Annals of Operations Research

, Volume 141, Issue 1, pp 193–210 | Cite as

Optimal multi-threshold control by the BMAP/SM/1 retrial system

  • Che Soong Kim
  • Valentina Klimenok
  • Alexander Birukov
  • Alexander Dudin


A single server retrial system having several operation modes is considered. The modes are distinguished by the transition rate of the batch Markovian arrival process (BMAP), kernel of the semi-Markovian (SM) service process and the intensity of retrials. Stationary state distribution is calculated under the fixed value of the multi-threshold control strategy. Dependence of the cost criterion, which includes holding and operation cost, on the thresholds is derived. Numerical results illustrating the work of the computer procedure for calculation of the optimal values of thresholds are presented.


Batch Markovian arrival process Controlled operation modes Cost criterion Optimal control 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Artalejo J.R. and Gomez-Corral A. (1997). “Steady State Solution of a Single-Server Queue with Linear Repeated Requests.” Journal of Applied Probability 34, 223–233.CrossRefGoogle Scholar
  2. Breuer L., Dudin A.N. and Klimenok V.I. (2002). “A retrial BMAP|PH|N system.” Queueing Systems 40, 433–457.CrossRefGoogle Scholar
  3. Chakravarthy S.R. (2001). “The batch Markovian arrival process: A review and future work.” in: Advances in Probability Theory and Stochastic Processes, eds. A. Krishnamoorthy, et al.(Notable Publications) pp. 21–39.Google Scholar
  4. Choi B.D., Chung Y.H., Dudin A.N. (2001). “The BMAP|SM|1 retrial queue with controllable operation modes.” European Journal of Operational Research 131, 16–30.CrossRefGoogle Scholar
  5. Cinlar E. (1975). Introduction to Stochastic Processes (Prentice-Hall).Google Scholar
  6. Dudin A.N. and Chakravarthy S.R. (2002). “Optimal hysteretic control for the BMAP |G| 1 system with single and group service modes.” Annals of Operations Research 112, 153–169.CrossRefGoogle Scholar
  7. Dudin A.N. and Klimenok V.I. (1999). “Multi-dimensional quasitoeplitz Markov chains.” Journal of Applied Mathematics and Stochastic Analysis 12, 393–415.Google Scholar
  8. Dudin A.N. and Klimenok V.I. (2000). “A retrial BMAP/SM/1 system with linear repeated requests.” Queueing Systems 34, 47–66.CrossRefGoogle Scholar
  9. Dudin A.N., Klimenok V.I., Klimenok I.A., et al. (2000). “Software “SIRIUS+” for evaluation and optimization of queues with the BMAP-input.” in: Advances in Matrix Analytic Methods for Stochastic Models, eds. G. Latouche and P. Taylor (Notable Publications, Inc., New Jersey) pp. 115–133.Google Scholar
  10. Kemeni J., Shell J. and Knapp A. (1966). “Van Nostrand, New York.” Denumerable Markov chains.Google Scholar
  11. Klimenok V.I. and Dudin A.N. (2003). “Application of censored Markov chains for calculating the stationary distribution of the multi-dimensional left-skip-free Markov chains.” Queues: flows, systems, networks 17, 121–128.Google Scholar
  12. Klimenok V.I. (2000). “About stationary distribution existence conditions in queueing sytems with the MAP and retrials.” Reports of Belarusian Academy of Science 39, 128–132 (in Russian).Google Scholar
  13. Lucantoni D.M. (1991). “New results on the single server queue with a batch markovian arrival process.” Communications in Statistics-Stochastic Models 7, 1–46.Google Scholar
  14. Neuts M.F. (1989). Structured Stochastic Matrices of M/G/1 type and their applications (Marcel Dekker).Google Scholar
  15. Tijms H.C. (1976). “On the optimality of a switch-over policy for controlling the queue size in an M/G/1 queue with variable service rate.” Lecture Notes in Computer Sciences 40, 736–742.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  • Che Soong Kim
    • 1
  • Valentina Klimenok
    • 2
  • Alexander Birukov
    • 2
  • Alexander Dudin
    • 2
  1. 1.Department of Industrial EngineeringSangji UniversityWonju, KangwonKorea
  2. 2.Department of Applied Mathematics and Computer ScienceBelarusian State UniversityMinsk

Personalised recommendations