The Annals of Applied Probability

Supermarket model on graphs

Amarjit Budhiraja, Debankur Mukherjee, and Ruoyu Wu

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We consider a variation of the supermarket model in which the servers can communicate with their neighbors and where the neighborhood relationships are described in terms of a suitable graph. Tasks with unit-exponential service time distributions arrive at each vertex as independent Poisson processes with rate $\lambda$, and each task is irrevocably assigned to the shortest queue among the one it first appears and its $d-1$ randomly selected neighbors. This model has been extensively studied when the underlying graph is a clique in which case it reduces to the well-known power-of-$d$ scheme. In particular, results of Mitzenmacher (1996) and Vvedenskaya et al. (1996) show that as the size of the clique gets large, the occupancy process associated with the queue-lengths at the various servers converges to a deterministic limit described by an infinite system of ordinary differential equations (ODE). In this work, we consider settings where the underlying graph need not be a clique and is allowed to be suitably sparse. We show that if the minimum degree approaches infinity (however slowly) as the number of servers $N$ approaches infinity, and the ratio between the maximum degree and the minimum degree in each connected component approaches $1$ uniformly, the occupancy process converges to the same system of ODE as the classical supermarket model. In particular, the asymptotic behavior of the occupancy process is insensitive to the precise network topology. We also study the case where the graph sequence is random, with the $N$th graph given as an Erdős–Rényi random graph on $N$ vertices with average degree $c(N)$. Annealed convergence of the occupancy process to the same deterministic limit is established under the condition $c(N)\to\infty$, and under a stronger condition $c(N)/\ln N\to\infty$, convergence (in probability) is shown for almost every realization of the random graph.

Article information

Ann. Appl. Probab., Volume 29, Number 3 (2019), 1740-1777.

Received: December 2017
Revised: September 2018
First available in Project Euclid: 19 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60K25: Queueing theory [See also 68M20, 90B22] 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx]
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Load balancing on network power-of-d scheme functional limit theorems many-server asymptotics asymptotic decoupling McKean–Vlasov process


Budhiraja, Amarjit; Mukherjee, Debankur; Wu, Ruoyu. Supermarket model on graphs. Ann. Appl. Probab. 29 (2019), no. 3, 1740--1777. doi:10.1214/18-AAP1437.

