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A survey on retrial queues. (English) Zbl 0658.60124
A systematized survey of the analytic results for retrival queues is given by discussing a generalized model A/B/s/m/O/H. Here, A and B describe the interarrival and service time distribution, respectively, s denotes the number of waiting positions in the queue, O is the capacity of an orbit for the retrying customers and \(H=\{H_ k\), \(k\geq 0\}\) denotes the loss model by which a customer joins to the orbit with probability \(1-H_ k\) after the k th unsuccessful retrial.
The survey concentrates an single-server models of type M/G/i/m/O/H. For \(m=1\), \(O=\infty\), \(H=NL=\{H_ k=1\), \(k\geq 0\}\), the number N of customers in the system, the waiting time w, the number \(\eta\) of retrials, the length L of busy periods, the number M of customers served in such periods, the server idle time i and the distribution \(\pi\) of time intervals between consecutive departures are examined. For M/M/1/m/O/H with finite m, O and \(H=GL=\{H_ k=\alpha \leq 1\), \(k\geq 0\}\) the exact values of state probabilities p(i,j), i,j\(\geq 0\), are derived, where i and j are the numbers of customers in the waiting room and in the orbit respectively. With \(O=\infty\) and \(H=NL\) an iterative procedure to approximate p(i,j) is reported.
In the case of batch arrivals the M/G/1 models are discussed. For the multi-server retrial queues only some major models, techniques dealing with the problems arising, and some main results are presented. Different full-availability and non-full-availability systems are reported. It is shown that the decomposition property, which is a common feature of queues with server vacations, is also true in retrial queues of the model M/G/s/m.
Reviewer: J.Tanko

60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
Full Text: DOI
[1] A.M. Aleksandrov, A queueing system with repeated orders, Engineering Cybernetics Rev. 12, 3(1974)1.
[2] W. Bux, Local-area subnetworks: A performance comparison, IEEE Trans. Comm. 29(1981)1465. · doi:10.1109/TCOM.1981.1094893
[3] M.L. Chaudhry and J.G.C. Templeton,A First Course in Bulk Queues (Wiley, New York, 1983). · Zbl 0559.60073
[4] Q.H. Choo and B. Conolly, New results in the theory of repeated orders queueing systems, J. Appl. Prob. 16(1979)631. · Zbl 0418.60088 · doi:10.2307/3213090
[5] J.W. Cohen, Basic problems of telephone traffic theory and the influence of repeated calls, Philips Telecom. Rev. 18, 2(1957)49.
[6] N. Deul, Stationarity conditions for multi-server queueing systems with repeated calls, Elektr. Informationsverarbeit. Kybernet. 16(1980)607. · Zbl 0476.90027
[7] B.T. Doshi, Queueing systems with vacations ?A survey, Queueing Systems 1(1986)29. · Zbl 0655.60089 · doi:10.1007/BF01149327
[8] A. Elldin, Approach to the theoretical description of repeated call attempts, Ericsson Technics 23, 3(1967)345.
[9] G.I. Falin, Aggregate arrival of customers in one-line system with repeated calls, Ukrainian Math. J. 28(1976)437. · Zbl 0361.60086 · doi:10.1007/BF01101670
[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.
[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. · Zbl 0446.90034
[12] G.I. Falin, A single-line system with secondary orders, Engineering Cybernetics Rev. 17, 2(1979)76.
[13] G.I. Falin, Not completely accessible schemes with allowance for repeated calls, Engineering Cybernetics Rev. 18, 5(1980)56. · Zbl 0497.90022
[14] G.I. Falin, Communication systems with repeated calls, Prob. Inf. Trans. 16, 2(1980)83. · Zbl 0458.90024
[15] G.I. Falin, Repeated calls in structurally complex systems, Engineering Cybernetics Rev. 18, 6(1980)46. · Zbl 0478.90020
[16] G.I. Falin, Investigation of weakly loaded switching systems with repeated calls, Engineering Cybernetics Rev. 19, 3(1981)69. · Zbl 0512.90053
[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. · Zbl 0577.90030
[18] G.I. Falin, On sufficient conditions for ergodicity of multichannel queueing systems with repeated calls, Adv. Appl. Prob. 16(1984)447. · Zbl 0535.60087 · doi:10.2307/1427079
[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. · Zbl 0644.60099
[20] G.I. Falin, On the waiting-time process in a single-line queue with repeated calls, J. Appl. Prob. 23(1986)185. · Zbl 0589.60077 · doi:10.2307/3214127
[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. · Zbl 0402.90043
[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 generalrH? 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. · Zbl 0626.60092 · doi:10.2307/3214270
[25] O. Hashida and K. Kawashima, Buffer behavior with repeated calls, Electronics and Communication in Japan 62-B, 3(1979)27.
