Abstract
Mean field modeling is a popular approach to assess the performance of large scale computer systems. The evolution of many mean field models is characterized by a set of ordinary differential equations that have a unique fixed point. In order to prove that this unique fixed point corresponds to the limit of the stationary measures of the finite systems, the unique fixed point must be a global attractor. While global attraction was established for various systems in case of exponential job sizes, it is often unclear whether these proof techniques can be generalized to non-exponential job sizes. In this paper we show how simple monotonicity arguments can be used to prove global attraction for a broad class of ordinary differential equations that capture the evolution of mean field models with hyperexponential job sizes. This class includes both existing as well as previously unstudied load balancing schemes and can be used for systems with either finite or infinite buffers. The main novelty of the approach exists in using a Coxian representation for the hyperexponential job sizes and a partial order that is stronger than the componentwise partial order used in the exponential case.
- A. Karthik A. Mukhopadhyay and R. R. Mazumdar. Randomized assignment of jobs to servers in heterogeneous clusters of shared servers for low delay. Stochastic Systems, 6(1):90 -- 131, 2016.Google Scholar
Cross Ref
- M. Benaim and J. Le Boudec. On mean field convergence and stationary regime. CoRR, abs/1111.5710, Nov 24 2011.Google Scholar
- N. P. Bhatia and G. P. Szegö. Stability theory of dynamical systems. Springer Science & Business Media, 2002.Google Scholar
- A. Bobbio, A. Horváth, and M. Telek. Matching three moments with minimal acyclic phase type distributions. Stochastic Models, 21(2--3):303--326, 2005.Google Scholar
- M. Bramson, Y. Lu, and B. Prabhakar. Asymptotic independence of queues under randomized load balancing. Queueing Syst., 71(3):247--292, 2012. Google Scholar
Digital Library
- A. Braverman, JG Dai, and J. Feng. Stein's method for steady-state diffusion approximations: an introduction through the erlang-a and erlang-c models. Stochastic Systems, 6(2):301--366, 2017.Google Scholar
Cross Ref
- F Cecchi, SC Borst, and JSH van Leeuwaarden. Mean-field analysis of ultra-dense csma networks. ACM SIGMETRICS Performance Evaluation Review, 43(2):13--15, 2015. Google Scholar
Digital Library
- A. Cumani. On the canonical representation of homogeneous markov processes modelling failure - time distributions. Microelectronics Reliability, 22(3):583 -- 602, 1982.Google Scholar
Cross Ref
- S.N. Ethier and T.C. Kurtz. Markov processes: characterization and convergence. Wiley, 1986.Google Scholar
Cross Ref
- DG Feitelson. Workload Modeling for Computer Systems Performance Evaluation. Cambridge University Press, New York, NY, USA, 1st edition, 2015. Google Scholar
Digital Library
- A. Feldmann and W. Whitt. Fitting mixtures of exponentials to long-tail distributions to analyze network performance models. Performance Evaluation, 31(3):245 -- 279, 1998. Google Scholar
Digital Library
- A. Ganesh, S. Lilienthal, D. Manjunath, A. Proutiere, and F. Simatos. Load balancing via random local search in closed and open systems. SIGMETRICS Perform. Eval. Rev., 38(1):287--298, June 2010. Google Scholar
Digital Library
- N. Gast. Expected values estimated via mean-field approximation are 1/n-accurate. Proc. ACM Meas. Anal. Comput. Syst., 1(1):17:1--17:26, June 2017. Google Scholar
Digital Library
- N. Gast and B. Gaujal. A mean field model of work stealing in large-scale systems. SIGMETRICS Perform. Eval. Rev., 38(1):13--24, June 2010.Google Scholar
Digital Library
- N. Gast and B. Van Houdt. Transient and steady-state regime of a family of list-based cache replacement algorithms. In Proceedings of the 2015 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems, pages 123--136. ACM, 2015. Google Scholar
Digital Library
- REA Khayari, R. Sadre, and B. R. Haverkort. Fitting world-wide web request traces with the EM-algorithm. Performance Evaluation, 52(2):175--191, 2003. Google Scholar
Digital Library
- T. Kurtz. Approximation of population processes. Society for Industrial and Applied Mathematics, 1981.Google Scholar
Cross Ref
- M. Lin, B. Fan, J.C.S. Lui, and D. Chiu. Stochastic analysis of file-swarming systems. Performance Evaluation, 64(9):856--875, 2007. Google Scholar
Digital Library
- J. B. Martin and Yu. M. Suhov. Fast jackson networks. Ann. Appl. Probab., 9(3):854--870, 08 1999.Google Scholar
Cross Ref
- L. Massoulié and M. Vojnović. Coupon replication systems. SIGMETRICS Perform. Eval. Rev., 33(1):2--13, June 2005. Google Scholar
Digital Library
