Dobrushin’s Approach to Queueing Network Theory

Journal of Applied Mathematics and Stochastic Analysis

Volumn 9, Issue 4, 1996, Pages 373-397

  Karpelevich, F I ^a   Pechersky, E A ^b   Suhov, Yu M ^c  

^a RUSSIAN UNIVERSITY OF TRANSPORT   (Russian Federation)

^b INSTITUTE FOR INFORMATION TRANSMISSION PROBLEMS   (Russian Federation)

^c UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

Invariant Distribution; Jackson s Network; Large Deviations; Message Switching; Poissonian Conjecture; Queue Transformation; Queueing Network; Single Server Queue; Stability

Indexed keywords

EID: 0009413891     PISSN: 10489533     EISSN: 16872177     Source Type: Journal    
DOI: 10.1155/S1048953396000342     Document Type: Article

References (136)

1. Anantharam, V., Uniquenss of stationary ergodic fixed point for a • /M/k node, Ann. Appl. Prob. 3 (1993), 154–173.
2. Avram, F., Bertsimas, D. and Ricard, M., Fluid models of sequencing problems in open queueing networks: an optimal control approach, In: Stochastic Networks, The IMA volumes in Mathematics and its Applications (ed. by F.P. Kelly and R.J. Williams) 71, Springer-Verlag, New York (1995), 199–234.
3. Baccelli, F. and Brémaud, P., Elements of Queueing Theory, Springer-Verlag, Berlin 1994.
4. Baccelli, F., Cohen, G., Olsder, G.J. and Quadrat, J.P., Synchronization and Linearity, Wiley, New York 1992.
5. Baccelli, F. and Foss, S., Ergodicity of Jackson-type queueing networks, Queueing Sys., 17(1994), 5–72.
6. Baccelli, F. and Foss, S., On the saturation rule for the stability of queues, J. Appl. Prob. 32 (1995), 494–507.
7. Bacelli, F., Foss, S., and Mairess, J., Stationary ergodic Jackson networks: results and counter-examples, In: Stochastic Networks. Theory and Applications (ed. by F.P.Kelly, S.Zachary and I.Ziedins), RSS Lecture Note Series 4, Clarendon Press, Oxford (1996), 281–308.
8. Baccelli, F., Karpelevich, F.I., Kelbert, M.Ya., Puhalskii, A.A., Rybko, A.N., and Suhov, Yu.M. A mean-field limit for a class of queueing networks, J. Stat. Phys. 66 (1992), 803–825.
9. Baskett, F., Chandy, K.M., Muntz, R.R. and Palacios, F.G., Open, closed and mixed networks of queues with different classes of customers, J. ACM 22 (1975), 248–260.
10. Berezner, S.A. and Malyshev, V.A., The stability of infinite-servers networks with random routing, J. Appl. Prob. 26 (1989), 363–371.
11. Berezner, S.A., Rose, D.M., and Suhov, Yu.M., Starlike networks with synchronization constraints, In: Stochastic Networks, The IMA volumes in Mathematics and its Applications (ed. By F.P. Kelly and R.J. Williams), Springer-Verlag, New York 71 (1995), 313–332.
12. Boldrighini, C., Dobrushin, R.L. and Suhov, Yu.M., One dimensional hard rod caricature of hydrodynamics, J. Stat. Phys. 31 (1983), 577–615.
13. Borovkov, A.A., Stochastic Processes in Queueing Theory, Springer-Verlag, New York 1976.
14. Borovkov, A.A., Asymptotic Methods in Queueing Theory, Wiley, New York 1984.
15. Borovkov, A.A., Limit theorems for queueing networks I, II, Theory of Probability and its Applications 31:3 (1987), 413–427
16. Borovkov, A.A., Limit theorems for queueing networks I, II, Theory of Probability and its Application 32 (1987), 257–272.
17. Boxma, O.J., On a tandem queueing model with identical service times at both counters I, II, Adv. Appl. Prob. 11 (1979), 616–643.
18. Bramson, M., Instability of FIFO queueing networks, Ann. Appl. Prob. 4(1994), 414–431.
19. Bramson, M., Instability of FIFO queueing networks with quick service times, Ann. Appl. Prob. 4 (1994), 693–718.
