Skip to main content
SpringerLink
Log in

Queueing systems with correlated arrival flows and their applications to modeling telecommunication networks

  • Reviews
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

We give a brief survey of literature devoted to studying queueing systems with Markovian and batch Markovian arrival processes and their application to modeling telecommunication networks.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kleinrock, L., Queueing Systems, vol. II: Computer Applications, New York: Wiley, 1976.

    MATH  Google Scholar 

  2. Schwartz, M., Computer Communication Network. Design and Analysis, New Jersey: Prentice-Hall, 1977.

    MATH  Google Scholar 

  3. Basharin, G.P., Bocharov, P.P., and Kogan, Ya.A., Analiz ocheredei v vychislitel’nykh setyakh. Teoriya i metody rascheta (Analysis of Queues in Computational Networks. Theory and Computational Methods), Moscow: Nauka, 1989.

    MATH  Google Scholar 

  4. Vishnevskii, V.M. and Zhozhikashvili, V.A., Seti massovogo obsluzhivaniya: teoriya i primenenie v setyakh EVM (Queueing Networks: Theory and Applications in Computer Networks), Moscow: Radio i Svyaz’, 1988.

    Google Scholar 

  5. Vishnevskii, V.M., Teoreticheskie osnovy proektirovaniya komp’yuternykh setei (Theoretical Foundations of the Design of Computer Networks), Moscow: Tekhnosfera, 2003.

    Google Scholar 

  6. Vishnevskii, V.M., Portnoi, S.L., and Shakhnovich, I.V., Entsiklopediya WiMAX. Put’ k 4G (The Encyclopaedia of WIMAX: The Road to 4G), Moscow: Tekhnosfera, 2010.

    Google Scholar 

  7. Lakatos, L., Szeidl, L., and Telek, M., Introduction to Queueing Systems with Telecommunication Application, New York: Springer, 2013.

    Book  MATH  Google Scholar 

  8. Lema, M.A., Pardo, E., Galinina, O., Andreev, S., and Dohler, M., Flexible Dual-Connectivity Spectrum Aggregation for Decoupled Uplink and Downlink Access in 5G Heterogeneous Systems, IEEE J. Selected Areas Commun., 2016, no. 99, pp. 1–12.

    Google Scholar 

  9. Niknam, S., Nasir, A.A., Mehrpouyan, H., and Natarajan, B., A Multiband OFDMA Heterogeneous Network for Millimeter Wave 5G Wireless Applications, IEEE Access, 2016, vol. 4, pp. 640–648.

    Article  Google Scholar 

  10. Vishnevsky, V., Larionov, A., and Frolov, S., Design and Scheduling in 5G Stationary and Mobile Communication Systems Based on Wireless Millimeter-Wave Mesh Networks, Commun. Comput. Inform. Sci., 2014, vol. 279, pp. 11–27.

    Article  Google Scholar 

  11. Vishnevsky, V.M., Larionov, A.A., and Ivanov, R.E., Applying Graph-Theoretic Approach for Time- Frequency Resource Allocation in 5G MmWave Backhaul Network, IEEE Xplore Digital Library, 2016, pp. 71–78.

    Google Scholar 

  12. Leland, W.E., Taqqu Murad, S., Willinger, W., and Wilson, D.V., On the Self-Similar Nature of Ethernet Traffic, J. IEEE/ACM Trans. Networking, 1994, vol. 2, no. 1, pp. 1–15.

    Article  Google Scholar 

  13. Tsybakov, B.S., Model of Teletraffic Based on a Self-Congruent Random Process, Radiotekh., 1999, no. 5, pp. 24–31.

    Google Scholar 

  14. Neuts, M.F., Versatile Markovian Point Process, J. Appl. Probab., 1979, vol. 16, no. 4, pp. 764–779.

    Article  MATH  MathSciNet  Google Scholar 

  15. Ramaswami, V., The N/G/1 Queue and Its Detailed Analysis, Adv. Appl. Probab., 1980, vol. 12, no. 1, pp. 222–261.

    Article  MATH  MathSciNet  Google Scholar 

  16. Dudin, A.N., Listopad, N.I., and Tsarenkov, G.V., An Improved Optimization Algorithm for an Internet Node, in Problemy proektirovaniya informatsionno-telekommunikatsionnykh sistem (Problems of Designing Information and Telecommunication Systems), Minsk: BGU, 2001, pp. 28–43.

    Google Scholar 

  17. Vishnevskii, V.M., Minnikhanov, R.N., Dudin, A.N., et al., A New Generation of Security Systems on Car Roads and Its Application to Intelligent Transportation Systems, Inform. Tekhnol. Vychisl. Sist., 2013, no. 4, pp. 17–26.

    Google Scholar 

  18. Dudin, A.N., Vishnevsky, V.M., and Sinjugina, Ju.V., Analysis of the BMAP/G/1 Queue with Gated Service and Adaptive Vacations Duration, Telecommun. Syst. J., 2016, vol. 61, no. 3, pp. 403–415.

    Article  Google Scholar 

  19. Vishnevsky, V.M., Dudin, A.N., Semenova, O.V., and Klimenok, V.I., Performance Analysis of the BMAP/G/1 Queue with Gated Servicing and Adaptive Vacations, Perform. Evaluat., 2011, vol. 68, no. 5, pp. 446–462.

    Article  Google Scholar 

  20. Klimenok, V. and Vishnevsky, V., Unreliable Queueing System with Cold Redundancy, Commun. Comput. Inform. Sci., 2015, vol. 522, pp. 336–347.

    Article  Google Scholar 

  21. Dudin, A., Klimenok, V., and Vishnevsky, V., Analysis of Unreliable Single Server Queuing System with Hot Back-up Server, Commun. Comput. Inform. Sci., 2015, vol. 499, pp. 149–161.

    Article  Google Scholar 

  22. Vishnevskii, V.M, Semenova, O.V., and Sharov, S.Yu., Modeling and Analysis of a Hybrid Communication Channel Based on Free-Space Optical and Radio-Frequency Technologies, Autom. Remote Control, 2013, vol. 74, no. 3, pp. 521–528.

    Article  MATH  MathSciNet  Google Scholar 

  23. Kim, C.S., Klimenok, V.I., and Dudin, A.N., Optimization of Guard Channel Policy in Cellular Mobile Networks with Account of Retrials, Comput. Oper. Res., 2014, vol. 43, pp. 181–190.

    Article  MATH  MathSciNet  Google Scholar 

  24. Lee, M.H., Dudin, A., and Klimenok, V., Mathematical Analysis of the Multi-Server Queueing Model for Dynamic Channel Reservation in Wireless Networks, IEEE Commun. Lett., 2006, vol. 10, pp. 855–857.

    Article  Google Scholar 

  25. Do, T.V., Solution for a Retrial Queueing Problem in Cellular Networks with the Fractional Guard Channel Policy, Math. Comput. Modell., 2011, vol. 53, pp. 2059–2066.

    Article  MATH  Google Scholar 

  26. Zhou, Z. and Zhu, Y., Optimization of the (MAP1,MAP2)/(PH1, PH2)/N Retrial Queue Model of Wireless Cellular Networks with Channel Allocation, Comput. Electric. Eng., 2013, vol. 39, pp. 1637–1649.

    Google Scholar 

  27. Kim, C.S., Dudin, A., Dudin, S., and Dudina, O., Analysis and Optimization of Guard Channel Policy with Buffering in Cellular Mobile Networks, Comput. Networks., 2016, vol. 107, pp. 258–269.

    Article  Google Scholar 

  28. Vishnevskii, V.M., Samuilov, K.E., Naumov, V.A., and Yarkina, N.V., Model of an LTE Cell with Inter- Machine Traffic in the Form of a Multiservice Queueing System with Elastic and Stream customers and Markovian Arrival Process, Vestn. Ross. Univ. Druzhby Narodov, Ser. “Mat. Informatika. Fiz.,” 2016, no. 4. pp. 26–36.

    Google Scholar 

  29. Vishnevsky, V.M., Larionov, A.A., and Ivanov, R.E., An Open Queueing Network with a Correlated Input Arrival Process for Broadband Wireless Network Performance Evaluation, Inform. Technol. Math. Modell. Queueing Theory Appli., 2016, CCIS, vol. 638, pp. 354–365.

