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A survey of retrial queues. (English) Zbl 0709.60097
This survey of results on retrial queues supplements the recent surveys by the author [Optimization 17, 649-667 (1986; Zbl 0618.90033)] and T. Yang and J. G. C. Templeton [Queueing Syst. 2, No.3, 201-233 (1987; Zbl 0658.60124); Correction in ibid. 4, 94 (1989)].
Author’s abstract: We concentrate on Markovian single and multi-channel systems. For the single channel case we consider the main model as well as models with batch arrivals, multiclasses, customer impatience, double connection, control devices, two-way communication and buffers. The stochastic processes arising from these models are considered in the stationary as well as the nonstationary regime. For multi-channel queues we survey numerical investigations of stationary distributions, limit theorems for high and low retrial intensities, and heavy and light traffic behaviour.
Reviewer: E.A.van Doorn

MSC:
60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
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[1] A.M. Aleksandrov, A queueing system with repeated orders, Eng. Cybernet. Rev. 12 (3) (1974) 1-4.
[2] Q.H. Choo, The interaction of theory and simulation in queueing analysis, Ph.D. Thesis, Chelsea College, University of London (1978).
[3] Q.H. Choo and B. Conolly, New results in the theory of repeated orders queueing systems, J. Appl. Probab. 16 (1979) 631-640. · Zbl 0418.60088 · doi:10.2307/3213090
[4] J.W. Cohen, Basic problems of telephone traffic theory and the influence of repeated calls, Philips Telecom. Rev. 18 (2) (1957) 49-100.
[5] B.W. Conolly, Letter to the Editor, J. Appl. Probab. 19 (1982) 904-905.
[6] B.N. Dimitrov and P.R. Ruskov, A discrete model of a single-line queue with repeated calls, in:Proc. 14th Spring Conf. of the Union of Bulgarian Mathematicians, Sunny Beach, April 6-9, 1985 (Sofia, Bulgarian Academy of Science, (1985) (in Russian). · Zbl 0585.90037
[7] A.N. Dudin, On a queue with repeated calls and changing operating conditions, Paper #293-85, All-Union Institute for Scientific and Technical Information, Moscow (1985) (in Russian).
[8] G.I. Falin, Multi-phase servicing in a single-channel system for automation of experiments with repeated calls, in:Problems of Automation of Scientific Investigations in Radio Engineering and Electronics (USSR Academy of Science, Moscow, 1975).
[9] G.I. Falin, Aggregate arrival of customers in one-line system with repeated calls, Ukrainian Math. J. 28 (1976) 437-440. · 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) 83-87.
[11] G.I. Falin, The output flow of a single-line queueing system when there are secondary orders, Eng. Cybernet. Rev. 16 (5) (1978) 64-67. · Zbl 0446.90034
[12] G.I. Falin, Model of coupled switching in the presence of recurrent calls, Eng. Cybernet. Rev. 17 (1) (1979) 53-59. · Zbl 0438.94039
[13] G.I. Falin, A single-line system with secondary orders, Eng. Cybernet. Rev. 17 (2) (1979) 76-83. · Zbl 0437.60073
[14] G.I. Falin, Effect of the recurrent calls on output flow of a single channel system of mass service, Eng. Cybernet. Rev. 17 (4) (1979) 99-103. · Zbl 0437.60074
[15] G.I. Falin, AnM/M/1 queue with repeated calls in the presence of persistence function, Paper #1606-80, All-Union Institute for Scientific and Technical Information, Moscow (1980) (in Russian).
[16] G.I. Falin, AnM/G/1 system with repeated calls in heavy traffic, Vestnik Moscow Univ. Ser. 1, Math. Mech. 6 (1980) 48-50. · Zbl 0512.60091
[17] G.I. Falin, Computation of a traffic of a telephone used by many subscribers, Vestnik Moscow Univ. Ser. 15, Comput. Math. Cybernet. 2 (1981) 59-62.
[18] G.I. Falin, Functioning under nonsteady conditions of a single-channel system with group arrival of requests and repeated calls, Ukrainian Math. J. 33 (1981) 429-432. · Zbl 0476.60088 · doi:10.1007/BF01085753
[19] G.I. Falin, The influence of inhomogeneity of the composition of subscribers on the functioning of telephone systems with repeated calls, Eng. Cybernet. Rev. 21 (6) (1983) 21-25. · Zbl 0577.90030
[20] G.I. Falin, Asymptotic properties of the number of demands distribution in anM/G/1/? queueing system with repeated calls, Paper #5418-83, All-Union Institute for Scientific and Technical Information, Moscow (1983) (in Russian).
