Advertisement

On Numerical Computations of Some Discrete-Time Queues

  • Mohan L. Chaudhry
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 24)

Abstract

Since Erlang did pioneering work in the application of queues to telephony, a lot has been written about queues in continuous time (see, for example [As-mussel, 1987, Bacelli and Bremaud, 1994, Bhat and Basawa, 1992, Boxma and Syski, 1988, Bunday, 1986, Bunday, 1996, Chaudhry and Templeton, 1983, Cohen, 1982, Cooper, 1981, Daigle, 1992, Gnedenko and Kovalenko, 1989, Gross and Harris, 1985, Kalashnikov, 1994, Kashyap and Chaudhry, 1988, Kleinrock, 1975, Lipsky, 1992, Medhi, 1991, Prabhu, 1965, Robertazzi, 1990, Srivastava and Kashyap, 1982, Tijms, 1986, White et al., 1975]). In comparison to that large body of literature, not much has been written about queues in discrete time.

Keywords

Service Time Busy Period Queueing System Interarrival Time Tail Probability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Akar and Arikan, 1996]
    Akar, N. and Arikan, E. (1996). A numerically efficient method for the MAP/D/1/K queue via rational approximations. Queueing Systems, 22: 97–120.CrossRefGoogle Scholar
  2. [Alfa, 1982]
    Alfa, A. (1982). Time-inhomogeneous bulk server queue in discrete time: A transportation type problem. Oper. Res., 30: 650–658.CrossRefGoogle Scholar
  3. [Alfa, 1995a]
    Alfa, A. (1995a). A discrete MAP/PH/1 queue with vacations and exhaustive time-limited service. Oper. Res. Lett., 18: 31–40.CrossRefGoogle Scholar
  4. [Alfa, 199513]
    Alfa, A. (1995b). Modelling traffic queues at a signalized intersection with vehicle-actuated control and Markovian arrival processes. Comput. Math. Applic.$130:105–119. Errata: (1996), 31: 137.Google Scholar
  5. [Alfa, 1998]
    Alfa, A. (1998). Matrix-geometric solution of discrete time MAP/PH/1 priority queue. Naval Res. Logist., 45: 23–50.CrossRefGoogle Scholar
  6. [Alfa and Chakravarthy, 1994]
    Alfa, A. and Chakravarthy, S. (1994). A discrete queue with the Markovian arrival process and phase type primary and secondary services. Commun. Statist. Stochastic Models, 10: 437–451.CrossRefGoogle Scholar
  7. [Alfa et al., 1995]
    Alfa, A., Dolhun, K., and Chakravarthy, S. (1995). A discrete single server queue with Markovian arrivals and phase type group services. J. of Applied Math. and Stochastic Analysis, 8: 151–176.CrossRefGoogle Scholar
  8. [Alfa and Neuts, 1995]
    Alfa, A. and Neuts, M. (1995). Modelling vehicular traffic using the discrete time Markovian arrival process. Transportation Science, 29: 109–117.CrossRefGoogle Scholar
  9. [Alfa and Frigui, 1996]
    Alfa, A. and Frigui, I. (1996). Discrete NT-policy single server queue with Markovian arrival process and phase type service. European J. Opl. Res., 88: 599–613.CrossRefGoogle Scholar
  10. [Asmussen, 1987]
    Asmussen, S. (1987). Applied Probability and Queues. John Wiley, NY.Google Scholar
  11. [Atkinson, 1995]
    Atkinson, J. (1995). The general two-server queueing loss system: Discrete-time analysis and numerical approximation of continuous-time systems. J. Opl. Res. Soc., 46: 386–397.Google Scholar
  12. [Bacelli and Bremaud, 1994]
    Bacelli, F. and Bremaud, P. (1994). Elements of Queueing Theory: Palm-Martingale Calculus and Stochastic Recurrences. Springer-Verlag, NY.Google Scholar
  13. [Bhat, 1968]