    Google Scholar 

  30. Vishnevsky, V.M., Dudin, A.N., Kozyrev, D.V., and Larionov, A.A., Methods of Performance Evaluation of Broadband Wireless Networks Along the Long Transport Routes, Commun. Comput. Inform. Sci., 2016, vol. 601, pp. 72–85.

    Article  MATH  Google Scholar 

  31. Vishnevsky, V.M., Krishnamoorthy, A., Kozyrev, D.V., and Larionov, A.A., Review of Methodology and Design of Broadband Wireless Networks with Linear Topology, Indian J. Pure Appl. Math., 2016, vol. 47, no. 2, pp. 329–342.

    Article  MATH  MathSciNet  Google Scholar 

  32. Vishnevsky, V.M., Larionov, A.A., and Smolnikov, R.V., Optimization of Topological Structure of BroadbandWireless Networks Aiong the Long Traffic Routes, Distributed Comput. Commun. Networks, 2016, vol. 601, pp. 30–39.

    Article  MATH  Google Scholar 

  33. Dongxiu Ou, Yuchen Yang, Lixia Xue, and Decun Dong, Optimal Connectivity-Based Deployment of Roadside Units for Vehicular Networks in Urban Areas, Transportat. Res. Record: J. Transport. Res. Board, 2016, vol. 2559, pp. 46–56.

    Article  Google Scholar 

  34. Cristiano M. and Meira, Wagner, An Architecture Integrating Stationary and Mobile Roadside Units for Providing Communication on Intelligent Transportation Systems, IEEE/IFIP NOMS (Network Oper. Management Sympos.), Istanbul, Turkey, 2016.

    Google Scholar 

  35. Lucantoni, D.M., New Results on the Single Server Queue with a Batch Markovian Arrival Process, Commun. Statist.: Stoch. Models, 1991, vol. 7, no. 1, pp. 1–46.

    MATH  MathSciNet  Google Scholar 

  36. Lucantoni, D.M., The BMAP/G/1 Queue: A Tutorial, in Models and Techniques for Performance Evaluation of Computer and Communications Systems, Donatiello, L. and Nelson, R., Eds., London, Springer-Verlag, 1993, pp. 330–358.

    Chapter  Google Scholar 

  37. Chakravarthy, S.R., The Batch Markovian Arrival Process: A Review and Future Work, in Advances Probab. Theory Stochast. Proc., New Jersey: Notable Publicat., 2001, pp. 21–49.

    Google Scholar 

  38. Dudin, A.N., Optimal Multithreshold Control for the BMAP/G/1 Queue with N Service Modes, Queueing Syst., 1998, vol. 30, pp. 41–55.

    Article  MATH  MathSciNet  Google Scholar 

  39. He, Q.M., Queues with Marked Customers, Adv. Appl. Probab., 1996, vol. 28, pp. 567–587.

    Article  MATH  MathSciNet  Google Scholar 

  40. Asmussen, S., Matrix-Analytic Models and Their Analysis, Scandinav. J. Statist., 2000, vol. 27, no. 2, pp. 193–226.

    Article  MATH  MathSciNet  Google Scholar 

  41. Buchholz, P., Kemper, P., and Kriege, J., Multi-ClassMarkovian Arrival Processes and Their Parameter Fitting, Perform. Evaluat., 2010, vol. 67, pp. 1092–1106.

    Article  Google Scholar 

  42. Buchholz, P. and Panchenko, A., Two-Step EMAlgorithm for MAP Fitting, Lecture Notes Comput. Sci., 2004, vol. 3280, pp. 217–272.

    Article  Google Scholar 

  43. Meszaros, A., Papp, J., and Telek, M., Fitting Traffic with Discrete Canonical Phase Type Distribution and Markov Arrival Processes, Int. J. Appl. Math. Comput. Sci., 2014, vol. 24, no. 3, pp. 453–470.

    Article  MATH  MathSciNet  Google Scholar 

  44. Panchenko, A. and Thummler, A., Efficient Phase-Type Fitting with Aggregated Traffic Traces, Perform. Evaluat., 2007, vol. 64, nos. 7–8, pp. 629–645.

    Article  Google Scholar 

  45. Ryden, T., Statistical Inference for Markov-Modulated Poisson Processes and Markovian Arrival Processes, in Advances Algorith. Methods Stochast. Models, Notable Publicat., New Jersey: Neshanic Station, 2000, pp. 329–350.

    Google Scholar 

  46. Diamond, J.E. and Alfa, A.S., On Approximating Higher Order MAPs with MAPs of Order two, Queueing Syst., 2000, vol. 34, nos. 1–4, pp. 269–288.

    Article  MATH  MathSciNet  Google Scholar 

  47. Artalejo, J.R., Gomez-Corral, A., and He, Q.-M., Markovian Arrivals in Stochastic Modelling: A Survey and Some New Results, SORT, 2010, vol. 34, no. 2, pp. 101–144.

    MATH  MathSciNet  Google Scholar 

  48. Neuts, M.F., Matrix-Geometric Solutions in Stochastic Models, Baltimore: The Johns Hopkins Univ. Press, 1981.

    MATH  Google Scholar 

  49. Ozawa, T., Sojourn Time Distributions in the Queue Defined by a General QBD Process, Queueing Syst., 2006, vol. 53, pp. 203–211.

    Article  MATH  MathSciNet  Google Scholar 

  50. Neuts, M.F., Structured Stochastic Matrices of M/G/1 Type and Their Applications, New York: Marcel Dekker, 1989.

    MATH  Google Scholar 

  51. Klimenok, V.I. and Dudin, A.N., Multi-Dimensional Asymptotically Quasi-Toeplitz Markov Chains and Their Application in Queueing Theory, Queueing Syst., 2006, vol. 54, pp. 245–259.

    Article  MATH  MathSciNet  Google Scholar 

  52. Klimenok, V.I., On the Modification of Rouche’s Theorem for the Queueing Theory Problems, Queueing Syst., 2001, vol. 38, no. 4, pp. 431–434.

    Article  MATH  MathSciNet  Google Scholar 

  53. Dudin, A., Klimenok, V., and Moon Ho Lee, Recursive Formulas for the Moments of Queue Length in the BMAP/G/1 Queue, IEEE Commun. Lett., 2009, vol. 13, no. 5, pp. 351–353.

    Article  Google Scholar 

  54. Gail, H.R., Hantler, S.L., and Taylor, B.A., Spectral Analysis of M/G/1 and G/M/1 Type Markov Chains, Adv. Appl. Probab., 1996, vol. 28, no. 1, pp. 114–165.

    Article  MATH  MathSciNet  Google Scholar 

  55. Klimenok, V.I., Quasi-Toeplitz and Asymptotically Quasi-Toeplitz Markov Chains and Their Applications to the Analysis of Queueing Systems, Doctoral Dissertation, BGU, 2002.

    Google Scholar 

  56. Dudin, A.N. and Klimenok, V.I., Calculating the Characteristics of a Queue in a Synchronous Random Markov Environment, Autom. Remote Control, 1997, vol. 58, no. 1, pp. 58–66.

    MATH  Google Scholar 

  57. Dudin, A.N. and Klimenok, V.I., Multidimensional Quasi-Toeplitz Markov Chains, J. Appl. Math. Stochast. Anal., 1999, vol. 12, no. 4, pp. 393–415.

    Article  MATH  Google Scholar 

  58. Klimov, G.P., Stokhasticheskie sistemy obsluzhivaniya (Stochastic Queueing Systems), Moscow: Nauka, 1966.

    Google Scholar 

  59. Kemeni, J.G., Snell, J.L., and Knapp, A.W., Denumerable Markov Chains, New York: Van Nostrand, 1966.

    MATH  Google Scholar 

  60. Ramaswami, V. and Lucantoni, D., Algorithm for the Multi-Server Queue with Phase-Type Service, Commun. Statist.: Stoch. Models, 1985, vol. 1, pp. 393–417.

    MATH  Google Scholar 

  61. Cinlar, E., Introduction to Stochastic Processes, New Jersey: Prentice Hall, 1975.

    MATH  Google Scholar 

  62. Klimov, G.P., Teoriya verioyatnostei i matematicheskaya statistika (Probability Theory and Mathematical Statistics), Moscow: Mosk. Gos. Univ., 1983.

    MATH  Google Scholar 

  63. Rykov, V.V. and Yastrebenetskii, M.A., On Regenerating Processes with Several Types of Regeneration Points, Kibernetika, 1971, no. 3, pp. 82–86.

