Queueing Systems

, Volume 7, Issue 2, pp 169–189 | Cite as

Numerical investigation of a multiserver retrial model

  • Marcel F. Neuts
  • B. M. Rao
Invited Paper


We consider a queueing model in which customers arrive in a Poisson stream to be served by one ofc servers. Each arriving customer enters a pool of active customers and starts generating requests for service at exponentially distributed time intervals at rate σ until he finds a free server and begins service. An analytical solution of this model is difficult and does not lend itself to numerical implementation. In this paper, we make a simplifying approximation, based on understanding of the physical behavior of the system, which yields an infinitesimal generator with a modified matrix-geometric equilibrium probability vector. That vector can be very efficiently computed even for high congestion levels. Illustrative numerical examples demonstrate the effectiveness of the approximation as well as the effect of the retrial rate on the system behavior for various levels of congestion. This study shows how numerical results for analytically intractable systems can be obtained by combining intuition with efficient algorithmic methods.


Queues with retrials matrix-geometric solutions algorithmic probability 


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  1. [1]
    A.M. Alexsandrov, A queueing system with repeated orders, Eng. Cybernet. 12 (1974) 1–4.Google Scholar
  2. [2]
    J.W. Cohen, Basic problems of telephone traffic theory and the influence of repeated calls, Philips Telecomm. Rev. 18 (1957) 49–99.Google Scholar
  3. [3]
    G.I. Falin, A single-line system with secondary orders, Eng. Cybernet. 17 (1979) 76–83.Google Scholar
  4. [4]
    G.I. Falin, Double channel queueing system with repeated calls, Paper #4221-84, All-Union Institute for Scientific and Technical Information, Moscow, USSR (1984).Google Scholar
  5. [5]
    G.I. Falin, On sufficient conditions for ergodicity of multichannel queueing systems with repeated calls, Adv. Appl. Probab. 16 (1984) 447–448.Google Scholar
  6. [6]
    G.I. Falin, Multichannel queueing systems with repeated calls under high intensity of repetition, J. Inform. Proc. Cybernet. 1 (1987) 37–47.Google Scholar
  7. [7]
    G.I. Falin, Error estimates in the approximation of countable Markov chains connected with the models of repeated calls, Vestnik Moskov Univ. Ser. 1, Mat. Mekh. 2 (1987) 12–15 (in Russian).Google Scholar
  8. [8]
    A.A. Fredericks and G.A. Reisner, Approximations to stochastic service systems, with an application to a retrial model, Bell Syst. Techn. J. 58 (1979) 557–576.Google Scholar
  9. [9]
    B. Greenberg, An upper bound on the performance of queues with returning customers, J. Appl. Probab. 24 (1987) 466–475.Google Scholar
  10. [10]
    T. Hanschke, Explicit formulas for the characteristics of theM/M/2/2 queue with repeated attempts, J. Appl. Probab. 24 (1987) 486–494.Google Scholar
  11. [11]
    G.L. Jonin and J.J. Sedol, Telephone systems with repeated calls,6th Int. Teletraffic Congress, Munich (1970) pp. 435.1–453.5.Google Scholar
  12. [12]
    L. Kaufman, Matrix methods for queueing systems, SIAM J. Sci. Statist. Comput. 4 (1983) 525–552.Google Scholar
  13. [13]
    J. Keilson, J. Cozzolino and M. Young, A service with unfilled requests repeated, Oper. Res. 16 (1968) 1126–1137.Google Scholar
  14. [14]
    V.G. Kulkarni, Letter to the Editor, J. Appl. Probab. 19 (1982) 901–904.Google Scholar
  15. [15]
    V.G. Kulkarni, On queueing systems with retrials, J. Appl. Probab. 20 (1983) 380–389.Google Scholar
  16. [16]
    M.F. Neuts,Matrix Geometric Solutions in Stochastic Models (The Johns Hopkins University Press, Baltimore, MD 1981).Google Scholar
  17. [17]
    M.F. Neuts and M.F. Ramalhoto, A service model in which the server is required to search for customers, J. Appl. Probab. 21 (1984) 157–166.Google Scholar
  18. [18]
    M.F. Neuts, The caudal characteristic curve of queues, Adv. Appl. Probab. 18 (1986) 221–254.Google Scholar
  19. [19]
    C.E.M. Pearce, On the problem of re-attempted calls in teletraffic, Stochastic Models 3 (1987) 393–407.Google Scholar
  20. [20]
    J. Riordan,Stochastic Service Systems (Wiley, New York, 1962).Google Scholar
  21. [21]
    R.I. Wilkinson, Theories for toll traffic engineering in the USA, Bell Syst. Techn. J. 35 (1956) 421–514.Google Scholar
  22. [22]
    T. Yang and J.G.C. Templeton, A survey on retrial queues, Queueing Systems 2 (1987) 201–233.Google Scholar

Copyright information

© J.C. Baltzer A.G. Scientific Publishing Company 1990

Authors and Affiliations

  • Marcel F. Neuts
    • 1
  • B. M. Rao
    • 2
  1. 1.Department of Systems and Industrial EngineeringUniversity of ArizonaTucsonUSA
  2. 2.Department of Applied Statistics and Operations ResearchBowling Green State UniversityBowling GreenUSA

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