47
Views
0
CrossRef citations to date
0
Altmetric
Research Article

On the law of iterated logarithm for extreme queue length in an open queueing network

&
Pages 220-235 | Received 18 Apr 2020, Accepted 06 Aug 2021, Published online: 06 Sep 2021

References

  • S. Asmussen, Extreme value theory for queues via cycle maxima, Extremes 1(2) (1998), pp. 137–168.
  • A. Berger and W. Whitt, Maximum values in queueing processes, Probab. Eng. Informational Sci. 9 (1995), pp. 375–409.
  • P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968.
  • P. Billingsley, Convergence of Probability Measures, 2nd ed., Wiley, New York, 1999.
  • A. Borovkov, Stochastic Processes in Queueing Theory, Nauka, Moscow, 1972. p. 367 (in Russian).
  • A. Borovkov, Asymptotic Methods in Theory of Queues, Nauka, Moscow, 1980. p. 352 (in Russian).
  • A.A. Borovkov, Limit theorems for queueing networks, Theory Probab. Appl. 31 (1986), pp. 413–427.
  • A. Braverman, J.G. Dai and M. Miyazawa, Heavy traffic approximation for the stationary distribution of a generalized Jackson network: the BAR approach, Stoch. Syst. 7 (1) (2017), pp. 143–196.
  • J. Cohen, Asymptotic relations in queueing theory, Stoch. Processes Appl. 1 (1973), pp. 107–124.
  • G. Falin and S. Yu, Singularly perturbed equations and an asymptotic study of the stationary characteristics of systems with repeated calls, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 5 (7-10) (1988), pp. 103.
  • D. Gamarnik and A.L. Stolyar, Multiclass multi-server queueing system in the Halfin-Whitt heavy traffic regime: asymptotics of the stationary distribution, Queueing Syst. 71 (1-2) (2012), pp. 25–51.
  • P.W. Glynn and W. Whitt, Heavy-traffic extreme-value limits for queues, Oper. Res. Lett. 18 (1995), pp. 107–111.
  • I. Gurvich, Validity of Heavy-Traffic Steady-State approximations in multiclass queueing networks: the case of Queue-Ratio disciplines, Math. Oper. Res. 39 (1) (2013), pp. 121–162.
  • J. Huang and I. Gurvich, Beyond Heavy-Traffic regimes: universal bounds and controls for the Single-Server queue, Oper. Res. 66(4) (2018), pp. 1168–1188.
  • D.L. Iglehart, Extreme values in the GI/G/1 queue, Ann. Math. Stat. 43(1972), pp. 627–635.
  • J.R. Jackson, Jobshop-like queueing systems, Manage. Sci. 10(1963), pp. 131–142.
  • F.P. Kelly, Reversibility and Stochastic Networks, Wiley, New York, 1987.
  • P. Konstantopoulps and J. Walrand, Stationarity and stability of fork-join networks, J. Appl. Probab.26(1989), pp. 604–614.
  • P. Konstantopoulps and J. Walrand, On the ergodicity of networks of -/GI/l/N queues, Adv. Appl. Probab. 22(1990), pp. 263–267.
  • C. Kuo-Hwa. Extreme and high-level sojourns of the single server queue in heavy traffic, Queueing Syst.: Theory Appl. 27 (1-2) (1997), pp. 17–35.
  • S.T. Maguluri, S.K. Burle and R. Srikant, Optimal Heavy-Traffic Queue Length Scaling in an Incompletely Saturated Switch, SIGMETRICS '16 Proceedings of the 2016 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Science, (2016) pp. 13 – 24.
  • M.G. Markakis, E. Modiano and J.N. Tsitsiklis, Max-Weight scheduling in queueing networks with Heavy-Tailed traffic, IEEE/ACM Trans. Networking 22(1) (2014), pp. 257–270.
  • W. McCormick and P.Y. Sung, Approximating the distribution of the maximum queue length for M/M/s queues, Queueing and related models, Oxford Statist. Sci. Ser., Vol. 9, Oxford Univ. Press, New York, 1992, pp. 240–261.
  • S. Minkevičius, Compound transient phenomena in multiphase queueing systems, Lit. Mat. J.37(1999), pp. 392–401.
  • S. Minkevičius, On a law of the iterated logarithm in multiserver open queueing networks, Stochastics 86(2014), pp. 46–59.
  • S. Minkevičius and V. Dolgopolov, A mixed-component open Jackson network in heavy traffic, submitted, Informatica. 2019.
  • M.I. Reiman, Open queueing networks in heavy traffic, Math. Oper. Res. 9(1984), pp. 441–459.
  • L.L. Sakalauskas and S. Minkevičius, On the law of the iterated logarithm in open queueing networks, Eur. J. Oper. Res. 120(2000), pp. 632–640.
  • R. Serfozo, Extreme values of birth and death processes and queues, Stoch. Processes Appl. 27(1988), pp. 291–306.
  • R. Serfozo, Extreme values of queue length in M/G/1 and GI/M/1 systems, Math. Oper. Res. 13(2) (2000), pp. 349–357.
  • V. Strassen, An invariance principle for the law of the iterated logarithm, Zeit. Fur Wahr. 3(1964), pp. 211–226.
  • P.Y. Sung, Asymptotic distributions of maximum queue lengths for M/G/1 and GI/M/1 systems, J. Korean Statist. Soc. 24(1) (1994), pp. 19–29.
  • W. Wang, S.T. Maguluri, R. Srikant and L. Ying, Heavy-Traffic delay insensitivity in Connection-Level models of data transfer with proportionally fair bandwidth sharing, ACM SIGMETRICS Perform. Evaluation Rev. 45(3) (2017), pp. 232–245.
  • W. Wang, K. Zhu, L. Ying, J. Tan and L. Zhang, MapTask scheduling in map reduce with data locality: throughput and heavy-traffic optimality, IEEE/ACM Trans. Networking 24(1) (2016), pp. 190–203.
  • W Whitt, Heavy traffic limit theorems for queues: a survey. In: Lecture Notes in Economics and Mathematical Systems Vol. 98, Springer-Verlag, Berlin, Heidelberg, New York, 1974.
  • W. Whitt and W. You, Heavy-Traffic limit of the GI/GI/1 stationary departure process and its variance function, Stoch. Syst. 8(2) (2018), pp. 91–165.
  • R.J. Williams, Stochastic processing networks, Annu. Rev. Stat. Appl. 3(2016), pp. 323–345.
  • Y. Wu, L. Bui and R. Johari, Heavy traffic approximation of equilibria in resource sharing games, IEEE J. Selected Areas Commun. 30(11) (2013), pp. 2200–2209.
  • X. Zhou, J. Tan and N. Shroff, Flexible load balancing with multidimensional state-space collapse: throughput and heavy-traffic delay optimality, Perform. Evaluation 127–128(2018), pp. 176–193.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.