Queueing Systems

, Volume 7, Issue 2, pp 127–167 | Cite as

A survey of retrial queues

  • Gennadij Falin
Invited Paper

Abstract

We present a survey of the main results and methods of the theory of retrial queues, concentrating 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 buffer. 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.

Keywords

Retrial queues single and multi-channel queue-length waiting time busy period limit theorems numerical methods 

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References

On single-channel retrial queues and related work

  1. [1]
    A.M. Aleksandrov, A queueing system with repeated orders, Eng. Cybernet. Rev. 12 (3) (1974) 1–4.Google Scholar
  2. [2]
    Q.H. Choo, The interaction of theory and simulation in queueing analysis, Ph.D. Thesis, Chelsea College, University of London (1978).Google Scholar
  3. [3]
    Q.H. Choo and B. Conolly, New results in the theory of repeated orders queueing systems, J. Appl. Probab. 16 (1979) 631–640.Google Scholar
  4. [4]
    J.W. Cohen, Basic problems of telephone traffic theory and the influence of repeated calls, Philips Telecom. Rev. 18 (2) (1957) 49–100.Google Scholar
  5. [5]
    B.W. Conolly, Letter to the Editor, J. Appl. Probab. 19 (1982) 904–905.Google Scholar
  6. [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).Google Scholar
  7. [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).Google Scholar
  8. [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).Google Scholar
  9. [9]
    G.I. Falin, Aggregate arrival of customers in one-line system with repeated calls, Ukrainian Math. J. 28 (1976) 437–440.Google Scholar
  10. [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.Google Scholar
  11. [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.Google Scholar
  12. [12]
    G.I. Falin, Model of coupled switching in the presence of recurrent calls, Eng. Cybernet. Rev. 17 (1) (1979) 53–59.Google Scholar
  13. [13]
    G.I. Falin, A single-line system with secondary orders, Eng. Cybernet. Rev. 17 (2) (1979) 76–83.Google Scholar
  14. [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.Google Scholar
  15. [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).Google Scholar
  16. [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.Google Scholar
  17. [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.Google Scholar
  18. [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.Google Scholar
  19. [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.Google Scholar
  20. [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).Google Scholar
  21. [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.Google Scholar
  22. [22]
    G.I. Falin, Quasi-input process in theM/G/1/∞ queue, Adv. Appl. Probab. 16 (1984) 695–696.Google Scholar
  23. [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).Google Scholar
  24. [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).Google Scholar
  25. [25]
    G.I. Falin, On waiting time process in single-line queues with repeated calls, J. Appl. Probab. 23 (1986) 185–192.Google Scholar
  26. [26]
    G.I. Falin, On ergodicity of multichannel queueing systems with repeated calls, Sov. J. Comput. Syst. Sci. 25 (1) (1987) 60–65.Google Scholar
  27. [27]
    G.I. Falin, Single-line repeated orders queueing systems, Mathematische Operationsforschung und Statistik, Optimization 5 (1986) 649–667.Google Scholar
  28. [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.Google Scholar
  29. [29]
    G.I. Falin, On a multiclass batch arrival retrial queue, Adv. Appl. Probab. 20 (1988) 483–487.Google Scholar
  30. [30]
    G.I. Falin, On the quasi-input process for theM/G/1/∞ queueing system, Ukrainian Math. J. 40 (1988) 226–229.Google Scholar
  31. [31]
    G.I. Falin, On virtual waiting time in retrial queues, Vestnik Moscow Univ. Ser. 1, Math. Mech., to appear.Google Scholar
  32. [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).Google Scholar
  33. [33]
    B.S. Greenberg, Queueing systems with returning customers and the optimal order of tandem queues, Ph.D. Thesis, University of California, Berkeley (1986).Google Scholar
  34. [33a]
    B.S. Greenberg,M/G/1 queueing systems with returning customers, J. Appl. Probab. 26 (1989) 152.Google Scholar
  35. [34]
    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.Google Scholar
  36. [35]
    S.A. Greeschechkin, Branching processes and queues with repeated calls or random service, Theory of Probability and its Applications, to appear.Google Scholar
  37. [36]
    T. Hanschke, A model for planning switching networks, in:Operations Research Proceedings 1984 (Springer, Berlin/Heidelberg, 1985).Google Scholar
  38. [37]
    T. Hanschke, TheM/G/1/1 queue with repeated attempts and different types of feedback effects, OR Spektrum 7 (1985) 209–215.Google Scholar
  39. [38]
    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).Google Scholar
