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A survey on retrial queues

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Abstract

Queueing systems in which arriving customers who find all servers and waiting positions (if any) occupied may retry for service after a period of time are called retrial queues or queues with repeated orders. Retrial queues have been widely used to model many problems in telephone switching systems, telecommunication networks, computer networks and computer systems. In this paper, we discuss some important retrial queueing models and present their major analytic results and the techniques used. Our concentration is mainly on single-server queueing models. Multi-server queueing models are briefly discussed, and interested readers are referred to the original papers for details. We also discuss the stochastic decomposition property which commonly holds in retrial queues and the relationship between the retrial queue and the queue with server vacations.

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References

  1. A.M. Aleksandrov, A queueing system with repeated orders, Engineering Cybernetics Rev. 12, 3(1974)1.

    Google Scholar 

  2. W. Bux, Local-area subnetworks: A performance comparison, IEEE Trans. Comm. 29(1981)1465.

    Google Scholar 

  3. M.L. Chaudhry and J.G.C. Templeton,A First Course in Bulk Queues (Wiley, New York, 1983).

    Google Scholar 

  4. Q.H. Choo and B. Conolly, New results in the theory of repeated orders queueing systems, J. Appl. Prob. 16(1979)631.

    Google Scholar 

  5. J.W. Cohen, Basic problems of telephone traffic theory and the influence of repeated calls, Philips Telecom. Rev. 18, 2(1957)49.

    Google Scholar 

  6. N. Deul, Stationarity conditions for multi-server queueing systems with repeated calls, Elektr. Informationsverarbeit. Kybernet. 16(1980)607.

    Google Scholar 

  7. B.T. Doshi, Queueing systems with vacations —A survey, Queueing Systems 1(1986)29.

    Google Scholar 

  8. A. Elldin, Approach to the theoretical description of repeated call attempts, Ericsson Technics 23, 3(1967)345.

    Google Scholar 

  9. G.I. Falin, Aggregate arrival of customers in one-line system with repeated calls, Ukrainian Math. J. 28(1976)437.

    Google Scholar 

  10. G.I. Falin, On the waiting time in a single-channel queueing system with secondary calls, Vestnik Moscow Univ. Ser. 15. Comput. Math. Cybernet. 4(1977)83.

    Google Scholar 

  11. G.I. Falin, The output flow of a single-line queueing system when there are secondary orders, Engineering Cybernetics Rev. 16, 5(1978)64.

    Google Scholar 

  12. G.I. Falin, A single-line system with secondary orders, Engineering Cybernetics Rev. 17, 2(1979)76.

    Google Scholar 

  13. G.I. Falin, Not completely accessible schemes with allowance for repeated calls, Engineering Cybernetics Rev. 18, 5(1980)56.

    Google Scholar 

  14. G.I. Falin, Communication systems with repeated calls, Prob. Inf. Trans. 16, 2(1980)83.

    Google Scholar 

  15. G.I. Falin, Repeated calls in structurally complex systems, Engineering Cybernetics Rev. 18, 6(1980)46.

    Google Scholar 

  16. G.I. Falin, Investigation of weakly loaded switching systems with repeated calls, Engineering Cybernetics Rev. 19, 3(1981)69.

    Google Scholar 

  17. G.I. Falin, The influence of inhomogeneity of the composition of subscribers on the functioning of telephone systems with repeated calls, Engineering Cybernetics Rev. 21, 6(1983)21.

    Google Scholar 

  18. G.I. Falin, On sufficient conditions for ergodicity of multichannel queueing systems with repeated calls, Adv. Appl. Prob. 16(1984)447.

    Google Scholar 

  19. G.I. Falin, Continuous approximation for a single-server system with an arbitrary service time under repeated calls, Engineering Cybernetics Rev. 22, 2(1984)66.

    Google Scholar 

  20. G.I. Falin, On the waiting-time process in a single-line queue with repeated calls, J. Appl. Prob. 23(1986)185.

    Google Scholar 

  21. A.A. Fredericks and G.A. Reisner, Approximations to stochastic service systems with an application to a retrial model, Bell System Tech. J. 58, 3(1979)557.

