Skip to main content

Appendix

  • Chapter
  • First Online:
Queueing Networks with Discrete Time Scale

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2046))

  • 424 Accesses

Abstract

In this appendix we collect for ease of reference some facts which are used in the preceding chapters. Some of these are standard, sometimes not explicitly stated in the literature, some are proved for special requirements in the present text, but their proof is not essential for understanding the main developments.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 44.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Bibliography

  1. B. van Arem, E. A. van Doorn, and T.M.J. Meijer. Queueing analysis of a discrete closed-loop conveyor with service facilities. Queueing System, 4:95–114, 1989.

    Article  MATH  Google Scholar 

  2. F. Baccelli and P. Bremaud. Elements of queueing theory. Springer, New York, 1994.

    MATH  Google Scholar 

  3. A. D. Barbour and R. Schassberger. Insensitive average residence times in generalized semi-markov-processes. Advances of Applied Probability, 13:720–735, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  4. F. Baskett, M. Chandy, R. Muntz, and F.G. Palacios. Open, closed and mixed networks of queues with different classes of customers. Journal of the Association for Computing Machinery, 22:248–260, 1975.

    MATH  MathSciNet  Google Scholar 

  5. G. Bolch, S. Greiner, H. de Meer, and K. S. Trivedi. Queueing networks and Markov chains. John Wiley, New York, 1998.

    MATH  Google Scholar 

  6. R.J. Boucherie. Product form in queueing networks. PhD thesis, Vrije Universiteit Amsterdam, 1992.

    Google Scholar 

  7. R.J. Boucherie and N.M. van Dijk. Spatial birth-death processes with multiple changes and applications to batch service networks and clustering processes. Advances of Applied Probability, 22:433–455, 1990.

    Article  MATH  Google Scholar 

  8. R.J. Boucherie and N.M. van Dijk. Product forms for queueing networks with state-dependent multiple job transitions. Advances of Applied Probability, 23:152–187, 1991.

    Article  MATH  Google Scholar 

  9. O. J. Boxma and H. Daduna. Sojourn times in queueing networks. In H. Takagi, editor, Stochastic Analysis of Computer and Communication Systems, pages 401–450, Amsterdam, 1990. IFIP, North-Holland.

    Google Scholar 

  10. O. J. Boxma, F. P. Kelly, and A.-G. Konheim. The product form for sojourn time distributions in cyclic exponential queues. Journal of the Association for Computing Machinery, 31:128–133, 1984.

    MATH  MathSciNet  Google Scholar 

  11. O.J. Boxma and J.A.C. Resing. Tandem queues with deterministic service times. Operations Research, 49:221–239, 1994.

    Article  MATH  Google Scholar 

  12. E. Brockmeyer, H.L. Halstrom, and A. Jensen. The life and the work of A.K.Erlang, volume 2 of Transactions of the Danish Academy of Techn. Science. Danish Academy of Science, Copenhagen, 1948.

    Google Scholar 

  13. S.C. Bruell and G. Balbo. Computational algorithms for closed queueing networks. North-Holland, New York, 1980.

    MATH  Google Scholar 

  14. H. Bruneel and Byung G. Kim. Discrete-Time Models for Communication Systems including ATM. Kluwer Academic Publications, Boston, 1993.

    Google Scholar 

  15. H. Bruneel, B. Steyaert, E. Desmet, and G.H. Petit. An analytical technique for the derivation of the delay performance of ATM switches with multiverser output queues. Intern. Journ. of Digital and Analog Communication Systems, 5:193–201, 1992.

    Article  Google Scholar 

  16. P.J. Burke. The output of a queueing system. Operations Research, 4:699–704, 1956.

    MathSciNet  Google Scholar 

  17. J.P. Buzen. Computational algorithms for closed queueing networks with exponential servers. Communications of the ACM, 16:527–531, 1973.

    Article  MATH  MathSciNet  Google Scholar 

  18. X. Chao, M. Miyazawa, and M. Pinedo. Queueing Networks — Customers, Signals, and Product Form Solutions. Wiley, Chichester, 1999.