    Google Scholar 

  64. Machihara, F., A New Approach to the Fundamental Period of a Queue with Phase-Type Markov Renewal Arrivals, Commun. Statist.: Stoch. Models, 1990, vol. 6, no. 3, pp. 551–560.

    MATH  MathSciNet  Google Scholar 

  65. Bocharov, P.P. and Pechinkin, A.V., Teoriya massovogo obsluzhivaniya (Queueing Theory), Moscow: Ross. Univ. Druzhby Narodov, 1995.

    Google Scholar 

  66. Dudin, A.N. and Klimenok, V.I., Sistemy massovogo obsluzhivaniya s korrelirovannymi potokami (Queueing Systems with Correlated Flows), Minsk: BGU, 2000.

    Google Scholar 

  67. Lee, G.M. and Jeon, J.W., A New Approach to an N/G/1 Queue, Queueing Syst., 2000, vol. 35, pp. 317–322.

    Article  MATH  MathSciNet  Google Scholar 

  68. Choi, B.D., Hwang, G.U., and Han, D.H., Supplementary Variable Method Applied to the MAP/G/1 Queueing System, J. Aust. Math. Soc., 1998, ser. B. 40, pp. 86–96.

    Article  MATH  MathSciNet  Google Scholar 

  69. Takine, T. and Takahashi, Y., On the Relationship between Queue Lengths at a Random Instant and at a Departure in the Stationary Queue with BMAP Arrivals, Commun. Statist.: Stoch. Models, 1998, vol. 14, no. 3, pp. 601–610.

    MATH  MathSciNet  Google Scholar 

  70. Takine, T., A New Recursion for the Queue Length Distribution in the Stationary BMAP/G/1 Queue, Commun. Statist.: Stoch. Models, 2000, vol. 16, no. 2, pp. 335–341.

    MATH  MathSciNet  Google Scholar 

  71. Lee, H.W., Park, N.I., and Jeon, J.W., A New Approach to the Queue Length and Waiting Time of BMAP/G/1 Queues, Comput. Oper. Res., 2003, vol. 30, pp. 2021–2045.

    Article  MATH  MathSciNet  Google Scholar 

  72. Chang, S.H., Takine, T., Chae, K.C., and Lee, H.W., A Unified Queue Length Formula for BMAP/G/1 Queue with Generalized Vacations, Commun. Statist.: Stoch. Models, 2002, vol. 18, no. 3, pp. 369–386.

    MATH  MathSciNet  Google Scholar 

  73. Singh, G., Gupta, U.C., and Chaudhry, M.L., Analysis of Queueing-Time Distributions forMAP/DN/1 Queue, Int. J. Comput. Math., 2014, vol. 91, no. 9, pp. 1911–1930.

    Article  MATH  Google Scholar 

  74. Gupta, U.C., Singh, G., and Chaudhry, M.L., An Alternative Method for Computing System-Length Distributions of BMAP/R/1 and BMAP/D/1 Queues Using Roots, Performance Evaluat., 2016, vol. 95, pp. 60–79.

    Article  Google Scholar 

  75. Samanta, S.K., Chaudhry, M.L., and Pacheco, A., Analysis of BMAP/MSP/1 Queue, Method. Comput. Appl. Probab., 2016, vol. 18, pp. 419–440.

    Article  MATH  MathSciNet  Google Scholar 

  76. Nishimura, S., A MAP/G/1 Queue with an Underlying Birth-Death Process, Stochast. Models, 2003, vol. 19, no. 4, pp. 425–447.

    Article  MATH  MathSciNet  Google Scholar 

  77. Maity, A. and Gupta, U.C., A Comparative Numerical Study of the Spectral Theory Approach of Nishimura and the Roots Method based on the Analysis of BDMMAP/G/1 Queue, Int. J. Stochast. Anal., 2015, vol. 2015, article ID 958730.

    MATH  MathSciNet  Google Scholar 

  78. Milovanova, T.A., BMAP/G/1/8 System with Last Come First Served Probabilistic Priority, Autom. Remote Control, 2009, vol. 70, no. 5, pp. 885–896.

    Article  MATH  MathSciNet  Google Scholar 

  79. Lucantoni, D.M. and Neuts, M.F., Some Steady-State Distributions for the MAP/SM/1 Queue, Commun. Statist.: Stoch. Models, 1994, vol. 10, pp. 575–598.

    MATH  MathSciNet  Google Scholar 

  80. Graham, A., Kronecker Products and Matrix Calculus with Applications, Cichester: Ellis Horwood, 1981.

    MATH  Google Scholar 

  81. Machihara, F., A BMAP/SM/1 Queue with Service Times Depending on the Arrival Process, Queueing Syst., 1999, vol. 33, no. 4, pp. 277–291.

    Article  MATH  MathSciNet  Google Scholar 

  82. Blondia, C., The N/G/1 Finite Capacity Queue, Commun. Statist.: Stoch. Models, 1989, vol. 5, no. 1, pp. 273–294.

    MATH  MathSciNet  Google Scholar 

  83. Blondia, C., Finite Capacity Vacation Models with Non-Renewal Input, J. Appl. Probab., 1991, vol. 28, no. 1, pp. 174–197.

    Article  MATH  MathSciNet  Google Scholar 

  84. Baiocchi, A., Analysis of the Loss Probability of the MAP/G/1/K Queue. Part 1: Asymptotic Theory, Commun. Statist.: Stoch. Models, 1994, vol. 10, no. 4, pp. 867–893.

    MATH  Google Scholar 

  85. Baiocchi, A. and Blefari-Melezzi, N., Analysis of the Loss Probability of the MAP/G/1/K Queue. Part 2, Commun. Statist.: Stoch. Models, 1994, vol. 10, no. 4, pp. 895–925.

    MATH  Google Scholar 

  86. Gouweleeuw, F.N., Calculating the Loss Probability in a BMAP/G/1/N+1 Queue, Commun. Statist.: Stoch. Models, 1996, vol. 12, no. 2, pp. 473–492.

    MATH  MathSciNet  Google Scholar 

  87. Dudin, A.N. and Nishimura, S., Optimal Hysteretic Control for a BMAP/SM/1/N Queue with Two Operation Modes, Math. Probl. Eng., 2000, vol. 5, pp. 397–420.

    Article  MATH  Google Scholar 

  88. Dudin, A.N., Klimenok, V.I., and Tsarenkov, G.V., A Single-Server Queueing System with Batch Markov Arrivals, Semi-Markov Service, and Finite Buffer: Its Characteristics, Autom. Remote Control, 2002, vol. 63, no. 8, pp. 1285–1297.

    MATH  Google Scholar 

  89. Dudin, A.N., Shaban, A.A., and Klimenok, V.I., Analysis of a Queue in the BMAP/G/1/N System, Int. J. Simulat.: Syst. Sci. Technol., 2005, vol. 6, nos. 1–2, pp. 13–23.

    Google Scholar 

  90. Dudin, A.N. and Shaban, A.A., Analysis of the BMAP/SM/1/N Type System with Randomized Choice of Customers Admission Discipline, Commun. Comput. Inform. Sci., 2016, vol. 638, pp. 44–56.

    Article  Google Scholar 

  91. Niu, Z., Shua, T., and Takahashi, Y., A Vacation Queue with Setup and Close-Down Times and Batch Markovian Arrival Processes, Perform. Evaluat., 2003, vol. 54, pp. 225–248.

    Article  Google Scholar 

  92. Rusek, K., Janowski, L., and Papir, Z., Transient and Stationary Characteristics of a Packet Buffer Modelled as an MAP/SM/1/B System, Int. J. Appl. Math. Comput. Sci., 2014, vol. 24, no. 2, pp. 429–442.

    Article  MATH  MathSciNet  Google Scholar 

  93. Chydzinski, A., Queue Size in a BMAP Queue with Finite Buffer, Lecture Notes Comput. Sci., 2006, vol. 4003, pp. 200–210.

    Article  Google Scholar 

  94. Bocharov, P.P., A MAP/G/1/r System with a Large Service Time Variation Coefficient, Autom. Remote Control, 2005, vol. 66, no. 11, pp. 1782–1790.

    Article  MATH  MathSciNet  Google Scholar 

  95. Cohen, J.W., Basic Problems of Telephone Traffic and the Influence of Repeated Calls, Philips Telecom. Rev., 1957, vol. 18, no. 2, pp. 49–100.