[21] G.I. Falin, Continuous approximation for a single server system with an arbitrary service time under repeated calls, Eng. Cybernet. Rev. 22 (2) (1984) 66-71. · Zbl 0644.60099
[22] G.I. Falin, Quasi-input process in theM/G/1/? queue, Adv. Appl. Probab. 16 (1984) 695-696. · Zbl 0544.60086 · doi:10.2307/1427298
[23] G.I. Falin, A probabilistic model for investigation of load of subscriber’s lines with waiting places, in:Probability Theory, Stochastic Processes and Functional Analysis (Moscow State University, Moscow, 1985).
[24] G.I. Falin and Yu.I. Sukharev, On single-line queues with double connections, Paper #6582-85, All-Union Institute for Scientific and Technical Information, Moscow (1985)(in Russian).
[25] G.I. Falin, On waiting time process in single-line queues with repeated calls, J. Appl. Probab. 23 (1986) 185-192. · Zbl 0589.60077 · doi:10.2307/3214127
[26] G.I. Falin, On ergodicity of multichannel queueing systems with repeated calls, Sov. J. Comput. Syst. Sci. 25 (1) (1987) 60-65. · Zbl 0655.60094
[27] G.I. Falin, Single-line repeated orders queueing systems, Mathematische Operationsforschung und Statistik, Optimization 5 (1986) 649-667. · Zbl 0618.90033
[28] G.I. Falin, Estimations of error in approximation of countable Markov chains associated with models of repeated calls, Vestnik Moscov. Univ. Ser. 1, Math. Mech. 2 (1987) 12-15. · Zbl 0625.60113
[29] G.I. Falin, On a multiclass batch arrival retrial queue, Adv. Appl. Probab. 20 (1988) 483-487. · Zbl 0672.60089 · doi:10.2307/1427403
[30] G.I. Falin, On the quasi-input process for theM/G/1/? queueing system, Ukrainian Math. J. 40 (1988) 226-229. · Zbl 0664.90033 · doi:10.1007/BF01056485
[31] G.I. Falin, On virtual waiting time in retrial queues, Vestnik Moscow Univ. Ser. 1, Math. Mech., to appear. · Zbl 0743.60096
[32] G. Fayolle, A simple telephone exchange with delayed feedbacks, in:Teletraffic Analysis and Computer Performance Evaluation, eds. O.J. Boxma, J.W. Cohen and H.C. Tijms (Elsevier Science, 1986).
[33] B.S. Greenberg, Queueing systems with returning customers and the optimal order of tandem queues, Ph.D. Thesis, University of California, Berkeley (1986).
[34] B.S. Greenberg,M/G/1 queueing systems with returning customers, J. Appl. Probab. 26 (1989) 152. · Zbl 0672.60094 · doi:10.2307/3214325
[35] B.S. Greenberg and R.W. Wolff, An upper bound on the performance of queues with returning customers, J. Appl. Probab. 24 (1987) 466-475. · Zbl 0626.60092 · doi:10.2307/3214270
[36] S.A. Greeschechkin, Branching processes and queues with repeated calls or random service, Theory of Probability and its Applications, to appear.
[37] T. Hanschke, A model for planning switching networks, in:Operations Research Proceedings 1984 (Springer, Berlin/Heidelberg, 1985). · Zbl 0557.90028
[38] T. Hanschke, TheM/G/1/1 queue with repeated attempts and different types of feedback effects, OR Spektrum 7 (1985) 209-215. · Zbl 0572.60093 · doi:10.1007/BF01720172
[39] T. Hanschke, A computational procedure for the variance of the waiting time in theM/M/1/1 queue with repeated calls, in:Operations Research Proceedings 1985 (Springer, Berlin/Heidelberg, 1986). · Zbl 0572.60093
[40] I.I. Homitchkov, A model of a route of a circuit switching network with repeated calls, in:Mathematics and Software for Systems of Automatic Design of Networks (Mari State University, Oshkar-Ola, 1985)(in Russian).
[41] I.I. Homitchkov, Generating functions of state probabilities of a single-line queue with repeated calls, Vestnik Beloruss. Univ. Ser. 1, 1 (1987) 51-55 (in Russian).