    Google Scholar 

  22. G. Gosztony, Comparison of calculated and simulated results for trunk group with repeated attempts,Proc 8th Int. Teletraffic Congress, I, Melbourne (1976)321/1–11.

  23. G. Gosztony, A generalrHβ formula of call repetition: Validity and constraints,Proc. 11th Int. Teletraffic Congress (1985)1010–1016.

  24. B.S. Greenberg and R.W. Wolff, An upper bound on the performance of queues with returning customers, J. Appl. Prob. 24(1987)466.

    Google Scholar 

  25. O. Hashida and K. Kawashima, Buffer behavior with repeated calls, Electronics and Communication in Japan 62-B, 3(1979)27.

    Google Scholar 

  26. H. Inamori, M. Sawai, T. Endo and K. Tanabe, An automatically repeated call model in NTT public facsimile,Proc. 11th Int. Teletraffic Congress (1985) 1017–1023.

  27. G.L. Jonin and Y.Y. Sedol, Investigation of telephone systems in the case of repeated calls, Latvian Mathematical Yearbook 7(1970)71.

    Google Scholar 

  28. G.L. Jonin and N. Brezgunova, One-line system with repested calls in the case of Γ-distributed occupation time, Latvian Mathematical Yearbook 11(1972)65.

    Google Scholar 

  29. G.L. Jonin, The systems with repeated calls: models, measurements, and results,Proc. Third Int Seminar on Teletraffic Theory, Moscow (1984)197–208.

  30. J. Keilson, J. Cozzolino and H. Young, A service system with unfilled requests repeated, Oper. Res. 16(1968)1126.

    Google Scholar 

  31. F.P. Kelly, On auto-repeat facilities and telephone network performance, J.R. Statistical Society B48, 1(1986)123.

    Google Scholar 

  32. A. Kharkevich et al., Approximate analysis of systems with repeated calls and multiphase service,Proc. 11th Int. Teletraffic Congress (1985)1029–1035.

  33. L. Kleinrock,Queueing Systems, Vol. I:Theory (Wiley, New York, 1975) ch. 5.

    Google Scholar 

  34. A.G. de Kok, Algorithmic methods for single-server systems with repeated attempts, Statistica Neerlandica 38, 1(1984)23.

    Google Scholar 

  35. J.N. Kornishev, Waiting positions for overloaded trunks, Elektrosviaz 7(1974)32.

    Google Scholar 

  36. L. Kosten, On the influence of repeated calls in the theory of probabilities of blocking, De Ingenieur 59(1947)1.

    Google Scholar 

  37. A. Kuczura, Loss systems with mixed renewal and Poisson inputs, Oper. Res. 21(1973)787.

    Google Scholar 

  38. V.G. Kulkarni, Letter to the Editor, J. Appl. Prob. 19(1982)901.

    Google Scholar 

  39. V.G. Kulkarni, On queueing systems with retrials, J. Appl. Prob. 20(1983)380.

    Google Scholar 

  40. V.G. Kulkarni, A game theoretic model for two types of customers competing for service, Oper. Res. Lett. 2(1983)119.

    Google Scholar 

  41. V.G. Kulkarni, Expected waiting times in a multiclass batch arrival retrial queue, J. Appl. Prob. 23(1986)144.

    Google Scholar 

  42. P. Le Gall, General telecommunications traffic without delay,Proc. 8th Int. Teletraffic Congress (1976)125/1–8.

  43. P. Le Gall, The repeated callmodel and the queue with impatience,Proc. Third Int. Seminar on Teletraffic Theory, Moscow (1984)278–289.

  44. J. Lubacz and J. Roberts, A new approach to the single-server repeated attempt system with balking,Proc.Third Int. Seminar on Teletraffic Theory, Moscow (1984)290–293.

  45. G.E. Ridout, A study of retrial queueing systems with buffers, M.A.Sc. Thesis, Department of Industrial Engineering, University of Toronto (1984).

