Skip to main content
SpringerLink
Log in

Queueing Models with Multiple Waiting Lines

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

This paper discusses analytic solution methods for queueing models with multiple waiting lines. The methods are briefly illustrated, using key models like the 2×2 switch, the shortest queue and the cyclic polling system.

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. I.J.B.F. Adan, A compensation approach for queueing problems, Ph.D. thesis, Eindhoven University of Technology (1991).

  2. I.J.B.F. Adan and G. Hooghiemstra, The M/M/c with critical jobs, Math. Oper. Res. 47 (1998) 341-353.

    Google Scholar 

  3. I.J.B.F. Adan, G.J.J.A.N. van Houtum and J. van der Wal, Upper and lower bounds for the waiting time in the symmetric shortest queue system, Ann. Oper. Res. 48 (1994) 197-217.

    Google Scholar 

  4. I.J.B.F. Adan, A.G. de Kok and J.A.C. Resing, A multi-server queueing model with locking, European J. Oper. Res. 116 (1999) 16-26.

    Google Scholar 

  5. I.J.B.F. Adan and J.A.C. Resing, A class of Markov processes on a semi-infinite strip, in: Numerical Solution of Markov Chains (NSMC '99), eds. B. Plateau, W.J. Stewart and M. Silva (Prensas Universitarias de Zaragoza, Zaragoza, 1999) pp. 41-57.

    Google Scholar 

  6. I.J.B.F. Adan and J.A.C. Resing, Multi-server batch-service systems, Statistica Neerlandica 54 (2000) 202-220.

    Google Scholar 

  7. I.J.B.F. Adan and J. Wessels, Shortest expected delay routing for Erlang servers, Queueing Systems 23 (1996) 77-105.

    Google Scholar 

  8. I.J.B.F. Adan, J. Wessels and W.H.M. Zijm, Analysis of the symmetric shortest queue problem, Stochastic Models 6 (1990) 691-713.

    Google Scholar 

  9. I.J.B.F. Adan, J. Wessels, and W.H.M. Zijm, Analysis of the asymmetric shortest queue problem, Queueing Systems 8 (1991) 1-58.

    Google Scholar 

  10. I.J.B.F. Adan, J.Wessels, and W.H.M. Zijm, A compensation approach for two-dimensional Markov processes, Adv. Appl. Probab. 25 (1993) 783-817.

    Google Scholar 

  11. I.J.B.F. Adan, J. Wessels and W.H.M. Zijm, Matrix-geometric analysis of the symmetric shortest queue problem with threshold jockeying, Oper. Res. Lett. 13 (1993) 107-112.

    Google Scholar 

  12. F. Baccelli, Two parallel queues created by arrivals with two demands: The M/G/2 symmetrical case, Technical report 426, INRIA-Rocquencourt (1985).

  13. F. Baccelli, A.M. Makowski, and A. Shwartz, The fork-join queue and related systems with synchronization constraints: Stochastic ordering and computable bounds, Adv. Appl. Probab. 21 (1989) 629-660.

    Google Scholar 

  14. F. Baccelli and V. Schmidt, Taylor series expansions for Poisson driven (max,+)-linear systems, Ann. Appl. Probab. 6 (1996) 138-185.

    Google Scholar 

  15. H. Bavinck, G. Hooghiemstra and E. deWaard, An application of Gegenbauer polynomials in queueing theory, J. Comput. Appl. Math. 49 (1993) 1-10.

    Google Scholar 

  16. V.E. Bene\(\mathop {\text{s}}\limits^ \sim \), Mathematical Theory of Connecting Networks and Telephone Traffic (Academic Press, New York, 1965).

    Google Scholar 

  17. S.A. Berezner, C.F. Kriel and A.E. Krzesinski, Quasi-reversible multiclass queues with order independent departure rates, Queueing Systems 19 (1995) 345-359.