  40. [39]
    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).Google Scholar
  41. [40]
    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).Google Scholar
  42. [41]
    I.I. Homitchkov, A model of local area computer network with random multiple access, Automatics and Telemechanics 1 (1987) 58–62 (in Russian).Google Scholar
  43. [42]
    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).Google Scholar
  44. [43]
    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).Google Scholar
  45. [44]
    J.J. Hunter, The non-renewal nature of the quasi-input process in theM/G/1/∞ queue, J. Appl. Probab. 23 (1986) 803–811.Google Scholar
  46. [45]
    G.L. Jonin and J.J. Sedol, Investigation of telephone systems with repeated calls, Latvian Math. Yearbook 7 (1970) 71–83.Google Scholar
  47. [46]
    G.L. Jonin and J.J. Sedol,Tables of Probabilistic Characteristics of Fully Available Trunk Groups in the Case of Repeated Calls (Moscow, 1970).Google Scholar
  48. [47]
    G.L. Jonin and J.J. Sedol, Telephone systems with repeated calls,Proc. 6th Int. Teletraffic Congress (1970) pp. 435/1–435/5.Google Scholar
  49. [48]
    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).Google Scholar
  50. [49]
    G.L. Jonin and N.M. Brezgunova, One-line system with repeated calls in the case of Γ-distributed occupation time, Latvian Math. Yearbook 11 (1972) 65–71.Google Scholar
  51. [50]
    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).Google Scholar
  52. [51]
    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.Google Scholar
  53. [52]
    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).Google Scholar
  54. [53]
    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).Google Scholar
  55. [54]
    V.A. Kapyrin, A study of the stationary characteristics of a queueing system with recurring demands, Cybernetics 13 (1977) 584–590.Google Scholar
  56. [55]
    J. Keilson, J. Cozzolino and H. Young, A service system with unfilled requests repeated, Oper. Res. 16 (1968) 1126–1137.Google Scholar
  57. [56]
    A.G. de Kok, Computational methods for single server systems with repeated attempts, Report #89, Interfaculteit der Actuariële Wetenschappen en Econometrie, Amsterdam (1982).Google Scholar
  58. [57]
    A.G. de Kok, Algorithmic methods for single server systems with repeated attempts, Statistica Neerlandica 38 (1984) 23–32.Google Scholar
  59. [58]
    Y.N. Kornishov, Calculation of coupled switching, Trudy Utchebnih Institutov Svyasi 37 (1968) 96–104 (in Russian).Google Scholar
  60. [59]
    Y.N. Kornishov, Repeated calls in a trunk-line, Elektrosvyaz 1 (1974) 35–41 (in Russian).Google Scholar
  61. [60]
    Y.N. Kornishov, A single-line queue with repeated calls and advance service, Izv. ANSSSR. Tekhn. Kibernetika 2 (1977) 83–88 (in Russian).Google Scholar
  62. [61]
    Y.N. Kornishov and A.M. Zelinskiy, Analysis of subscriber's line states, in:Information Networks and its Analysis (Moscow, 1978) (in Russian).Google Scholar
  63. [62]
    Y.N. Kornishov, A single-line queue with heterogeneity repeated calls, in:Teletraffic Theory and Networks with Controlled Elements (Moscow, 1980)(in Russian).Google Scholar
  64. [63]
    V.G. Kulkarni, Letter to the Editor, J. Appl. Probab. 19 (1982) 901–904.Google Scholar
  65. [64]
    V.G. Kulkarni, On queueing systems with retrials, J. Appl. Probab. 20 (1983) 380–389.Google Scholar
  66. [65]
    V.G. Kulkarni, A game theoretic model for two types of customers competing for service, Oper. Res. Lett. 2 (1983) 119–122.Google Scholar
  67. [66]
    V.G. Kulkarni, Expected waiting times in a multiclass batch arrival retrial queue, J. Appl. Probab. 23 (1986) 144–159.Google Scholar
  68. [67]
    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.Google Scholar
  69. [68]
    B. Pourbabai, Analysis of aG/M/K/0 queueing system with heterogeneous servers and retrials, Int. J. Syst. Sci. 18 (1987) 985–992.Google Scholar
  70. [69]
    G.E. Ridout, A study of retrial queueing systems with buffers, M.A.Sc. Thesis, Department of Industrial Engineering, University of Toronto (1984).Google Scholar
  71. [70]
    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.Google Scholar
  72. [71]
    S.N. Stepanov, Moments of overload traffic for single-line queues with repeated calls, Izv. AN SSSR. Tekhn. Kibernetika 1 (1977) 88–93.Google Scholar
  73. [72]
    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).Google Scholar
  74. [73]
    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).Google Scholar
  75. [74]
    T. Yang and J.G.C. Templeton, A survey on retrial queues, Queueing Systems 2 (1987) 203–233.Google Scholar
  76. [75]
    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).Google Scholar
  77. [76]
    A.M. Zelinskiy and Y.N. Kornishov, Two models of a system with repeated calls, Elektrosvyaz 1 (1978) 60–63 (in Russian).Google Scholar