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  • [1] Aghajani, R. and Ramanan, K. (2017). The hydrodynamic limit of a randomized load balancing network. Available at arXiv:1707.02005.
  • [2] Azar, Y., Broder, A. Z., Karlin, A. R. and Upfal, E. (1994). Balanced allocations. In Proc. STOC ’94 593–602.
  • [3] Bhamidi, S., Budhiraja, A. and Wu, R. (2018). Weakly interacting particle systems on inhomogeneous random graphs. Stochastic Process. Appl. To appear. Available at arXiv:1612.00801.
  • [4] Bramson, M., Lu, Y. and Prabhakar, B. (2012). Asymptotic independence of queues under randomized load balancing. Queueing Syst. 71 247–292.
  • [5] Bramson, M., Lu, Y. and Prabhakar, B. (2013). Decay of tails at equilibrium for FIFO join the shortest queue networks. Ann. Appl. Probab. 23 1841–1878.
  • [6] Budhiraja, A. and Friedlander, E. (2017). Diffusion approximations for load balancing mechanisms in cloud storage systems. Available at arXiv:1706.09914.
  • [7] Chung, F. and Lu, L. (2006). Complex Graphs and Networks. CBMS Regional Conference Series in Mathematics 107. Amer. Math. Soc., Providence, RI.
  • [8] Delattre, S., Giacomin, G. and Luçon, E. (2016). A note on dynamical models on random graphs and Fokker–Planck equations. J. Stat. Phys. 165 785–798.
  • [9] Eschenfeldt, P. C. and Gamarnik, D. (2017). Supermarket queueing system in the heavy traffic regime. Short queue dynamics. Available at arXiv:1610.03522.
  • [10] Fricker, C. and Gast, N. (2016). Incentives and redistribution in homogeneous bike-sharing systems with stations of finite capacity. EURO J. Transp. Logist. 5 261–291.
  • [11] Gast, N. (2015). The power of two choices on graphs: The pair-approximation is accurate. In Proc. MAMA Workshop 2015 69–71.
  • [12] Graham, C. (2005). Functional central limit theorems for a large network in which customers join the shortest of several queues. Probab. Theory Related Fields 131 97–120.
  • [13] Kenthapadi, K. and Panigrahy, R. (2006). Balanced allocation on graphs. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms 434–443. ACM, New York.
  • [14] Kolokoltsov, V. N. (2010). Nonlinear Markov Processes and Kinetic Equations. Cambridge Tracts in Mathematics 182. Cambridge Univ. Press, Cambridge.
  • [15] Kurtz, T. G. and Xiong, J. (1999). Particle representations for a class of nonlinear SPDEs. Stochastic Process. Appl. 83 103–126.
  • [16] Luczak, M. J. and McDiarmid, C. (2006). On the maximum queue length in the supermarket model. Ann. Probab. 34 493–527.
  • [17] Luczak, M. J. and Norris, J. (2005). Strong approximation for the supermarket model. Ann. Appl. Probab. 15 2038–2061.
  • [18] Mitzenmacher, M. (2001). The power of two choices in randomized load balancing. IEEE Trans. Parallel Distrib. Syst. 12 1094–1104.
  • [19] Mitzenmacher, M., Prabhakar, B. and Shah, D. (2002). Load balancing with memory. In Proc. FOCS ’02 799–808.
  • [20] Mitzenmacher, M. D. (1996). The power of two choices in randomized load balancing. Ph.D. thesis, Univ. California, Berkeley.
  • [21] Mukherjee, D., Borst, S. C. and Van Leeuwaarden, J. S. H. (2018). Asymptotically optimal load balancing topologies. Proc. ACM Meas. Anal. Comput. Syst. 2 1–29.
  • [22] Mukherjee, D., Borst, S. C., van Leeuwaarden, J. S. H. and Whiting, P. A. (2018). Universality of power-of-d load balancing in many-server systems. Stoch. Syst. To appear. Available at arXiv:1612.00723.
  • [23] Peres, Y., Talwar, K. and Wieder, U. (2015). Graphical balanced allocations and the $(1+\beta)$-choice process. Random Structures Algorithms 47 760–775.
  • [24] Sznitman, A.-S. (1991). Topics in propagation of chaos. In École D’Été de Probabilités de Saint-Flour XIX—1989. Lecture Notes in Math. 1464 165–251. Springer, Berlin.
  • [25] Tsitsiklis, J. N. and Xu, K. (2012). On the power of (even a little) resource pooling. Stoch. Syst. 2 1–66.
  • [26] Tsitsiklis, J. N. and Xu, K. (2017). Flexible queueing architectures. Oper. Res. 65 1398–1413.
  • [27] Turner, S. R. E. (1998). The effect of increasing routing choice on resource pooling. Probab. Engrg. Inform. Sci. 12 109–124.
  • [28] Van der Boor, M., Borst, S. C., van Leeuwaarden, J. S. H. and Mukherjee, D. (2018). Scalable load balancing in networked systems: Universality properties and stochastic coupling methods. In Proc. ICM ’18.
  • [29] Vvedenskaya, N. D., Dobrushin, R. L. and Karpelevich, F. I. (1996). A queueing system with a choice of the shorter of two queues—An asymptotic approach. Problemy Peredachi Informatsii 32 20–34.
  • [30] Wang, W., Zhu, K., Ying, L., Tan, J. and Zhang, L. (2016). MapTask scheduling in MapReduce with data locality: Throughput and heavy-traffic optimality. IEEE/ACM Trans. Netw. 24 190–203.
  • [31] Wieder, U. (2016). Hashing, load balancing and multiple choice. Found. Trends Theor. Comput. Sci. 12 276–379.
  • [32] Xie, Q., Yekkehkhany, A. and Lu, Y. (2016). Scheduling with multi-level data locality: Throughput and heavy-traffic optimality. In Proc. INFOCOM ’16 1–9.
  • [33] Ying, L. (2017). Stein’s method for mean field approximations in light and heavy traffic regimes. Proc. ACM Meas. Anal. Comput. Syst. 1 12.