1932

Abstract

Stochastic processing networks arise as models in manufacturing, telecommunications, transportation, computer systems, the customer service industry, and biochemical reaction networks. Common characteristics of these networks are that they have entities—such as jobs, packets, vehicles, customers, or molecules—that move along routes, wait in buffers, receive processing from various resources, and are subject to the effects of stochastic variability through such quantities as arrival times, processing times, and routing protocols. The mathematical theory of queueing aims to understand, analyze, and control congestion in stochastic processing networks. In this article, we begin by summarizing some of the highlights in the development of the theory of queueing prior to 1990; this includes some exact analysis and development of approximate models for certain queueing networks. We then describe some surprises of the early 1990s and ensuing developments of the past 25 years related to the use of approximate models for analyzing the stability and performance of multiclass queueing networks. We conclude with a description of recent developments for more general stochastic processing networks and point to some open problems.

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2016-06-01
2024-06-25
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Literature Cited

  1. Anderson DF, Kurtz TG. 2011. Continuous-time Markov chain models for chemical reaction networks. Design and Analysis of Biomolecular Circuits H Koeppl, D Densmore, G Setti, M di Bernardo 3–42 New York: Springer [Google Scholar]
  2. Arazi A, Ben-Jacob E, Yechiali U. 2004. Bridging genetic networks and queueing theory. Phys. A Stat. Mech. Appl. 332:585–616 [Google Scholar]
  3. Asmussen S. 2003. Applied Probability and Queues. New York: Springer-Verlag, 2nd ed.. [Google Scholar]
  4. Ata B, Kumar S. 2005. Heavy traffic analysis of open processing networks with complete resource pooling: asymptotic optimality of discrete review policies. Ann. Appl. Probab. 15:331–91 [Google Scholar]
  5. Ata B, Lin W. 2008. Heavy traffic analysis of maximum pressure policies for stochastic processing networks with multiple bottlenecks. Queueing Syst. 59:191–235 [Google Scholar]
  6. Baskett F, Chandy KM, Muntz RR, Palacios FG. 1975. Open, closed and mixed networks of queues with different classes of customers. J. ACM 22:248–60 [Google Scholar]
  7. Bell SL, Williams RJ. 2001. Dynamic scheduling of a system with two parallel servers in heavy traffic with resource pooling: asymptotic optimality of a threshold policy. Ann. Appl. Probab. 11:608–49 [Google Scholar]
  8. Bell SL, Williams RJ. 2005. Dynamic scheduling of a parallel server system in heavy traffic with complete resource pooling: asymptotic optimality of a threshold policy. Electron. J. Probab. 10:1044–115 [Google Scholar]
  9. Bertsekas D, Gallager R. 1992. Data Networks. Englewood Cliffs, NJ: Prentice-Hall [Google Scholar]
  10. Bonald T. 2007. Insensitive traffic models for communication networks. Discret. Event Dyn. Syst. 17:405–21 [Google Scholar]
  11. Bonald T, Massoulié L. 2001. Impact of fairness on Internet performance. Proc. 2001 ACM SIGMETRICS Intl. Conf. Meas. Model. Comput. Syst.82–91 New York: ACM [Google Scholar]
  12. Borovkov A. 1964. Some limit theorems in the theory of mass service, I. Theory Probab. Appl. 9:550–65 [Google Scholar]
  13. Borovkov A. 1965. Some limit theorems in the theory of mass service, II. Theory Probab. Appl. 10:375–400 [Google Scholar]