    Bhat, N. (1968). A study of the queueing systems M/G/1 and GI/M/1. In Beckmann, M. and Künzi, H., editors, Lecture Notes in Operations Research and Mathematical Economics, volume 2. Springer-Verlag, NY.Google Scholar
  14. [Bhat and Basawa, 1992]
    Bhat, U. and Basawa, I. (1992). Queueing and Related Models. Clarendon, Oxford.Google Scholar
  15. [Blondia, 1989]
    Blondia, C. (1989). The N/G/1 finite capacity queue. Commun. Statist.— Stochastic Models, 5: 273–294.CrossRefGoogle Scholar
  16. [Blondia, 1993]
    Blondia, C. (1993). A discrete-time batch Markovian arrival process as B-ISDN traffic model. Belgian Journal of Operations Research, Statistics and Computer Science, 32 (3, 4): 3–23.Google Scholar
  17. [Blondia and Theimer, 1989]
    Blondia, C. and Theimer, T. (1989). A discrete-time model for ATM traffic. working paper. Philips Research Laboratory, Brussels, Belgium.Google Scholar
  18. [Böhm et al., 1997]
    Böhm, W., Krinik, A., and Mohanty, S. (1997). The cornbinatorics of birth-death processes and applications to queues. Queueing Systems, 26: 255–267.CrossRefGoogle Scholar
  19. [Böhm and Mohanty, 1993]
    Böhm, W. and Mohanty, S. (1993). The transient solution of M/M/1 queues under (m,n)-policy. A combinatorial approach. Journal of Statistical Planning and Inference, 34: 23–33.CrossRefGoogle Scholar
  20. [Böhm and Mohanty, 1994a]
    Böhm, W. and Mohanty, S. (1994a). On discrete-time Markovian n-policy queues involving batches. Sankhya, 56, Ser. A: 144–163.Google Scholar
  21. [Böhm and Mohanty, 1994b]
    Böhm, W. and Mohanty, S. (1994b). On random walks with barriers and their applications to queues. Studia Sci. Math. Hun-gar., 29: 397–413.Google Scholar
  22. Böhm and Mohanty, 1994c] Böhm, W. and Mohanty, S. (1994c). Transient analysis of M/M/1 queues in discrete time with general server vacations. J. Appl. Probab.,31A:115–129. In Studies in Probability (Essays in honour of Lajos Takdcs).Google Scholar
  23. [Böhm and Mohanty, 1994d]
    Böhm, W. and Mohanty, S. (1994d). Transient analysis of queues with heterogeneous arrivals. Queueing Systems, 18: 27–46.CrossRefGoogle Scholar
  24. Boucherie and van Dijk, 1991] Boucherie, R. and van Dijk, N. (1991). Product forms for queueing networks with state-dependent multiple job transitions. Adv. in Appl. Probab.,23:152–187.Google Scholar
  25. [Boxma and Syski, 1988]
    Boxma, O. and Syski, R. (1988). Queueing Theory and its Applications Liber Amicorum for J. W. Cohen. North-Holland, Amsterdam.Google Scholar
  26. [Bruneel, 1986]
    Bruneel, H. (1986). A general treatment of discrete-time buffers with one randomly interrupted output line. European J. Opl. Res., 27: 67–81.CrossRefGoogle Scholar
  27. [Bruneel, 1993]
    Bruneel, H. (1993). Performance of discrete-time queueing systems. Comput. Oper. Res., 20: 303–320.CrossRefGoogle Scholar
  28. [Bruneel and Kim, 1993]
    Bruneel, H. and Kim, B. (1993). Discrete-Time Models for Communication Systems Including ATM. Kluwer, Boston.CrossRefGoogle Scholar
  29. [Bruneel and Wuyts, 1994]
    Bruneel, H. and Wuyts, I. (1994). Analysis of discrete-time multiserver queueing models with constant service times. Oper. Res. Lett., 15: 231–236.CrossRefGoogle Scholar
  30. [Bunday, 1986]
    Bunday, B. (1986). Basic Queuing Theory. Edward Arnold, London.Google Scholar
  31. [Bunday, 1996]
