraction vector, in terms of a technique of tailed equations. Secondly, as N→ ∞ we use the operator semigroup to provide a mean-field limit for the sequence of Markov processes, which asymptotically approaches a single trajectory identified by the unique and global solution to the infinite-dimensional system of limiting differential equations. Thirdly, we provide an effective algorithm for computing the fixed point of the infinite-dimensional system of limiting differential equations, and use the fixed point to give performance analysis of this supermarket model, including the mean of stationary queue length in any server and the expected sojourn time that any arriving customer spends in this system. Finally, we use some numerical examples to analyze how the performance measures depend on some crucial factors of this supermarket model. Note that the method of this paper will be useful and effective for performance analysis of complicated supermarket models with respect to resource management in practical areas such as computer networks, manufacturing systems and transportation networks.

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Acknowledgements

The authors thank the two referees for their valuable comments and remarks, and acknowledge Professor Benny van Houdt for many valuable suggestions to sufficiently improve the system of differential equations derived in Section 2 of this paper. At the same time, Q.L. Li and Y. Wang thank that this research is partly supported by the National Natural Science Foundation of China (No. 71271187, No. 61001075) and the Hebei Natural Science Foundation of China (No. A2012203125).

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Authors and Affiliations

  1. School of Economics and Management Sciences, Yanshan University, Qinhuangdao, 066004, People’s Republic of China

    Quan-Lin Li & Guirong Dai

  2. Department of Computer Science & Engineering, The Chinese University of Hong Kong, Shatin, NT, Hong Kong

    John C. S. Lui

  3. Institute of Network Computing & Information Systems, Peking University, Beijing, 100871, People’s Republic of China

    Yang Wang

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  1. Quan-Lin Li
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  2. Guirong Dai
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  3. John C. S. Lui
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  4. Yang Wang
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Correspondence to Quan-Lin Li.

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Li, QL., Dai, G., Lui, J.C.S. et al. The mean-field computation in a supermarket model with server multiple vacations. Discrete Event Dyn Syst 24, 473–522 (2014). https://doi.org/10.1007/s10626-013-0171-5

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  • DOI: https://doi.org/10.1007/s10626-013-0171-5

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