    MATH  Google Scholar 

  19. X. Chao and M. Pinedo. On generalized networks of queues with positive and negative arrivals. Prob.Eng.Inf.Sci, 7:301–334, 1993.

    Google Scholar 

  20. M.L. Chaudhry and U.C. Gupta. Transient behaviour of the discrete time Geom/Geom/m/m Erlang loss model. In A.C. Borthakur and M.L. Choudhry, editors, Probability Models and Statistics, A.J. Medhi Festschrift, pages 133–145, New Delhi, 1996. New Age International Limited, Publishers.

    Google Scholar 

  21. J. W. Cohen. The Single Server Queue. North-Holland Publishing Company, Amsterdam-London, second edition, 1982.

    MATH  Google Scholar 

  22. J.W. Cohen. The multiple phase service network with generalized processor sharing. Acta Informatica, 12:245–284, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  23. Coleman, J.L., Henderson, W., Pearce, C.E.M., and Taylor, P.G. A correspondence between product-form batch-movement queueing networks and single-movement networks. Journal of Applied Probability, 34:160–175, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  24. R. B. Cooper. Queueing theory. In D. P. Heyman and M. J. Sobel, editors, Stochastic Models, volume 2 of Handbooks in Operations Research and Management Science, chapter 10, pages 469–518. North-Holland, Amsterdam, 1990.

    Google Scholar 

  25. H. Daduna. Spezielle Stochastische Prozesse. Institute of Mathematical Stochastics, University of Hamburg, 1984. Lecture notes.

    Google Scholar 

  26. H. Daduna. The cycle time distribution in a cycle of Bernoulli servers in discrete time. Mathematical Methods of Operations Research, 44:295–332, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  27. H. Daduna. Stochastische Prozesse. Vorlesungsmanuskript, Institut für Mathematische Stochastik der Universität Hamburg, 1996.

    Google Scholar 

  28. H. Daduna. Discrete time analysis of a state dependent tandem with different customer types. In Christian Freksa, Matthias Jantzen, and Rüdiger Valk, editors, Foundations of Computer Science, Potential-Theory-Cognition, volume 1337 of Lecture Notes in Computer Science, pages 287–296. Springer, Berlin, 1997.

    Google Scholar 

  29. H. Daduna. The joint distribution of sojourn times for a customer traversing an overtake-free series of queues: The discrete time case. Queueing Systems and Their Applications, 27:297–323, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  30. H. Daduna. Sojourn time distributions in non-product form queueing networks. In Jewgeni Dshalalow, editor, Frontiers in Queueing: Models and Applications in Science and Engineering, chapter 7, pages 197–224. CRC Press, Boca Raton, 1997.

    Google Scholar 

  31. H. Daduna. Some results for steady-state and sojourn time distributions in open and closed linear networks of Bernoulli servers with state-dependent service and arrival rates. Performance Evaluation, 30:3–18, 1997.

    Article  Google Scholar 

  32. H. Daduna and R. Schassberger. A discrete-time round-robin queue with bernoulli input and general arithmetic service time distributions. Acta Informatica, 15:251–263, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  33. H. Daduna and R. Schassberger. Networks of queues in discrete time. Zeitschrift fuer Operations Research ZOR, 27:159–175, 1983.

    MATH  MathSciNet  Google Scholar 

  34. H. Daduna and R. Schassberger. Delay time distributions and adjusted transfer rates for Jackson networks. Archiv für Elektronik und übertragungstechnik, 47:342–348, 1993.

    Google Scholar 

  35. H. Daduna and R. Szekli. Conditional job observer properties in multitype closed queueing networks. Preprint, Mathematical Institute of the University of Wroclaw, 1999.

    Google Scholar 

  36. J. N. Daigle and St. C. Tang. The queue length distribution for multiserver discrete time queues with batch Markovian arrivals. Comm.Statist.-Stochastic Models, 8:665–683, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  37. E. de Souza e Silva and R. R. Muntz. Queueing networks: Solutions and applications. In H. Takagi, editor, Stochastic Analysis of Computer and Communication Systems, pages 319–399, Amsterdam, 1990. IFIP, North-Holland.

    Google Scholar 

  38. B. Desert. Lineare stochastische Netzwerke in diskreter Zeit: Gleichgewichtsverhalten und Durchlaufzeitverteilungen, 1997. Diploma thesis.

