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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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References

  • Alfa AS (2003) Vacation models in discrete time. Queueing Syst 44:5–30

    Article  MATH  MathSciNet  Google Scholar 

  • Benaim M, Le Boudec JY (2008) A class of mean-field interaction models for computer and communication systems. Perform Eval 65:823–838

    Article  Google Scholar 

  • Bordenave C, McDonald D, Proutiere A (2009) A particle system in interaction with a rapidly varying environment: mean-field limits and applications. Available: arXiv:0701363v3

  • Le Boudec JY, McDonald D, Mundinger J (2007) A generic mean-field convergence result for systems of interacting objects. In: Proc. Conf. IEEE on the quantitative evaluation of systems, pp 3–18

  • Bramson M (2008) Stability of queueing networks. Springer-Verlag, New York

    MATH  Google Scholar 

  • Bramson M (2011) Stability of join the shortest queue networks. Ann Appl Probab 21:1568–1625

    Article  MATH  MathSciNet  Google Scholar 

  • Bramson M, Lu Y, Prabhakar B (2010) Randomized load balancing with general service time distributions. In: Proceedings of the ACM SIGMETRICS international conference on measurement and modeling of computer systems, pp 275–286

  • Bramson M, Lu Y, Prabhakar B (2011) Decay of tails at equilibrium for FIFO join the shortest queue networks. Available: arXiv:1106.4582

  • Bramson M, Lu Y, Prabhakar B (2012) Asymptotic independence of queues under randomized load balancing. Queueing Syst 71:247–292

    Article  MATH  MathSciNet  Google Scholar 

  • Doshi BT (1986) Queueing system with vacation: a survey. Queueing Syst 1:29-66

    Article  MATH  MathSciNet  Google Scholar 

  • Doshi BT (1990) Generalization of the stochastic decomposition results for single server queues with vacations. Stoch Model 6:307–333

    MATH  MathSciNet  Google Scholar 

  • Dshalalow JH (1995) Advances in queueing theory, methods and open problems. CRC Press, Boca Raton

    MATH  Google Scholar 

  • Dshalalow JH (1997) Frontiers in queueing models and applications in science and engineering. CRC Press, Boca Raton

    MATH  Google Scholar 

  • Ethier SN, Kurtz TG (1986) Markov processes: characterization and convergence. John Wiley & Sons, New York

    Book  MATH  Google Scholar 

  • Fleming W (1977) Functions of several variables, 2nd edn. Springer, New York

    Book  MATH  Google Scholar 

  • Foss S, Chernova N (1998) On the stability of a partially accessible multi-station queue with state-dependent routing. Queueing Syst 29:55–73

    Article  MATH  MathSciNet  Google Scholar 

  • Fuhrmann SW, Cooper RB (1985) Stochastic decomposition in the M/G/1 queue with generalized vacation. Oper Res 33:1117–1129

    Article  MATH  MathSciNet  Google Scholar 

  • Gast N, Gaujal B (2009) A mean-field approach for optimization in particles systems and applications. Technical Report N0 6877, INRIA Rocquencourt, Frence

  • Gast N, Gaujal B (2012) Markov chains with discontinuous drift have differential inclusions limits. Technical Report N0 7315, INRIA Rocquencourt, Frence

  • Gast N, Gaujal B, Le Boudec JY (2011) Mean-field for Markov decision processes: from discrete to continuous optimization. Technical Report N0 7239, INRIA Rocquencourt, Frence

  • Graham C (2000a) Kinetic limits for large communication networks. In: Bellomo N, Pulvirenti M (eds) Modelling in applied sciences. Birkhäuser, pp 317–370

  • Graham C (2000b) Chaoticity on path space for a queueing network with selection of the shortest queue among several. J Appl Probab 37:198–201

    Article  MATH  MathSciNet  Google Scholar 

  • Graham C (2004) Functional central limit theorems for a large network in which customers join the shortest of several queues. Probab Theory Relat Fields 131:97–120

    Article  Google Scholar 

  • Hale JK (1980) Ordinary differential equations. Roberte E. Krieger Publishing, Melbourne

    MATH  Google Scholar 

  • Jacquet P, Vvedenskaya ND (1998) On/off sources in an interconnection networks: performance analysis when packets are routed to the shortest queue of two randomily selected nodes. Technical Report N0 3570, INRIA Rocquencourt, Frence

  • Jacquet P, Suhov YM, Vvedenskaya ND (1999) Dynamic routing in the mean-field approximation. Technical Report N0 3789, INRIA Rocquencourt, Frence

