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Rate of convergence to stationarity of the system M/M/N/N+R

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Abstract

We consider the M/M/N/N+R service system, characterized by N servers, R waiting positions, Poisson arrivals and exponential service times. We discuss representations and bounds for the rate of convergence to stationarity of the number of customers in the system, and study its behaviour as a function of RN and the arrival rate λ, allowing λ to be a function of N.

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References

  • Chen MF (1996) Estimation of spectral gap for Markov chains. Acta Math Sin New Ser 12:337–360

    Article  Google Scholar 

  • Chen MF (1998) Estimate of exponential convergence rate in total variation by spectral gap. Acta Math Sin New Ser 14:9–16

    Article  Google Scholar 

  • Chihara TS (1978) An introduction to orthogonal polynomials. Gordon and Breach, New York

    Google Scholar 

  • Gamarnik G, Goldberg D (2010) On the rate of convergence to stationarity of the M/M/N queue in the Halfin–Whitt regime. Preprint arXiv:1003.2004

  • Granovsky BL, Zeifman AI (1997) The decay function of nonhomogeneous birth–death processes, with application to mean-field models. Stoch Process Appl 72:105–120

    Article  Google Scholar 

  • Ismail MEH, Muldoon ME (1991) A discrete approach to monotonicity of zeros of orthogonal polynomials. Trans Am Math Soc 323:65–78

    Article  Google Scholar 

  • Jagerman DL (1974) Some properties of the Erlang loss function. Bell Syst Tech J 53:525–551

    Google Scholar 

  • Karlin S, McGregor JL (1958) Many server queueing processes with Poisson input and exponential service times. Pac J Math 8:87–118

    Google Scholar 

  • Karlin S, McGregor JL (1965) Ehrenfest urn models. J Appl Probab 2:352–376

    Article  Google Scholar 

  • Keilson J, Ramaswamy R (1987) The relaxation time for truncated birth–death processes. Probab Eng Inf Sci 1:367–381

    Article  Google Scholar 

  • Kijima M (1990) On the largest negative eigenvalue of the infinitesimal generator associated with M/M/n/n queues. Oper Res Lett 9:59–64

    Article  Google Scholar 

  • Kijima M (1992) Evaluation of the decay parameter for some specialized birth–death processes. J Appl Probab 29:781–791

    Article  Google Scholar 

  • Kijima M (1997) Markov processes for stochastic modelling. Chapman & Hall, London

    Google Scholar 

  • Takács L (1962) Introduction to the theory of queues. Oxford University Press, New York

    Google Scholar 

  • van Doorn EA (1981) Stochastic monotonicity and queueing applications of birth–death processes. Lecture notes in statistics, vol 4. Springer, New York

    Book  Google Scholar 

  • van Doorn EA (1985) Conditions for exponential ergodicity and bounds for the decay parameter of a birth–death process. Adv Appl Probab 17:514–530

    Article  Google Scholar 

  • van Doorn EA, Zeifman AI (2009) On the speed of convergence to stationarity of the Erlang loss system. Queueing Syst 63:241–252

    Article  Google Scholar 

  • van Doorn EA, Zeifman AI, Panfilova TL (2010) Bounds and asymptotics for the rate of convergence of birth–death processes. Theory Probab Appl 54:97–113

    Article  Google Scholar 

  • Zeifman AI (1995) Upper and lower bounds on the rate of convergence for non-homogeneous birth and death processes. Stoch Process Appl 59:157–173

    Article  Google Scholar 

  • Zeifman AI, Bening VE, Sokolov IA (2008) Markov chains and models in continuous time. Elex-KM, Moscow (in Russian)

    Google Scholar 

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Correspondence to Erik A. van Doorn.

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van Doorn, E.A. Rate of convergence to stationarity of the system M/M/N/N+R . TOP 19, 336–350 (2011). https://doi.org/10.1007/s11750-011-0173-0

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  • DOI: https://doi.org/10.1007/s11750-011-0173-0

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