    Google Scholar 

  39. B. Desert and H. Daduna. Discrete time tandem networks of state dependent queues: The effect of different regulation schemes for simultaneous events on customer oriented performance measures. Preprint 99-07, Institut für Mathematische Stochastik der Universität Hamburg, 1999. submitted.

    Google Scholar 

  40. N. M. van Dijk. Queueing Networks and Product Forms — A Systems Approach. Wiley, Chichester, 1993.

    Google Scholar 

  41. M. El-Taha and S. Jr. Stidham. A filtered ASTA property. Queueing Systems and Their Applications, 11:211–222, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  42. M. El-Taha and S. Jr. Stidham. Sample-Path Analysis of Queueing Systems. Kluwer Academic Publisher, Boston, 1999.

    Google Scholar 

  43. M. El-Taha, S. Jr. Stidham, and R. Anand. Sample-path insensitivity of symmetric queues in discrete time. Nonlinear Analysis, Theory Methods and Applications, 30:1099–1110, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  44. W. Feller. An Introduction to Probability Theory and Its Applications, volume I. John Wiley and Sons, Inc., New York-London-Sidney, third edition, 1968.

    MATH  Google Scholar 

  45. H.-W. Ferng and J.-F. Chang. The departure process of discrete-time queueing systems with Markovian type input. Queueing Systems and Their Applications, 36:201–220, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  46. E. Gelenbe. Produkt form queueing networks with negative and positve customers. Journal of Applied Probability, 28:656–663, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  47. E. Gelenbe and G. Pujolle. Introduction to queueing networks. Wiley, Chichester, 1987.

    MATH  Google Scholar 

  48. P. Glasserman. Gradient estimation via Perturbation Analysis. Kluwer Academic Press, Boston, 1991.

    MATH  Google Scholar 

  49. B.W Gnedenko and D. König. Handbuch der Bedienungstheorie, volume 2. Akademie-Verlag, Berlin, 1984.

    MATH  Google Scholar 

  50. W.J. Gordon and G.F. Newell. Closed queueing networks with exponential servers. Operations Research, 15:252–267, 1967.

    Google Scholar 

  51. A. Gravey and G. Hebuterne. Simultaneity in discrete-time single server queues with Bernoulli inputs. Performance Evaluation, 14:123–131, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  52. S. Halfin. Batch delays versus customer delays. The Bell System Technical Journal, 62:2011–2015, 1983.

    Google Scholar 

  53. A. Harel, S. Namn, and J. Sturm. Simple bounds for closed queueing networks. Queueing Systems and Their Applications, 31:125–135, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  54. N. Harnpanichpun and G. Pujolle. Product-form discrete-time queues in series with batch transition: late arrival case. Performance Evaluation, 21:261–269, 1995.

    Article  MATH  Google Scholar 

  55. W. Henderson, B.S. Northcote, and P.G. Taylor. Triggered batch movement in queueing networks. Queueing Systems and Their Applications, 21:125–141, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  56. W. Henderson and P.G. Taylor. Product form in queueing networks with batch arrivals and batch services. Queueing Systems and Their Applications, 6:71–88, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  57. W. Henderson and P.G. Taylor. Embedded processes in stochastic Petrinets. IEEE Transactions on Software Engineering, 17:108–116, 1991.

    Article  MathSciNet  Google Scholar 

  58. W. Henderson and P.G. Taylor. Some new results on queueing networks with batch movements. Journal of Applied Probability, 28:409–421, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  59. W. Henderson and P.G. Taylor. Discrete-time queueing networks with geometric release probabilities. Advances of Applied Probability, 24:229–233, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  60. Henderson, W., Pearce, C.E.M., Taylor, P.G., and Dijk, N.M. van. Closed queueing networks with batch services. Queueing Systems and Their Applications, 6:59–70, 1990.