    Google Scholar 

  96. Wilkinson, R.I., Theories for Toll Traffic Engineering in the USA, The Bell Syst. Technic. J., 1956, vol. 35, no. 2, pp. 421–507.

    Article  Google Scholar 

  97. Artalejo, J.R., Accessible Bibliography on Retrial Queues: Progress in 2000–2009, Math. Comput. Modell., 2010, vol. 51, pp. 1071–1081.

    Article  MATH  MathSciNet  Google Scholar 

  98. Artalejo, J.R. and Gomez-Corral, A., Retrial Queueing Systems: A Computational Approach, Berlin: Springer, 2008.

    Book  MATH  Google Scholar 

  99. Falin, G., A Survey of Retrial Queues, Queueing Syst., Theory Appl., 1990, vol. 7, pp. 127–168.

    MATH  MathSciNet  Google Scholar 

  100. Falin, G. and Templeton, J., Retrial Queues, London: Chapman and Hall, 1997.

    Book  MATH  Google Scholar 

  101. Gomez-Corral, A., A Bibliographical Guide to the Analysis of Retrial Queues through Matrix Analytic Techniques, Ann. Oper. Res., 2006, vol. 141, pp. 163–191.

    Article  MATH  MathSciNet  Google Scholar 

  102. Kim, J. and Kim, B., A Survey of Retrial Queueing Systems, Ann. Oper. Res., 2016, vol. 247, no. 1. pp. 3–36.

    Article  MATH  MathSciNet  Google Scholar 

  103. Yang, T. and Templeton, J., A Survey on Retrial Queues, Queueing Syst., 1987, vol. 2, pp. 201–233.

    Article  MATH  MathSciNet  Google Scholar 

  104. Dudin, A. and Klimenok, V., Queueing System BMAP/G/1 with Repeated Calls, Math. Comput. Modell., 1999, vol. 30, nos. 3–4, pp. 115–128.

    Article  MATH  MathSciNet  Google Scholar 

  105. Dudin, A. and Klimenok, V., A Retrial BMAP/SM/1 System with Linear Repeated Requests, Queueing Syst., 2000, vol. 34, pp. 47–66.

    Article  MATH  MathSciNet  Google Scholar 

  106. Diamond, J.E. and Alfa, A.S., The MAP/PH/1 Retrial Queue, Commun. Statist.: Stoch. Models, 1998, vol. 14, no. 5, pp. 1151–1177.

    MATH  MathSciNet  Google Scholar 

  107. Dudin, A.N. and Klimenok, V.I., BMAP/SM/1 Model with Markov Modulated Retrials, TOP, 1999, vol. 7, no. 2, pp. 267–278.

    Article  MATH  MathSciNet  Google Scholar 

  108. Klimenok, V.I., ABMAP/SM/1 Queueing System with Hybrid Operation Mechanism, Autom. Remote Control, 2005, vol. 66, no. 5, pp. 779–790.

    Article  MATH  MathSciNet  Google Scholar 

  109. Dudin, A.N., Krishnamoorthy, A., Joshua, V.C., and Tsarenkov, G.V., Analysis of the BMAP/G/1 Retrial System with Search of Customers from the Orbit, Eur. J. Oper. Res., 2004, vol. 157, no. 1, pp. 169–179.

    Article  MATH  Google Scholar 

  110. Breuer, L., Dudin, A.N., and Klimenok, V.I., A Retrial BMAP/PH/N System, Queueing Syst., 2002, vol. 40, pp. 433–457.

    Article  MATH  MathSciNet  Google Scholar 

  111. Asmussen, S., Applied Probability and Queues, New York: Springer, 2003.

    MATH  Google Scholar 

  112. Sharma, R., Mathematical Analysis of Queue with Phase Service: An Overview, Adv. Oper. Res., 2014, vol. 2014, article ID 240926.

    MATH  MathSciNet  Google Scholar 

  113. Combe, M., Queueing Models with Dependence Structures, Amsterdam: CWI, 1994.

    Google Scholar 

  114. Gnedenko, B.V. and Kovalenko, I.N., Vvedenie v teoriyu massovogo obsluzhivaniya (Introduction to Queueing Theory), Moscow: Nauka, 1966.

    MATH  Google Scholar 

  115. Klimenok, V.I., Kim, C.S., Orlovsky, D.S., and Dudin, A.N., Lack of Invariant Property of Erlang BMAP/PH/N/0 Model, Queueing Syst., 2005, vol. 49, pp. 187–213.

    Article  MATH  MathSciNet  Google Scholar 

  116. He, Q.M., Li, H., and Zhao, Y.Q., Ergodicity of the BMAP/PH/S/S + K Retrial Queue with PHRetrial Times, Queueing Syst., 2000, vol. 35, pp. 323–347.

    Article  MATH  Google Scholar 

  117. Klimenok, V.I., A Multiserver Retrial Queueing System with Batch Markov Arrival Process, Autom. Remote Control, 2001, vol. 62, no. 8, pp. 1312–1322.

    Article  MATH  MathSciNet  Google Scholar 

  118. Breuer, L., Klimenok, V.I., Birukov, A.A., Dudin, A.N., and Krieger, U.R., Mobile Networks Modelling the Access to a Wireless Network, Eur. Transact. Telecommun., 2005, vol. 16, pp. 309–316.

    Article  Google Scholar 

  119. Klimenok, V.I., Orlovsky, D.S., and Dudin, A.N., A BMAP/PH/N System with Impatient Repeated Calls, Asia-Pacific J. Oper. Res., 2007, vol. 24, no. 3, pp. 293–312.

    Article  MATH  MathSciNet  Google Scholar 

  120. Kim, C.S., Mushko, V.V., and Dudin, A., Computation of the Steady State Distribution for Multi- Server Retrial Queues with Phase Type Service Process, Ann. Oper. Res., 2012, vol. 201, no. 1, pp. 307–323.

    Article  MATH  MathSciNet  Google Scholar 

  121. Ramaswami, V., Independent Markov Process in Parallel, Commun. Statist.: Stoch. Models, 1985, vol. 1, pp. 419–432.

    MATH  Google Scholar 

  122. Ramaswami, V., A Stable Recursion for the Steady State Vector in Markov Chains of M/G/1 Type, Commun. Statist.: Stoch. Models, 1988, vol. 4, no. 1, pp. 183–188.

    MATH  MathSciNet  Google Scholar 

  123. Klimenok, V.I., Orlovsky, D.S., and Kim, C.S., The BMAP/PH/N Retrial Queue with Markovian Flow of Breakdowns, Eur. J. Oper. Res., 2008, vol. 189, no. 3, pp. 1057–1072.

    Article  MATH  MathSciNet  Google Scholar 

  124. Dudina, O., Kim, Ch., and Dudin, S., Retrial Queueing System with Markovian Arrival Flow and Phase Type Service Time Distribution, Comput. Indust. Eng., 2013, vol. 66, pp. 360–373.

    Article  Google Scholar 

  125. Chakravarthy, S. and Dudin, A.N., A Multi-Server Retrial Queue with BMAP Arrivals and Group Services, Queueing Syst., 2002, vol. 42, pp. 5–31.

    Article  MATH  MathSciNet  Google Scholar 

  126. Dudin, A. and Klimenok, V., A Retrial BMAP/PH/N Queueing System with Markov Modulated Retrials, Proc. 2nd Baltic Congr. Future Internet Commun., BCFIC 2012, 2012, pp. 1–6.

    Google Scholar 

  127. Dudin, A. and Klimenok, V., Retrial Queue of BMAP/PH/N Type with Customers Balking, Impatience and Non-Persistence, 2013 Conf. Future Internet Commun. (CFIC), IEEE, Coimbra, pp. 1–6.

  128. Lian, Z. and Liu, L., A Tandem Network withMAP Inputs, Oper. Res. Lett., 2008, vol. 36, pp. 189–195.

    Article  MATH  MathSciNet  Google Scholar 

  129. Gomez-Corral, A., A Tandem Queue with Blocking and Markovian Arrival Process, Queueing Syst., 2002, vol. 41, pp. 343–370.