On multi-channel retrial queues

  1. [77]
    G. Bretschneider, Repeated calls with limited repetition probability,Proc. 6th Int. Teletraffic Congress, Munich (1970) pp. 431/1–434/5.Google Scholar
  2. [78]
    N. Deul, Stationary conditions for multiserver queueing systems with repeated calls, Elektronische Informationsverarbeitung und Kybernetik 10–12 (16) (1980) 607–613.Google Scholar
  3. [79]
    A. Elldin, Approach to the theoretical description of repeated call attempts, Ericsson Technics 23 (3) (1967) 346–407.Google Scholar
  4. [80]
    G.I. Falin, Not completely accessible schemes with allowance for repeated calls, Eng. Cybernet. Rev. 18 (5) (1980) 56–63.Google Scholar
  5. [81]
    G.I. Falin, Switching systems with allowance for repeated calls, Probl. Inform. Transmission 16(1980) 145–151.Google Scholar
  6. [82]
    G.I. Falin, Repeated calls in structurally complex systems, Eng. Cybernet. Rev. 18 (6) (1980) 46–51.Google Scholar
  7. [83]
    G.I. Falin, Investigation of weakly loaded switching systems with repeated calls, Eng. Cybernet. Rev. 19 (3) (1981) 69–73.Google Scholar
  8. [84]
    G.I. Falin, State consolidation in symmetrical partially accessible circuits, Probl. Control Inform. Theory 11 (1982) 3–12.Google Scholar
  9. [85]
    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.Google Scholar
  10. [86]
    G.I. Falin, On the accuracy of a numerical method of calculation of characteristics of systems with repeated calls, Elektrosvyaz 8 (1983) 35–36.Google Scholar
  11. [87]
    G.I. Falin, On sufficient conditions for ergodicity of multi-channel queueing systems with repeated calls, Adv. Appl. Probab. 16 (1984) 447–448.Google Scholar
  12. [88]
    G.I. Falin, Double-channel queueing system with repeated calls, Paper #4221-84, All-Union Institute for Scientific and Technical Information, Moscow (1984).Google Scholar
  13. [89]
    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.Google Scholar
  14. [90]
    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].Google Scholar
  15. [91]
    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).Google Scholar
  16. [92]
    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].Google Scholar
  17. [93]
    G.I. Falin, Multichannel queueing systems with repeated calls under high intensity of repetition, J. Inform. Processing Cybernet. 1 (1987) 37–47.Google Scholar
  18. [94]
    G.I. Falin, Comparability of migration processes, Probab. Theory Appl. 2 (1986) 392–396.Google Scholar
  19. [95]
    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.Google Scholar
  20. [96]
    G.I. Falin, Theorems of ergodicity and stability for retrial queues, Ukrainian Math. J., to appear.Google Scholar
  21. [97]
    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.Google Scholar
  22. [98]
    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).Google Scholar
  23. [99]
    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
  24. [100]
    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.Google Scholar
  25. [101]
    Yu.N. Kornishov, Waiting positions for overloading trunks, Elektrosvyaz 7 (1974) 32–39.Google Scholar
  26. [102]
    C.E.M. Pearce, On the problem of re-attempted calls in teletraffic, Commun. Statist.-Stochastic Models 3 (3) (1987) 393–407.Google Scholar
  27. [103]
    J. Riordan,Stochastic Service Systems (Wiley, New York, 1962).Google Scholar
  28. [104]
    S.N. Stepanov,Numerical Methods of Calculation for Systems with Repeated Calls (Nauka, Moscow, 1983).Google Scholar
  29. [105]
    S. Stepanov, Optimal calculation of characteristics of models with repeated calls,Proc. 12th Int. Teletraffic Congress, Torino (1988).Google Scholar
  30. [106]
    R.I. Wilkinson, Theories for toll traffic engineering in the USA, Bell Syst. Techn. J. 35 (2) (1956) 421–507.Google Scholar
  31. [107]
    R. Wilkinson and R. Radnik, Customers' retrials in toll circuit operation,IEEE Int. Conf. on Communications (1968).Google Scholar
  32. [108]
    R.W. Wolff,Stochastic Modeling and the Theory of Queues (Prentice-Hall, Englewood Cliffs, NJ, 1989).Google Scholar
  33. [109]
    A.M. Zelinskiy and Yu.N. Kornishev, Two models of a system with repeated calls, Elektrosvyaz 1 (1978) 60–63.Google Scholar

Copyright information

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

Authors and Affiliations

  • Gennadij Falin
    • 1
  1. 1.Department of Probability, Mechanics and Mathematics FacultyMoscow State UniversityMoscowUSSR

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