    Article  MATH  Google Scholar 

  61. Henderson, W., Pearce, C.E.M., Taylor, P.G., and Dijk, N.M. van. Insensitivity in discrete-time generalized semi-Markov processes allowing multiple events and probabilistic service scheduling. Annals of Applied Probability, 5:78–96, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  62. C. Herrmann. Stochastische Modelle für ATM-Konzepte. PhD thesis, RWTH Aachen, 1995. Aachener Beiträge zur Mobil-und Telekommunikation.

    Google Scholar 

  63. J. Hofmann, N. Müller, and K. Natarajan. Parallel versus sequential task processing: A new performance model in discrete time. Preprint, Department of Computer Science, University of Trier, 1996.

    Google Scholar 

  64. S.D. Hohl and P.J. Kühn. Approximate analysis of flow and cycle times in queueing networks. In L.F.M. de Moraes, E. de Souza e Silva, and L.F.G. Soares, editors, Proceedings of the 3rd International Conference on Data Communication Systems and Their Performance, pages 471–485, Amsterdam, 1988. North-Holland.

    Google Scholar 

  65. J. Hsu and P.J. Burke. Behaviour of tandem buffers with geometric input and markovian output. IEEE Transactions on Communications, 24:358–361, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  66. J. J. Hunter. Mathematical Techniques of Applied Probability, volume I: Discrete Time Models: Basic Theory. Academic Press, New York, 1983.

    Google Scholar 

  67. J. J. Hunter. Mathematical Techniques of Applied Probability, volume II: Discrete Time Models: Techniques and Applications. Academic Press, New York, 1983.

    Google Scholar 

  68. F. Ishizaki and T. Takine. Loss probability in a finite discrete-time queue in terms of the steady state distribution of an infinite queue. Queueing Systems and Their Applications, 31:317–326, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  69. J.R. Jackson. Networks of waiting lines. Operations Research, 5:518–521, 1957.

    MathSciNet  Google Scholar 

  70. S. Karlin and H. M. Taylor. A First Course in Stochastic Processes. Academic Press, New York-San Francisco-London, second edition, 1975.

    MATH  Google Scholar 

  71. S. Karlin and H. M. Taylor. A Second Course in Stochastic Processes. Academic Press, New York-San Francisco-London, 1981.

    MATH  Google Scholar 

  72. U. S. Karmarkar. Manufacturing lead times, order release and capacity loading. In S.C. Graves, A.H.G. Rinnooy Kan, and P.H. Zipkin, editors, Logistics of Production and Inventory, volume 4 of Handbooks in Operations Research and Management Science, chapter 6, pages 287–329. North-Holland, Amsterdam, 1993.

    Google Scholar 

  73. J. Keilson. Markov chain models — Rarity and exponentiality. Springer, New York, 1979.

    MATH  Google Scholar 

  74. F. Kelly. Networks of queues. Advances of Applied Probability, 8:416–432, 1976.

    Article  MATH  Google Scholar 

  75. F. P. Kelly. Reversibility and Stochastic Networks. John Wiley and Sons, Chichester-New York-Brisbane-Toronto, 1979.

    MATH  Google Scholar 

  76. F. P. Kelly and Phillip Pollett. Sojourn times in closed queueing networks. Advances of Applied Probability, 15:638–653, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  77. J. G. Kemeny, J. L. Snell, and A. W. Knapp. Denumerable Markov Chaines. Springer-Verlag, New York-Heidelberg-Berlin, 1976. Reprint of the book published in 1966 by Van Nostrand, Princeton.

    Google Scholar 

  78. L. Kleinrock. Analysis of a time-shared processor. Naval Research Logistics Quarterly, 10(II):59–73, 1964.

    Article  MathSciNet  Google Scholar 

  79. L. Kleinrock. Communication nets — Stochastic message flow and delay. McGraw-Hill, New York, 1964.

    Google Scholar 

  80. L. Kleinrock. Time-shared systems: A theoretical treatment. Journal of the Association for Computing Machinery, 14(2):242–261, 1967.

    MATH  MathSciNet  Google Scholar 

  81. L. Kleinrock. Queueing Theory, volume I. John Wiley and Sons, New York, 1975.

    MATH  Google Scholar 

  82. L. Kleinrock. Queueing Theory, volume II. John Wiley and Sons, New York, 1976.

    Google Scholar 

  83. H. Kobayashi. Stochastic modeling: Queueing networks. In G. Louchard and G. Latouche, editors, Probability Theory and Computer Science, International Lecture Series in Computer Science, chapter Part II, pages 53–121. Academic Press, London,Orlando, 1983.

