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Threshold policies for controlled retrial queues with heterogeneous servers

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Abstract

Retrial queues are an important stochastic model for many telecommunication systems. In order to construct competitive networks it is necessary to investigate ways for optimal control. This paper considers K -server retrial systems with arrivals governed by Neut' Markovian arrival process, and heterogeneous service time distributions of general phase-type. We show that the optimal policy which minimizes the number of customers in the system is of a threshold type with threshold levels depending on the states of the arrival and service processes. An algorithm for the numerical evaluation of an optimal control is proposed on the basis of Howar's iteration algorithm. Finally, some numerical results will be given in order to illustrate the system dynamics.

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References

  • Asmussen, S., O. Nerman, and M. Olsson. (1996), “Fitting Phase-Type Distributions Via the EM Algorithm.” Scand. J. Stat. 23(4), 419–441.

    Google Scholar 

  • Breuer, L. (2001). “The Periodic BMAP/PH/c Queue.” Queueing Systems 38(1), 67–76.

    Article  Google Scholar 

  • Breuer, L. (2002). ‘An EM Algorithm for Batch Markovian Arrival Processes and its Comparison to a Simpler Estimation Procedure.” Annals of Operations Research 112, 123–138.

    Article  Google Scholar 

  • Breuer, L., A. Dudin, and V. Klimenok. (2002). “A Retrial BMAP/PH/N System.” Queueing Systems 40(4), 431–455.

    Article  Google Scholar 

  • Efrosinin, D.V. (2004). Controlled Queueing Systems with Heterogeneous Servers. Ph.D. Dissertation, Trier University, Germany.

  • Howard, R.A. (1960). Dynamic Programming and Markov Processes. Wiley.

  • Kitaev, M.Y. and V.V. Rykov. (1995). Controlled Queueing Systems. CRC Press: New York.

    Google Scholar 

  • Lerma, O.H. and J.B. Lassere. (1996). “Discrete-Time Markov Control Processes.” Applications of Mathematics, New-York.

  • Lin, W. and P.R. Kumar. (1984). “Optimal Control of a Queueing System with two Heterogeneous Servers.” IEEE Trans. on Autom. Control 29, 696–703.

    Article  Google Scholar 

  • Lucantoni, D.M. (1991). “New Results on the Single Server Queue with a Batch Markovian Arrival Process.” Commun. Stat., Stochastic Models 7(1), 1–46.

    Google Scholar 

  • Luh, H.P. and I. Viniotis. (1990). Optimality of Threshold Policies for Heterogeneous Server Systems. Raleign, North Carolina State University.

  • Naoumov, V., U.R. Krieger, and D. Wagner. (1997). Analysis of a Multiserver Delay-Loss System with a General Markovian Arrival Process. In Matrix-analytic methods in stochastic models (Flint, MI), pp. 43–66. Dekker, New York.

  • Neuts, M.F. (1978). “Markov Chains with Applications in Queueing Theory, Which Have a Matrix-Geometric Invariant Probability Vector.” Adv. Appl. Probab. 10, 185–212.

    Article  Google Scholar 

  • Neuts, M.F. (1981). Matrix-Geometric Solutions in Stochastic Models. Baltimore, London: The Johns Hopkins University Press.

    Google Scholar 

  • Neuts, M.F. (1989). Structured Stochastic Matrices of M/G/1 Type and Their Applications. New York etc.: Marcel Dekker.

    Google Scholar 

  • Nobel, R. and H.C. Tijms. 2000. “Optimal Control of a Queueing System with Heterogeneous Servers.” IEEE Transactions on Autom. Control 45(4).

  • Puterman, M.L. (1994). Markov Decision Process. Wiley series in Probability and Mathematical Statistics.

  • Rosberg, Z. and A.M. Makowski. (1990). “Optimal Routing to Parallel Heterogeneous Servers—Small Arrival Rates.” Transactions on automatic control, 35(7), 789–796.

    Article  Google Scholar 

  • Ross, Sh. (1970). Applied Probability Models with Optimization Applications. Holden Day, San Francisco.

    Google Scholar 

  • Rykov, V.V. (1975). Controllable Queueing Systems. In Itogi nauki i techniki. Teoria verojatn. Matem. Sftatist. Teoretich. Kibern. 12, 45–152 (In Russian). There is an English translation in Journ. of Soviet Math.

  • Rykov, V.V. (2001). “Monotone Control of Queueing Systems with Heterogeneous Servers.” QUESTA 37: 391–403.

    Google Scholar 

  • Rykov, V.V. (1966) “Markov Decision Processes with Finite State and Decision Spaces.” Probability Theory and its Applications 11(2), 302–311.

    Article  Google Scholar 

  • Rykov, V.V. and D.V. Efrosinin. (2002). “Numerical Analysis of Optimal Control Polices for Queueing Systems with Heterogeneous Servers.” Information Processes 2(2), 252–256.

    Google Scholar 

  • Sennott, L. (1999). Stochastic Dynamic Programming and Control of Queueing Systems. New York, Wiley, 328.

    Google Scholar 

  • Tijms, H.C. (1994). Stochastic Models. An Algorithmic Approach. John Wiley and Sons.

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Authors and Affiliations

  1. FB IV, Informatik, University Trier, Trier, 54286, Germany

    Dimitri frosinin & L. Breuer

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  1. Dimitri frosinin
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  2. L. Breuer
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Correspondence to Dimitri frosinin.

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AMS subject classification: 60K25 93E20

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frosinin, D., Breuer, L. Threshold policies for controlled retrial queues with heterogeneous servers. Ann Oper Res 141, 139–162 (2006). https://doi.org/10.1007/s10479-006-5297-5

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  • DOI: https://doi.org/10.1007/s10479-006-5297-5

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