    Bunday, B. (1996). An Introduction To Queueing Theory. Edward Arnold, London.Google Scholar
  32. [Chan and Maa, 1978]
    Chan, W. and Maa, D. (1978). The GI/Geom/N queue in discrete time. INFOR, 16: 232–252.Google Scholar
  33. [Chang and Chang, 1984]
    Chang, J. and Chang, R. (1984). The application of the residue theorem to the study of a finite queue with batch Poisson arrivals and synchronous servers. SIAM J. Appl. Math., 44: 646–656.CrossRefGoogle Scholar
  34. [Chaudhry, 1965]
    Chaudhry, M. (1965). Correlated queueing. J. of the Canad. Operat. Res. Soc., 3: 142–151.Google Scholar
  35. Chaudhry, 1992a] Chaudhry, M. (1992c). QPACK Software Package. AandA Publications, 395 Carrie Crescent, Kingston, Ontario K7M 5X7, Canada.Google Scholar
  36. Chaudhry, 1992b] Chaudhry, M. (1992b). QROOT Software Package. AandA Publications, 395 Carrie Crescent, Kingston, Ontario K7M 5X7, Canada.Google Scholar
  37. [Chaudhry, 1997]
    Chaudhry, M. (1997). On the discrete-time bulk-arrival queues with finite-and infinite-waiting spaces. Working Paper. Dept. of Math. and Computer Science, Royal Military College of Canada, Kingston, Ontario, Canada.Google Scholar
  38. [Chaudhry and Gupta, 1996a]
    Chaudhry, M. and Gupta, U. (1996a). On the analysis of the discrete-time Geom(n)/G(n)/1/N queue. Probab. Engrg. Inform. Sci., 10: 59–68.CrossRefGoogle Scholar
  39. [Chaudhry and Gupta, 1996b]
    Chaudhry, M. and Gupta, U. (1996b). Performance analysis of the discrete-time GI/Geom/1/N queue. J. Appl. Probab., 33: 239–255.CrossRefGoogle Scholar
  40. [Chaudhry and Gupta, 1997c]
    Chaudhry, M. and Gupta, U. (1997c). Queue-length and waiting-time distributions of discrete-time GIX /Geom/1 queueing systems with early and late arrivals. Queueing Systems, 25: 307–334.CrossRefGoogle Scholar
  41. [Chaudhry and Gupta, 1997d]
    Chaudhry, M. and Gupta, U. (1997d). Algorithmic discussions of distributions of numbers of busy channels for GI/Geom/m/m queues. Working Paper. Dept. of Math. and Computer Science, Royal Military College of Canada, Kingston, Ontario, Canada.Google Scholar
  42. [Chaudhry and Gupta, 1998]
    Chaudhry, M. and Gupta, U. (1998). Performance analysis of discrete-time finite-buffer batch-arrival GIX/Geom/1/N queues. Discrete Event Dynamic Systems, 8: 55–70.CrossRefGoogle Scholar
  43. [Chaudhry et al., 1997]
    Chaudhry, M., Gupta, U. and Goswami, V. (1997). Relations among state probabilities in discrete-time bulk-service queues with finite-and infinite-waiting spaces. Working Paper. Dept. of Math. and Computer Science, Royal Military College of Canada, Kingston, Ontario, Canada.Google Scholar
  44. [Chaudhry et al., 1990]
    Chaudhry, M., Harris, C., and Marchai, W. (1990). Robustness of rootfinding in single-server queueing models. ORSA J. Computing, 2: 273–286.CrossRefGoogle Scholar
  45. [Chaudhry and Templeton, 1968]
    Chaudhry, M. and Templeton, J. (1968). On the discrete-time queue length distribution in a bulk service considering correlated arrivals. Canad. Operat. Res. Soc., 2: 79–88.Google Scholar
  46. [Chaudhry and Templeton, 1983]
    Chaudhry, M. and Templeton, J. (1983). A First Course on Bulk Queues. NY: John Wiley.Google Scholar
  47. [Chaudhry et al., 1996]
    Chaudhry, M., Templeton, J. and Gupta, U. (1996). Analysis of the discrete-time GI/Geom(n)/1/N queue. Comput. Math. Applic., 31: 59–68.CrossRefGoogle Scholar