[42] I.I. Homitchkov, A model of local area computer network with random multiple access, Automatics and Telemechanics 1 (1987) 58-62 (in Russian).
[43] I.I. Homitchkov, Single-line queue with repeated calls and Cox input process of second order, Vestnik Belorus. Univ. 1 (1988) 70-71 (in Russian).
[44] I.I. Homitchkov, Computing the characteristics of a queueing system with repeated units and twin connections, Automatics and Telemechanics 4 (1988) 77-84 (in Russian).
[45] J.J. Hunter, The non-renewal nature of the quasi-input process in theM/G/1/? queue, J. Appl. Probab. 23 (1986) 803-811. · Zbl 0624.60109 · doi:10.2307/3214017
[46] G.L. Jonin and J.J. Sedol, Investigation of telephone systems with repeated calls, Latvian Math. Yearbook 7 (1970) 71-83.
[47] G.L. Jonin and J.J. Sedol,Tables of Probabilistic Characteristics of Fully Available Trunk Groups in the Case of Repeated Calls (Moscow, 1970).
[48] G.L. Jonin and J.J. Sedol, Telephone systems with repeated calls,Proc. 6th Int. Teletraffic Congress (1970) pp. 435/1-435/5.
[49] G.L. Jonin, A single-line system with repeated calls, in:Scientific and Technical Conf. for Problems of Information Networks and Automatic Switching, Thesis of Reports (Moscow, 1971).
[50] G.L. Jonin and N.M. Brezgunova, One-line system with repeated calls in the case of ?-distributed occupation time, Latvian Math. Yearbook 11 (1972) 65-71.
[51] G.L. Jonin, J.J. Sedol and A.V. Kibild, General queueing model with repeated calls, in:Information Networks and Automatic Switching, 3rd All-Union Scientific and Technical Conference, Thesis of Reports (Moscow, 1975)(in Russian).
[52] G.L. Jonin, An investigation of single-line queues with repeated calls under independent discrete check of channel state, Latvian Math. Yearbook 24 (1980) 204-209.
[53] G.L. Jonin, An investigation of single-line queues with repeated calls under service without interruption and with independent discrete check of channel state, in:Models of Information Networks and Switching Systems (Moscow, 1982).
[54] G.L. Jonin, Determination of probabilistic characteristics of single-line queues with double connections and repeated calls, in:Models of Systems of Distribution of Information and Its Analysis (Moscow, 1982).
[55] V.A. Kapyrin, A study of the stationary characteristics of a queueing system with recurring demands, Cybernetics 13 (1977) 584-590. · Zbl 0368.60111
[56] J. Keilson, J. Cozzolino and H. Young, A service system with unfilled requests repeated, Oper. Res. 16 (1968) 1126-1137. · Zbl 0165.52703 · doi:10.1287/opre.16.6.1126
[57] A.G. de Kok, Computational methods for single server systems with repeated attempts, Report #89, Interfaculteit der Actuariële Wetenschappen en Econometrie, Amsterdam (1982).
[58] A.G. de Kok, Algorithmic methods for single server systems with repeated attempts, Statistica Neerlandica 38 (1984) 23-32. · Zbl 0547.60098 · doi:10.1111/j.1467-9574.1984.tb01094.x
[59] Y.N. Kornishov, Calculation of coupled switching, Trudy Utchebnih Institutov Svyasi 37 (1968) 96-104 (in Russian).
[60] Y.N. Kornishov, Repeated calls in a trunk-line, Elektrosvyaz 1 (1974) 35-41 (in Russian).
[61] Y.N. Kornishov, A single-line queue with repeated calls and advance service, Izv. ANSSSR. Tekhn. Kibernetika 2 (1977) 83-88 (in Russian).
[62] Y.N. Kornishov and A.M. Zelinskiy, Analysis of subscriber’s line states, in:Information Networks and its Analysis (Moscow, 1978) (in Russian).
[63] Y.N. Kornishov, A single-line queue with heterogeneity repeated calls, in:Teletraffic Theory and Networks with Controlled Elements (Moscow, 1980)(in Russian).