    Google Scholar 

  84. D. Koenig and V. Schmidt. EPSTA: The coincidence of time-stationary and customer-stationary distributions. Queueing Systems and Their Applications, 5:247–264, 1989.

    Article  MATH  Google Scholar 

  85. E. Koenigsberg. Twenty five years of cyclic queues and closed queue networks: A review. Journal of the Operational Research Society, 33:605–619, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  86. K.H. Kook and R.F. Serfozo. Travel and sojourn times in stochastic networks. Annals of Applied Probability, 3:228–252, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  87. D. Kouvatsos. Performance Evaluation and Applications of ATM Networks, volume 557 of The Kluwer International Series in Engineering and Computer Science. Kluwer, Boston, 2000.

    Google Scholar 

  88. D. D. Kouvatsos, N. M. Tabet-Aouel, and S. G. Denazis. ME-based approximations for general discrete-time queueing models. Performance Evaluation, 21:81–109, 1994.

    Article  MATH  Google Scholar 

  89. M. Krüger. Geschlossene Warteschlangen-Netzwerke unter doppelt-stochastischen Bediendisziplinen. Diplomarbeit Technische Universität Berlin, Fachbereich Mathematik, 1983.

    Google Scholar 

  90. P. R. Kumar. Re-entrant lines. Queueing Systems and Their Applications, 13:87–110, 1993.

    Article  MATH  Google Scholar 

  91. P. R. Kumar. Scheduling manufacturing systems of re-entrant lines. In D. D. Yao, editor, Stochastic Modeling and Analysis of Manufacturing Systems, Springer Series in Operations Research, chapter 8, pages 325–360. Springer, New York, 1994.

    Google Scholar 

  92. K. Laevens. The round-robin service discipline in discrete time for phase-type distributed packet-lengths. Preprint, SMAC Research Group, University of Ghent, 1996.

    Google Scholar 

  93. K. Laevens and H. Bruneel. Discrete-time queueing models with feedback for input-buffered ATM switches. Performance Evaluation, 27,28:71–87, 1996.

    Google Scholar 

  94. S. S. Lavenberg and M. Reiser. Stationary state probabilities at arrival instants for closed queueing networks with multiple types of customers. Journal of Applied Probability, 17:1048–1061, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  95. Louchard, G. and Latouche, G., editors. Probability theory and computer science. International Lecture Series in Computer Science. Academic Presss, New York, 1983.

    MATH  Google Scholar 

  96. A. Makowski, B. Melamed, and W. Whitt. On averages seen by arrivals in discrete time. In IEEE Conference on Decision and Control, Vol. 28, pages 1084–1086, Tampa, FL., 1989.

    MathSciNet  Google Scholar 

  97. Benjamin Melamed and Wy Whitt. On arrivals that see time averages. Operations Research, 38:156–172, 1990.

    MATH  MathSciNet  Google Scholar 

  98. M. Miyazawa. On the characterisation of departure rules for discrete-time queueing networks with batch movements and its applications. Queueing Systems and Their Applications, 18:149–166, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  99. M. Miyazawa. A note on my paper: On the characterisation of departure rules for discrete-time queueing networks with batch movements and its applications. Queueing Systems and Their Applications, 19:445–448, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  100. M. Miyazawa. Stability of discrete-time Jackson networks with batch movements. In Paul Glasserman, Karl Sigman, and David D. Yao, editors, Stochastic Networks: Stability and Rare Events, volume 117 of Lecture Notes in Statistics, chapter 5, pages 76–93. Springer, New York, 1996.

    Google Scholar 

  101. M. Miyazawa and H. Takagi. Editorial introduction to: Advances in discrete time queues, (Special issue of Queueing Systems, Theory and Applications). Queueing Systems and Their Applications, 18:1–3, 1994.