  14. Bramson M. 1994. Instability of FIFO queueing networks. Ann. Appl. Probab. 4:414–31 [Google Scholar]
  15. Bramson M. 1996a. Convergence to equilibria for fluid models of FIFO queueing networks. Queueing Syst. 22:5–45 [Google Scholar]
  16. Bramson M. 1996b. Convergence to equilibria for fluid models of head-of-the-line proportional processor sharing queueing networks. Queueing Syst. 23:1–26 [Google Scholar]
  17. Bramson M. 1998. State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Syst. 30:89–148 [Google Scholar]
  18. Bramson M. 2008. Stability of Queueing Networks. Lect. Notes Math 1950 Heidelberg: Springer-Verlag [Google Scholar]
  19. Bramson M. 2010. Network stability under max-min fair bandwidth sharing. Ann. Appl. Probab. 20:1126–76 [Google Scholar]
  20. Bramson M, Dai JG. 2001. Heavy traffic limits for some queueing networks. Ann. Appl. Probab. 11:49–90 [Google Scholar]
  21. Bramson M, Williams RJ. 2000. On dynamic scheduling of stochastic networks in heavy traffic and some new results for the workload process. Proc. 39th IEEE Conf. Decis. Control516–521 New York: IEEE [Google Scholar]
  22. Bramson M, Williams RJ. 2003. Two workload properties for Brownian networks. Queueing Syst. 45:191–221 [Google Scholar]
  23. Brown L, Gans N, Mandelbaum A, Sakov A, Shen H. et al. 2005. Statistical analysis of a telephone call center: a queueing science perspective. J. Am. Stat. Assoc. 100:36–50 [Google Scholar]
  24. Budhiraja A, Chen J, Rubenthaler S. 2014. A numerical scheme for invariant distributions of constrained diffusions. Math. Oper. Res. 39:262–89 [Google Scholar]
  25. Budhiraja A, Ghosh AP. 2005. A large deviations approach to asymptotically optimal control of crisscross network in heavy traffic. Ann. Appl. Probab. 15:1887–935 [Google Scholar]
  26. Budhiraja A, Ghosh AP. 2012. Controlled stochastic networks in heavy traffic: convergence of value functions. Ann. Appl. Probab. 22:734–91 [Google Scholar]
  27. Burke PJ. 1956. The output of a queueing system. Oper. Res. 4:699–704 [Google Scholar]
  28. Chen H, Mandelbaum A. 1991. Stochastic discrete flow networks: diffusion approximations and bottlenecks. Ann. Probab. 19:1463–519 [Google Scholar]
  29. Chao X, Miyazawa M, Pinedo M. 1999. Queueing Networks: Customers, Signals and Product Form Solutions Chichester, UK: Wiley [Google Scholar]
  30. Dai JG. 1995. On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models. Ann. Appl. Probab. 5:49–77 [Google Scholar]
  31. Dai JG, He S. 2012. Many server queues with customer abandonment: a survey of fluid and diffusion approximations. J. Syst. Sci. Syst. Eng. 21:1–36 [Google Scholar]
  32. Dai JG, Kurtz TG. 1995. A multiclass station with Markovian feedback in heavy traffic. Math. Oper. Res. 20:721–42 [Google Scholar]
  33. Dai JG, Lin W. 2005. Maximum pressure policies in stochastic processing networks. Oper. Res. 53:197–218 [Google Scholar]
  34. Dai JG, Lin W. 2008. Asymptotic optimality of maximum pressure policies in stochastic processing networks. Ann. Appl. Probab. 18:2239–99 [Google Scholar]
  35. Dai JG, Prabhakar B. 2000. The throughput of data switches with and without speedup. Proc. IEEE INFOCOM 2556–64 New York: IEEE [Google Scholar]
  36. Dai JG, Wang Y. 1993. Nonexistence of Brownian models of certain multiclass queueing networks. Queueing Syst. 13:41–46 [Google Scholar]
  37. de Veciana G, Lee TJ, Konstantopoulos T. 2001. Stability and performance analysis of networks supporting elastic services. IEEE/ACM Trans. Networking 9:2–14 [Google Scholar]
