Queueing Systems

, Volume 82, Issue 1–2, pp 7–28 | Cite as

Delay analysis of a queue with re-sequencing buffer and Markov environment



There are simple service disciplines where the system time of a tagged customer depends only on the customers arriving in the system earlier (for example first-in-first-out (FIFO)) or later (for example LIFO) than the tagged one. In this paper we consider a single-server queueing system with two infinite queues in which the system time of a tagged customer may depend on both the customers arriving in the system earlier and later than the tagged one. New regular customers arrive in the system according to Markov arrival process (MAP) flow, occupy one place in the buffer and receive service in FIFO order. External re-sequencing signals also arrive at the system according to (different) MAP flow. Each re-sequencing signal transforms one regular customer into a delayed one by moving it to another queue (re-sequencing buffer), wherefrom it is served with lower priority than the regular ones. Service times of customers from both queues also have MAP distribution different from those which govern arrivals. Queueing system with memoryless ingredients (arrival, service, resequencing) has already been a subject of extensive research. In this paper we investigate how the essential analytical properties of scalar functions, which made the analysis of the memoryless system feasible, can be extended to the case of a Markov environment.


Delay analysis Re-sequencing buffer Matrix analytic methods Kronecker expansion 

Mathematics Subject Classification

68M20 60K25 



R. Razumchik thanks the support of Russian Foundation for Basic Research (Grant 13-07-00223). M. Telek thanks the support of OTKA Grant K101150.


  1. 1.
    Bini, D.A., Latouche, G., Meini, B.: Numerical Methods for Structured Markov Chains. Oxford University Press, New York, NY (2005)CrossRefGoogle Scholar
  2. 2.
    Graham, A.: Kronecker Products and Matrix Calculus: With Applications. Wiley, New York, NY (1982)Google Scholar
  3. 3.
    He, Q.-M.: Fundamentals of Matrix-Analytic Methods. Springer, New York (2013)Google Scholar
  4. 4.
    Higham, N.J., Kim, H.-M.: Numerical analysis of a quadratic matrix equation. IMA J. Numer. Anal. 20(4), 499–519 (2000)CrossRefGoogle Scholar
  5. 5.
    Horváth, G.: Efficient analysis of the queue length moments of the MMAP/MAP/1 preemptive priority queue. Perform. Eval. 69(12), 684–700 (2012)CrossRefGoogle Scholar
  6. 6.
    Horváth, G., Van Houdt, B.: Departure process analysis of the multi-type MMAP[K]/PH[K]/1 FCFS queue. Perform. Eval. 70(6), 423–439 (2013)CrossRefGoogle Scholar
  7. 7.
    Horváth, G., Van Houdt, B., Telek, M.: Commuting matrices in the queue length and sojourn time analysis of MAP/MAP/1 queues. Stoch. Models 30(4), 554–575 (2014)CrossRefGoogle Scholar
  8. 8.
    Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. Society for Industrial and Applied Mathematics, Philadelphia (1999)CrossRefGoogle Scholar
  9. 9.
    Neuts, M.F.: Matrix Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore (1981)Google Scholar
  10. 10.
    Neuts, M.F.: Structured Stochastic Matrices of M/G/1 Type and Their Applications. Marcel Dekker, New York (1989)Google Scholar
  11. 11.
    Ozawa, T.: Sojourn time distributions in the queue defined by a general QBD process. Queueing Syst. Theory Appl. 53(4), 203–211 (2006)CrossRefGoogle Scholar
  12. 12.
    Pechinkin, A.V., Razumchik, R.V.: On temporal characteristics in an exponential queueing system with negative claims and a bunker for ousted claims. Autom. Remote Control 72(12), 2492–2504 (2011)CrossRefGoogle Scholar
  13. 13.
    Steeb, W.H., Hardy, Y.: Matrix Calculus and Kronecker Product: A Practical Approach to Linear and Multilinear Algebra. World Scientific, River Edge, NJ (2011)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institute of Informatics Problems, Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences and Peoples’ Friendship University of RussiaMoscowRussia
  2. 2.MTA-BME Information Systems Research Group and Technical University of BudapestBudapestHungary

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