    Google Scholar 

  18. D. Bertsimas and X.A. Papaconstantinou, Analysis of the stationary Ek/C2/s queueing system, European J. Oper. Res. 37 (1998) 272-287.

    Google Scholar 

  19. D. Bertsimas, I.Ch. Paschalidis and J.N. Tsitsiklis, Large deviations analysis of the generalized processor sharing policy, Report Boston University (1997).

  20. J.P.C. Blanc, Application of the theory of boundary value problems in the analysis of a queueing model with paired services, Ph.D. thesis, University of Utrecht (1982).

  21. J.P.C. Blanc, A note on waiting times in systems with queues in parallel, J. Appl. Probab. 24 (1987) 540-546.

    Google Scholar 

  22. J.P.C. Blanc, A numerical approach to cyclic-service queueing models, Queueing Systems6 (1990) 173-188.

    Google Scholar 

  23. J.P.C. Blanc, The power-series algorithm applied to the shortest-queue model, Oper. Res. 40 (1992) 157-167.

    Google Scholar 

  24. J.P.C. Blanc, Performance analysis and optimization with the power-series algorithm, in: Performance Evaluation of Computer and Communication Systems, eds. L. Donatiello and R.D. Nelson (North-Holland, Amsterdam, 1993) pp. 53-80.

    Google Scholar 

  25. B. Blaszczyszyn, A. Frey and V. Schmidt, Light traffic aproximations for Markov-modulated multiserver queues, Stochastic Models11 (1995) 423-445.

    Google Scholar 

  26. S.C. Borst and O.J. Boxma, Polling systems with and without switchover times, Oper. Res. 45 (1997) 536-543.

    Google Scholar 

  27. O.J. Boxma, Workloads and waiting times in single-server systems with multiple customer classes, Queueing Systems 5(1989) 185-214.

    Google Scholar 

  28. O.J. Boxma and W.P. Groenendijk, Two queues with alternating service and switching times, in: Queueing Theory and its Applications, eds. O.J. Boxma and R. Syski (North-Holland, Amsterdam, 1988) pp. 261-288.

    Google Scholar 

  29. O.J. Boxma and G.J. van Houtum, The compensation approach applied to a 2 x 2 switch, Probab. Engrg. Inform. Sci. 7 (1993) 171-193.

    Google Scholar 

  30. O.J. Boxma, G.M. Koole and Z. Liu, Queueing-theoretic solution methods for models of parallel and distributed systems, in: Performance Evaluation of Parallel and Distributed Systems-Solution Methods, eds. O.J. Boxma and G.M. Koole (CWI Tract 105, CWI, Amsterdam, 1994) pp. 1-24.

    Google Scholar 

  31. G.L. Choudhury and W. Whitt, Computing distributions and moments in polling models by numerical transform inversion, Performance Evaluation 25 (1996) 267-292.

    Google Scholar 

  32. A. Cobham, Priority assignment in waiting line problems, Oper. Res. 2 (1954) 70-76.

    Google Scholar 

  33. J.W. Cohen, Certain delay problems for a full availability trunk group loaded by two traffic sources, Communication News 16 (1956) 105-113.

    Google Scholar 

  34. J.W. Cohen, On the M/G/2 queueing model, Stochastic Process. Appl. 12 (1982) 231-248.

    Google Scholar 

  35. J.W. Cohen, The Single Server Queue (North-Holland, Amsterdam, 1982).

    Google Scholar 

  36. J.W. Cohen, On a functional relation in three complex variables; three coupled processors, Report 359, Mathematical Institute, University of Utrecht (1984).

  37. J.W. Cohen, On the analysis of parallel, independent processors, Report 374, Mathematical Institute, University of Utrecht (1985).

  38. J.W. Cohen, A two-queue, one-server model with priority for the longer queue, Queueing Systems 2 (1987) 261-283.

    Google Scholar 

  39. J.W. Cohen, Boundary value problems in queueing theory, Queueing Systems 3 (1988) 97-128.

    Google Scholar 

  40. J.W. Cohen, Analysis of Random Walks (IOS Press, Amsterdam, 1992).

    Google Scholar 

  41. J.W. Cohen, On a class of two-dimensional nearest neighbouring random walks, in: Studies in Applied Probability Theory, eds. J. Gani and J. Galambos, Special Volume of J. Appl. Probab. 31A (1994) 207-237.

  42. J.W. Cohen, Analysis of a two-dimensional algebraic nearest-neighbour random walk (queue with paired services), Technical report BS-R9437, CWI, Amsterdam (1994).

    Google Scholar 

  43. J.W. Cohen, Two-dimensional nearest-neighbour queueing models, a review and an example, in: Quantitative Methods in Parallel Systems, eds. F. Baccelli, A. Jean-Marie and I. Mitrani (Springer, Berlin, 1995) pp. 141-152.

    Google Scholar 

  44. J.W. Cohen, On the determination of the stationary distribution of a symmetric clocked buffered switch, in: Teletraffic Contributions for the Information Age, P roc. ITC-15, eds. V. Ramaswami and P.E. Wirth (North-Holland, Amsterdam, 1997) pp. 297-307.

    Google Scholar 

  45. J.W. Cohen, On the asymmetric clocked buffered switch, Queueing Systems 30 (1998) 385-404.

    Google Scholar 

  46. J.W. Cohen, Analysis of the asymmetrical shortest two-server queueing model, J. Appl. Math. Stochastic Anal.11 (1998) 115-162.

    Google Scholar 

  47. J.W. Cohen and O.J. Boxma, The M/G/1 queue with alternating service formulated as a Riemann–Hilbert problem, in: Performance '81, ed. F.J. Kylstra (North-Holland, Amsterdam, 1981) pp. 181-199.

    Google Scholar 

  48. J.W. Cohen and O.J. Boxma, Boundary Value Problems in Queueing System Analysis (North-Holland, Amsterdam, 1983).

    Google Scholar 

  49. B.W. Conolly, The autostrada queueing problem, J. Appl. Probab. 21 (1984) 394-403.

    Google Scholar 

  50. R.L. Cruz, A calculus for network delay, Part I: Network elements in isolation, IEEE Trans. Inform. Theory 37 (1991) 114-131.

    Google Scholar 

  51. R.L. Cruz, A calculus for network delay, Part II: Network analysis, IEEE Trans. Inform. Theory 37 (1991) 132-141.

    Google Scholar 

  52. J.N. Daigle and D.M. Lucantoni, Queueing systems having phase-dependent arrival and service rates, in: Numerical Solution of Markov Chains, ed. W.J. Stewart (Marcel Dekker, New York, 1991) pp. 161-202.

    Google Scholar 

  53. R.H. Davis, Waiting-time distribution of a multi-server, priority queuing system, Oper. Res. 14 (1966) 133-136.

    Google Scholar 

  54. A. Dukhovny, Applications of vector Riemann boundary value problems to analysis of queueing systems, in: Advances in Queueing: Theory, Methods and Open Problems, ed. J.H. Dshalalow (CRC Press, Boca Raton, 1995) pp. 353-376.

    Google Scholar 

  55. P. Dupuis and K. Ramanan, A Skorokhod problem formulation and large deviation analysis of a processor sharing model, Queueing Systems 28 (1998) 109-124.

    Google Scholar 

  56. N.M. van Dijk, A formal proof for the insensitivity of simple bounds for finite multi-server nonexponential tandem queues, Stoch. Process. Appl. 27 (1988) 261-277.

    Google Scholar 

  57. N.M. van Dijk, Simple bounds for queueing systems with breakdowns, Performance Evaluation8(1988) 117-128.

    Google Scholar 

  58. N.M. van Dijk, Queueing Networks and Product Forms: A Systems Approach (Wiley, Chichester, 1993).

    Google Scholar 

  59. N.M. van Dijk and B.F. Lamond, Simple bounds for finite single-server exponential tandem queues, Oper. Res. 36 (1988) 470-477.

    Google Scholar 

  60. N.M. van Dijk and J. van der Wal, Simple bounds and monotonicity results for finite multi-server exponential tandem queues, Queueing Systems 4 (1989) 1-16.

    Google Scholar 

  61. M. Eisenberg, Two queues with alternating service, SIAM J. Appl. Math. 36 (1979) 287-303.

    Google Scholar 

  62. A.I. Elwalid, D. Mitra and T.E. Stern, A theory of statistical multiplexing of Markovian sources: spectral expansions and algorithms, in: Numerical Solution of Markov Chains, ed. W.J. Stewart (Marcel Dekker, New York, 1991) pp. 223-238.

    Google Scholar 

  63. G. Fayolle, Methodes analytiques pour les files d'attente couplées, Ph.D. thesis, Université de Paris VI, Paris (1979).

    Google Scholar 

  64. G. Fayolle and R. Iasnogorodski, Two coupled processors: The reduction to a Riemann–Hilbert problem, Z. Wahrscheinlichkeitsth. verw. Geb. 47 (1979) 325-351.

    Google Scholar 

  65. G. Fayolle, R. Iasnogorodski and V. Malyshev, Random Walks in the Quarter-Plane (Springer, Berlin, 1999).

    Google Scholar 

  66. G. Fayolle, P.J. King and I. Mitrani, The solution of certain two-dimensional Markov models, Adv. Appl. Probab. 14 (1982) 295-308.

    Google Scholar 

  67. L. Flatto, The longer queue model, Probab. Engrg. Inform. Sci. 3 (1989) 537-559.

    Google Scholar 

  68. L. Flatto and S. Hahn, Two parallel queues created by arrivals with two demands, SIAM J. Appl. Math. 44 (1984) 1041-1053.

    Google Scholar 

  69. L. Flatto and H.P. McKean, Two queues in parallel, Comm. Pure Appl. Math. 30 (1977) 255-263.

    Google Scholar 

  70. G.J. Foschini and J. Salz, A basic dynamic routing problem and diffusion, IEEE Trans. Commun. 26 (1978) 320-327.

    Google Scholar 

  71. H.R. Gail, S.L. Hantler and B.A. Taylor, Analysis of a non-preemptive priority multiserver queue, Adv. Appl. Probab. 20 (1988) 852-879.

    Google Scholar 

  72. H.R. Gail, S.L. Hantler and B.A. Taylor, On a preemptive Markovian queue with multiple servers and two priority classes, Math. Oper. Res.17 (1992) 365-391.

    Google Scholar 

  73. F.D. Gakhov, Boundary Value Problems(Pergamon Press, Oxford, 1966).

    Google Scholar 

  74. P.R. Garabedian, Partial Differential Equations (Wiley, Chichester, UK, 1967).

    Google Scholar 

  75. I. Gertsbakh, The shorter queue problem: A numerical study using the matrix geometric solution, European J. Oper. Res. 15 (1984) 374-381.

    Google Scholar 

  76. W.B. Gong and J.Q. Hu, The MacLaurin series for the GI/G/1 queue, J. Appl. Probab. 29 (1992) 176-184.

    Google Scholar 

  77. D. Grillo, Polling mechanism models in communication systems–some application examples, in: Stochastic Analysis of Computer and Communication Systems, ed. H. Takagi (North-Holland, Amsterdam, 1990) pp. 659-698.

    Google Scholar 

  78. S. Halfin, The shortest queue problem, J. Appl. Probab. 22 (1985) 865-878.

    Google Scholar 

  79. G. Hooghiemstra, M. Keane, and S. van de Ree, Power series for stationary distributions of coupled processor models, SIAM J. Appl. Math. 48 (1988) 1159-1166.

    Google Scholar 

  80. W.B. van den Hout, The power-series algorithm, Ph.D. thesis, Tilburg University (1996).

  81. W.B. van den Hout and J.P.C. Blanc, The power-series algorithm for Markovian queueing networks, in: Computations with Markov Chains, ed. W.J. Stewart (Kluwer Academic Publishers, Boston, 1995) pp.321-338.

    Google Scholar 

  82. G.J.J.A.N. van Houtum, I.J.B.F. Adan, J. Wessels, and W.H.M. Zijm, The compensation approach for three and more dimensional random walks, Technical Report COSOR 92-39, Eindhoven University of Technology, Eindhoven (1992).

    Google Scholar 

  83. G.J.J.A.N. van Houtum, New approaches for multi-dimensional queueing systems, Ph.D. thesis, Eindhoven University of Technology (1995).

  84. G.J.J.A.N. van Houtum, W.H.M. Zijm, I.J.B.F. Adan and J. Wessels, Bounds for performance characteristics: a systematic approach via cost structures, Stochastic Models 14 (1998) 205-224.

    Google Scholar 

  85. G.J.J.A.N. van Houtum, I.J.B.F. Adan and J. van der Wal, The symmetric longest queue system, Stochastic Models 13 (1997) 105-120.

    Google Scholar 

  86. J.Q. Hu, Analyticity of single-server queues in light traffic, Queueing Systems 19 (1995) 63-80.

    Google Scholar 

  87. R. Iasnogorodski, Problèmes-frontières dans les files d'attente, Ph.D. thesis, Université de Paris VI, Paris (1979).

    Google Scholar 

  88. S. Jaffe, The equilibrium distribution for a clocked buffered switch, Probab. Engrg. Inform. Sci. 6 (1992) 425-438.

    Google Scholar 

  89. S. Jaffe, Equilibrium results for a pair of coupled discrete-time queues, Ultracomputer Note, NYA Ultracomputer Research Lab, Courant Institute of Mathematical Sciences, New York (1989).

    Google Scholar 

  90. N.K. Jaiswal, Priority Queues (Academic Press, New York, 1968).

    Google Scholar 

  91. O. Kella and U. Yechiali, Waiting times in the non-preemptive priority M/M/c queue, Stochastic Models 1 (1985) 257-262.

    Google Scholar 

  92. J.F.C. Kingman, Two similar queues in parallel, Ann. Math. Statistics 32 (1961) 1314-1323.

    Google Scholar 

  93. S.J. de Klein, Fredholm integral equations in queueing analysis, Ph.D. thesis, University of Utrecht (1988).

  94. L. Kleinrock, Communication Nets; Stochastic Message Flow and Delay (McGraw-Hill, London, 1964).

    Google Scholar 

  95. L. Kleinrock, Queueing Systems, Vol. 2: Computer Applications (Wiley, New York, 1976).

    Google Scholar 

  96. C. Knessl, B.J. Matkowsky, Z. Schuss, and C. Tier, Two parallel queues with dynamic routing, IEEE Trans. Commun. 34 (1986) 1170-1176.

    Google Scholar 

  97. C. Knessl, B.J. Matkowsky, Z. Schuss, and C. Tier, Two parallel M/G/1 queues where arrivals join the system with the smaller buffer content, IEEE Trans. Commun. 35 (1987) 1153-1158.

    Google Scholar 

  98. A.G. Konheim, I. Meilijson, A. Melkman, Processor-sharing of two parallel lines, J. Appl. Probab.18 (1981) 952-956.

    Google Scholar 

  99. G.M. Koole, On the power series algorithm, in: Performance Evaluation of Parallel and Distributed Systems-Solution Methods, eds. O.J. Boxma and G.M. Koole, CWI Tract 105 & 106 (CWI, Amsterdam, 1994) pp. 139-155.

    Google Scholar 

  100. G. Latouche and V. Ramaswami, A logarithmic reduction algorithm for quasi-birth-and-deatch processes, J. Appl. Probab.30 (1993) 650-674.

    Google Scholar 

  101. ] D.-S. Lee, Analysis of a two-queue model with Bernoulli schedules, J. Appl. Probab. 34 (1997) 176-191.

    Google Scholar 

  102. K.K. Leung, Waiting time distributions for token-passing systems with limited-k service via discrete Fourier transforms, in: Performance '90, eds. P.J.B. King, I. Mitrani and R.J. Pooley (North-Holland, Amsterdam, 1990) pp. 333-347.

    Google Scholar 

  103. H. Levy and M. Sidi, Polling systems: Applications, modeling and optimization, IEEE Trans. Commun. 38 (1990) 1750-1760.

    Google Scholar 

  104. J.C.S. Lui, R.R. Muntz and R. Richard, Algorithmic approach to bounding the mean response time of a minimum expected delay routing system, Performance Evaluation Rev. 20 (1992) 140-151.

    Google Scholar 

  105. J.C.S. Lui, R.R. Muntz and D. Towsley, Bounding the mean response time of a minimum expected delay routing system: an algorithmic approach, CMPSCI Technical report 93-68, University of Massachusetts (1993).

  106. V. Malyshev, An analytic method in the theory of two-dimensional random walks, Siberic Math. J. 13 (1972) 1314-1329.

    Google Scholar 

  107. L. Massoulié, Large deviations for polling and weighted fair queueing service systems, Advances in Performance Analysis 2 (1999) 103-128.

    Google Scholar 

  108. I. Mitrani, The spectral expansion solution method for Markov processes on lattice strips, in: Advances in Queueing: Theory, Methods and Open Problems, ed. J.H. Dshalalow (CRC Press, Boca Raton, 1995) pp. 337-352.

    Google Scholar 

  109. I. Mitrani and B. Avi-Itzhak, A many server queue with service interruptions, Oper. Res. 16 (1968) 628-638.

    Google Scholar 

  110. I. Mitrani and D. Mitra, A spectral expansion method for random walks on semi-infinite strips, in: IterativeMethods in Linear Algebra, eds. R. Beauwens and P. de Groen (North-Holland, Amsterdam, 1992) pp. 141-149.

    Google Scholar 

  111. N.I. Mushkelishvili, Singular Integral Equations (Noordhoff, Groningen, 1953).

    Google Scholar 

  112. R.D. Nelson and T.K. Philips, An approximation to the mean response time for shortest queue routing, Performance Evaluation Rev. 7 (1989) 181-189.

    Google Scholar 

  113. R.D. Nelson and T.K. Philips, An approximation for the mean response time for shortest queue routing with general interarrival and service times, Performance Evaluation 17 (1993) 123-139.

    Google Scholar 

  114. R. Nelson and A.N. Tantawi, Approximate analysis of fork/join synchronization in parallel queues, IEEE Trans. Computers 37 (1988) 739-743.

    Google Scholar 

  115. R. Nelson and A.N. Tantawi, Approximating task response times in fork/join queues, in: High Performance Computer Systems ed. E. Gelenbe (Elsevier, Amsterdam, 1988) pp. 157-167.

    Google Scholar 

  116. M.F. Neuts, Matrix-Geometric Solutions in Stochastic Models (Johns Hopkins, Baltimore, 1981).

    Google Scholar 

  117. A.K. Parekh and R.G. Gallager, A generalized processor sharing approach to flow control in integrated services networks: The single node case, IEEE/ACM Trans. Networking 1 (1993) 344-357.

    Google Scholar 

  118. A.K. Parekh and R.G. Gallager, A generalized processor sharing approach to flow control in integrated services networks: The multiple node case, IEEE/ACMTrans. Networking 2 (1994) 137-150.

    Google Scholar 

  119. V. Ramaswami and G. Latouche, A general class of Markov processes with explicit matrix-geometric solutions, OR Spektrum 8 (1986) 209-218.

    Google Scholar 

  120. B.M. Rao and M.J.M. Posner, Algorithmic and approximation analysis of the shorter queue model, Naval Res. Logist. 34 (1987) 381-398.

    Google Scholar 

  121. J.A.C. Resing, Polling systems and multitype branching processes, Queueing Systems 13 (1993) 409-426.

    Google Scholar 

  122. S. Shapiro, The M-server queue with Poisson input and Gamma-distributed service of order two, Oper. Res.14 (1966) 685-694.

    Google Scholar 

  123. A. Shwartz and A. Weiss, Induced rare events: Analysis via large deviations and time reversal, Adv. Appl. Probab.25 (1993) 667-689.

    Google Scholar 

  124. J.H.A. De Smit, The queue GI/M/s with customers of different types or the queue GI/Hm/s, Adv. Appl. Probab. 15 (1983) 392-419.

    Google Scholar 

  125. J.H.A. De Smit, A numerical solution for the multi-server queue with hyper-exponential service times, Oper. Res. Lett. 2 (1983) 217-224.

    Google Scholar 

  126. J.H.A. de Smit, ExplicitWiener-Hopf factorizations for the analysis of multidimensional queues, in: Advances in Queueing: Theory, Methods and Open Problems, ed. J.H. Dshalalow(CRC Press, Boca Raton, 1995) pp. 293-311.

    Google Scholar 

  127. H. Takagi, Queueing analysis of polling models: An update, in: Stochastic Analysis of Computer and Communication Systems, ed. H. Takagi (North-Holland, Amsterdam, 1990) pp. 267-318.

    Google Scholar 

  128. H. Takagi, Applications of polling models to computer networks, Computer Networks and ISDN Systems 22 (1991) 193-211.

    Google Scholar 

  129. H. Takagi, Queueing analysis of polling models: progress in 1990–1994, in: Frontiers in Queueing: Models and Applications in Science and Engineering, ed. J.H. Dshalalow (CRC Press, Boca Raton, 1997) pp. 119-146.

    Google Scholar 

  130. R.W. Wolff, Stochastic Modeling and the Theory of Queues (Prentice Hall, Englewood Cliffs, 1989).

    Google Scholar 

  131. P.E. Wright, Two parallel processors with coupled inputs, Adv. Appl. Probab. 24 (1992) 986-1007.

    Google Scholar 

  132. O. Yaron and M. Sidi, Performance and stability of communication networks via robust exponential bounds, IEEE/ACM Trans. Networking 1 (1993) 372-385.

    Google Scholar 

  133. Z.-L. Zhang, D. Towsley and J. Kurose, Statistical analysis of the generalized processor sharing scheduling discipline, IEEE J. Selected Areas Commun.13 (1995) 1071-1080.

    Google Scholar 

  134. Y. Zhao and W.K. Grassmann, A numerically stable algorithm for two server queue models, Queueing Systems 8 (1991) 59-79.

    Google Scholar 

  135. Y. Zhao and W.K. Grassmann, Queueing analysis of a jockeying model, Oper. Res. 43 (1995) 520-529.

    Google Scholar 

  136. Y. Zhu and H. Li, The MacLaurin expansion for a G/G/1 queue with Markov-modulated arrivals and services, Queueing Systems 14 (1993) 125-134.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    I.J.B.F. Adan, O.J. Boxma & J.A.C. Resing

Authors
  1. I.J.B.F. Adan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O.J. Boxma
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J.A.C. Resing
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Adan, I., Boxma, O. & Resing, J. Queueing Models with Multiple Waiting Lines. Queueing Systems 37, 65–98 (2001). https://doi.org/10.1023/A:1011040100856

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1011040100856

Search

Navigation