  38. Dieker AB. 2010. Reflected Brownian motion. Encyclopedia of Operations Research and Management Science JJ Cochran, LA Cox Jr., P Kiskinocak, JP Kharoufeh, JC Smith 1–7 New York: Wiley [Google Scholar]
  39. Doytchinov B, Lehoczky J, Shreve S. 2001. Real-time queues in heavy traffic with earliest-deadline-first queue discipline. Ann. Appl. Probab. 11:332–78 [Google Scholar]
  40. Dupuis P, Williams RJ. 1994. Lyapunov functions for semimartingale reflecting Brownian motions. Ann. Prob. 22:680–702 [Google Scholar]
  41. Elgart V, Jia T, Kulkarni RV. 2010. Applications of Little's law to stochastic models of gene expression. Phys. Rev. E 82:021901 [Google Scholar]
  42. Erlang AK. 1917. Solution of some problems in the theory of probabilities of significance in automatic telephone exchanges. Elektroteknikeren 13:5–13 [Google Scholar]
  43. Ganesh A, O'Connell N, Wischik D. 2004. Big Queues Lect. Notes Math 1838 Berlin: Springer-Verlag [Google Scholar]
  44. Gans N, Koole G, Mandelbaum A. 2003. Telephone call centers: tutorial, review, and research prospects. Manuf. Serv. Oper. Manag. 5:79–141 [Google Scholar]
  45. Goss PJE, Peccoud J. 1998. Quantitative modeling of stochastic systems in molecular biology by using stochastic Petri nets. PNAS 95:6750–55 [Google Scholar]
  46. Gromoll HC. 2004. Diffusion approximation for a processor sharing queue in heavy traffic. Ann. Appl. Probab. 14:555–611 [Google Scholar]
  47. Gromoll HC, Kruk Ł. 2007. Heavy traffic limit for a processor sharing queue with soft deadlines. Ann. Appl. Probab. 17:1049–101 [Google Scholar]
  48. Gromoll HC, Kruk L, Puha AL. 2011. Diffusion limits for shortest remaining processing time queues. Stoch. Syst. 1:1–16 [Google Scholar]
  49. Gromoll HC, Williams RJ. 2008. Fluid model for a data network with α-fair bandwidth sharing and general document size distributions: two examples of stability. Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz253–65 Inst. Math. Stat. Collect 4 Beachwood, OH: Inst. Math. Statist. [Google Scholar]
  50. Gromoll HC, Williams RJ. 2009. Fluid limits for networks with bandwidth sharing and general document size distributions. Ann. Appl. Probab. 19:243–80 [Google Scholar]
  51. Harrison JM. 1978. The diffusion approximation for tandem queues in heavy traffic. Adv. Appl. Probab. 10:886–905 [Google Scholar]
  52. Harrison JM. 1996. The BIGSTEP approach to flow management in stochastic processing networks. Stochastic Networks: Theory and Applications FP Kelly, S Zachary, I Ziedins 57–90 R. Stat. Soc. Lect. Note Ser 4 Oxford, UK: Oxford Univ. Press [Google Scholar]
  53. Harrison JM. 1998. Heavy traffic analysis of a system with parallel servers: asymptotic analysis of discrete-review policies. Ann. Appl. Probab. 8:822–48 [Google Scholar]
  54. Harrison JM. 2000. Brownian models of open processing networks: canonical representation of workload. Ann. Appl. Probab. 10:75–103 Correction 13:390–93 [Google Scholar]
  55. Harrison JM. 2002. Stochastic networks and activity analysis. Analytic Methods in Applied Probability: In Memory of Fridrikh Karpelevich Y Suhov 53–76 Providence, RI: Am. Math. Soc. [Google Scholar]
  56. Harrison JM. 2003. A broader view of Brownian networks. Ann. Appl. Probab. 13:1119–50 [Google Scholar]
  57. Harrison JM, López MJ. 1999. Heavy traffic resource pooling in parallel-server systems. Queueing Syst. 33:339–68 [Google Scholar]
  58. Harrison JM, Mandayam CV, Shah D, Yang Y. 2014. Resource sharing networks: overview and an open problem. Stoch. Syst. 4:524–55 [Google Scholar]