  48. [Chaudhry et al., 1992]
    Chaudhry, M., Templeton, J. and Medhi, J. (1992). Computational analysis of multiserver bulk-arrival queues with constant service time M X /D/c. ORSA, 40 (Supp. 2): 229–238.CrossRefGoogle Scholar
  49. [Chaudhry and van Ommeren, 1997]
    Chaudhry, M. and van Ommeren, J. (1997). Analytically explicit results for the transient solutions of one-sided skip-free finite Markov chains. Working Paper. Dept. of Math. and Computer Science, Royal Military College of Canada, Kingston, Ontario, Canada.Google Scholar
  50. [Chaudhry and Zhao, 1994]
    Chaudhry, M. and Zhao, Y. (1994). First-passagetime and busy period distributions of discrete-time Markovian queues: Geom(n)/Geom(n)/1/N. Queueing Systems, 18: 5–26.CrossRefGoogle Scholar
  51. [Chaudhry and Zhao, 1997]
    Chaudhry, M. and Zhao, Y. (1997). Transient solutions for Markov chains having lower Hessenberg transition probability matrices and their applications. Working Paper. Dept. of Math. and Computer Science, Royal Military College of Canada, Kingston, Ontario, Canada.Google Scholar
  52. [Choneyko et al., 1993]
    Choneyko, I., Mohanty, S., and Sen, K. (1993). Enumeration of restricted three-dimensional lattice paths with fixed number of turns and an application. J. Statistical Planning and Inference, 34: 57–62.CrossRefGoogle Scholar
  53. [Choukri, 1993]
    Choukri, T. (1993). The transient blocking probabilities in M/M/N systems via large deviations. Advances in Applied Probab., 25: 483–486.CrossRefGoogle Scholar
  54. [Cohen, 1982]
    Cohen, J. (1982). The Single Server Queue. North-Holland, Amsterdam, second edition.Google Scholar
  55. [Cooper, 1981]
    Cooper, R. (1981). Introduction to Queueing Theory. North-Holland, Amsterdam, second edition.Google Scholar
  56. [Daduna, 1996]
    Daduna, H. (1996). Discrete time queueing networks: Recent developments. Tutorial Lecture Notes, Performance `96, 8:1–42. Lausanne.Google Scholar
  57. [Dafermos et al., 1971]
    Dafermos, S., Stella, C., and Neuts, M. (1971). A single server queue in discrete time. Cahiers du Centre de Rech. Opér, 13: 23–40.Google Scholar
  58. [Daigle, 1992]
    Daigle, J. (1992). Queueing Theory for Telecommunications. Addison-Wesley, NY.Google Scholar
  59. [Feller, 1968]
    Feller, W. (1968). An Introduction to Probability Theory and Its Applications, volume 1. John Wiley, NY, third edition.Google Scholar
  60. [Frigui et al., 1997]
    Frigui, I., Alfa, A, and Xu, X. (1997). Algorithms for computing waiting time distributions under different queue disciplines for the D-BMAP/PH/1. Naval Res. Logist., 44: 559–576.CrossRefGoogle Scholar
  61. [Garcia and Casals, 1989]
    Garcia, J. and Casals, O. (1989). A discrete time queueing model to evaluate cell delay variation in an ATM network. Working paper. Computer Architecture Dept., Polytechnic University of Catalonia, Spain.Google Scholar
  62. [Gnedenko and Kovalenko, 1989]
    Gnedenko, B. and Kovalenko, I. (1989). Introduction to Queueing Theory. Birkhäuser, Boston, MA.Google Scholar
  63. [Gopinath and Morrison, 1977]
    Gopinath, R. and Morrison, J. (1977). Discrete-time single server queues with correlated inputs. In Chandy, K. and Reiser, M., editors, Computer Performance, pages 263–278. North-Holland, Amsterdam.Google Scholar