    Article  MATH  MathSciNet  Google Scholar 

  130. Gomez-Corral, A., On a Tandem G-Network with Blocking, Adv. Appl. Probab., 2002, vol. 34, no. 3, pp. 626–661.

    Article  MATH  MathSciNet  Google Scholar 

  131. Ferng, H.W. and Chang, J.F., Departure Processes of BMAP/G/1 Queues, Queueing Syst., 2001, vol. 39, pp. 109–135.

    Article  MATH  MathSciNet  Google Scholar 

  132. Ferng, H.W. and Chang, J.F., Connection-Wise End-to-End Performance Analysis of Queueing Networks with MMPP Inputs, Perform. Evaluat., 2001, vol. 43, pp. 39–62.

    Article  MATH  Google Scholar 

  133. Heindl, A., Decomposition of General Tandem Networks with MMPP Input, Perform. Evaluat., 2001, vol. 44, pp. 5–23.

    Article  MATH  Google Scholar 

  134. Heindl, A., Decomposition of General Queue Networks with MMPP Inputs and Customer Losses, Perform. Evaluat., 2003, vol. 51, pp. 117–136.

    Article  Google Scholar 

  135. Shioda, S., Departure Process of the MAP/SM/1 Queue, Queueing Syst., 2003, vol. 44, pp. 31–50.

    Article  MATH  MathSciNet  Google Scholar 

  136. Klimenok, V.I., Breuer, L., Tsarenkov, G.V., and Dudin, A.N., The BMAP/G/1/N PH/1/M -1 Tandem Queue with Losses, Perform. Evaluat., 2005, vol. 61, pp. 17–60.

    Article  Google Scholar 

  137. Breuer, D., Dudin, A.N., Klimenok, V.I., and Tsarenkov, G.V., A Two-Phase BMAP/G/1/N PH/1/M - 1 System with Blocking, Autom. Remote Control, 2004, vol. 65, no. 1, pp. 104–115.

    Article  MATH  MathSciNet  Google Scholar 

  138. Klimenok, V., Kim, C.S., Tsarenkov, G.V., Breuer, L., and Dudin, A.N., The BMAP/G/1 ·/ PH/1/M Tandem Queue with Feedback and Losses, Perform. Evaluat., 2007, vol. 64, pp. 802–818.

    Article  Google Scholar 

  139. Bocharov, P.P., Manzo, R., and Pechinkin, A.V., Analysis of a Two-Phase Queueing System with a Markov Arrival Process and Losses, J. Math. Sci., 2005, vol. 131, no. 3, pp. 5606–5613.

    Article  MATH  MathSciNet  Google Scholar 

  140. Bocharov, P.P., Manzo, R., and Pechinkin, A.V., Analysis of a Two-Phase Queueing System with a Markov Arrival Process and Blocking, J. Math. Sci., 2006, vol. 132, no. 5, pp. 578–589.

    Article  MATH  MathSciNet  Google Scholar 

  141. Kim, C.S., Park, S.H., Dudin, A., Klimenok, V., and Tsarenkov, G., Investigaton of the BMAP/G/1 •/PH/1/M Tandem Queue with Retrials and Losses, Appl. Math. Modell., 2010, vol. 34, no. 10, pp. 2926–2940.

    Article  MATH  Google Scholar 

  142. Klimenok, V.I. and Taramin, O.S., Tandem Service System with Batch Markov Flow and Repeated Calls, Autom. Remote Control, 2010, vol. 71, no. 1, pp. 1–13.

    Article  MATH  MathSciNet  Google Scholar 

  143. Kim, C.S., Klimenok, V.I., Taramin, O.S., and Dudin, A., A Tandem BMAP/G/1 ? •/M/N/0 Queue with Heterogeneous Customers, Math. Probl. Eng., 2012, vol. 2012, rticle ID 324604.

    MATH  Google Scholar 

  144. Kim, C.S., Klimenok, V., and Taramin, O., A Tandem Retrial Queueing System with Two Markovian Flows and Reservation of Channels, Comput. Oper. Res., 2010, vol. 37, no. 7, pp. 1238–1246.

    Article  MATH  MathSciNet  Google Scholar 

  145. Kim, Ch., Dudin, A., and Klimenok, V., Tandem Retrial Queueing System with Correlated Arrival Flow and Operation of the Second Station Described by a Markov Chain, Communicat. Comput. Inform. Sci., 2012, vol. 291, pp. 370–382.

    Article  Google Scholar 

  146. Dudin, A., Dudin, S., and Dudina, O., Tandem Queueing System MAP/M/N/K - N ? •M/R/8 with Impatient Customers as a Model of Remote Technical Support, Proc. 2 Baltic Congr. Future Internet Commun., BCFIC 2012, 2012, pp. 1–6.

    Google Scholar 

  147. Kim, C.S., Dudin, A., Dudin, S., and Dudina, O., Tandem Queueing System with Impatient Customers as a Model of Call Center with Interactive Voice Response, Performance Evaluat., 2013, vol. 70, pp. 440–453.

    Article  MATH  Google Scholar 

  148. Dudin, S. and Dudina, O., A Two-Phase MAP/PH/N/K - N ? •/PH/R/8 System as a Model of an Information and Technical Support Center, Probl. Peredachi Inf., 2013, vol. 49, no. 1, pp. 58–72.

    MATH  Google Scholar 

  149. Kim, C.S., Dudin, A., Dudina, O., and Dudin, S., Tandem Queueing System with Infinite and Finite Intermediate Buffers and Generalized Phase-Type Service Time Distribution, Eur. J. Oper. Res., 2014, vol. 235, pp. 170–179.

    Article  MATH  MathSciNet  Google Scholar 

  150. Kim, C.S., Klimenok, V., and Dudin, A., Priority Tandem Queueing System with Retrials and Reservation of Channels as a Model of Call Center, Comput. Indust. Eng., 2016, vol. 96, pp. 61–71.

    Article  Google Scholar 

  151. Buchholz, P., Bounding Stationary Results of Tandem Networks with MAP Input and PH Service Time Distributions, Perform. Evaluat. Rev., 2006, vol. 34, no. 1, pp. 191–202.

    Article  Google Scholar 

  152. Klimenok, V., Dudin, A., and Vishnevsky, V., On the Stationary Distribution of Tandem Queue Consisting of a Finite Number of Stations, Commun. Comput. Inform. Sci., 2012, vol. CCIS 291, pp. 383–392.

    Article  Google Scholar 

  153. Klimenok, V., Dudin, A., and Vishnevsky, V., Tandem Queueing System with Correlated Input and Cross-Traffic, Commun. Comput. Inform. Sci., 2013, vol. CCIS 370, pp. 416–425.

    Article  Google Scholar 

  154. Li, Q.-L., Ying, Y., and Zhao, Y.Q., A BMAP/G/1 Retrial Queue with a Server Subject to Breakdowns and Repairs, Ann. Oper. Res., 2006, vol. 141, pp. 233–270.

    Article  MATH  MathSciNet  Google Scholar 

  155. Li, Q.-L., Constructive Computation in Stochastic Models with Applications: The RG-Factorizations, Berlin: Springer, 2011.

    Google Scholar 

  156. Al-Begain, K., Dudin, A., Klimenok, V., and Dudin, S., Generalised Survivability Analysis of Systems with Propagated Failures, Comput. Math. Appl., 2012, vol. 64, no. 12, pp. 3777–3791.

    Article  MATH  MathSciNet  Google Scholar 

  157. Gelenbe, E., Glynn, P., and Sigman, K., Queues with Negative Arrivals, J. Appl. Probab., 1991, vol. 28, pp. 245–250.

    Article  MATH  MathSciNet  Google Scholar 

  158. Bocharov, P.P. and Vishnevskii, V.M., G-Networks: Development of the Theory of Multiplicative Networks, Autom. Remote Control, 2003, vol. 64, no. 5, pp. 714–739.

    Article  MATH  MathSciNet  Google Scholar 

  159. Artalejo, J.R., G-Networks: A Versatile Approach for Work Removal in Queueing Networks, Eur. J. Oper. Res., 2000, vol. 126, pp. 233–249.

    Article  MATH  MathSciNet  Google Scholar 

  160. Kim, Ch.S., Klimenok, V.I., and Orlovskii, D.S., Multi-Server Queueing System with a Batch Markovian Arrival Process and Negative Customers, Autom. Remote Control, 2006, vol. 67, no. 12, pp. 1958–1973.

