Lei Ying
Abstract
Mean-field analysis is an analytical method for understanding large-scale stochastic systems such as large-scale data centers and communication networks. The idea is to approximate the stationary distribution of a large-scale stochastic system using the equilibrium point (called the mean-field limit) of a dynamical system (called the mean-field model). This approximation is often justified by proving the weak convergence of stationary distributions to its mean-field limit. Most existing mean-field models concerned the light-traffic regime where the load of the system, denote by ρ, is strictly less than one and is independent of the size of the system. This is because a traditional mean-field model represents the limit of the corresponding stochastic system. Therefore, the load of the mean-field model is ρ= limN→ ∞ ρ(N), where ρ(N) is the load of the stochastic system of size N. Now if ρ(N)→ 1 as N→ ∞ (i.e., in the heavy-traffic regime), then ρ=1. For most systems, the mean-field limits when ρ=1 are trivial and meaningless. To overcome this difficulty of traditional mean-field models, this paper takes a different point of view on mean-field models. Instead of regarding a mean-field model as the limiting system of large-scale stochastic system, it views the equilibrium point of the mean-field model, called a mean-field solution, simply as an approximation of the stationary distribution of the finite-size system. Therefore both mean-field models and solutions can be functions of N. This paper first outlines an analytical method to bound the approximation error based on Stein's method and the perturbation theory. We further present two examples: the M/M/N queueing system and the supermarket model under the power-of-two-choices algorithm. For both applications, the method enables us to characterize the system performance under a broad range of traffic loads. For the supermarket model, this is the first paper that rigorously quantifies the steady-state performance of the-power-of-two-choices in the heavy-traffic regime. These results in the heavy-traffic regime cannot be obtained using the traditional mean-field analysis and the interchange of the limits.
- R. Atar. A diffusion regime with nondegenerate slowdown. Operations Research, 60(2):490--500, 2012. Google Scholar
Digital Library
- A. D. Barbour and L. H. Chen. An Introduction to Stein's Method, volume 4. World Scientific, 2005.Google Scholar
- C. Bordenave, D. Mcdonald, and A. Proutiere. A particle system in interaction with a rapidly varying environment: Mean field limits and applications. Networks and Heterogeneous Media (NHM), 2010.Google ScholarCross Ref
- L. Bortolussi, J. Hillston, D. Latella, and M. Massink. Continuous approximation of collective system behaviour: A tutorial. Perform. Eval., 70(5):317--349, 2013. Google ScholarDigital Library
- A. Braverman and J. G. Dai. High order steady-state diffusion approximation of the Erlang-C system. arXiv preprint arXiv:1602.02866, 2016.Google Scholar
- A. Braverman and J. G. Dai. Stein's method for steady-state diffusion approximations of M/Ph/nGoogle Scholar
- M$ systems. Adv. in Appl. Probab., accepted.Google Scholar
- A. Braverman, J. G. Dai, and J. Feng. Stein's method for steady-state diffusion approximations: An introduction through the Erlang-A and Erlang-C models. Stoch. Syst., 6(2):301--366, 2016.Google ScholarCross Ref
- F. Cecchi, S. C. Borst, J. S. van Leeuwaarden, and P. A. Whiting. Mean-field limits for large-scale random-access networks. arXiv preprint arXiv:1611.09723, 2016.Google Scholar
- A. Chaintreau, J.-Y. Le Boudec, and N. Ristanovic. The age of gossip: spatial mean field regime. In Proc. Ann. ACM SIGMETRICS Conf., pages 109--120, Seattle, Washington, USA, 2009. Google ScholarDigital Library
- J. G. Dai. On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models. Ann. Appl. Probab., 5(1):49--77, 1995.Google ScholarCross Ref
- P. Eschenfeldt and D. Gamarnik. Join the shortest queue with many servers. the heavy traffic asymptotics. arXiv preprint arXiv:1502.00999, 2015.Google Scholar
- P. Eschenfeldt and D. Gamarnik. Supermarket queueing system in the heavy traffic regime. Short queue dynamics. arXiv preprint arXiv:1610.03522, 2016.Google Scholar
- N. Gast and B. Gaujal. A mean field approach for optimization in discrete time. Discrete Event Dynamic Systems, 21(1):63--101, 2011. Google ScholarDigital Library
- P. W. Glynn and A. Zeevi. Bounding stationary expectations of Markov processes. In Markov processes and related topics: a Festschrift for Thomas G. Kurtz, pages 195--214. Institute of Mathematical Statistics, 2008.Google Scholar
- I. Gurvich et al. Diffusion models and steady-state approximations for exponentially ergodic markovian queues. Adv. in Appl. Probab., 24(6):2527--2559, 2014.Google ScholarCross Ref
- S. Halfin and W. Whitt. Heavy-traffic limits for queues with many exponential servers. Operations research, 29(3):567--588, 1981. Google ScholarDigital Library
- H. K. Khalil. Nonlinear systems. Prentice Hall, 2001.Google Scholar
- M. Mitzenmacher. The Power of Two Choices in Randomized Load Balancing. PhD thesis, University of California at Berkeley, 1996. Google ScholarDigital Library
- D. Mukherjee, S. C. Borst, J. S. van Leeuwaarden, and P. A. Whiting. Universality of power-of-d load balancing in many-server systems. arXiv preprint arXiv:1612.00723, 2016.Google Scholar
- R. Srikant and L. Ying. Communication Networks: An Optimization, Control and Stochastic Networks Perspective. Cambridge University Press, 2014. Google ScholarDigital Library
- C. Stein. A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. Sixth Berkeley Symp. Math. Stat. Prob., pages 583--602, 1972.Google Scholar
- C. Stein. Approximate computation of expectations. Lecture Notes-Monograph Series, 7:i--164, 1986.Google Scholar
- A. Stolyar. Pull-based load distribution in large-scale heterogeneous service systems. Queueing Syst., 80(4):341--361, 2015. Google ScholarDigital Library
- A. Stolyar. Tightness of stationary distributions of a flexible-server system in the Halfin-Whitt asymptotic regime. Stoch. Syst., 5(2):239--267, 2015.Google ScholarCross Ref
- J. N. Tsitsiklis and K. Xu. On the power of (even a little) resource pooling. Stoch. Syst., 2(1):1--66, 2012.Google ScholarCross 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(1):20--34, 1996.Google Scholar
- W. Whitt. How multiserver queues scale with growing congestion-dependent demand. Operations Research, 51(4):531--542, 2003. Google ScholarDigital Library
- Q. Xie, X. Dong, Y. Lu, and R. Srikant. Power of d choices for large-scale bin packing: A loss model. In Proc. Ann. ACM SIGMETRICS Conf., 2015. Google ScholarDigital Library
- L. Ying. On the approximation error of mean-field models. In Proc. Ann. ACM SIGMETRICS Conf., Antibes Juan-les-Pins, France, 2016. Google ScholarDigital Library
- L. Ying, R. Srikant, and X. Kang. The power of slightly more than one sample in randomized load balancing. Mathematics of Operations Research, 2017.Google ScholarDigital Library
Index Terms
- Stein's Method for Mean Field Approximations in Light and Heavy Traffic Regimes
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