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Heavy Traffic Limit Theorems for Queues: A Survey

  • Conference paper
Mathematical Methods in Queueing Theory

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 98))

Abstract

Heavy traffic research is part of a general program to obtain simple descriptions and useful approximations for queueing models. Throughout this paper, I try to put heavy traffic research into this broader perspective. I discuss the two principal objectives of heavy traffic research, namely, (1) to describe unstable queueing systems and (2) to approximate stable queueing systems. These objectives are each related to more general themes. In an unstable queueing system the queueing processes are growing processes, so that descriptions of queueing processes in unstable queueing systems are similar to descriptions of other growing processes associated with queues. In the same way, heavy traffic approximations for stable queues are part of a large class of approximations for queueing systems. I relate heavy traffic research to these more general activities.

Partially supported by NSF Grant GK-38149.

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References

  1. ARJAS, E. and DE SMIT, J. H. A. (1973) On the total waiting time during a busy period of the single server queue. Discussion Paper No. 7312, Center for Operations Research and Econometrics, Louvain, Belgium.

    Google Scholar 

  2. AVI-ITZHAK, B. (1971) Heavy traffic characteristics of a circular data network. Bell System Tech. J. 50 2521–2549.

    MATH  MathSciNet  Google Scholar 

  3. BATHER, J. A. (1966) A continuous time inventory model. J. Appl. Prob. 3 538–549.

    MATH  MathSciNet  Google Scholar 

  4. BATHER, J. A. (1968) A diffusion model for the control of a dam. J. Appl. Prob. 5 55–71.

    MATH  MathSciNet  Google Scholar 

  5. BATHER, J. A. (1969) Diffusion models in stochastic control theory. J. Roy. Stat. Soc. Ser. A. 132 335–352.

    MathSciNet  Google Scholar 

  6. BILLINGSLEY, P. (1968) Convergence of Probability Measures. John Wiley and Sons, New York.

    MATH  Google Scholar 

  7. BILLINGSLEY, P. (1971) Weak Convergence of Measures: Applications in Probability. Regional Conference Series in Applied Mathematics, SIAM, Philadelphia.

    Google Scholar 

  8. BLOMQVIST, N. (1973) A heavy traffic result for the finite dam. J. Appl. Prob. 10 223–228.

    MATH  MathSciNet  Google Scholar 

  9. BLOOMFIELD, P. and COX, D. R. (1972) A low traffic approximation for queues. J. Appl. Prob. 9 832–840.

    MATH  MathSciNet  Google Scholar 

  10. BOROVKOV, A. (1964) Some limit theorems in the theory of mass service, I. Theor. Probability Appl. 9 550–565.

    Google Scholar 

  11. BOROVKOV, A. (1965) Some limit theorems in the theory of mass service, II. Theor. Probability Appl. 10 375–400.

    Google Scholar 

  12. BOROVKOV, A. (1967a) On limit laws for service processes in multi-channel systems. Siberian Math. J. 8 746–763.

    MATH  Google Scholar 

  13. BOROVKOV, A. (1967b) Convergence of weakly dependent processes to the Wiener process. Theor. Probability Appl. 12 159–174.

    MATH  Google Scholar 

  14. BOROVKOV, A. (1967c) On the convergence to diffusion processes. Theor. Probability Appl. 12 405–431.

    MATH  Google Scholar 

  15. BOROVKOV, A. (1972) Stochastic Processes in the Theory of Mass Service. Science Publishers, Moscow (in Russian).

    Google Scholar 

  16. BRODY, S. (1963) On a limit theorem in the theory of mass service. Ukrain. Mat. Zh. 15 76–79 (in Russian).

    Google Scholar 

  17. BRUNELLE, S. L. (1971) Some inequalities for parallel server queues. Opns. Res. 19 402–413.

    Google Scholar 

  18. BRUNELLE, S. L. (1973) Bounds on the wait in a GI/M/k queue. Man. Sci. 19 773–777.

    Google Scholar 

  19. CHERNOFF, H. (1968) Optimal stochastic control. Sankhya 30 221–252.

    MATH  MathSciNet  Google Scholar 

  20. CHUNG, K. L. (1968) A Course in Probability Theory. Harcourt, Brace and World, New York.

    Google Scholar 

  21. Q,INLAR, E. (1972) Superposition of point processes. Stochastic Point Processes. Ed. P. A. W. Lewis, John Wiley and Sons, New York, 549–606.

    Google Scholar 

  22. COHEN, J. W. (1969) The Single Server Queue. North Holland Publ. Co., Amsterdam.

    Google Scholar 

  23. COHEN, J. W. (1972) Asymptotic relations in queueing theory. Stoch. Proc. Appl. 2 107–124.

    Google Scholar 

  24. CRANE, M. A. (1971) Limit theorems for queues in transportation systems Ph.D. thesis and TR No. 16, Department of Operations Research, Stanford University.

    Google Scholar 

  25. CRANE, M. A. and IGLEHART, D. L. (1973) Simulating stable sto- chastic systems, I-III. J. Assoc. Comput. Mach. To appear.

    Google Scholar 

  26. DALEY, D. J. (1969) The total waiting time in a busy period of a stable single server-queue, I. J. Appl. Prob. 6 550–564.

    MATH  MathSciNet  Google Scholar 

  27. DALEY, D. J. and JACOBS, D. R., Jr. (1969) The total waiting time in a busy period of a stable single-server queue, II. J. Appl. Prob. 6 565–572.

    MATH  MathSciNet  Google Scholar 

  28. DARLING, D. (1956) The maximum of sums of stable random variables. Trans. Amer. Math. Soc. 83 164–169.

    MATH  MathSciNet  Google Scholar 

  29. DISNEY, R. L. (1973) Some Topics in Queueing Network Theory. These proceedings.

    Google Scholar 

  30. DUDLEY, R. M. (1972) Speeds of metric probability convergence. Z. Wahrscheinlichkeitstheorie verw. Geb. 22 323–332.

    MATH  MathSciNet  Google Scholar 

  31. ERDÖS, P. and KAC, M. (1946) On certain limit theorems in the theory of probability. Bull. Amer. Math. Soc. 52 292–302.

    MATH  MathSciNet  Google Scholar 

  32. FAHADY, K. S., QUINE, M. P., and VERE-JONES, D. (1971) Heavy traffic approximations for the Galton-Watson process. Adv. Appl. Prob. 3 282–300.

    MATH  Google Scholar 

  33. FISHMAN, G. (1972a) Output analysis for queueing simulations. TR No. 56, Department of Administrative Sciences, Yale University.

    Google Scholar 

  34. FISHMAN, G. (1972b) Estimation in multiserver queueing simula- tions. TR No. 58, Department of Administrative Sciences, Yale University.

    Google Scholar 

  35. FISHMAN, G. (1973) Statistical analysis in multiserver queue- ing simulations. TR No. 64, Department of Administrative Sciences, Yale University.

    Google Scholar 

  36. FREEDMAN, D. (1971) Brownian Motion and Diffusion. Holden-Day, San Francisco.

    MATH  Google Scholar 

  37. GAFARIAN, A. V., MUNJAL, P. K., and PAHL, J. (1971) An experimental validation of two Boltzmann-type statistical models for multi-lane traffic flow. Transp. Res. 5 211–224.

    Google Scholar 

  38. GAVER, D. P., Jr. (1968) Diffusion approximations and models for certain congestion problems. J. Appl. Prob. 5 607–623.

    MATH  MathSciNet  Google Scholar 

  39. GAVER, D. P., Jr. (1969) Highway delays resulting from flow-stopping incidents. J. Appl. Prob. 6 137–153.

    MATH  MathSciNet  Google Scholar 

  40. GAVER, D. P., Jr. (1971) Analysis of remote terminal backlogs under heavy demand conditions. J. ACM. 18 405–415.

    MATH  MathSciNet  Google Scholar 

  41. GAVER, D. P., Jr. and SHEDLER, G. S. (1973) Processor utilization in multi-programming systems via diffusion approximations. Opns. Res. 21 569–576.

    Google Scholar 

  42. GIMON, J. G. (1967) A continuous time inventory model. TR No. 29, Department of Statistics, Stanford University.

    Google Scholar 

  43. GNEDENKO, B. and KOVALENKO, I. (1968) Introduction to Oueueinq Theory. Israel Program for Scientific Translations, Ltd., Jerusalem.

    Google Scholar 

  44. HARRISON, J. M. (1970) Queueing models for assembly-like systems. Ph.D. thesis and TR No. 7, Department of Operations Research, Stanford University.

    Google Scholar 

  45. HARRISON, J. M. (1971a) Assembly-like queues. Graduate School of Business, Stanford University. (J. Appl. Prob. 10 354–367 )

    Google Scholar 

  46. HARRISON, J. M. (1971b) The heavy traffic approximation for single server queues in series. TR No. 24, Department of Operations Research, Stanford University. (J. Appl. Prob. 10 613–629 )

    Google Scholar 

  47. HARRISON, J. M. (1973) A limit theorem for priority queues in heavy traffic. J. App. Prob. 10. To appear.

    Google Scholar 

  48. HEATHCOTE, C. (1965) Divergent single server queues. Proc. of the Symp. on Congestion Theory. Eds. W. Smith and W. Wilkinson, The University of North Carolina Press, Chapel Hill, 108–129.

    Google Scholar 

  49. HEATHCOTE, C. and WINER, P. (1969) An approximation for the mo- ments of waiting times. Opns. Res. 17 175–186.

    MATH  Google Scholar 

  50. HEYDE, C. C. (1967) A limit theorem for random walks with drift. J. Appl. Prob. 4 144–150.

    MATH  MathSciNet  Google Scholar 

  51. HEYDE, C. C. (1969a) On extended rate of convergence results for the invariance principle. Ann. Math Rtat;ct A, 2178–2179.

    Google Scholar 

  52. HEYDE, C. C. (1969b) On the maximum of sums of random vari- ables and the supremum functional for stable processes. J. Appl. Prob. 6 419–429.

    MATH  MathSciNet  Google Scholar 

  53. HEYDE, C. C. (1970) On some mixing sequences in queueing theory. Opns. Res. 18 312–315.

    MATH  MathSciNet  Google Scholar 

  54. HEYDE, C. C. (1971) On the growth of the maximum queue length in a stable queue. Opns. Res. 19 447–452.

    MATH  MathSciNet  Google Scholar 

  55. HEYDE, C. C. and SCOTT, D. J. (1969) A weak convergence approach to some limit results with mixing which have applications in queueing theory. Australian National University.

    Google Scholar 

  56. HEYMAN, D. P. (1974) An approximation for the busy period of the M/G/1 queue using a diffusion model. J. Appl. Prob. 11. To appear.

    Google Scholar 

  57. HEYMAN, D. P. (1972) Diffusion approximations for congestion models. Bell Telephone Laboratories, Holmdel, New Jersey.

    Google Scholar 

  58. HOOKE, J. A. (1969) Some Limit Theorems for Priority Queues. Ph.D. thesis and TR No. 91, Department of Operations Research, Cornell University.

    Google Scholar 

  59. HOOKE, J. A. (1970) On some limit theorems for the GI/G/1. J. Appl. Prob. 7 634–640.

    MATH  MathSciNet  Google Scholar 

  60. HOOKE, J. A. (1972a) A priority queue with low-priority arriv- als general. Opns. Res. 20 373–380.

    MATH  MathSciNet  Google Scholar 

  61. HOOKE, J. A. (1972b) Some heavy-traffic limit theorems for a priority queue with general arrivals. Opns. Res. 20 381–388.

    MATH  MathSciNet  Google Scholar 

  62. HOOKE, J. A. and PRABHU, N. U. (1971) Priority queues in heavy traffic. Opsearch 8 1–9.

    MathSciNet  Google Scholar 

  63. IGLEHART, D. L. (1965a) Limit diffusion approximations for the many-server queue and the repairman problem. J. Appl. Prob. 2 429–441.

    MATH  MathSciNet  Google Scholar 

  64. IGLEHART, D. L. (1965b) Limit theorems for queues with traffic intensity one. Ann. Math Statist. 36 1437–1449.

    MATH  MathSciNet  Google Scholar 

  65. IGLEHART, D. L. (1967) Diffusion approximations in applied proba- bility. Lectures in Applied Mathematics, Vol. 12: Mathematics of the Decision Sciences, Part 2, 234–254.

    Google Scholar 

  66. IGLEHART, D. L. (1969) Diffusion approximations in collective risk theory. J. Appl. Prob. 6 285–292.

    MATH  MathSciNet  Google Scholar 

  67. IGLEHART, D. L. (1971a) Multiple channel queues in heavy traffic, IV: law of the iterated logarithm. Z. Wahrscheinlichkeitstheorie verw. Geb. 17 168–180.

    MathSciNet  Google Scholar 

  68. IGLEHART, D. L. (1971b) Functional limit theorems for the queue GI/G/1 in light traffic. Adv. Appl. Prob. 3 269–281.

    MATH  MathSciNet  Google Scholar 

  69. IGLEHART, D. L. (1972a) Extreme values in the GI/G/1 queue. Ann. Math. Statist. 43 627–635.

    MATH  MathSciNet  Google Scholar 

  70. IGLEHART, D. L. (1972b) Weak convergence in applied probability. TR No. 26, Department of Operations Research, Stanford University.

    Google Scholar 

  71. IGLEHART, D. L. (1972c) Weak convergence in queueing theory. TR No. 27, Department of Operations Research, Stanford University.

    Google Scholar 

  72. IGLEHART, D. L. (1973a) Weak convergence of compound stochastic processes. Stochastic Processes and Their Applications 1 11–31.

    MATH  MathSciNet  Google Scholar 

  73. IGLEHART, D. L. (1973b) Functional central limit theorems for ran- dom walks conditioned to stay positive. TR No. 28, Department of Operations Research, Stanford University.

    Google Scholar 

  74. IGLEHART, D. L. and KENNEDY, D. P. (1970) Weak convergence of the average of flag processes. J. Appl. Prob. 7 747–753.

    MATH  MathSciNet  Google Scholar 

  75. IGLEHART, D. L. and LEMOINE, A. J. (1972) Approximations for the repairman problem with two repair facilities, I and II. TR No. 266–2, 4, Control Analysis Corporation, Palo Alto, California.

    Google Scholar 

  76. IGLEHART, D. L. and TAYLOR, H. M. (1968) Weak convergence of a sequence of quickest detection problems. Ann. Math. Statist. 39 2149–2153.

    MATH  MathSciNet  Google Scholar 

  77. IGLEHART, D. L. and WHITT, W. (1970a) Multiple channel queues in heavy traffic, I. Adv. Appl. Prob. 2 150–177.

    MATH  MathSciNet  Google Scholar 

  78. IGLEHART, D. L. (1970b) Multiple channel queues in heavy traffic, II: sequences, networks and batches. Adv. Appl. Prob. 2 355–369.

    MATH  MathSciNet  Google Scholar 

  79. JACOBS, D. R., Jr. and SCHACH, S. (1972) Stochastic order relationships between GI/G/k systems. Ann. Math. Statist. 43 1623–1633.

    MATH  MathSciNet  Google Scholar 

  80. KARLIN, S. and McGREGOR, J. (1964) On some stochastic models in genetics. Stochastic Models in Medicine and Biology, ed. J. Gurland, University of Wisconsin Press, Madison, 245–279.

    Google Scholar 

  81. KENDALL, D. G. (1957) Some problems in the theory of dams. J. Roy. Stat. Soc. Ser. B 19 207–212.

    MathSciNet  Google Scholar 

  82. KENDALL, D. G. (1964) Some recent work and further problems in the theory of queues. Theor. Probability Appl. 9 1–13.

    MathSciNet  Google Scholar 

  83. KENNEDY, D. P. (1972a) The continuity of the single server queue. J. Appl. Prob. 9 370–381.

    MATH  Google Scholar 

  84. KENNEDY, D. P. (1972b) Rates of convergence for queues in heavy traffic, I and II. Adv. Appl. Prob. 4 357–391.

    MATH  Google Scholar 

  85. KENNEDY, D. P. (1973) Limit theorems for finite dams. Stochastic Processes and Their Applications 1 269–278.

    MATH  MathSciNet  Google Scholar 

  86. KHINTCHINE, A. Y. (1960) Mathematical Methods in the Theory of Queueing. Griffin, London.

    MATH  Google Scholar 

  87. KINGMAN, J. F. C. (1961) The single server queue in heavy traffic. Proc. Camb. Phil. Soc. 57 902–904.

    MATH  MathSciNet  Google Scholar 

  88. KINGMAN, J. F. C. (1962) On queues in heavy traffic. J. Roy. Statist. Soc., Ser. B 25 383–392.

    MathSciNet  Google Scholar 

  89. KINGMAN, J. F. C. (1965a) The heavy traffic approximation in the theory of queues. Eds. W. Smith and W. Wilkinson, Proc. of the Symp. on Congestion Theory. The University of North Carolina Press, Chapel Hill, 137–159.

    Google Scholar 

  90. KINGMAN, J. F. C. (1965b) Approximations for queues in heavy traffic. Queueing Theory: Recent Developments and Applications. Ed. R. Cruon, Elsevier Publishers, New York.

    Google Scholar 

  91. KINGMAN, J. F. C. (1970) Inequalities in the theory of queues. J. Roy. Stat. Soc., Ser. B 32 102–110.

    MATH  MathSciNet  Google Scholar 

  92. KYPRIANOU, E. K. (1971a) On the quasi-stationary distribution of virtual waiting time in queues with Poisson arrivals. J. Appl. Prob. 8 494–507.

    MATH  MathSciNet  Google Scholar 

  93. KYPRIANOU, E. K. (1971b) The virtual waiting time of the GI/G/1 queue in heavy traffic. Adv. Appl. Prob. 3 249–268.

    MATH  MathSciNet  Google Scholar 

  94. KYPRIANOU, E. K. (1972a) On the quasi-stationary distribution of the GI/M/1 queue. J. Appl. Prob. 9 117–128.

    MATH  MathSciNet  Google Scholar 

  95. KYPRIANOU, E. K. (1972b) The quasi-stationary distributions of queues in heavy traffic. J. Appl. Prob. 9 821–831.

    MATH  MathSciNet  Google Scholar 

  96. LALCHANDANI, A. (1967) Some limit theorems in queueing theory. Ph.D. thesis and TR No. 29, Department of Operations Research, Cornell University.

    Google Scholar 

  97. LE GALL, P. (1962) Les systemes avec ou sans attente et les processus stochastiques. Tome I, Dunod. (in French).

    MATH  Google Scholar 

  98. LEWIS, P. A. W. (1972) Stochastic Point Processes: Statistical Analysis, Theory, and Applications. John Wiley and Sons, New York.

    Google Scholar 

  99. LOULOU, R. J. (1971) Weak convergence for multichannel queues in heavy traffic. Ph.D. thesis and TR No. 71–31, Operations Research Center, College of Engineering, University of California, Berkeley.

    Google Scholar 

  100. LOULOU, R. J. (1973a) Multichannel queues in heavy traffic. J. Appl. Prob. 10. To appear.

    Google Scholar 

  101. LOULOU, R. J. (1973b) On the extension of some heavy-traffic theorems to multiple-channel systems. These proceedings.

    Google Scholar 

  102. LOYNES, R. (1962) The stability of a queue with non-independent interarrival and service times. Proc. Camb. Phil. Soc. 58 497–520.

    MATH  MathSciNet  Google Scholar 

  103. LOYNES, R. (1965) Extreme values in uniformly mixing sta- tionary stochastic processes. Ann. Math. Statist. 36 993–999.

    MATH  MathSciNet  Google Scholar 

  104. LOYNES, R. (1970) Stopping times on Brownian motion: some properties of Root’s construction. Z. Wahrscheinlichkeitstheorie verw. Geb. 16 211–218.

    MATH  MathSciNet  Google Scholar 

  105. MANDL, P. (1968) Analytic Treatment of One-Dimensional Markov Processes. Springer-Verlag, Berlin and New York.

    Google Scholar 

  106. MAXUMDAR, S. (1970) On priority queues in heavy traffic. J. Roy. Stat. Soc. Ser. B. 32 111–114.

    Google Scholar 

  107. NAGAEV, S. V. (1970) On the speed of convergence in a boundary problem, I, II. Theor. Probability Appl. 15 179–199 and 419–441.

    Google Scholar 

  108. NEWELL, G. F. (1965) Approximation methods for queues with application to the fixed-cycle traffic light. SIAM Review 7 223–239.

    MATH  MathSciNet  Google Scholar 

  109. NEWELL, G. F. (1968a) Queues with time-dependent arrival rates; I. The transition through saturation. J. Appl. Prob. 5 436–451.

    MATH  MathSciNet  Google Scholar 

  110. NEWELL, G. F. (1968b) Queues with time-dependent arrival rates; II, The maximum queue and the return to equilibrium. J. Appl. Prob. 5 579–590.