  • Kurtz TG (1981) Approximation of population processes. SIAM

  • Li QL (2010) Constructive computation in stochastic models with applications: the RG-factorizations. Springer and Tsinghua Press

  • Li QL (2011) Super-exponential solution in Markovian supermarket models: framework and challenge. Available: arXiv:1106.0787

  • Li QL, Dai G, Lui JCS, Wang Y (2012) Matrix-structured super-exponential tail in the supermarket model with PH service times. http://www.liquanlin.com/_d274639327.htm

  • Li QL, Lui JCS (2010) Doubly exponential solution for randomized load balancing models with Markovian arrival processes and PH service times. Available: arXiv:1105.4341

  • Li QL, Lui JCS, Wang Y (2011) A matrix-analytic solution for randomized load balancing models with PH service times. In: Performance evaluation of computer and communication systems: milestones and future challenges. Lecture notes in computer science, vol 6821, pp 240–253

  • Luczak MJ, McDiarmid C (2006) On the maximum queue length in the supermarket model. Ann Probab 34:493–527

    Article  MATH  MathSciNet  Google Scholar 

  • Luczak MJ, McDiarmid C (2007) Asymptotic distributions and chaos for the supermarket model. Electron J Probab 12:75–99

    Article  MATH  MathSciNet  Google Scholar 

  • Luczak MJ, Norris JR (2005) Strong approximation for the supermarket model. Ann Appl Probab 15:2038–2061

    Article  MATH  MathSciNet  Google Scholar 

  • Martin JB (2001) Point processes in fast Jackson networks. Ann Appl Probab 11:650–663

    Article  MATH  MathSciNet  Google Scholar 

  • Martin JB, Suhov YM (1999) Fast Jackson networks. Ann Appl Probab 9:854–870

    Article  MATH  MathSciNet  Google Scholar 

  • Mitzenmacher MD (1996) The power of two choices in randomized load balancing. PhD thesis, Department of Computer Science, University of California at Berkeley, USA

  • Mitzenmacher MD (1999) On the analysis of randomized load balancing schemes. Theory Comput Syst 32:361–386

    Article  MATH  MathSciNet  Google Scholar 

  • Mitzenmacher MD, Richa A, Sitaraman R (2001) The power of two random choices: a survey of techniques and results. In: Handbook of randomized computing, vol 1, pp 255–312

  • Mitzenmacher MD, Upfal E (2005) Probability and computing: randomized algorithms and probabilistic analysis. Cambridge University Press

  • Suhov YM, Vvedenskaya ND (2002) Fast Jackson networks with dynamic routing. Probl Inf Transm 38:136–153

    Article  MATH  Google Scholar 

  • Sznitman A (1989) Topics in propagation of chaos. In: Springer-Verlag lecture notes in mathematics 1464. École d’Été de Probabilités de Saint-Flour XI, pp 165–251

  • Takagi H (1991) Queueing analysis: a foundation of performance evaluation, vacation and priority systems, part 1. North-Holland, Amsterdam

  • Taylor ME (1996) Partial differential equations, I Basic Theory. Springer-Verlag, New York

    Google Scholar 

  • Tian NS, Li QL, Cao J (1999) Conditional stochastic decompositions in the M/M/C queue with server vacations. Stoch Model 15:367–378

    Article  MATH  MathSciNet  Google Scholar 

  • Tian NS, Zhang ZG (2006) Vacation queueing models: theory and applications. Springer, New York

    Google Scholar 

  • Tsitsiklis JN, Xu K (2012) On the power of (even a little) resource pooling. Stochastic Systems 2:1–66

    Article  MATH  MathSciNet  Google Scholar 

  • Turner SRE (1996) Resource pooling in stochastic networks. Ph.D. Thesis, Statistical Laboratory, Christ’s College, University of Cambridge

  • Turner SRE (1998) The effect of increasing routing choice on resource pooling. Probability in the Engineering and Informational Sciences 12:109–124

    Article  MATH  MathSciNet  Google Scholar 

  • Vvedenskaya ND, Dobrushin RL, Karpelevich FI (1996) Queueing system with selection of the shortest of two queues: an asymptotic approach. Probl Inf Transm 32:15–27

    MATH  MathSciNet  Google Scholar 

  • Vvedenskaya ND, Suhov YM (1997) Dobrushin’s mean-field approximation for a queue with dynamic routing. Markov Processes and Related Fields 3:493–526

    MATH  MathSciNet  Google Scholar 

  • Vvedenskaya ND, Suhov YM (2005) Dynamic routing queueing systems with vacations. Information Processes, Electronic Scientific Journal. The Keldysh Institute of Applied Mathematics. The Institute for Information Transmission Problems, vol 5, pp 74–86

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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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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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