[64] V.G. Kulkarni, Letter to the Editor, J. Appl. Probab. 19 (1982) 901-904. · doi:10.2307/3213849
[65] V.G. Kulkarni, On queueing systems with retrials, J. Appl. Probab. 20 (1983) 380-389. · Zbl 0518.90023 · doi:10.2307/3213810
[66] V.G. Kulkarni, A game theoretic model for two types of customers competing for service, Oper. Res. Lett. 2 (1983) 119-122. · Zbl 0523.60095 · doi:10.1016/0167-6377(83)90019-6
[67] V.G. Kulkarni, Expected waiting times in a multiclass batch arrival retrial queue, J. Appl. Probab. 23 (1986) 144-159. · Zbl 0589.60073 · doi:10.2307/3214123
[68] J. Lubacz and J. Roberts, A new approach to the single server repeat attempts system with balking,Proc. 3rd Int. Seminar on Teletraffic Theory, Moscow (1984) pp. 290-293.
[69] B. Pourbabai, Analysis of aG/M/K/0 queueing system with heterogeneous servers and retrials, Int. J. Syst. Sci. 18 (1987) 985-992. · Zbl 0611.90050 · doi:10.1080/00207728708964025
[70] G.E. Ridout, A study of retrial queueing systems with buffers, M.A.Sc. Thesis, Department of Industrial Engineering, University of Toronto (1984).
[71] P. Ruskov, K. Yanev, B. Dimitrov and K. Boyanov, A model for investigating local area computer networks, Control Systems and Machines 5 (1984) 37-40.
[72] S.N. Stepanov, Moments of overload traffic for single-line queues with repeated calls, Izv. AN SSSR. Tekhn. Kibernetika 1 (1977) 88-93.
[73] S.N. Stepanov, The correlation function of a single-line queue with repeated attempts and its application to load measurement, in:Methods and Structures of Teletraffic Systems (Moscow, 1979)(in Russian).
[74] Yu.I. Sukcharev, Calculation of probabilistic characteristics ofM/G/1/? queues with repeated calls in the presence of network blocking, Paper #6258-84, All-Union Institute for Scientific and Technical Information, Moscow (1984)(in Russian).
[75] T. Yang and J.G.C. Templeton, A survey on retrial queues, Queueing Systems 2 (1987) 203-233. · Zbl 0658.60124 · doi:10.1007/BF01158899
[76] A.M. Zelinskiy and Y.N. Kornishov, Equivalent models of a system with repeated calls, Trudy Utchebnih. Institutov Svyazi 80 (1976) 37-42 (in Russian).
[77] A.M. Zelinskiy and Y.N. Kornishov, Two models of a system with repeated calls, Elektrosvyaz 1 (1978) 60-63 (in Russian).
[78] G. Bretschneider, Repeated calls with limited repetition probability,Proc. 6th Int. Teletraffic Congress, Munich (1970) pp. 431/1-434/5.
[79] N. Deul, Stationary conditions for multiserver queueing systems with repeated calls, Elektronische Informationsverarbeitung und Kybernetik 10-12 (16) (1980) 607-613. · Zbl 0476.90027
[80] A. Elldin, Approach to the theoretical description of repeated call attempts, Ericsson Technics 23 (3) (1967) 346-407.
[81] G.I. Falin, Not completely accessible schemes with allowance for repeated calls, Eng. Cybernet. Rev. 18 (5) (1980) 56-63. · Zbl 0497.90022
[82] G.I. Falin, Switching systems with allowance for repeated calls, Probl. Inform. Transmission 16(1980) 145-151. · Zbl 0458.90024
[83] G.I. Falin, Repeated calls in structurally complex systems, Eng. Cybernet. Rev. 18 (6) (1980) 46-51. · Zbl 0478.90020
[84] G.I. Falin, Investigation of weakly loaded switching systems with repeated calls, Eng. Cybernet. Rev. 19 (3) (1981) 69-73. · Zbl 0512.90053
[85] G.I. Falin, State consolidation in symmetrical partially accessible circuits, Probl. Control Inform. Theory 11 (1982) 3-12. · Zbl 0483.90043
[86] G.I. Falin, Calculation of probabilistic characteristics of a multi-channel queue with repeated calls, Vestnik Mosk. Univ. Ser. 15, Vychisl. Mat. Cybernet. 1 (1983) 35-41.
[87] G.I. Falin, On the accuracy of a numerical method of calculation of characteristics of systems with repeated calls, Elektrosvyaz 8 (1983) 35-36.
[88] G.I. Falin, On sufficient conditions for ergodicity of multi-channel queueing systems with repeated calls, Adv. Appl. Probab. 16 (1984) 447-448. · Zbl 0535.60087 · doi:10.2307/1427079
[89] G.I. Falin, Double-channel queueing system with repeated calls, Paper #4221-84, All-Union Institute for Scientific and Technical Information, Moscow (1984).