- W. Minnebo and B. Van Houdt. A fair comparison of pull and push strategies in large distributed networks. IEEE/ACM Transactions on Networking, 22:996--1006, 2014. Google Scholar
Digital Library
- M. Mitzenmacher. The power of two choices in randomized load balancing. IEEE Trans. Parallel Distrib. Syst., 12:1094--1104, October 2001. Google Scholar
Digital Library
- C. O'Cinneide. On non-uniqueness of representations of phase-type distributions. Communications in Statistics. Stochastic Models, 5(2):247--259, 1989.Google Scholar
Cross Ref
- Takayuki Osogami and Mor Harchol-Balter. Closed form solutions for mapping general distributions to quasi-minimal ph distributions. Perform. Eval., 63(6):524--552, June 2006.Google Scholar
Digital Library
- A. Riska, V. Diev, and E. Smirni. An EM-based technique for approximating long-tailed data sets with ph distributions. Performance Evaluation, 55(1):147 -- 164, 2004. Google Scholar
Digital Library
- Grégory Roth and William H Sandholm. Stochastic approximations with constant step size and differential inclusions. SIAM Journal on Control and Optimization, 51(1):525--555, 2013.Google Scholar
Digital Library
- D. Starobinski and M. Sidi. Modeling and analysis of power-tail distributions via classical teletraffic methods. Queueing Systems, 36(1--3):243--267, 2000.Google Scholar
- Dietrich Stoyan and Daryl J Daley. Comparison methods for queues and other stochastic models. JOHN WILEY & SONS, INC., 605 THIRD AVE., NEW YORK, NY 10158, USA, 1983, 1983.Google Scholar
- H.C. Tijms. Stochastic models: an algorithmic approach. Wiley series in probability and mathematical statistics. John Wiley & Sons, 1994.Google Scholar
- B. Van Houdt. A mean field model for a class of garbage collection algorithms in flash-based solid state drives. ACM SIGMETRICS Perform. Eval. Rev., 41(1):191--202, 2013. Google Scholar
Digital Library
- B. Van Houdt. Performance of garbage collection algorithms for flash-based solid state drives with hot/cold data. Perform. Eval., 70(10):692--703, 2013. Google Scholar
Digital Library
- B. Van Houdt. Randomized work stealing versus sharing in large-scale systems with non-exponential job sizes. arXiv preprint, 2018. arXiv:1810.13186.Google Scholar
- T. Vasantam, A. Mukhopadhyay, and R. R. Mazumdar. Mean-field analysis of loss models with mixed-erlang distributions under power-of-d routing. In 2017 29th International Teletraffic Congress (ITC 29), volume 1, pages 250--258, Sept 2017.Google Scholar
Cross Ref
- N.D. Vvedenskaya, R.L. Dobrushin, and F.I. Karpelevich. Queueing system with selection of the shortest of two queues: an asymptotic approach. Problemy Peredachi Informatsii, 32:15--27, 1996.Google Scholar
- W. Whitt. Approximating a point process by a renewal process, i: Two basic methods. Oper. Res., 30(1):125--147, February 1982. Google Scholar
Digital Library
- Q. Xie, X. Dong, Y. Lu, and R. Srikant. Power of d choices for large-scale bin packing: A loss model. SIGMETRICS Perform. Eval. Rev., 43(1):321--334, June 2015. Google Scholar
Digital Library
- L. Ying. On the approximation error of mean-field models. In Proceedings of the 2016 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Science, SIGMETRICS '16, pages 285--297, New York, NY, USA, 2016. ACM. Google Scholar
Digital Library
- L. Ying, R. Srikant, and X. Kang. The power of slightly more than one sample in randomized load balancing. In 2015 IEEE Conference on Computer Communications (INFOCOM), pages 1131--1139, April 2015.Google Scholar
Cross Ref
Index Terms
Global Attraction of ODE-based Mean Field Models with Hyperexponential Job Sizes
Recommendations
Mean Field Analysis of Join-Below-Threshold Load Balancing for Resource Sharing Servers
SIGMETRICSLoad balancing plays a crucial role in many large scale computer systems. Much prior work has focused on systems with First-Come-First-Served (FCFS) servers. However, servers in practical systems are more complicated. They serve multiple jobs at once, ...
Global Attraction of ODE-based Mean Field Models with Hyperexponential Job Sizes
SIGMETRICS '19: Abstracts of the 2019 SIGMETRICS/Performance Joint International Conference on Measurement and Modeling of Computer SystemsMean field modeling is a popular approach to assess the performance of large scale computer systems. The evolution of many mean field models is characterized by a set of ordinary differential equations that have a unique fixed point. In order to prove ...
Global Attraction of ODE-based Mean Field Models with Hyperexponential Job Sizes
Mean field modeling is a popular approach to assess the performance of large scale computer systems. The evolution of many mean field models is characterized by a set of ordinary differential equations that have a unique fixed point. In order to prove ...






Comments