    Article  MATH  MathSciNet  Google Scholar 

  161. Bocharov, P.P., d’Apice, C., Manzo, R., and Pechinkin, A.V., Analysis of the Multi-Server Markov Queuing System with Unlimited Buffer and Negative Customers, Autom. Remote Control, 2007, vol. 68, no. 1, pp. 85–94.

    Article  MATH  MathSciNet  Google Scholar 

  162. Klimenok, V.I. and Dudin, A.N., A BMAP/PH/N Queue with Negative Customers and Partial Protection of Service, Commun. Statist. Simulat. Comput., 2012, vol. 41, no. 7, pp. 1062–1082.

    Article  MATH  MathSciNet  Google Scholar 

  163. Krishnamoorthy, A., Pramod, P.K., and Chakravarthy, S.R., Queues with Interruptions: A Survey, Top., 2014, vol. 22, pp. 290–320.

    Article  MATH  MathSciNet  Google Scholar 

  164. Dudin, A., Jacob, V., and Krishnamoorthy, A., A Multi-Server Queueing System with Service Interruption, Partial Protection and Repetition of Service, Ann. Oper. Res., 2015, vol. 233, no. 1, pp. 101–121.

    Article  MATH  MathSciNet  Google Scholar 

  165. Dudin, A. and Nishimura, S., A BMAP/SM/1 Queueing System with Markovian Arrival Input of Disasters, J. Appl. Probab., 1999, vol. 36, no. 3, pp. 868–881.

    Article  MATH  MathSciNet  Google Scholar 

  166. Dudin, A. and Karolik, A.V., BMAP/SM/1 Queue with Markovian Input of Disasters and Non- Instantaneous Recovery, Perform. Evaluat., 2001, vol. 45, no. 1, pp. 19–32.

    Article  MATH  Google Scholar 

  167. Dudin, A. and Semenova, O.V., Stable Algorithm for Stationary Distribution Calculation for a BMAP/SM/1 Queueing System with Markovian Input of Disasters, J. Appl. Probab., 2004, vol. 42, no. 2, pp. 547–556.

    Article  MATH  MathSciNet  Google Scholar 

  168. Dudin, A.N., Kim, C.S., and Klimenok, V.I., Markov Chains with Hybrid Repeated Rows—Upper- Hessenberg Quasi-Toeplitz Structure of Block Transition Probability Matrix, J. Appl. Probab., 2008, vol. 45, no. 1, pp.211–225.

    Article  MATH  MathSciNet  Google Scholar 

  169. Shin, Y.W., Multi-Server Retrial Queue with Negative Customers and Disasters, Queueing Syst., 2007, vol. 55, pp. 223–237.

    Article  MATH  MathSciNet  Google Scholar 

  170. Semenova, O.V., A Queueing System with Two Operation Modes and a Disaster Flow: Its Stationary State Probability Distribution, Autom. Remote Control, 2002, vol. 63, no. 10, pp. 1597–1608.

    Article  MATH  MathSciNet  Google Scholar 

  171. Semenova, O.V., An Optimal Threshold Control for a BMAP/SM/1 System with MAP Disaster Flow, Autom. Remote Control, 2003, vol. 64, no. 9, pp. 1442–1454.

    Article  MATH  MathSciNet  Google Scholar 

  172. Semenova, O.V., A Stable Algorithm for Computing a Stationary Distribution of a BMAP/SM/1 Servicing System with Markov Flow of Disasters and Two Operation Modes, Avtom. Vychisl. Tekhn., 2004, no. 1, pp. 75–84.

    Google Scholar 

  173. Semenova, O.V., Optimal Control for a BMAP/SM/1 Queue with MAP-Input of Disasters and Two Operation Modes, RAIRO—Oper. Res., 2004, vol. 38, no. 2, pp. 153–171.

    Article  MATH  MathSciNet  Google Scholar 

  174. Semenova, O.V., Hysteresis Control for aBMAP/G/1 Queueing System withMarkov Flow of Disasters, Avtom. Vychisl. Tekhn., 2006, no. 2, pp. 63–72.

    Google Scholar 

  175. Semenova, O.V., Optimal Hysteresis Control for BMAP/SM/1 Queue with MAP-Input of Disasters, Quality Technol. Quantitat. Manage., 2007, vol. 4, no. 3, pp. 395–405.

    Article  MathSciNet  Google Scholar 

  176. Semenova, O.V., Multithreshold Control of the BMAP/G/1 Queuing System with MAP Flow of Markovian Disasters, Autom. Remote Control, 2007, vol. 68, no. 1, pp. 95–108.

    Article  MATH  MathSciNet  Google Scholar 

  177. Semenova, O.V. and Dudin, A.N., Optimal Multi-Threshold Control for a BMAP/SM/1 Retrial Queue with Disasters, J. Appl. Math. Anal. Appl., 2008, vol. 3, nos. 1–2, pp. 25–43.

    MATH  MathSciNet  Google Scholar 

  178. Klimenok, V., Dudin, A., and Vishnevsky, V., Performance Analysis of Unreliable Queue with Back-Up Server, Commun. Comput. Inform. Sci., 2015, vol. CCIS 564, pp. 226–239.

    Article  Google Scholar 

  179. Klimenok, V. and Vishnevsky, V., A Dual Tandem Queue with Multi-Server Stations and Losses, Commun. Comput. Inform. Sci., vol. CCIS 608, pp. 316–325.

  180. Tian, N. and Zhang, Z.G., Vacation Queueing Models—Theory and Applications, Heidelberg: Springer, 2006.

    Book  MATH  Google Scholar 

  181. Vishnevskii, V.M. and Semenova, O.V., Mathematical Methods to Study the Polling Systems, Autom. Remote Control, 2006, vol. 67, no. 2, pp. 173–220.

    Article  MATH  MathSciNet  Google Scholar 

  182. Vishnevskii, V.M. and Semenova, O.V., Sistemy pollinga. Teoriya i primenenie v shirokopolosnykh besprovodnykh setyakh (Polling Systems. Theory and Applications in Broadband Wireless Networks), Moscow: Tekhnosfera, 2007.

    Google Scholar 

  183. Vishnevsky, V. and Semenova, O., Polling Systems: Theory and Applications for Broadband Wireless Networks, London: LAMBERT, 2012.

    Google Scholar 

  184. Blondia, C., Finite Capacity Vacation Model with Non-Renewal Input, J. Appl. Probab., 1991, vol. 28, pp. 174–197.

    Article  MATH  MathSciNet  Google Scholar 

  185. Ferrandiz, J.M., The BMAP/G/1 Queue with Server Set-Up Times and Server Vacations, Adv. Appl. Probab., 1993, vol. 25, pp. 235–254.

    Article  MATH  MathSciNet  Google Scholar 

  186. Matendo, S.K., Some Performance Measures for Vacation Models with a Batch Markovian Arrival Process, J. Appl. Math. Stochast. Anal., 1994, vol. 7, pp. 111–124.

    Article  MATH  MathSciNet  Google Scholar 

  187. Kasahara, S., Takine, T., Takahashi, Y., and Hasegawa, T., MAP/G/1 Queues under N-Policy with and without Vacations, J. Oper. Res. Soc. Jpn., 1996, vol. 39, no. 2, pp. 188–212.

    Article  MATH  MathSciNet  Google Scholar 

  188. Banik, A.D., The Infinite-Buffer Single Server Queue with a Variant of Multiple Vacation Policy and Batch Markovian Arrival Process, Appl. Math. Modell., 2009, vol. 33, pp. 3025–3039.

    Article  MATH  MathSciNet  Google Scholar 

  189. Baek, J.W., Lee, H.W., Lee, S.W., and Ahn, S., A Workload Factorization for BMAP/G/1 Vacation Queues under Variable Service Speed, Oper. Res. Lett., 2014, vol. 42, pp. 58–63.

    Article  MathSciNet  MATH  Google Scholar 

  190. Peng, Y. and Yang, X.-G., On a BMAP/G/1 G-Queue with Setup Times and Multiple Vacations, Acta Math. Appl. Sinica, 2011, vol. 27, no. 4, pp. 625–638.

    Article  MATH  MathSciNet  Google Scholar 

  191. Banik, A.D., Gupta, U.C., and Pathak, S.S., BMAP/G/1/N Queue with Vacations and Limited Service Discipline, Appl. Math. Comput., 2006, vol. 180, pp. 707–721.