  59. Harrison JM, Nguyen V. 1990. The QNET method for two-moment analysis of open queueing networks. Queueing Syst. 6:1–32 [Google Scholar]
  60. Harrison JM, Nguyen V. 1993. Brownian models of multiclass queueing networks: current status and open problems. Queueing Syst. 13:5–40 [Google Scholar]
  61. Harrison JM, Reiman MI. 1981. Reflected Brownian motion on an orthant. Ann. Prob. 9:302–8 [Google Scholar]
  62. Harrison JM, Wein LM. 1989. Scheduling networks of queues: heavy traffic analysis of a simple open network. Queueing Syst. 5:265–80 [Google Scholar]
  63. Hunt P, Kurtz T. 1994. Large loss networks. Stoch. Proc. Appl. 53:363–78 [Google Scholar]
  64. Iglehart DL, Whitt W. 1970a. Multiple channel queues in heavy traffic I. Adv. Appl. Probab. 2:150–77 [Google Scholar]
  65. Iglehart DL, Whitt W. 1970b. Multiple channel queues in heavy traffic II: sequences, networks, and batches. Adv. Appl. Probab. 2:355–69 [Google Scholar]
  66. Jackson JR. 1957. Networks of waiting lines. Oper. Res. 5:518–21 [Google Scholar]
  67. Jackson JR. 1963. Jobshop-like queueing systems. Manag. Sci. 10:131–42 [Google Scholar]
  68. Jia T, Kulkarni RV. 2011. Intrinsic noise in stochastic models of gene expression with molecular memory and bursting. Phys. Rev. Lett. 106:058102 [Google Scholar]
  69. Johnson DP. 1983. Diffusion approximations for optimal filtering of jump processes and for queueing networks PhD Thesis, Univ. Wisconsin [Google Scholar]
  70. Kang WN, Kelly FP, Lee NH, Williams RJ. 2009. State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy. Ann. Appl. Probab. 19:1719–80 [Google Scholar]
  71. Kang WN, Williams RJ. 2012. Diffusion approximation for an input-queued switch operating under a maximum weight matching policy. Stoch. Syst. 2:277–321 [Google Scholar]
  72. Kelly FP. 1979. Reversibility and Stochastic Networks Chichester, UK: Wiley [Google Scholar]
  73. Kelly FP. 1991. Loss networks. Ann. Appl. Probab. 1:319–78 [Google Scholar]
  74. Kelly FP. 1997. Charging and rate control for elastic traffic. Eur. Trans. Telecommun. 8:33–37 [Google Scholar]
  75. Kelly FP, Laws CN. 1993. Dynamic routing in open queueing networks: Brownian models, cut constraints and resource pooling. Queueing Syst. 13:47–86 [Google Scholar]
  76. Kelly FP, Yudovina E. 2014. Stochastic Networks. Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  77. Kendall DG. 1953. Stochastic processes occurring in the theory of queues and their analysis by the method of the imbedded Markov chain. Ann. Math. Stat. 24:338–54 [Google Scholar]
  78. Keslassy I, McKeown N. 2001. Analysis of scheduling algorithms that provide 100% throughput in input-queued switches. Proc. 39th Annu. Allerton Conf. Commun. Control Comput., Monticello, IL [Google Scholar]
  79. Khintchine AY. 1932. Mathematical theory of a stationary queue. Mat. Sb. 39:73–84 [Google Scholar]
  80. Kingman JFC. 1961. The single server queue in heavy traffic. Proc. Camb. Philos. Soc. 57:902–4 [Google Scholar]
  81. Kingman JFC. 1962. On queues in heavy traffic. J. R. Stat. Soc. Ser. B 24:383–92 [Google Scholar]
  82. Kingman JFC. 1965. The heavy traffic approximation in the theory of queues. Proc. Symp. Congest. Theory WL Smith, WE Wilkinson 137–59 Chapel Hill: Univ. North Carolina Press [Google Scholar]
  83. Koole G. 2013. Call Center Optimization Amsterdam: MG Books [Google Scholar]
  84. Koopmans TC. 1951. Activity Analysis of Production and Allocation. New York: Wiley [Google Scholar]