20. Bramson, M., Two badly behaved queueing networks, In: Stochastic Networks, The IMA volumes in Mathematics and its Applications (ed. By F.P. Kelly and R.J. Williams), Springer-Verlag, New York 71 (1995), 105–116.
21. Bramson, M., Convergence to equilibria for fluid models of certain FIFO and processor sharing queueing networks, In: Stochastic Networks. Theory and Applications, RSS Lecture Note Series (ed. by F.P. Kelly, S. Zachary and I. Ziedins), Clarendon Press, Oxford 4 (1996), 1–18.
22. Bramson, M., Convergence to equilibria for fluid models of FIFO queueing networks, Queueing Sys., to appear.
23. Bramson, M., Convergence to equilibria for fluid models of certain processor sharing queueing networks, Queueing Sys., to appear.
24. Brandt, A., On stationary waiting times and limiting behavior of queues with many servers, I. The general G/G/m/∞ case; II. The G/GI/m/∞ case, Elektron. Informations-verarb. Kybern. 21 (1985), 47–64;
25. Brandt, A., On stationary waiting times and limiting behavior of queues with many servers, I. The general G/G/m/∞ case; II. The G/GI/m/∞ case, Elektron. Informations-verarb. Kybern. 21 (1985), 151–162.
26. Brandt, A., Franken, P. and Liesek, B., Stationary Stochastic Models, Wiley, Chichester 1990.
27. Brown, T.G. and Pollett, P.K., Some distributional approximations in Markovian queueing networks, Adv. Appl. Prob. 14 (1982), 654–671.
28. Burke, P.J., The output of a queueing system, Oper. Res. 4 (1956), 699–704.
29. Burke, P.J., Output processes and tandem queues, In: Proc. Symp. on Computer-Communication Networks and Teletraffic (ed. by J. Fox), Polytechnic Press, Brooklyn (1972), 419–428.
30. Chang, C.S., On the input-output map of a G/G/1 queue, J. Appl. Prob. 31 (1994), 1128-1133.
31. Chen, H., Fluid approximations and stability of multiclass queueing networks: work-conserving disciplines, Ann. Appl. Prob. 5 (1995), 637–665.
32. Chen, H. and Yao, D., Dynamic scheduling of a multiclass fluid network, Oper. Res. 41(1993), 1104–1115.
33. Chen, H. and Zhang, H., Stability of multiclass queueing networks under FIFO service discipline, Math. Oper. Res. (1996), to appear.
34. Cohen, J.W., The Single Server Queue, North-Holland, Amsterdamand Wiley, New York 1969.
35. Dai, J.G., Stability of open multiclass queueing networks via fluid models, In: Stochastic Networks, The IMA volumes in Mathematics and its Applications (ed. By F.P. Kelly and R.J. Williams), Springer-Verlag, New York 71 (1995), 71–90.
36. Dai, J.G., On the positive Harris recurrence for multiclass queueing networks: a unified approach via fluid models, Ann. Appl. Probl. 5 (1995), 49–77.
37. Dai, J.G. and Meyn, S., Stability and convergence of moments for multiclass queueing networks via fluid models, IEEE Trans. Autom. Control 40 (1995), 1889–1904.
38. Dai, J.G. and Vande-Vate, J., The stability region for two station queueing networks (1995), preprint.
39. Dai, J.G. and Weiss, G., Stability and instability of fluid models for re-entrant lines, Math. Oper. Res. 21 (1996), 115–134.
40. Daley, D.J., Queueing output processes, Adv. Appl. Prob. 8 (1976), 395–415.
41. Dobrushin, R.L., On Poisson laws for the distributions of particles in space, Ukrainskii Mathematicheskii Zhurnal 8 (1958), 127–134. [Russian]
42. Dobrushin, R.L., The description of a random field by means of conditional probabilities and conditions of its regularity, Theory Probab. Its Appl. 13 (1968), 197–213.
43. Dobrushin, R.L., Gibbsian random fields for lattice systems with pairwise interactions, Functional Analysis and its Applications 2 (1968), 292–301.
44. Dobrushin, R.L., The problem of uniqueness of a Gibbsian random field and the problem of phase transition, Functional Analysis and Its Applications 2 (1968), 302–312.
45. Dobrushin, R.L., Prescribing a system of random variables by conditional distributions, Theory of Probability and Its Applications 15 (1970), 458–486.
46. Dobrushin, R.L., Markov processes with a large number of locally interacting components: existence of a limit processes and their ergodicity, Problems of Information Transmission 7 (1971), 149–164.
47. Dobrushin, R.L., Markov processes with many locally interacting components – the reversible case and some generalizations, Problems of Information Transmission 7 (1971), 235–241.
48. Dobrushin, R.L., The Vlasov equation, Functional Analysis and Its Applications 13 (1979), 115–123.
49. Dobrushin, R.L., Switching networks, Gibbsian fields-interconnections, In: Proc. 1st World Congress of the Bernoulli Society, Tashkent 1986, VN Sci. Press, Utrecht (1987), 377–393.
50. Dobrushin, R.L., On the way to the mathematical foundations of statistical mechanics, In: R.L. Dobrushin and S. Kusuoka, Statistical Mechanics and Fractals, Lecture Notes in Math 1567, Springer-Verlag, Berlin (1993), 1–37.
51. Dobrushin, R.L., Kelbert, M.Ya., Rybko, A.N., and Suhov, Yu.M., Qualitative methods of queueing network theory, In: Stochastic Cellular Systems: Ergodicity, Memory, Morphogenesis (ed. By R.L. Dobrushin, V.M. Kryukov and A.L. Toom), University Press, Manchester (1990), 183–224.
52. Dobrushin, R.L. and Pechersky, E.A., Large deviations for tandem queueing systems, J. Appal. Math. Stoch. Anal. 7 (1994), 301–330.
53. Dobrushin, R.L., Pellegrinotti, A., and Suhov, Yu.M., One-dimensional harmonic lattice caricature of hydrodynamics. The higher corrections, J. Stat. Phys. 61 (1990), 387–402.
54. Dobrushin, R.L., Pellegrinotti, A., Suhov, Yu.M., and Triolo, L., One-dimensional harmonic lattice caricature of hydrodynamics, J. Stat. Phys. 43 (1986), 571–608.
55. Dobrushin, R.L., Pellegrinotti, A., Suhov, Yu.M., and Triolo, L., One-dimensional harmonic lattice caricature of hydrodynamics: Second approximation, J. Stat. Phys. 52 (1988), 423–439.
56. Dobrushin, R.L. and Prelov, V.V., Asymptotic approach to the investigation of message switching networks of a linear structure with a large number of centers, Problems of Information Transmission 15 (1979), 46–55.
57. Dobrushin, R.L. and Siegmund-Schultze, R., The hydrodynamic limit for systems of particles with independent evolution, Math. Nachr. 105 (1982), 199–224.
58. Dobrushin, R.L., Sinai, Ya.G. and Sukhov, Yu.M., Dynamical systems of statistical mechanics, In: Dynamical Systems II, Encyclopedia of Math. Sci. 2 (ed. by Ya.G. Sinai), Springer-Verlag, Berlin (1989), 208–254.
59. Dobrushin, R.L. and Sukhov, Yu.M., On the problem of the mathematical foundation of the Gibbs postulate in classical statistical mechanics, In: Math. Problems in Theoretical Phys. (ed. by G. Dell’Antonio, S. Doplicher and G. Iona-Lasinio), Lecture Notes in Physics, Springer-Verlag, Berlin 80 (1978), 325–340.
60. Dobrushin, R.L. and Sukhov, Yu.M., Asymptotical investigation of star-shape message switching networks with a large number of radial rays, Problems of Information Transmission 12 (1976), 49–65.
61. Dobrushin, R.L. and Sukhov, Yu.M., Time asymptotics for some degenerate models of time evolution for system with infinitely many particles, J. Soviet Math. 16 (1981), 12771340.
62. Dumas, V., Unstable cycles in fluid Bramson networks, Rapport de Recherche 2138, INRIA August 1994.
63. Dupuis, P. and Ellis, R.S., The large deviation principle for a general class of queueing systems, Trans. Amer. Math. Soc., to appear.
64. Dupuis, P. and Ellis, R.S., Large deviation analysis of queueing systems, In: Stoch. Networks, The IMA Volumes in Mathematics and its Applications (ed. by F.P. Kelly and R.J. Williams), Springer-Verlag, New York 71 (1995), 347–365.
65. Ethier, S.N. and Kurtz, T.G., Markov Processes: Characterization and Convergence, Wiley, New York 1986.
66. Foss, S.G., Some properties of open queueing networks, Problems of Information Transmission 25 (1989), 241–247.
67. Foss, S.G., Ergodicity of queueing networks, Sib. Math. J. 32 (1991), 184–203.
68. Foss, S. and Rybko, A., Stability of multiclass Jackson-type networks, (1995), preprint.
69. Franken, P., König, D., Arndt, U. and Schmidt, V., Queues and Point Processes, Wiley, Chichester, 1982.
70. Georgii, H.-O., Gibbs Measures and Phase Transitions, De Gruyter, Berlin, 1988.
71. Gross, D. and Harris, C.M., Fundamentals of Queueing Theory, Wiley, Chichester 1985.
72. Ignatyuk, I.A. and Malyshev, V.A., Locally interacting processes and communication networks, Problems of Information Transmission 25 (1989), 50–60.
73. Jackson, J.R., Networks of waiting times, Oper. Res. 5 (1957), 518–527.
74. Jackson, J.R., Jobshop-like queueing systems, Management Sci. 10 (1965), 131–142.
75. Karpelevich, F.I., Kelbert, M.Ya., and Suhov, Yu.M., Higher-order Lindley equations, Stoch. Proc. Appal. 53 (1994), 65–96.
76. Karpelevich, F.I. and Kreinin, A.Ya., Characteristics of two-phase systems M/M/1→ M/1/∞, Izvestia AN SSR, Ser. Tekhn. Kibern. 2 (1985), 182–184. [Russian]
77. Karpelevich, F.I. and Kreinin, A.Ya., Asymptotic analysis of telegraphic message switching systems, Problems of Information Transmission 26 (1990), 261–274.
78. Karr, A.F., Point Processes and Their Statistical Inference, Marcel-Dekker, New York 1986.
79. Kelbert, M. Ya., On the existence of limiting distributions in some networks of commutation of messages, In: Multi-component Random Systems (ed. by R.L. Dobrushin and Ya.G. Sinai), Advances in Probability and Related Topics, Marcel-Dekker, New York 6 (1980), 397–421.
80. Kelbert, M. Ya., Poisson limit theorem for message-switched networks with small intensities of transit flows, Problem of Information Transmission 29:1 (1993), 92–98.
81. Kelbert, M. Ya., Kontsevich, M.A. and Rybko, A.N., On Jackson networks on denumerable graphs, Theory of Probability and Its Applications 33 (1989), 358–361.
82. Kelbert, M.Ya., Kopeika, R.P., Shamsiev, R.N., and Suhov, Y.M., Perturbation theory approach for a class of hybrid switching networks with small transit flows, Adv. Appal. Prob. 22 (1990), 211–229.
83. Kelbert, M.Ya. and Suhov, Yu.M., Existence and uniqueness conditions for a random field describing the state of a switching network, Problems of Information Transmission 19 (1983), 289–304.
84. Kelbert, M.Ya. and Suhov, Yu.M., Weak dependence of the random field describing state of a switching network with small transit flows, Problems of Information Transmission 21 (1985), 237–245.
85. Kelbert, M.Ya. and Suhov, Yu.M., A Poisson limit theorem for hybrid star networks: a mean-field approximation, Problems of Information Transmission 25 (1989), 60–67.
86. Kelbert, M.Ya. and Suhov, Yu.M., Circuit-switching networks with FIFO plus justifying blockout service on infinite graphs, Queueing Sys., 4 (1989), 157–170.
87. Kelbert, M.Y. and Suhov, Yu.M., Mathematical problems in the theory of queueing networks, J. Soviet Math. 50 (1990), 1527–1600.
88. Kelbert, M.Ya. and Suhov, Yu.M., On a class of packet-switching networks on infinite graphs, Queueing Sys., 5 (1990).
89. Kelbert, M.Ya. and Suhov, Yu.M., On the Dobrushin mean-field conjecture for queueing networks (1996), in preparation.
90. Kelly, F.P., Networks of queues, Adv. Appal. Probl. 8:2 (1976), 416–432.
91. Kelly, F.P., Reversibility and Stochastic Networks, Wiley, New York 1979.
92. Kelly, F.P., Loss networks, Ann. Appal. Prob. 1 (1991), 319–378.
93. Kendall, D.G., Some problems in the theory of queues, J. Roy. Stat. Soc. B13 (1951), 151185.
94. Kiefer, J. and Wolfowitz, J., On the theory of queues with many servers, Trans. Amer. Math. Soc. 78 (1955), 1–18.
95. Kleinrock, L., Communication Networks. Stochastic Flows and Delay of Messages, McGraw-Hill, Montreal 1960.
96. Kumar, P.R., Re-entrant lines, Queueing Sys., 13 (1993), 87–110.
97. Kumar, P.R., Scheduling manufacturing systems of re-entrant lines, In: Stochastic Modelling and Analysis of Manufacturing Systems (ed. by D.D.Yao), Springer-Verlag, New York (1994), 325–360.
98. Kumar, P.R., Scheduling queueing networks: stability, performance analysis and design, In: Stochastic Networks, The IMA volumes in Mathematics and its Applications (ed. by F.P. Kelly and R.J.Williams), Springer-Verlag, New York 71 (1995), 21–70.
99. Kumar, S. and Kumar, P.R., Performance bounds for queueing networks and scheduling policies, IEEE Trans. Autom. Control 39 (1994), 1600–1611.
100. Kumar, P.R. and Meyn, S.P., Stability of queueing networks and scheduling policies, IEEE Trans. Autom. Control 40 (1995), 251–260.
101. Kumar, P.R. and Seidman, T.I., Dynamic instabilities and stabilization methods in distributed real-time scheduling of manufacturing systems, IEEE Trans. Autom. Control 35 (1990), 289–298.
102. Liggett, T., Interacting Particle Systems, Springer-Verlag, New York 1985.
103. Lindley, D.V., The theory of queues with a single server, Proc. Cambridge Phil Soc. 48 (1952), 277–289.
104. Loynes, R.M., The stability of a queue with non-independent interarrival and service times, Proc. Cambridge Phil. Soc. 58 (1962), 497–520.
105. Lu, S.H. and Kumar, P.R., Distributed scheduling based on due dates and buffer priorities, IEEE Trans. Autom. Control 36 (1991), 1406–1416.
106. Malyshev, V.A. and Tsaregradskii, I.P., Queueing systems with local interaction, Theory Prob. Its Appal 27 (1983), 579–583.
107. Malyshev, V.A. and Yakovlev, A.V., Condensation in a large closed Jackson networks, Ann. Appal Prob. 6 (1996), 92–115.
108. Marbukh, V.V., Asymptotic investigation of a complete communications network with a large number of nodes and bypass routes, Problems of Information Transmission 17(1981), 212–216.
109. Marbukh, V.V., Fully connected message-switching network with a large number of nodes, bypass routes and a limited number of waiting places at nodes, Problems of Information Transmission 21 (1985), 154–161.
110. Meyn, S. and Down, D., Stability of generalized Jackson networks, Ann. Appal Probl. 4 (1994), 124–149.
111. Mountford, T.S. and Prabhakar, B., On the weak convergence of departures from an infinite sequence of • /M/1 queue, Ann. Appal Prob. 5 (1995), 121–127.
112. Mountford, T. and Prabhakar, B., The Cesaro limit from certain • /GI/1 queueing tandems, In: Stochastic Networks. Theory and Applications, RSS Lecture Note Series (ed. by F.P. Kelly, S. Zachary and I. Ziedins), Clarendon Press, Oxford 4 (1996), 309–322.
113. Mountford, T. and Prabhakar, B., in preparation.
114. Prabhakar, B., Uniqueness of fixed points for the • /GI/1 queueing operator (1996), in preparation.
115. Rybko, A.N., Stationary distributions of time homogeneous Markov processes modeling message switching communication networks, Problems of Information Transmission 17 (1981), 49–63.
116. Rybko, A.N., Existence conditions for the stationary regime for two types of communication networks with message switching, Problems of Information Transmission 18 (1982), 76–84.
117. Rybko, A.N. and Mikhailov, V.A., Capacity region for circuit-switched networks, Problems of Information Transmission 22 (1986), 51–67.
118. Rybko, A.N. and Stolyar, A.L., On the ergodicity of random processes describing open service networks, Problems of Information Transmission 28 (1992), 199–220.
119. Shamsijev, R.N., Circuit-switching networks with priorities on a one-dimensional lattice, J. Appal Prob. 26 (1989), 901–905.
120. Shwartz, A. and Weiss, A., Large Deviations for Performance Analysis: Queues, Communication and Computing, Chapman and Hill, New York 1995.
121. Sigman, K., The stability of open queueing networks, Stoch. Proc. Appl. 35 (1989), 11–25.
122. Sigman, K., Stationary Marked Point Processes. An Intuitive Approach, Chapman and Hall, New York 1995.
123. Spohn, H., Large Scale Dynamics of Interacting Particles, Springer-Verlag, New York 1991.
124. Stolyar, A., Asymptotic behavior of the stationary distribution of a closed queueing system, Problems of Information Transmission 25 (1989), 321–332.
125. Stolyar, A., On the stability of multiclass queueing networks, In: Proc. of the 2nd Conference on Telecommunication Systems – Modelling and Analysis, Nashville, TN, March 24–27 (1994), 1020–1028.
126. Suhov, Yu.M. and Rose, D., The Poisson-independence hypothesis for infinitely-growing fully-connected packet-switching networks, In: Stochastic Networks. Theory and Applications, RSS Lecture Note Series (ed. by F.P. Kelly, S. Zachary and I. Ziedins), Clarendon Press, Oxford 4 (1996), 323–338.
127. Suhov, Yu.M., Statistical mechanics of complex queueing networks, In: Proc. of the IX International Congress on Math. Physics (ed. by B. Simon, A. Truman and I.M. Davies), Adam Higler, Bristol (1989), 388–391.
128. Suhov, Yu.M. and Shamsijev, R.N., Hybrid switching networks on an infinite one-dimensional lattice, Problems of Information Transmission 23 (1987), 312–322.
129. Tsoucas, P., Rare events in series of queues, J. Appl. Prob. 29 (1992), 168–175.
130. Vinogradov, O.P., Two-phase queueing system with identical service, Math. Zametki 33:1 (1983), 141–146. [Russian]
131. Vinogradov, O.P., On some asymptotic properties of the waiting time in a queueing system with an identical service-time and large number of servers, Theory Prob. Its Appl. 39 (1994), 714–718.
132. Vvedenskaya, N.D., Dobrushin, R.L., and Karpelevich, F.I., A queueing system with selection of the shortest of two queues: an asymptotical approach, Problems of Information Transmission 32 (1996), 15–27.
133. Vvedenskaya, N.D. and Suhov, Y.M., The limiting departure flow in an infinite series of queues, Problems of Information Transmission 31 (1995), 100–111.
134. Weiss, G., On optimal draining of re-entrant fluid lines, In: Stochastic Networks, The IMA volumes in Math and its Appl. (ed. by F.P. Kelly and R.J. Williams), Springer-Verlag, New York 71 (1995), 91–106.
135. Weiss, G., Optimal draining of fluid re-entrant lines: some solved examples, In: Stochastic Networks. Theory and Applications, RSS Lecture Note Series (ed. by F.P. Kelly, S. Zachary and I. Ziedins), Clarendon Press, Oxford 4 (1996), 19–34.
136. Wolff, R.N., Stochastic Modeling and the Theory of Queues, Prentice-Hall, Englewood Cliffs, New Jersey 1989.