    MATH  MathSciNet  Google Scholar 

  192. Saffer, Z. and Telek, M., Analysis of BMAP Vacation Queue and its Application to IEEE 802.16e Sleep Mode, J. Indust. Manage. Optim., 2010, vol. 6, pp. 661–690.

    Article  MATH  MathSciNet  Google Scholar 

  193. Saffer, Z. and Telek, M., Unified Analysis of BMAP/G/1 Cyclic Polling Models, Queueing Syst., 2010, vol. 64, pp. 69–102.

    Article  MATH  MathSciNet  Google Scholar 

  194. Krieger, U., Klimenok, V.I., Kazimirsky, A.V., et al., A BMAP/PH/1 Queue with Feedback Operating in a Random Environment, Math. Comput. Modell., 2005, vol. 41, nos. 8–9, pp. 867–882.

    Article  MATH  MathSciNet  Google Scholar 

  195. Dudin, A.N., Kazimirsky, A.V., Klimenok, V.I., et al., Queueing Model MAP/PH/1/N with Feedback Operating in the Markovian Random Environment, Aust. J. Statist., 2005, vol. 32, no. 2, pp. 101–110.

    MATH  Google Scholar 

  196. Kim, C.S., Klimenok, V., Lee, S.C., and Dudin, A., The BMAP/PH/1 Retrial Queueing System Operating in Random Environment, J. Statist. Plann. Inference, 2007, vol. 137, pp. 3904–3916.

    Article  MATH  MathSciNet  Google Scholar 

  197. Kim, Ch., Dudin, A., Klimenok, V., and Khramova, V., Erlang Loss Queueing System with Batch Arrivals Operating in a Random Environment, Comput. Oper. Res., 2009, vol. 36, no. 3, pp. 674–697.

    Article  MATH  MathSciNet  Google Scholar 

  198. Kim, Ch., Klimenok, V., Mushko, V., and Dudin, A., The BMAP/PH/N Retrial Queueing System Operating in Markovian Random Environment, Comput. Oper. Res., 2010, vol. 37, no. 7, pp. 1228–1237.

    Article  MATH  MathSciNet  Google Scholar 

  199. Wu, J., Liu, Z., and Yang, G., Analysis of the Finite Source MAP/PH/N Retrial G-Queue Operating in a Random Environment, Appl. Math. Modell., 2011, vol. 35, pp. 1184–1193.

    Article  MATH  MathSciNet  Google Scholar 

  200. Yang, G., Yao, L.G., and Ouyang, Z., The MAP/PH/N Retrial Queue in a Random Environment, Acta Math. Appl. Sinica, 2013, vol. 29, pp. 725–738.

    Article  MATH  MathSciNet  Google Scholar 

  201. Kim, C.S., Dudin, A., Dudin, S., and Dudina, O., Analysis of MMAP/PH1, PH2/N/8 Queueing System Operating in a Random Environment, Int. J. Appl. Math. Comput. Sci., 2014, vol. 24, no. 3, pp. 485–502.

    MATH  Google Scholar 

  202. Kim, C.S., Dudin, A., Dudina, O., and Kim, J.S., Queueing System Operating in Random Environment as a Model of a Cell Operation, Indust. Eng. Manage. Syst., 2016, vol. 15, no. 2, pp. 131–142.

    Article  Google Scholar 

  203. Kim, C., Dudin, A., Dudin, S., Dudina, O., Multi-Server Queueing System MAP/M/N(r)/8 Operating in Random Environment, Commun. Comput. Inform. Sci., 2015, vol. CCIS 522, pp. 306–315.

    Article  Google Scholar 

  204. Dudin, A.N. and Nazarov, A.A., Analysis of a MMAP/M/R/0 Queueing System with Device Backup Operating in a Random Environment, Probl. Peredachi Inf., 2015, vol. 51, no. 3, pp. 117–128.

    MATH  Google Scholar 

  205. Dudina, O. and Dudin, S., Queueing System MAP/M/N/N + K Operating in Random Environment as a Model of Call Center, Commun. Comput. Inform. Sci., 2013, vol. CCIS 356, pp. 83–92.

    Article  MATH  Google Scholar 

  206. Dudin, A., Kim, C.S., Dudin, S., and Dudina, O., Priority Retrial Queueing Model Operating in Random Environment with Varying Number and Reservation of Servers, Appl. Math. Comput., 2015, vol. 269, pp. 674–690.

    MathSciNet  Google Scholar 

  207. Dudin, A. and Dudina, O., Analysis of Multi-Server Retrial Queueing System with Varying Capacity and Parameters, Math. Probl. Eng., 2015, ID 180481, pp. 1–12.

    Article  MATH  Google Scholar 

  208. Kim, C.S. and Dudin, S., Analysis of BMAP(R)/M(R)/N(R) Type Queueing System Operating in Random Environment, J. Korean Instit. Industr. Eng., 2016, vol. 42, no. 1, pp. 30–37.

    Article  Google Scholar 

  209. Dudin, A.N., OptimalMulti-Threshold Control for a BMAP/G/1 Queue with N Service Modes, Queueing Syst., 1998, vol. 30, no. 3–4, pp. 273–287.

    Article  MATH  MathSciNet  Google Scholar 

  210. Dudin, A.N. and Nishimura, S., Optimal Control for a BMAP/G/1 Queue with Two Service Modes, Math. Probl. Eng., 1999, vol. 5, no. 3, pp. 255–273.

    Article  MATH  Google Scholar 

  211. Dudin, A.N. and Nishimura, S., Optimal Hysteretic Control for a BMAP/SM/1/N Queue with Two Operation Modes, Math. Probl. Eng., 2000, vol. 5, no. 5, pp. 397–420.

    Article  MATH  Google Scholar 

  212. Choi, B.D., Chung, Y., and Dudin, A.N., The BMAP/SM/1 Retrial Queue with Controllable Operation Modes, Eur. J. Oper. Res., 2001, vol. 131, no. 1, pp. 16–30.

    Article  MATH  MathSciNet  Google Scholar 

  213. Chakravarthy, S. and Dudin, A.N., A Multi-Server Retrial Queue with BMAP Arrivals and Group Services, Queueing Syst., 2002, vol. 42, pp. 5–31.

    Article  MATH  MathSciNet  Google Scholar 

  214. Dudin, A.N. and Chakravarthy, S., Optimal Hysteretic Control for the BMAP/G/1 System with Single and Group Service Modes, Ann. Oper. Res., 2002, vol. 111, pp. 153–169.

    Article  MATH  MathSciNet  Google Scholar 

  215. Dudin, A.N., Optimal Hysteresis Control for an Unreliable BMAP/SM/1 System with Two Operation Modes, Autom. Remote Control, 2002, vol. 63, no. 10, pp. 1585–1596.

    Article  MATH  MathSciNet  Google Scholar 

  216. Chakravarthy, S. and Dudin, A.N., Multi-Threshold Control for the BMAP/SM/1/K Queue with Group Services, J. Appl. Math. Stochast. Anal., 2003, vol. 16, no. 4, pp. 327–348.

    Article  MATH  MathSciNet  Google Scholar 

  217. Dudin, A.N., Kim, Ch.S., and Semenova, O.V., An Optimal Multithreshold Control for the Input Flow of the GI/PH/1 Queueing System with a BMAP flow of Negative Customers, Autom. Remote Control, 2004, vol. 65, no. 9, pp. 1417–1428.

    Article  MATH  MathSciNet  Google Scholar 

  218. Kim, C.S., Klimenok, V.I., Birukov, A.V., and Dudin, A.N., Optimal Multi-Threshold Control by the BMAP/SM/1 Retrial System, Ann. Oper. Res., 2006, vol. 141, no. 1, pp. 193–210.

    Article  MATH  MathSciNet  Google Scholar 

  219. Efrosinin, D. and Breuer, L., Threshold Policies for Controlled Retrial Queues with Heterogeneous Servers, Ann. Oper. Res., 2006, vol. 141, pp. 139–162.

    Article  MATH  MathSciNet  Google Scholar 

  220. Dudin, A.N. and Sung, B., A Multi-Server MAP/PH/N System with Controllable Broadcast Servicing of Unreliable Devices, Avtom. Vychisl. Tekhn., 2009, vol. 43, no. 5. C. 32–43.

    Google Scholar 

  221. Dudin, A.N. and Sung, B., Unreliable Milti-Server System with Controllable Broadcasting Service, Autom. Remote Control, 2009, vol. 70, no. 12, pp. 2073–2085.