    MATH  MathSciNet  Google Scholar 

  111. NEWELL, G. F. (1968c) Queues with time-dependent arrival rates; III: A mild rush hour. J. Appl. Prob. 5 591–606.

    MATH  MathSciNet  Google Scholar 

  112. NEWELL, G. F. (1971) Applications of Queueing Theory. Chapman and Hall, London.

    Google Scholar 

  113. NEWELL, G. F. (1973) Approximate stochastic behavior of n-server service systems with large n. Lecture Notes in Economics and Mathematical Systems No. 87. Springer-Verlag, Berlin, Heidelberg, and New York.

    Google Scholar 

  114. PARTHASARATHY, K. R. (1967) Probability Measures in Metric Spaces. Academic Press, New York.

    Google Scholar 

  115. PIPES, L. A. (1968) Topics in the hydrodynamic theory of traffic flow. Transp. Res. 2 143–149.

    Google Scholar 

  116. PLEDGER, G. and SERFLING, R. (1971) The waiting time of a vehicle in queue. TR No. M203, Department of Statistics, Florida State University.

    Google Scholar 

  117. PRABHU, N. U. (1968) Some new results in storage theory. J. Appl. Prob. 5 452–460.

    MATH  MathSciNet  Google Scholar 

  118. PRABHU, N. U. (1969) The simple queue in non-equilibrium. Opsearch 6 118–128.

    MathSciNet  Google Scholar 

  119. PRABHU, N. U. (1970a) The queue GI/M/1 with traffic intensity one. Studia Sci. Math. Hung. 5 89–96.

    MATH  MathSciNet  Google Scholar 

  120. PRABHU, N. U. (1970b) Limit theorems for the single server queue with traffic intensity one. J. Appl. Prob. 7 227–233.

    MATH  MathSciNet  Google Scholar 

  121. PRABHU, N. U. and STIDHAM, S., Jr. (1973) Optimal control of queueing systems. These proceedings.

    Google Scholar 

  122. PRESMAN, E. (1965) On the waiting time for many-server queueing systems. Theor. Probability Appl. 10 63–73.

    Google Scholar 

  123. PROHOROV, Yu. (1956) Convergence of random processes and limit theorems in probability theory. Theor. Probability Appl. 1 157–214.

    MathSciNet  Google Scholar 

  124. PROHOROV, Yu. (1963) Transient phenomena in processes of mass service. Litovsk. Mat. Sb. 3 199–205. (in Russian)

    MathSciNet  Google Scholar 

  125. PUTERMAN, M. L. (1972) On the optimal control of diffusion processes. Ph.D. thesis and TR NO. 14, Department of Operations Research, Stanford University.

    Google Scholar 

  126. PUTERMAN, M. L. (1973) A diffusion process model for a produc- tion facility. Graduate School of Business, University of Massachusetts. To appear in Man. Sci.

    Google Scholar 

  127. RATH, J. (1973) Limit theorems for controlled queues. Ph.D. thesis, Department of Operations Research, Stanford University.

    Google Scholar 

  128. ROSENKRANTZ, W. (1967) On rates of convergence for the invariance principle. Trans. Amer. Math. Soc. 129 542–552.

    MATH  MathSciNet  Google Scholar 

  129. ROSS, S. M. (1973) Bounds on the delay distribution in GI/G/1 queues. Operations Research Center, University of California, Berkeley.

    Google Scholar 

  130. SAMADAROV, E. (1963) Service systems in heavy traffic. Theor. Probability Appl. 8 307–309.

    Google Scholar 

  131. SAWYER, S. (1972) Rates of convergence for some functionals in probability. Ann. Math. Statist. 43 273–284.

    MATH  MathSciNet  Google Scholar 

  132. SAWYER, S. (1973a) The Skorohod Representation. Department of Mathematics, Belfer Graduate School, Yeshiva University, New York.

    Google Scholar 

  133. SCHASSBERGER, R. (1970) On the waiting time in the queueing system GI/G/1. Ann. Math. Statist. 41 182–187.

    MATH  MathSciNet  Google Scholar 

  134. SCHASSBERGER, R. (1972) On the work load process in a general preemptive resume priority queue. J. Appl. Prob. 9 588–603.

    MATH  MathSciNet  Google Scholar 

  135. SCHASSBERGER, R. (1973b) Forthcoming book, Springer, Berlin. (in German)

    Google Scholar 

  136. SKOROHOD, A. V. (1965) Studies in the Theory of Random Processes. Addison-Wesley, Reading, Massachusetts.

    Google Scholar 

  137. SMITH, W. L. (1955) Regenerative stochastic processes. Proc. Roy. Soc. A 232 6–31.

    Google Scholar 

  138. SMITH, W. L. (1958) Renewal theory and its ramifications. J. Roy. Stat. Soc., Ser. B. 20 243–302.

    Google Scholar 

  139. SOBEL, M. J. (1973) Optimal operation in queues. These proceedings.

    Google Scholar 

  140. SPEED, T. P. (1973) Some remarks on a result of Blomqvist. J. Appl. Prob. 10 229–232.

    MATH  MathSciNet  Google Scholar 

  141. STIDHAM, S., Jr. (1970) On the optimality of single-server queueing systems. Opns. Res. 18 708–732.

    MATH  MathSciNet  Google Scholar 

  142. STOYAN, D. (1972) Monotonicity in stochastic models. ZAMM 52 23–30. (in German)

    Google Scholar 

  143. STRAF, M. L. (1971) Weak convergence of stochastic processes with several parameters. Proc. Sixth Berk. Symp. Math. Stat. Prob. 2 187–221.

    Google Scholar 

  144. SUZUKI, T. and YOSHIDA, Y. (1970) Inequalities for many-server queues and other queues. J. Opns. Res. Soc. of Japan 13 59–77

    MATH  MathSciNet  Google Scholar 

  145. TAKACS, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. John Wiley and Sons, New York.

    MATH  Google Scholar 

  146. TAKACS, L. (1973) Occupation time problems in the theory of queues. These proceedings.

    Google Scholar 

  147. TOMKO, J. (1972) The rate of convergence in limit theorems for service systems with finite queueing capacity. J. Appl. Prob. 9 87–102.

    MATH  MathSciNet  Google Scholar 

  148. VERVAAT, W. (1972) Functional central limit theorems for processes with positive drift and their inverses. Z. Wahscheinlichkeitstheorie verw. Geb. 23 245–253.

    MATH  MathSciNet  Google Scholar 

  149. VERVAAT, W. (1973) Limit theorems for sample maxima and record values: a review. To appear.

    Google Scholar 

  150. VISKOV, O. (1964) Two asymptotic formulae in the theory of mass service. Theor. Probability Appl. 9 177–178.

    MATH  MathSciNet  Google Scholar 

  151. The probability of loss of calls in-heavy traffic. Theor. Probability Appl. 9 99–104.

    Google Scholar 

  152. WHITT, W. (1968) Weak convergence theorems for queues in heavy traffic. Ph.D. thesis, Department of Operations Research, Cornell University and TR. 2, Department of Operations Research, Stanford University.

    Google Scholar 

  153. WHITT, W. (1970a) Multiple channel queues in heavy traffic, III: random server selection. Adv. Appl. Prob. 2 370–375.

    MATH  MathSciNet  Google Scholar 

  154. WHITT, W. (1970b) A guide to the application of limit theorems for sequences of stochastic processes. Opns. Res. 18 1207–1213.

    Google Scholar 

  155. WHITT, W. (1971a) Weak convergence theorems for priority queues: preemptive-resume discipline. J. Appl. Prob. 8 74–94.