[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.
[28] G.L. Jonin and N. Brezgunova, One-line system with repested calls in the case of ?-distributed occupation time, Latvian Mathematical Yearbook 11(1972)65.
[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. · Zbl 0165.52703 · doi:10.1287/opre.16.6.1126
[31] F.P. Kelly, On auto-repeat facilities and telephone network performance, J.R. Statistical Society B48, 1(1986)123. · Zbl 0595.60088
[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. · Zbl 0334.60045
[34] A.G. de Kok, Algorithmic methods for single-server systems with repeated attempts, Statistica Neerlandica 38, 1(1984)23. · Zbl 0547.60098
[35] J.N. Kornishev, Waiting positions for overloaded trunks, Elektrosviaz 7(1974)32.
[36] L. Kosten, On the influence of repeated calls in the theory of probabilities of blocking, De Ingenieur 59(1947)1.
[37] A. Kuczura, Loss systems with mixed renewal and Poisson inputs, Oper. Res. 21(1973)787. · Zbl 0278.90030 · doi:10.1287/opre.21.3.787
[38] V.G. Kulkarni, Letter to the Editor, J. Appl. Prob. 19(1982)901. · doi:10.2307/3213849
[39] V.G. Kulkarni, On queueing systems with retrials, J. Appl. Prob. 20(1983)380. · Zbl 0518.90023 · doi:10.2307/3213810
[40] V.G. Kulkarni, A game theoretic model for two types of customers competing for service, Oper. Res. Lett. 2(1983)119. · Zbl 0523.60095 · doi:10.1016/0167-6377(83)90019-6
[41] V.G. Kulkarni, Expected waiting times in a multiclass batch arrival retrial queue, J. Appl. Prob. 23(1986)144. · Zbl 0589.60073 · doi:10.2307/3214123
[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). · Zbl 0106.33601
[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.
[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.
[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. · Zbl 0527.94001
[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. · Zbl 0477.60086
[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. · Zbl 0532.60097
[53] S.N. Stepanov, Properties of probabilistic characteristics of a communication network with repeated calls, Prob. Inf. Trans. 19, 1(1983)69. · Zbl 0522.90026
[54] S.N. Stepanov, Analysis of full-available group with repeated calls and waiting positions, Elektrosviaz 6(1983)9.
[55] S.N. Stepanov,Numerical Methods of Calculation for Systems with Repeated Calls (Nauka, Moscow, 1983). · Zbl 0517.90032
[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. · Zbl 0583.90037
[57] S.N. Stepanov, Numerical calculation accuracy of communication models with repeated calls, Problems of Control and Information Theory 13, 6(1984)371.
[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. · Zbl 0572.60090
[60] R.I. Wilkinson, Theories for toll traffic engineering in the U.S.A., Bell System Tech. J. 35(1956)421.
[61] D.W. Sabo, Closure methods for the single-server retrial queue, M.Sc. Thesis, Department of Mathematics, University of British Columbia, Vancouver(1987).
[62] T. Hanschke, Explicit formulas for the characteristics of the M/M/2/2 queue with repeated attempts, J. Appl. Prob. 24(1987)486. · Zbl 0624.60110 · doi:10.2307/3214272
[63] S.N. Stepanov, Increasing the efficiency of numerical methods for models with repeat calls, Prob. Inf. Trans. 22, 4(1987)313.
[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. · Zbl 0616.90020
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