    Article  MATH  MathSciNet  Google Scholar 

  222. Taramin, O.S. and Klimenok, V.I., A Two-Phase Servicing System with a Threshold Strategy of Retrials, Avtom. Vychisl. Tekhn., 2009, vol. 43, no. 6, pp. 5–17.

    Google Scholar 

  223. Chakravarthy, S. and Dudin, A.N., Analysis of a Retrial Queueing Model with MAP Arrivals and Two Types of Customers, Math. Comput. Modell., 2003, vol. 37, pp. 343–364.

    Article  MATH  Google Scholar 

  224. Avrachenkov, K., Dudin, A., Klimenok, V., et al., Optimal Threshold Control by the Robots of Web Search Engines with Obsolescence of Documents, Comput. Networks, 2011, vol. 55, pp. 1880–1893.

    Article  Google Scholar 

  225. Sun, B., Lee, M.H., Dudin, S.A., and Dudin, A.N., Analysis of Multiserver Queueing System with Opportunistic Occupation and Reservation of Servers,Math. Probl. Eng., 2014, ID 178108, pp. 1–13.

    MathSciNet  Google Scholar 

  226. Sun, B., Lee, M.H., Dudin, S.A., and Dudin, A.N., MAP +MAP/M2/N/8 Queueing System with Absolute Priority and Reservation of Servers, Math. Probl. Eng., 2014, ID 813150, pp. 1–15.

    Google Scholar 

  227. Dudin, A., Lee, M., Dudina, O., and Lee, S., Analysis of Priority Retrial Queue with Many Types of Customers and Servers Reservation as a Model of Cognitive Radio System, IEEE Trans. Commun., 2017, vol. 65, no. 1, pp. 186–199.

    Google Scholar 

  228. Dudin, A.N., Piscopo, R., and Manzo, R., Queue with Group Admission of Customers, Comput. Oper. Res., 2015, vol. 61, pp. 89–99.

    Article  MATH  MathSciNet  Google Scholar 

  229. Brugno, A., Dudin, A.N., and Manzo, R., Analysis of a Strategy of Adaptive Group Admission of Customers to Single Server Retrial System, J. Ambient Intell. Humanized Comput., 2017, vol. 8.

    Google Scholar 

  230. Kim, C.S., Dudin, A., Dudin, S., and Dudina, O., Hysteresis Control by the Number of Active Servers in Queueing System with Priority Service, Perform. Evaluat., 2016, vol. 101, pp. 20–33.

    Article  MATH  Google Scholar 

  231. Dudin, A.N., Lee, M.H., and Dudin, S.A., Optimization of Service Strategy in Queueing System with Energy Harvesting and Customers Impatience, Appl. Math. Comput. Sci., 2016, vol. 26, no. 2, pp. 367–378.

    MATH  MathSciNet  Google Scholar 

  232. Klimenok, V. and Dudina, O., Retrial Tandem Queue with Controlled Strategy of Repeated Attempts, Quality Technol. Quantitat. Manage., 2017, vol. 14, no. 1, pp. 74–93.

    Article  Google Scholar 

  233. http://www.radio-electronics.com/info/cellulartelecomms/cellularconcepts/handover handoff.php

  234. Guerin, R., Queueing-Blocking System with Two Arrival Streams and Guard Channels, IEEE Transact. Commun., 1998, vol. 36, pp. 153–163.

    Article  MATH  MathSciNet  Google Scholar 

  235. Tran-Gia, P. and Mandjes, M., Modeling of Customer Retrial Phenomenon in CellularMobile Networks, IEEE J. Selected Areas Commun., 1997, vol. 15, pp. 1406–1414.

    Article  Google Scholar 

  236. Mandjes, M. and Tutschku, K., Efficient Call Handling Procedures in Cellular Mobile Networks, Forschungsbericht, Preprint-Reihe Nr 14X, Universität Würzburg, Institut für Informatik, 1996.

    Google Scholar 

  237. Choi, B.D. and Chang, Y., MAP1,MAP2/M/c Retrial Queue with the Retrial Group of Finite Capacity and Geometric Loss, Math. Comput. Modell., 1999, vol. 30, pp. 99–114.

    Article  MATH  Google Scholar 

  238. Alfa, A.S. and Li, W., PCS Networks with Correlated Arrival Process and Retrial Phenomenon, IEEE Transact. Wireless Commun., 2002, vol. 1, pp. 630–637.

    Article  Google Scholar 

  239. Choi, B.D., Melikov, A., and Velibekov, A., A Simple Numerical Approximation of Joint Probabilities of Calls in Service and Calls in the Retrial Group in a Picocell, Appl. Computat. Math., 2008, vol. 7, pp. 21–30.

    MATH  MathSciNet  Google Scholar 

  240. Dudin, S.A., A MAP/M/N Servicing System with Batch Arrivals of Customers and Repeated Calls, Probl. Peredachi Inf., 2009, vol. 45, no. 3, pp. 85–97.

    MathSciNet  Google Scholar 

  241. Kim, C.S., Dudin, S., and Klimenok, V., The MAP/PH/1/N Queue with Flows of Customers as Model for Traffic Control in Telecommunication Networks, Perform. Evaluat., 2009, vol. 66, nos. 9–10, pp. 564–579.

    Article  Google Scholar 

  242. Dudin, S.A. and Klimenok, V.I., A Servicing System with Phase Process of Customer Arrivals in a Batch, Avtom. Vychisl. Tekhn., 2009, vol. 43, no. 3, pp. 5–17.

    Google Scholar 

  243. Dudin, S.A., The MAP + MAP/PH/1/N Queuing System with Single and Batch Arrivals of Customers, Autom. Remote Control, 2009, vol. 70, no. 5, pp. 872–884.

    Article  MATH  MathSciNet  Google Scholar 

  244. Dudin, S.A., The servicing system MAP(PH) +MAP/PH/N/R as a model of optimizing an HTTP server with blockings, Autom. Remote Control, 2010, vol. 71, no. 1, pp. 28–38.

    Article  MATH  MathSciNet  Google Scholar 

  245. Kim, C.S., Dudin, A., Dudin, S., and Klimenok, V.I., Queueing System with Batch Arrival of Customers in Sessions, Comput. Industr. Eng., 2012, vol. 62, no. 4, pp. 890–897.

    Article  Google Scholar 

  246. Dudin, S. and Dudina, O., Tandem Queueing System with Batch Session Arrivals, Commun. Comput. Inform. Sci., 2013, vol. CCIS 356, pp. 59–68.

    Article  MATH  Google Scholar 

  247. Dudin, S., Dudin, A., and Dudina, O., Queueing System MAP/PH/N/R with Session Arrivals Operating in Random Environment, Commun. Comput. Inform. Sci., 2013, vol. CCIS 370, pp. 406–415.

    Article  MATH  Google Scholar 

  248. Buchholz, P., Kriege, J., and Felko, I., Input Modeling with Phase-Type Distributions and Markov Models: Theory and Applications, Berlin: Springer, 2014.

    Book  MATH  Google Scholar 

  249. Singh, G., Gupta, U.C., and Chaudhry, M.L., Detailed Computational Analysis of Queueing-Time Distributions of the BMAP/G/1 Queue Using Roots, J. Appl. Probab., 2016, vol. 53, pp. 1078–1097.

    Article  MATH  MathSciNet  Google Scholar 

  250. Heyman, D.P. and Lucantoni, D., Modeling Multiple IP Traffic Streams with Rate Limits, IEEE/ACM Transact. Networking, 2003, vol. 11, no. 6, pp. 948–958.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    V. M. Vishnevskii

  2. Belarusian State University, Minsk, Belarus

    A. N. Dudin

Authors
  1. V. M. Vishnevskii
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. N. Dudin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. M. Vishnevskii.

Additional information

Original Russian Text © V.M. Vishnevskii, A.N. Dudin, 2017, published in Avtomatika i Telemekhanika, 2017, No. 8, pp. 3–59.

This paper was recommended for publication by A.I. Kibzun, a member of the Editorial Board

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vishnevskii, V.M., Dudin, A.N. Queueing systems with correlated arrival flows and their applications to modeling telecommunication networks. Autom Remote Control 78, 1361–1403 (2017). https://doi.org/10.1134/S000511791708001X

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S000511791708001X

Keywords

Search

Navigation