[90] G.I. Falin, Multilinear completely accessible systems with repeated calls in heavy traffic, Vestnik Moskov Univ. Ser. 15, Vychisl. Mat. Kibernet. 3 (1984) 66-69. · Zbl 0561.90037
[91] G.I. Falin, Asymptotic investigation of fully available switching systems with high repetition intensity of blocked calls, Mosc. Univ. Math. Bull. 39 (6) (1984) 72-77 [Transl. from Vestn. Mosc. Univ. Ser. 1, no. 6 (1984) 49-53]. · Zbl 0572.60089
[92] G.I. Falin, Limit theorems for queueing systems with repeated calls,4th Int. Vilnius Conf. on Probability Theory and Mathematical Statistics, Abstracts of Communications, Vol. 3, Vilnius, USSR (1985).
[93] G.I. Falin, On heavily loaded systems with repeated calls, Sov. J. Comput. Syst. Sci. 24 (4) (1986) 124-128 [Transi, from Izv. Akad. Nauk SSSR. Tekn. Kibern. (1986) 180-184]. · Zbl 0619.60090
[94] G.I. Falin, Multichannel queueing systems with repeated calls under high intensity of repetition, J. Inform. Processing Cybernet. 1 (1987) 37-47. · Zbl 0616.90020
[95] G.I. Falin, Comparability of migration processes, Probab. Theory Appl. 2 (1986) 392-396. · Zbl 0649.60087
[96] G.I. Falin and Yu.I. Suharev, Singular perturbed equations and asymptotic investigation of stationary characteristics of retrial queues, Vestnik Moskow Univ. Ser. 1, Math. Mech. 5 (1988) 7-10.
[97] G.I. Falin, Theorems of ergodicity and stability for retrial queues, Ukrainian Math. J., to appear.
[98] B.S. Greenberg and R.W. Wolff, An upper bound on the performance of queues with returning customers, J. Appl. Probab. 24 (1987) 466-475. · Zbl 0626.60092 · doi:10.2307/3214270
[99] T. Hanschke, Die von Bretschneider, Cohen und Schwartzbart/Puri entwickelte Warteschlangenmodelle mit wiederholten Versuchen: eine Methode zur Berechnung der ergodischen Projektion ihrer Markovschen Warteprozesse und die Simulation der Wartezeiten, Fakultät für Mathematik der Universität Karlsruhe (1978).
[100] T. Hanschke, Explicit formulas for the characteristics of theM/M/2/2 queue with repeated attempts, J. Appl. Probab. 24 (1987) 486-494. · Zbl 0624.60110 · doi:10.2307/3214272
[101] G.L. Jonin and J.J. Sedol, Full-availability groups with repeated calls and time of advanced service,Proc. 7th Int. Teletraffic Congress, Stockholm (1973) pp. 137/1-137/4.
[102] Yu.N. Kornishov, Waiting positions for overloading trunks, Elektrosvyaz 7 (1974) 32-39.
[103] C.E.M. Pearce, On the problem of re-attempted calls in teletraffic, Commun. Statist.-Stochastic Models 3 (3) (1987) 393-407. · Zbl 0641.90041 · doi:10.1080/15326348708807063
[104] J. Riordan,Stochastic Service Systems (Wiley, New York, 1962). · Zbl 0106.33601
[105] S.N. Stepanov,Numerical Methods of Calculation for Systems with Repeated Calls (Nauka, Moscow, 1983). · Zbl 0517.90032
[106] S. Stepanov, Optimal calculation of characteristics of models with repeated calls,Proc. 12th Int. Teletraffic Congress, Torino (1988).
[107] R.I. Wilkinson, Theories for toll traffic engineering in the USA, Bell Syst. Techn. J. 35 (2) (1956) 421-507.
[108] R. Wilkinson and R. Radnik, Customers’ retrials in toll circuit operation,IEEE Int. Conf. on Communications (1968).
[109] R.W. Wolff,Stochastic Modeling and the Theory of Queues (Prentice-Hall, Englewood Cliffs, NJ, 1989). · Zbl 0701.60083
[110] A.M. Zelinskiy and Yu.N. Kornishev, Two models of a system with repeated calls, Elektrosvyaz 1 (1978) 60-63.
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