  64. [Gouweleeuw, 1994]
    Gouweleeuw, F. (1994). The loss probability in finite-buffer queues with batch arrivals and complete rejection. Probab. Engrg. Inform. Sci, 8: 221–227.CrossRefGoogle Scholar
  65. [Grassmann and Jain, 1989]
    Grassmann, W. and Jain, J. (1989). Numerical solutions of the waiting time distribution and idle time distribution of the arithmetic GI/G/1 queue. Oper. Res., 37: 141–150.CrossRefGoogle Scholar
  66. [Gravey and Hébuterne, 1992]
    Gravey, A. and Hébuterne, G. (1992). Simultaneity in discrete time single server queues with Bernoulli inputs. Perf. Eval., 14: 123–131.CrossRefGoogle Scholar
  67. [Gravey et al., 1990]
    Gravey, A., Louvion, J., and Boyer, P. (1990). On the Geo/D/1 and Geo/D/1/n queues. Perf. Eval., 11: 117–125.CrossRefGoogle Scholar
  68. [Gross and Harris, 1985]
    Gross, D. and Harris, C. (1985). Fundamentals of Queueing Theory. John Wiley, NY, second edition.Google Scholar
  69. [Hashida et al., 1991]
    Hashida, O., Takahashi, Y., and Shimogawa, S. (1991). Switched batch Bernoulli process (SBBP) and the discrete-time SBBP/G/1 queue with applications to statistical multiplexer performance. IEEE J. Selected Areas Commun., 9: 394–401.CrossRefGoogle Scholar
  70. [Hasslinger, 1995]
    Hasslinger, G. (1995). A polynomial factorization approach to the discrete time GI/G/1/(N) queue size distribution. Perf. Eval., 23: 217240.Google Scholar
  71. [Hasslinger and Rieger, 1996]
    Hasslinger, G. and Rieger, E. (1996). Analysis of open discrete time queueing networks: A refined decomposition approach. J. Opl. Res. Soc., 47: 640–653.Google Scholar
  72. [Heimann and Neuts, 1973]
    Heimann, D. and Neuts, M. (1973). The single server queue in discrete time. Numerical analysis IV. Nay. Res. Logist., 20: 753–766.CrossRefGoogle Scholar
  73. [Henderson et al., 1995]
    Henderson, W., Pearce, C., Taylor, P., and van Dijk, N. (1995). Insensitivity in discrete-time generalized semi-Markov processes allowing multiple events and probabilistic service scheduling. Ann. Appl. Probab., 5: 78–96.CrossRefGoogle Scholar
  74. [Heyman and Sobel, 1982]
    Heyman, D. and Sobel, M.J. (1982). Stochastic Models in Operations Research, volume 1. McGraw-Hill, NY.Google Scholar
  75. [Hunter, 1983]
    Hunter, J. (1983). Discrete Time Models: Techniques and Applications, volume 2 of Mathematical Techniques of Applied Probability. Academic Press, NY.Google Scholar
  76. [Ishizaki et al., 1994a]
    Ishizaki, F., Takine, T., and Hasegawa, T. (1994a). Analysis of a discrete-time queue with a gate. In Labetoulle, J. and Roberts, J., editors, The Fundamental Role of Teletraffic in the Evolution of Telecommunications Networks, pages 169–178. Elsevier, Amsterdam.Google Scholar
  77. [Ishizaki et al., 1994b]
    Ishizaki, F., Takine, T., Takahashi, Y., and Hasegawa, T. (1994b). A generalized SBBP/G/1 queue and its applications. Perf. Eval., 21: 163–181.CrossRefGoogle Scholar
  78. [Johnson and Narayana, 1996]
    Johnson, M. and Narayana, S. (1996). Descriptors of arrival-process burstiness with applications to the discrete Markovian arrival process. Queueing Systems, 23: 107–130.CrossRefGoogle Scholar
  79. [Kalashnikov, 1994]
    Kalashnikov, V. (1994). Mathematical Methods in Queuing Theory. Kluwer, Amsterdam.Google Scholar
  80. [Kashyap and Chaudhry, 1988]