  85. Kruk Ł, Lehoczky J, Ramanan K, Shreve S. 2011. Heavy traffic analysis for EDF queues with reneging. Ann. Appl. Probab. 21:484–545 [Google Scholar]
  86. Kruk Ł, Lehoczky J, Shreve S, Yeung SN. 2004. Earliest-deadline-first service in heavy-traffic acyclic networks. Ann. Appl. Probab. 14:1306–52 [Google Scholar]
  87. Kumar PR, Meyn SP. 1995. Stability of queueing networks and scheduling policies. IEEE Trans. Autom. Control 40:251–60 [Google Scholar]
  88. Kumar PR, Seidman TI. 1990. Dynamic instabilities and stabilization methods in distributed real-time scheduling of manufacturing systems. IEEE Trans. Autom. Control 35:289–98 [Google Scholar]
  89. Kumar S, Kumar PR. 2001. Queueing network models in the design and analysis of semiconductor wafer fabs. IEEE Trans. Robot. Autom. 17:548–61 [Google Scholar]
  90. Kushner HJ. 2001. Heavy Traffic Analysis of Controlled Queueing and Communication Networks New York: Springer-Verlag [Google Scholar]
  91. Kushner HJ, Chen YN. 2000. Optimal control of assignment of jobs to processors under heavy traffic. Stoch. Stoch. Rep. 68:177–228 [Google Scholar]
  92. Kushner HJ, Dupuis P. 2001. Numerical Methods for Stochastic Control Problems in Continuous Time New York: Springer-Verlag, 2nd ed.. [Google Scholar]
  93. Lakshmikantha A, Beck CL, Srikant R. 2004. Connection level stability analysis of the Internet using the sum of squares (SoS) techniques. Proc. 38th Conf. Inf. Sci. Syst., Princeton, NJ [Google Scholar]
  94. Laws CN. 1992. Resource pooling in queueing networks with dynamic routing. Adv. Appl. Probab. 24:699–726 [Google Scholar]
  95. Laws CN, Louth GM. 1990. Dynamic scheduling of a four-station queueing networks. Prob. Eng. Inf. Sci. 4:131–56 [Google Scholar]
  96. Lee NH. 2008. A sufficient condition for stochastic stability of an Internet congestion control model in terms of fluid model stability PhD Thesis, Univ. California, San Diego [Google Scholar]
  97. Levine E, Hwa T. 2007. Stochastic fluctuations in metabolic pathways. PNAS 104:9224–29 [Google Scholar]
  98. Limic V. 2000. On the behavior of LIFO preemptive resume queues in heavy traffic. Electron. Commun. Probab. 5:13–27 [Google Scholar]
  99. Little JDC. 1961. A proof for the queuing formula. Oper. Res. 9:383–87 [Google Scholar]
  100. Lu SH, Kumar PR. 1991. Distributed scheduling based on due dates and buffer priorities. IEEE Trans. Autom. Control 36:1406–16 [Google Scholar]
  101. Maglaras C. 2003. Continuous-review tracking policies for dynamic control of stochastic networks. Queueing Syst. 43:43–80 [Google Scholar]
  102. Maguluri ST, Srikant R. 2015. Queue length behavior in a switch under the maxweight algorithm. arXiv:1503.05872 [math.PR]
  103. Mandelbaum A, Stolyar AL. 2004. Scheduling flexible servers with convex delay costs: heavy traffic optimality of the generalized cμ-rule. Oper. Res. 52:836–55 [Google Scholar]
  104. Martins LF, Shreve SE, Soner HM. 1996. Heavy traffic convergence of a controlled, multi-class queueing system. SIAM J. Control Optim. 34:2133–71 [Google Scholar]
  105. Massoulié L. 2007. Structural properties of proportional fairness: stability and insensitivity. Ann. Appl. Probab. 17:809–39 [Google Scholar]
  106. Massoulié L, Roberts JW. 2000. Bandwidth sharing and admission control for elastic traffic. Telecommun. Syst. 15:185–201 [Google Scholar]
  107. Mather WH, Cookson NA, Hasty J, Tsimring LS, Williams RJ. 2010. Correlation resonance generated by coupled enzymatic processing. Biophys. J. 99:3172–81 [Google Scholar]