    MATH  MathSciNet  Google Scholar 

  156. WHITT, W. (1971b) Classical limit theorems for queues. Department of Administrative Sciences, Yale University.

    Google Scholar 

  157. WHITT, W. (1972a) Complements to heavy traffic limit theo- rems for the GI/G/1 queue. J. Appl. Prob. 9 185–191.

    MATH  MathSciNet  Google Scholar 

  158. WHITT, W. (1972b) Embedded renewal processes in the GI/G/s queue. J. Appl. Prob. 9 650–658.

    MATH  MathSciNet  Google Scholar 

  159. WHITT, W. (1971e) Heavy traffic limit theorems for priority queues.

    Google Scholar 

  160. WHITT, W. (1972a) Complements to heavy traffic limit theo- rems for the GI/G/1 queue. J. Appl. Prob. 9 185–191.

    MATH  MathSciNet  Google Scholar 

  161. WHITT, W. (1972b) Embedded renewal processes in the GI/G/s queue. J. Appl. Prob. 9 650–658.

    MATH  MathSciNet  Google Scholar 

  162. WHITT, W. (1972c) Continuity of several functions on the function space D. To appear in Ann. Prob.

    Google Scholar 

  163. WHITT, W. (1972d) On the quality of Poisson approximations. To appear in Z. Wahrscheinlichkeitstheorie verw. Geb.

    Google Scholar 

  164. WHITT, W. (1972e) Preservation of rates of convergence under mappings. To appear in Z. Wahr.

    Google Scholar 

  165. WHITT, W. (1973a) A converse to the continuous mapping theorem for composition.

    Google Scholar 

  166. WHITT, W. (1973b) A broad structural analysis of multi- server queueing models.

    Google Scholar 

  167. WHITT, W. (1973c) Deterministic inventory and production models.

    Google Scholar 

  168. WHITT, W. (1973d) A diffusion model for a queue with removable server.

    Google Scholar 

  169. WHITT, W. (1973e) Exponential heavy traffic approximations for multi-server queues.

    Google Scholar 

  170. WICHURA, M. J. (1973) On the functional form of the law of the iterated logarithm for the partial maxima of independent identically distributed random variables. Department of Statistics, University of Chicago.

    Google Scholar 

  171. WORTHINGTON, F. (1967) Ladder processes in continuous time. Ph.D. thesis and TR No. 34, Department of Operations Research, Cornell University.

    Google Scholar 

  172. YU, O. S. (1972) On the steady-state solution of a GI/Ek/r queue with heterogeneous servers. TR No. 38, Department of Operations Research, Stanford University.

    Google Scholar 

  173. YU, O. S. (1973) Stochastic bounding relations for a heterogeneous-server queue with Erlang service times. Stanford Research Institute. Menlo Park, California.

    Google Scholar 

  174. BARTFAI, P. (1970) Limsuperior theorems for the queueing model. Studia Sci. Math. Hung. 5 317–325.

    MathSciNet  Google Scholar 

  175. KÖLLERSTRÖM, J. (1973) Heavy traffic theory for queues with several servers, I. Department of Mathematics, The University of Oxford.

    Google Scholar 

  176. LEMOINE, A. J. (1972) Delayed random walks and limit theorems for generalized single server queues. Ph.D. thesis and Technical Report No. 21, Department of Operations Research, Stanford University.

    Google Scholar 

  177. LEMOINE, A. J. (1973) Limit theorems for generalized single server queues: the exceptional system. Department of Mathematics, Clemson University.

    Google Scholar 

  178. McNEIL, D. R. (1973) Diffusion limits for congestion models. J. Appl. Prob. 10 368–376.

    MATH  MathSciNet  Google Scholar 

  179. STOYAN, D. (1972) Continuity theorem for single-server queues. Math. Operationsforsch. Statist. 3 103–111.

    MathSciNet  Google Scholar 

  180. STOYAN, D. (1973) A continuity theorem for queue size. Bull. Acad. Sci. Polon. To appear.

    Google Scholar 

  181. STOYAN, D. (1974) Some bounds for many-server systems. Math. Operationsforsch. Statist. To appear.

    Google Scholar 

  182. STOYAN, H. (1973) Monotonicity and continuity properties of many server queues. Math. Operationsforsch. Statist. 4 155–163.

    MathSciNet  Google Scholar 

  183. ROLSKI, T. and STOYAN, D. (1973) Two classes of semi-orderings and their application to queueing theory. ZAMM. To appear.

    Google Scholar 

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Authors and Affiliations

  1. Yale University, USA

    Ward Whitt

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  1. Ward Whitt
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  1. Department of Mathematics, Western Michigan University, 49001, Kalamazoo, MI, USA

    A. Bruce Clarke

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Whitt, W. (1974). Heavy Traffic Limit Theorems for Queues: A Survey. In: Clarke, A.B. (eds) Mathematical Methods in Queueing Theory. Lecture Notes in Economics and Mathematical Systems, vol 98. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-80838-8_15

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  • DOI: https://doi.org/10.1007/978-3-642-80838-8_15

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