    Kashyap, B. and Chaudhry, M. (1988). An Introduction to Queueing Theory. AandA Publications, Kingston, Ontario.Google Scholar
  81. [Kinney, 1962]
    Kinney, J. (1962). A transient discrete time queue with finite storage. Ann. Math. Stat., 33: 130–106.CrossRefGoogle Scholar
  82. [Kleinrock, 1975]
    Kleinrock, L. (1975). Queueing Systems: Theory, volume 1. John Wiley, NY.Google Scholar
  83. [Klimko and Neuts, 1973]
    Klimko, E. and Neuts, M. (1973). The single server queue in discrete time. Numerical analysis II. Nay. Res. Logist., 20: 305–319.CrossRefGoogle Scholar
  84. [Knessl, 1990]
    Knessl, C. (1990). On the transient behavior of the M/M/m/m loss model. Communications in Statistics- Stochastic Models, 6: 749–776.CrossRefGoogle Scholar
  85. [Kobayashi, 1983]
    Kobayashi, H. (1983). Discrete-time queueing systems. In Louchard, G. and Latouche, G., editors, Probability Theory and Computer Science, pages 53–121. Academic Press, NY.Google Scholar
  86. [Konheim, 1975]
    Konheim, A. (1975). An elementary solution of the queueing system G/G/l. SIAM J. Comput., 4: 540–545.CrossRefGoogle Scholar
  87. [Kouvatsos, 1994]
    Kouvatsos, D. (1994). Entropy maximisation and queueing network models. Ann. Oper. Res., 48: 63–126.CrossRefGoogle Scholar
  88. Kouvatsos, 1995] Kouvatsos, D. (1995). Performance Modelling and Evaluation of ATM Networks,volume 1. Chapman and Hall, London.Google Scholar
  89. [Kouvatsos, 1996]
    Kouvatsos, D. (1996). Performance Modelling and Evaluation of ATM Networks, volume 2. Chapman and Hall, London.Google Scholar
  90. [Kouvatsos and Fretwell, 1995]
    Kouvatsos, D. and Fretwell, R. (1995). Closed form performance distributions of a discrete time GIG/D/1/N queue with correlated traffic. In Proceedings of the Sixth IFIP WG6.3 Conference of Computer Networks, pages 141–163, Istanbul, Turkey.Google Scholar
  91. [Kuehn, 1979]
    Kuehn, P. (1979). Approximate analysis of general queueing networks by decomposition. IEEE Trans. Commun., 27: 113–126.CrossRefGoogle Scholar
  92. [Lipsky, 1992]
    Lipsky, L. (1992). Queueing Theory. Macmillan, NY.Google Scholar
  93. [Liu and Mouftah, 1997]
    Liu, X. and Mouftah, H. (1997). Queueing performance of copy networks with dynamic cell splitting for multicast ATM switching. IEEE Trans. Selected Areas Commun., 45: 464–472.Google Scholar
  94. [Liu and Neuts, 1994]
    Liu, D. and Neuts, M. (1994). A queueing model for an ATM control scheme. Telecommunication Systems, 2: 321–348.CrossRefGoogle Scholar
  95. [Lucantoni, 1991]
    Lucantoni, D. (1991). New results on the single server queue with a batch Markovian arrival process. Commun. Statist.- Stochastic Models, 7: 1–46.CrossRefGoogle Scholar
  96. [Madill et al., 1985]
    Madill, B., Chaudhry, M., and Buckholtz, P. (1985). On the diophantine queueing system, D/Da,b/1. J. Opl. Res. Soc., 36: 531–535.Google Scholar
  97. [Medhi, 1991]
    Medhi, J. (1991). Stochastic Models in Queueing Theory. Academic Press, NY.Google Scholar
  98. [Meisling, 1958]
    Meisling, T. (1958). Discrete-time queueing theory. Oper. Res., 6: 96–105.CrossRefGoogle Scholar
  99. [Mitra and Weiss, 1988]
    Mitra, D. and Weiss, A. (1988). On the transient behavior in Erlang’s model for large trunk groups and various traffic conditions. In Proceedings of the 12th International Teletraffic Congress, volume 5.1B4.1–5.1B4. 8. Torino, Italy.Google Scholar