    Article  Google Scholar 

  102. M. Miyazawa and Y. Takahashi. Rate conservation principle for discrete-time queues. Queueing Systems and Their Applications, 12:215–230, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  103. J. A. Morrison. Two discrete time queues in tandem. IEEE Transactions on Communications, 27(3):563–573, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  104. H. Osawa. Quasi-reversibility of a discrete-time queue and related models. Queueing Systems and Their Applications, 18:133–148, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  105. H. G. Perros. Approximation algorithms for open queueing networks with blocking. In H. Takagi, editor, Stochastic Analysis of Computer and Communication Systems, pages 451–498. North-Holland, Amsterdam, 1990.

    Google Scholar 

  106. V. Pestien and S. Ramakrishnan. Asymptotic behavior of large discrete-time cyclic queueing networks. Annals of Applied Probability, 4:591–606, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  107. V. Pestien and S. Ramakrishnan. Features of some discrete-time cyclic queueing networks. Queueing Systems and Their Applications, 18:117–132, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  108. S. Peterson. General batch service disciplines — A product-form batch processing network with customer coalescence. Mathematical Methods of Operations Research, 52:79–97, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  109. G. Pujolle, J.P. Claude, and D. Seret. A discrete queueing system with a product form solution. In Hasegawa, T., Takagi, H., and Takahashi, Y., editors, Proceedings of the IFIP WG 7.3 International Seminar on Computer Networking and Performance Evaluation, pages 139–147, Amsterdam, 1986. Elsevier Science Publisher.

    Google Scholar 

  110. M. Reiser. A queueing network analysis of computer communication networks with window flow control. IEEE Transactions in Communications, COM-27:1199–1209, 1979.

    Article  MathSciNet  Google Scholar 

  111. M. Reiser. Performance evaluation of data communication systems. Proceedings of the IEEE, 70:171–196, 1982.

    Article  Google Scholar 

  112. J.-F. Ren, J. W. Mark, and J.W. Wong. Performance analysis of a leaky-bucket controlled ATM multiplexer. Performance Evaluation, 19:73–101, 1994.

    Article  MATH  Google Scholar 

  113. Y. Sakai, Y. Takahashi, and T. Hasegawa. Discrete time multi-class feedback queue with vacations and close time under random order of service discipline. Journal of the Operartions Research Society of Japan, 41:589–609, 1998.

    MATH  MathSciNet  Google Scholar 

  114. M. Sakata, S. Noguchi, and J. Oizumi. An analysis of the M/G/1 queue under round-robin scheduling. Operations Research, 19:371–385, 1971.

    MATH  Google Scholar 

  115. R. Schassberger. Insensitivity of steady-state distributions of generalized semi-Markov processes, part I. Ann. Prob., 5:87–99, 1977.

    Article  MATH  MathSciNet  Google Scholar 

  116. R. Schassberger. Insensitivity of steady-state distributions of generalized semi-Markov processes, part II. Ann. Prob., 6:85–93, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  117. R. Schassberger. The doubly stochastic server: A time-sharing model. Zeitschrift fuer Operations Research ZOR, 25:179–189, 1981.

    MATH  MathSciNet  Google Scholar 

  118. R. Schassberger and H. Daduna. A discrete-time technique for solving closed queueing network models of computer systems. In Paul J. Kühn and K.M. Schulz, editors, Messung,Modellierung und Bewertung von Rechensystemen, pages 122–134, Berlin, 1983. Informatik-Fachberichte 61, Springer.

    Google Scholar 

  119. R. Schassberger and H. Daduna. The time for a roundtrip in a cycle of exponential queues. Journal of the Association for Computing Machinery, 30:146–150, 1983.

    MATH  MathSciNet  Google Scholar 

  120. R. F. Serfozo. Queueing networks with dependent nodes and concurrent movements. Queueing Systems and Their Applications, 13:143–182, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  121. R. F. Serfozo. Introduction to Stochastic Networks, volume 44 of Applications of Mathematics. Springer, New York, 1999.

    MATH  Google Scholar 

  122. R. F. Serfozo and B. Yang. Markov network processes with string transitions. Annals of Applied Probability, 8:793–821, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  123. K. C. Sevcik and I. Mitrani. The distribution of queueing network states at input and output instants. Journal of the Association for Computing Machinery, 28:358–371, 1981.