  46. J. Riordan,Stochastic Service Systems (Wiley, New York, 1962).

    Google Scholar 

  47. J.T. Runnenburg, On the use of method of collective marks in queueing theory, in:Proc. Symp. on Congestion Theory, ed. W.L. Smith and W.E. Wilkinson (The University of North Carolina Press, Chapel Hill, North Carolina, 1964)399–418.

    Google Scholar 

  48. E.I. Skolny and G.I. Ialkapov, Systems with repeated calls and waiting positions,Information Systems and Their Analysis (Nauka, Moscow, 1982)23–37.

    Google Scholar 

  49. M. Stastny, On the substitution of the basic retrial model for a complex loss model with retrials,Proc. Third Int. Seminar on Teletraffic Theory, Moscow (1984)395–399.

  50. S.N. Stepanov, Integral equilibrium relations of non-full-access systems with repeated calls and their applications, Prob. Inf. Trans. 16, 4(1980)88.

    Google Scholar 

  51. S.N. Stepanov, Probabilistic characteristics of an incompletely accessible multi-phase service system with several types of repeated calls, Problems of Control and Information Theory 10(1981)387.

    Google Scholar 

  52. S.N. Stepanov, Asymptotic formulae and estimations for probabilistic characteristics of full-available group with absolutely persistent subscribers, Problems of Control and Information Theory 12, 5(1983)361.

    Google Scholar 

  53. S.N. Stepanov, Properties of probabilistic characteristics of a communication network with repeated calls, Prob. Inf. Trans. 19, 1(1983)69.

    Google Scholar 

  54. S.N. Stepanov, Analysis of full-available group with repeated calls and waiting positions, Elektrosviaz 6(1983)9.

    Google Scholar 

  55. S.N. Stepanov,Numerical Methods of Calculation for Systems with Repeated Calls (Nauka, Moscow, 1983).

    Google Scholar 

  56. S.N. Stepanov, Probabilistic characteristics of an incompletely accessible service system with repeated calls for arbitrary values of subscriber persistent function, Problems of Control and Information Theory 13, 2(1984)69.

    Google Scholar 

  57. S.N. Stepanov, Numerical calculation accuracy of communication models with repeated calls, Problems of Control and Information Theory 13, 6(1984)371.

    Google Scholar 

  58. S.N. Stepanov, Estimation of characteristics of multilinear systems with repeated calls,Proc. Third Int. Seminar on Teletraffic Theory, Moscow (1984) 400–409.

  59. S.N. Stepanov and I.I. Tsitovich, The model of a full-available group with repeated calls and waiting positions in the case of extreme load, Problems of Control and Information Theory 14, 1(1985)25.

    Google Scholar 

  60. R.I. Wilkinson, Theories for toll traffic engineering in the U.S.A., Bell System Tech. J. 35(1956)421.

    Google Scholar 

  61. D.W. Sabo, Closure methods for the single-server retrial queue, M.Sc. Thesis, Department of Mathematics, University of British Columbia, Vancouver(1987).

    Google Scholar 

  62. T. Hanschke, Explicit formulas for the characteristics of the M/M/2/2 queue with repeated attempts, J. Appl. Prob. 24(1987)486.

    Google Scholar 

  63. S.N. Stepanov, Increasing the efficiency of numerical methods for models with repeat calls, Prob. Inf. Trans. 22, 4(1987)313.

    Google Scholar 

  64. B.S. Greenberg, M/G/1 queueing systems with returning customers, Working Paper 87/88-3-1, Department of Management Science and Information Systems, University of Texas at Austin (1987).

  65. G.I. Falin, Multichannel queueing systems with repeated calls under high intensity of repetition, Elektr. Informationsverarbeit. Kybernet. 23, 1(1987)37.

    Google Scholar 

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Authors and Affiliations

  1. Department of Industrial Engineering, University of Toronto, M5S 1A4, Toronto, Canada

    Tao Yang & J. G. C. Templeton

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  1. Tao Yang
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Yang, T., Templeton, J.G.C. A survey on retrial queues. Queueing Syst 2, 201–233 (1987). https://doi.org/10.1007/BF01158899

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  • DOI: https://doi.org/10.1007/BF01158899

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