  100. [Miyazawa and Takagi, 1994]
    Miyazawa, M. and Takagi, H. (1994). Advances in discrete time queues. Queueing Systems, 18 (1–2).Google Scholar
  101. [Mohanty, 1991]
    Mohanty, S. (1991). On the transient behavior of a finite discrete time birth-death process. Assam Statistical Review, 5: 1–7.Google Scholar
  102. [Mohanty and Panny, 1989]
    Mohanty, S. and Panny, W. (1989). A discrete-time analogue of the M/M/1 queue and the transient solution. An analytic approach. In Revesz, P., editor, Coll. Math. Soc. Jdnos Bolyai, volume 57 of Limit Theorems in Probab. and Statist., pages 417–424.Google Scholar
  103. [Mohanty and Panny, 1990]
    Mohanty, S. and Panny, W. (1990). A discrete-time analogue of the M/M/1 queue and the transient solution: A geometric approach. Sankhya, 52 (Ser. A): 364–370.Google Scholar
  104. [Murata and Miyahara, 1991]
    Murata, H. and Miyahara, H. (1991). An analytic solution of the waiting time distribution for the discrete-time GI/G/1 queue. Perf. Eval., 13: 87–95.CrossRefGoogle Scholar
  105. [Neuts, 1973]
    Neuts, M. (1973). The single server queue in discrete time. Numerical analysis I. Nay. Res. Logist., 20: 297–304.CrossRefGoogle Scholar
  106. [Neuts and Klimko, 1973]
    Neuts, M. and Klimko, E. (1973). The single server queue in discrete time. Numerical analysis III. Nay. Res. Logist., 20: 557–567.CrossRefGoogle Scholar
  107. [Park and Perros, 1994]
    Park, D. and Perros, H. (1994). m-MMBP characterization of the departure process of an m-MMBP/Geo/1/K queue. ITC, 14: 75–84.Google Scholar
  108. [Park et al., 1994]
    Park, D., Perros, H., and Yamahita, H. (1994). Approximate analysis of discrete-time tandem queueing networks with bursty and correlated input and customer loss. Oper. Res., 15: 95–104.Google Scholar
  109. [Pestien and Ramakrishnan, 1994]
    Pestien, V. and Ramakrishnan, S. (1994). Features of some discrete-time cyclical queueing networks. Queueing Systems, 18: 117–132.CrossRefGoogle Scholar
  110. [Prabhu, 1965]
    Prabhu, N. (1965). Queues and Inventories: A Study of Their Basic Stochastic Processes. John Wiley, NY.Google Scholar
  111. [Pujolle et al., 1986]
    Pujolle, G., Claude, J., and Seret, D. (1986). A discrete queueing system with a product form solution. Computer Networking and Performance Evaluation, pages 139–147.Google Scholar
  112. [Robertazzi, 1990]
    Robertazzi, T. (1990). Computer Networks and Systems: Queueing Theory and Performance Evaluation. Springer-Verlag, NY.CrossRefGoogle Scholar
  113. [Sharma, 1990]
    Sharma, O. (1990). Markovian Queues. Ellis Harwood, NY.Google Scholar
  114. [Srivastava and Kashyap, 1982]
    Srivastava, H. and Kashyap, B. (1982). Special Functions in Queueing Theory and Related Stochastic Processes. Academic Press, NY.Google Scholar
  115. [Takâcs, 1971]
    Takâcs, L. (1971). Discrete queues with one server. J. Appl. Probab., 8: 691–707.CrossRefGoogle Scholar
  116. [Takagi, 1993]
    Takagi, H. (1993). Queueing Analysis - A Foundation of Performance Evaluation: Discrete-Time Systems, volume 3. North-Holland, Amsterdam.Google Scholar
  117. [Tijms, 1986]
    Tijms, H. (1986). Stochastic Modelling and Analysis: A Computational Approach. John Wiley, NY.Google Scholar
  118. [Tjhie and Rzehak, 1996]
    Tjhie, D. and Rzehak, H. (1996). Analysis of discrete-time TES/G/1 and TES/D/1-K queueing systems. Perf. Eval., 27and28: 367–390.Google Scholar
  119. [Tran-Gia et al., 1995]
    Tran-Gia, P., Blondia, C., and Towsley, D. (1995). Discrete-time models and analysis methods. Perf. Eval., 21 (1–2).Google Scholar
  120. [Tsuchiya and Takahashi, 1993]
    Tsuchiya, T. and Takahashi, Y. (1993). On discrete-time single-server queues with Markov modulated batch arrival Bernoulli input and finite capacity. J. Oper. Res. Soc. Japan, 36: 22–45.Google Scholar
  121. Ushakumari, 1995] Ushakumari, P. (1995). Analysis of Some Infinite-Server Queues and Related Optimization Problems. PhD thesis, The Cochin University of Science and Technology, Cochin - 682 022, India.Google Scholar
  122. [van Dijk, 1990]
    van Dijk, N. (1990). An insensitive product form for discrete-time communication networks. Performance ‘80, pages 77–89.Google Scholar
  123. [van Doom and Schrijner, 1996]
    van Doom, E. and Schrijner, P. (1996). Limit theorems for discrete-time Markov chains of the nonnegative integers conditioned on recurrence to zero. Communi Statists.- Stochastic Models, 12: 77102.Google Scholar
  124. [van Ommeren, 1991]
    van Ommeren, J. (1991). The discrete-time single-server queue. Queueing Systems, 8: 279–294.CrossRefGoogle Scholar
  125. [Wally and Viterbi, 1996]
    Wally, S. and Viterbi, A. (1996). A tandem of discrete-time queues with arrivals and departures at each stage. Queueing Systems, 23: 157–176.CrossRefGoogle Scholar
  126. [White et al., 1975]
    White, J., Schmidt, J., and Bennett, C. (1975). Analysis of Queueing Systems. Academic Press, NY.Google Scholar
  127. [Wolf, 1982]
    Wolf, D. (1982). Approximating the stationary waiting time distribution function of GI/G/1-queues with arithmetic interarrival time and service time distribution function. Oper. Res. Spektrum, 4: 135–148.CrossRefGoogle Scholar
  128. [Woodward, 1994]
    Woodward, M. (1994). Communication and Computer Networks: Modelling with Discrete-Time Queues. California IEEE Computer Society Press, Los Alamitos, CA.Google Scholar
  129. [Xie and Knessl, 1993]
    Xie, M. and Knessl, C. (1993). On the transient behavior of the Erlang loss model: Heavy usage asymptotics. SIAM J. Appl. Math., 53: 555–599.CrossRefGoogle Scholar
  130. [Yang and Chaudhry, 1996]
    Yang, T. and Chaudhry, M. (1996). On the steady-state queue size distributions of the discrete-time GI/G/1 queue. Adv. Appl. Probab., 28: 1177–1200.CrossRefGoogle Scholar
  131. [Yang and Li, 1995]
    Yang, T. and Li, H. (1995). On the steady-state queue size distribution of the discrete-time Geo/G/1 queue with repeated customers. Queueing Systems, 21: 199–215.CrossRefGoogle Scholar
  132. [Yoshimoto et al., 1993]
    Yoshimoto, M., Takine, T., Takahashi, Y., and Hasegawa, T. (1993). Waiting time and queue length distributions for go-back-n and selective-repeat ARQ protocols. IEEE Trans. Commun., 41: 1687–1693.CrossRefGoogle Scholar
  133. [Zhao and Campbell, 1996]
    Zhao, Y. and Campbell, L. (1996). Equilibrium probability calculations for a discrete-time bulk queue model. Queueing Systems, 22: 189–198.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Mohan L. Chaudhry
    • 1
  1. 1.Department of Mathematics and Computer ScienceRoyal Military College of CanadaKingstonCanada

Personalised recommendations