    MATH  MathSciNet  Google Scholar 

  124. V. Sharma. Open queueing networks in discrete time — some limit theorems. Queueing Systems and Their Applications, 14:159–175, 1993.

    Article  MATH  Google Scholar 

  125. V. Sharma and N. D. Gangadhar. Some algorithms for discrete time queues with finite capacity. Queueing Systems and Their Applications, 25:281–305, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  126. K. Sohraby and J. Zhang. Spectral decomposition approach for transient analysis of multi-server discrete-time queues. Performance Evaluation, 21:131–150, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  127. H. Takagi. Stochastic Analysis of Computer and Communication Systems. North-Holland, Amsterdam, 1990.

    MATH  Google Scholar 

  128. H. Takagi. Queueing Analysis: A Foundation of Performance Analysis, volume 3. North-Holland, New York, 1993. Discrete-Time Systems.

    Google Scholar 

  129. H. Tempelmeier. Inventory control using a service constraint on the expected customer order waiting time. European Journal of Operational Research, 19:313–323, 1983.

    Article  MathSciNet  Google Scholar 

  130. H. Tempelmeier. Inventory service-levels in the customer supply chain. Operations Research Spektrum, 22:361–380, 2000.

    MATH  MathSciNet  Google Scholar 

  131. D. D. Tjhie. Zeitkritischer Verkehr in Wartesystemen von Hochgeschwindigkeitsnetzen: Modellbildung und Mathematische Analyse. Herbert Utz Verlag Wissenschaft, Munchen, 1996.

    Google Scholar 

  132. P. Tran-Gia. Analytische Leistungsbewertung verteilter Systeme. Springer, Berlin, 1996.

    MATH  Google Scholar 

  133. P. Tran-Gia, C. Blondia, and D. Towsley. Editorial introduction to: Discrete-time models and analysis methods, (Special issue of Performance Evaluation). Performance Evaluation, 21:1–2, 1994.

    Article  Google Scholar 

  134. P. Tran-Gia and R. Dittmann. A discrete-time analysis of the cyclic reservation multiple access protocol. Performance Evaluation, 16:185–200, 1992.

    Article  MATH  Google Scholar 

  135. J. Walrand. A discrete-time queueing network. Journal of Applied Probability, 20:903–909, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  136. J. Walrand. An introduction to queueing networks. Prentice-Hall International Editions, Englewood Cliffs, 1988.

    MATH  Google Scholar 

  137. J. Walrand. Queueing networks. In D. P. Heyman and M. J. Sobel, editors, Stochastic Models, volume 2 of Handbooks in Operations Research and Management Science, chapter 11, pages 519–603. North-Holland, Amsterdam, 1990.

    Google Scholar 

  138. W. Whitt. An overview of Brownian and non-Browninan FCLTs for the single-server queue. Queueing Systems and Their Applications, 36:39–70, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  139. P. Whittle. Systems in Stochastic Equilibrium. Wiley, Chichester, 1986.

    MATH  Google Scholar 

  140. S. Wittevrongel and H. Bruneel. Discrete-time ATM queues with independent and correlated arrival streams. In D. D. Kouvatsos, editor, Performance evaluation and applications of ATMnetworks, volume 557 of The Kluwer international series in engineering and computer science, chapter 16, pages 387–412. Kluwer, Boston, 2000.

    Chapter  Google Scholar 

  141. R.W. Wolff. Poisson arrivals see time averages. Operations Research, 30:223–231, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  142. M.E. Woodward. Communication and Computer Networks: Modelling with Discrete-Time Queues. IEEE Computer Society Press, Los Alamitos, CA, 1994.

    Google Scholar 

  143. S.F. Yashkov. Properties of invariance of probabilistic models of adaptive scheduling in shared-use systems. Automatic control and computer science, 14:46–51, 1980.

    MATH  Google Scholar 

Download references

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

(2001). Appendix. In: Queueing Networks with Discrete Time Scale. Lecture Notes in Computer Science, vol 2046. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44592-7_7

Download citation

  • DOI: https://doi.org/10.1007/3-540-44592-7_7

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42357-7

  • Online ISBN: 978-3-540-44592-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics