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Analysis of Markovian Arrivals and Service with Applications to Intermittent Overload (2405.04102v4)

Published 7 May 2024 in cs.PF and math.PR

Abstract: In many important real-world queueing settings, arrival and service rates fluctuate over time. We consider the MAMS system, where the arrival and service rates each vary according to an arbitrary finite-state Markov chain, allowing intermittent overload to be modeled. This model has been extensively studied, and we derive results matching those found in the literature via a somewhat novel framework. We derive a characterization of mean queue length in the MAMS system, with explicit bounds for all arrival and service chains at all loads, using our new framework. Our bounds are tight in heavy traffic. We prove even stronger bounds for the important special case of two-level arrivals with intermittent overload. Our framework is based around the concepts of relative arrivals and relative completions, which have previously been used in studying the MAMS system, under different names. These quantities allow us to tractably capture the transient correlational effect of the arrival and service processes on the mean queue length.

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References (34)
  1. An efficient periodic resource supply model for workloads with transient overloads. In 2013 25th Euromicro Conference on Real-Time Systems, pages 249–258, 2013. doi: 10.1109/ECRTS.2013.34.
  2. Queues with time-varying arrivals and inspections with applications to hospital discharge policies. Operations Research, 65(2):469–495, 2017. doi: 10.1287/opre.2016.1536.
  3. Inho Cho. Mitigating Compute Congestion for Low Latency Datacenter RPCs. PhD thesis, Massachusetts Institute of Technology, 2023.
  4. Asymptotically tight steady-state queue length bounds implied by drift conditions. Queueing Systems, 72:311–359, 2012.
  5. A transient overload generator for web servers. In 2008 IEEE International Performance, Computing and Communications Conference, pages 119–126, 2008. doi: 10.1109/PCCC.2008.4745140.
  6. Autoscale: Dynamic, robust capacity management for multi-tier data centers. ACM Trans. Comput. Syst., 30(4), nov 2012. ISSN 0734-2071. doi: 10.1145/2382553.2382556. URL https://doi.org/10.1145/2382553.2382556.
  7. Bounding stationary expectations of Markov processes. Markov processes and related topics: a Festschrift for Thomas G. Kurtz, 4:195–214, 2008.
  8. The RESET and MARC techniques, with application to multiserver-job analysis. Performance Evaluation, 162:102378, 2023. ISSN 0166-5316. doi: https://doi.org/10.1016/j.peva.2023.102378. URL https://www.sciencedirect.com/science/article/pii/S0166531623000482.
  9. Fundamental characteristics of queues with fluctuating load. In Proceedings of the Joint International Conference on Measurement and Modeling of Computer Systems, SIGMETRICS ’06/Performance ’06, page 203–215, New York, NY, USA, 2006. Association for Computing Machinery. ISBN 1595933190. doi: 10.1145/1140277.1140301. URL https://doi.org/10.1145/1140277.1140301.
  10. Madhav Erraguntla Guy L. Curry, Hiram Moya and Amarnath Banerjee. Transient queueing analysis for emergency hospital management. IISE Transactions on Healthcare Systems Engineering, 12(1):36–51, 2022. doi: 10.1080/24725579.2021.1933655.
  11. Bruce Hajek. Hitting-time and occupation-time bounds implied by drift analysis with applications. Advances in Applied Probability, 14(3):502–525, 1982. doi: 10.2307/1426671.
  12. Overload management in RTDBS. In Real-Time Database Systems: Architecture and Techniques, pages 125–139. Springer, 2001.
  13. Mor Harchol-Balter. Performance modeling and design of computer systems: Queueing theory in action. Cambridge University Press, 2013.
  14. Cloud energy-efficient load balancing: A green cloud survey. In 2022 11th International Conference on System Modeling & Advancement in Research Trends (SMART), pages 581–585, 2022. doi: 10.1109/SMART55829.2022.10046686.
  15. Dave Kearns. CPU utilization: When to start getting worried. https://www.networkworld.com/article/2341220/cpu-utilization–when-to-start-getting-worried.html, 2003. Accessed: 2023-10-16.
  16. Introduction to matrix analytic methods in stochastic modeling. Society for Industrial and Applied Mathematics, 1999.
  17. Breaking the Transience-Equilibrium nexus: A new approach to datacenter packet transport. In 18th USENIX Symposium on Networked Systems Design and Implementation (NSDI 21), pages 47–63, 2021.
  18. William A Massey. The analysis of queues with time-varying rates for telecommunication models. Telecommunication Systems, 21:173–204, 2002.
  19. Heavy traffic queue length behaviour in a switch under Markovian arrivals, 2020.
  20. Marcel F. Neuts. The single server queue with poisson input and semi-Markov service times. Journal of Applied Probability, 3(1):202–230, 1966. doi: 10.2307/3212047.
  21. Marcel F Neuts. The M/M/1 queue with randomly varying arrival and service rates. Management Science, 15(4):139–157, 1978.
  22. MF Neuts. Matrix-geometric solutions in stochastic models: An algorithmic approach. Johns Hopkins University Press, 1981.
  23. Peter Purdue. The M/M/1 queue in a Markovian environment. Operations Research, 22(3):562–569, 1974. doi: 10.1287/opre.22.3.562.
  24. Martin L Puterman. Markov decision processes. Wiley Series in Probability and Statistics, 1994.
  25. Auto-scaling web applications in clouds: A taxonomy and survey. 51(4), jul 2018. ISSN 0360-0300. doi: 10.1145/3148149. URL https://doi.org/10.1145/3148149.
  26. V. Ramaswami. The N/G/1 queue and its detailed analysis. Advances in Applied Probability, 12(1):222–261, 1980. doi: 10.2307/1426503.
  27. Web servers under overload: How scheduling can help. ACM Trans. Internet Technol., 6(1):20–52, feb 2006. ISSN 1533-5399. doi: 10.1145/1125274.1125276. URL https://doi.org/10.1145/1125274.1125276.
  28. Performance analysis of time-dependent queueing systems: Survey and classification. Omega, 63:170–189, 2016. ISSN 0305-0483. doi: https://doi.org/10.1016/j.omega.2015.10.013. URL https://www.sciencedirect.com/science/article/pii/S0305048315002170.
  29. Scaling properties of queues with time-varying load processes: Extensions and applications. Probability in the Engineering and Informational Sciences, 36(3):690–731, 2022. doi: 10.1017/S0269964821000048.
  30. Ward Whitt. Time-varying queues. Queueing models and service management, 1(2), 2018.
  31. Ronald W. Wolff. Poisson arrivals see time averages. Operations Research, 30(2):223–231, 1982. doi: 10.1287/opre.30.2.223.
  32. U. Yechiali and P. Naor. Queuing problems with heterogeneous arrivals and service. Operations Research, 19(3):722–734, 1971. doi: 10.1287/opre.19.3.722.
  33. Erhan Çinlar. Queues with semi-Markovian arrivals. Journal of Applied Probability, 4(2):365–379, 1967a. doi: 10.2307/3212030.
  34. Erhan Çinlar. Time dependence of queues with semi-Markovian services. Journal of Applied Probability, 4(2):356–364, 1967b. doi: 10.2307/3212029.
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