  108. Mather WH, Hasty J, Tsimring LS, Williams RJ. 2011. Factorized time-dependent distributions for certain multiclass queueing networks and an application to enzymatic processing networks. Queueing Syst. 69:313–28 [Google Scholar]
  109. McKeown N, Anantharam V, Walrand J. 1996. Achieving 100% throughput in an input-queued switch. Proc. IEEE INFOCOM ‘96296–302 New York: IEEE [Google Scholar]
  110. Meyn S. 2008. Control Techniques for Complex Networks Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  111. Mo J, Walrand J. 2000. Fair end-to-end window-based congestion control. IEEE/ACM Trans. Netw. 8:556–67 [Google Scholar]
  112. Muntz RR. 1972. Poisson departure process and queueing networks IBM Res. Rep. RC 4145, IBM Thomas J. Watson Res. Cent. Yorktown Heights, NY: [Google Scholar]
  113. Paganini F, Tang A, Ferragut A, Lachlan LH. 2012. Network stability under alpha fair bandwidth allocation with general file size distribution. IEEE Trans. Autom. Control 57:579–91 [Google Scholar]
  114. Pesic V, Williams RJ. 2016. Dynamic scheduling for parallel server systems in heavy traffic: graphical structure, decoupled workload matrix and some sufficient conditions for solvability of the Brownian control problem. Stoch. Syst.6 [Google Scholar]
  115. Peterson WP. 1991. A heavy traffic limit theorem for networks of queues with multiple customer types. Math. Oper. Res. 16:90–118 [Google Scholar]
  116. Pollaczek F. 1930. Über eine Aufgabe der Wahrscheinlichkeitstheorie. Math. Z. 32:64–100 [Google Scholar]
  117. Prohorov Y. 1963. Transient phenomena in processes of mass service. Litovsk. Mat. Sb. 3:199–205 [Google Scholar]
  118. Puha AL. 2015. Diffusion limits for shortest remaining processing time queues under nonstandard spatial scaling. Ann. Appl. Probab. 25:3381–404 [Google Scholar]
  119. Reiman MI. 1984a. Open queueing networks in heavy traffic. Math. Oper. Res. 9:441–58 [Google Scholar]
  120. Reiman MI. 1984b. Some diffusion approximations with state space collapse. Modeling and Performance Evaluation Methodology F Baccelli, G Fayolle 209–40 Berlin: Springer [Google Scholar]
  121. Reiman MI. 1988. A multiclass feedback queue in heavy traffic. Adv. Appl. Probab. 20:179–207 [Google Scholar]
  122. Rybko AN, Stolyar AL. 1992. Ergodicity of stochastic processes describing the operation of open queueing networks. Probl. Inf. Transm. 28:199–220 [Google Scholar]
  123. Seidman TI. 1994. ‘First come, first served’ can be unstable!. IEEE Trans. Autom. Control 39:2166–71 [Google Scholar]
  124. Sevastyanov BA. 1957. An ergodic theorem for Markov processes and its application to telephone systems with refusals. Theory Probab. Appl. 2:104–12 [Google Scholar]
  125. Shah D, Tsitsiklis JN, Zhong Y. 2011. Optimal scaling of average queue sizes in an input-queued switch: an open problem. Queueing Syst. 68:375–84 [Google Scholar]
  126. Shah D, Tsitsiklis JN, Zhong Y. 2016. On queue-size scaling for input-queued switches. Stoch. Syst.6 [Google Scholar]
  127. Shah D, Walton NS, Zhong Y. 2014. Optimal queue-size scaling in switched networks. Ann. Appl. Probab. 24:2207–45 [Google Scholar]
  128. Shah D, Wischik D. 2006. Optimal scheduling algorithms for input-queued switches. Proc. IEEE INFOCOM 20061810–20 New York: IEEE [Google Scholar]
  129. Shah D, Wischik D. 2012. Switched networks with maximum weight policies: fluid approximation and multiplicative state space collapse. Ann. Appl. Probab. 22:70–127 [Google Scholar]
  130. Srikant R. 2004. The Mathematics of Internet Congestion Control. Syst. Control Found. Appl Boston: Birkhäuser [Google Scholar]
  131. Stidham S. 1974. A last word on . Oper. Res. 22:417–21 [Google Scholar]
  132. Stolyar AL. 1995. On the stability of multiclass queueing networks: a relaxed sufficient condition via limiting fluid processes. Markov Process. Related Fields 1:491–512 [Google Scholar]
  133. Stolyar AL. 2004. Maxweight scheduling in a generalized switch: state space collapse and equivalent workload minimization in heavy traffic. Ann. Appl. Probab. 14:1–53 [Google Scholar]
  134. Sutton C, Jordan MI. 2011. Bayesian inference for queueing networks and modeling of Internet services. Ann. Appl. Stat. 5:254–82 [Google Scholar]
  135. Tassiulas L, Ephremides A. 1992. Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks. IEEE Trans. Autom. Control 37:1936–48 [Google Scholar]
  136. Taylor LM, Williams RJ. 1993. Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant. Probab. Theory Relat. Fields 96:283–317 [Google Scholar]
  137. Vlasiou M, Zhang J, Zwart B. 2014. Insensitivity of proportional fairness in critically loaded bandwidth sharing networks. arXiv:1411.4841 [math.PR]
  138. Wein L. 1990. Scheduling networks of queues: heavy traffic analysis of a two-station network with controllable inputs. Oper. Res. 38:1065–78 [Google Scholar]
  139. Whitt W. 1971. Weak convergence theorems for priority queues: preemptive-resume discipline. J. Appl. Probab. 8:74–94 [Google Scholar]
  140. Whitt W. 2002. Stochastic Process Limits New York: Springer [Google Scholar]
  141. Whittle P. 1968. Equilibrium distributions for an open migration process. J. Appl. Prob. 5:567–71 [Google Scholar]
  142. Williams RJ. 1995. Semimartingale reflecting Brownian motions in the orthant. Stochastic Networks FP Kelly, RJ Williams 125–37 IMA Vol. Math. Appl. 71 New York: Springer [Google Scholar]
  143. Williams RJ. 1996. On the approximation of queueing networks in heavy traffic. Stochastic Networks: Theory and Applications FP Kelly, S Zachary, I Ziedins 35–56 R. Stat. Soc. Lect. Note Ser 4 Oxford, UK: Oxford Univ. Press [Google Scholar]
  144. Williams RJ. 1998a. An invariance principle for semimartingale reflecting Brownian motions in an orthant. Queueing Syst. 30:5–25 [Google Scholar]
  145. Williams RJ. 1998b. Diffusion approximations for open multiclass queueing networks: Sufficient conditions involving state space collapse. Queueing Syst. 30:27–88 [Google Scholar]
  146. Williams RJ. 1998c. Reflecting diffusions and queueing networks. Proc. Int. Congr. Math., Berlin, Aug. 18–27, Vol. III321–330 [Google Scholar]
  147. Williams RJ. 2000. On dynamic scheduling of a parallel server system with complete resource pooling. Analysis of Communication Networks: Call Centres, Traffic and Performance DR McDonald, SRE Turner 49–71 Toronto: Fields Inst. Commun. [Google Scholar]
  148. Yao DD. 1994. Stochastic Modeling and Analysis of Manufacturing Systems. Springer Ser. Oper. Res New York: Springer [Google Scholar]
  149. Ye H, Yao DD. 2012. A stochastic network under proportional fair resource control—diffusion limit with multiple bottlenecks. Oper. Res. 60:716–38 [Google Scholar]
  150. Zachary S. 2007. A note on insensitivity in stochastic networks. J. Appl. Prob. 44:238–48 [Google Scholar]
  151. Zachary S, Ziedins I. 2002. A refinement of the Hunt-Kurtz theory of large loss networks, with an application to virtual partitioning. Ann. Appl. Probab. 12:1–22 [Google Scholar]
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