The Annals of Applied Probability

The hydrodynamic limit of a randomized load balancing network

Reza Aghajani and Kavita Ramanan

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Randomized load balancing networks arise in a variety of applications, and allow for efficient sharing of resources, while being relatively easy to implement. We consider a network of parallel queues in which incoming jobs with independent and identically distributed service times are assigned to the shortest queue among a subset of $d$ queues chosen uniformly at random, and leave the network on completion of service. Prior work on dynamical properties of this model has focused on the case of exponential service distributions. In this work, we analyze the more realistic case of general service distributions. We first introduce a novel particle representation of the state of the network, and characterize the state dynamics via a countable sequence of interacting stochastic measure-valued evolution equations. Under mild assumptions, we show that the sequence of scaled state processes converges, as the number of servers goes to infinity, to a hydrodynamic limit that is characterized as the unique solution to a countable system of coupled deterministic measure-valued equations. As a simple corollary, we also establish a propagation of chaos result that shows that finite collections of queues are asymptotically independent. The general framework developed here is potentially useful for analyzing a larger class of models arising in diverse fields including biology and materials science.

Article information

Ann. Appl. Probab., Volume 29, Number 4 (2019), 2114-2174.

Received: October 2017
Revised: July 2018
First available in Project Euclid: 23 July 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 90B15: Network models, stochastic 90B22: Queues and service [See also 60K25, 68M20] 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx] 60F17: Functional limit theorems; invariance principles

Randomized load balancing hydrodynamic limit fluid limit power of two choices stochastic network measure-valued processes interacting particle system propagation of chaos


Aghajani, Reza; Ramanan, Kavita. The hydrodynamic limit of a randomized load balancing network. Ann. Appl. Probab. 29 (2019), no. 4, 2114--2174. doi:10.1214/18-AAP1444.

Export citation


  • [1] Agarwal, P. and Ramanan, K. (2018). Invariant states of the hydrodynamic limit of the $\operatorname{SQ}(d)$ model with general service times. Preprint.
  • [2] Aghajani, R. (2017). Infinite-dimensional scaling limits of stochastic networks Ph.D. thesis, Brown Univ., Providence, RI.
  • [3] Aghajani, R., Li, X. and Ramanan, K. (2015). Mean-field dynamics of load-balancing networks with general service distributions. Available at arXiv:1512.05056 [math.PR].
  • [4] Aghajani, R., Li, X. and Ramanan, K. (2017). The PDE method for the analysis of randomized load balancing networks. Proc. ACM Meas. Anal. Comput. Syst. 1 38:1–38:28.
  • [5] Aghajani, R. and Ramanan, K. (2017). The hydrodynamic limit of a randomized load balancing network. Extended version. Available at arXiv:1707.02005 [math.PR].
  • [6] Asmussen, S. (2003). Applied Probability and Queues: Stochastic Modelling and Applied Probability, 2nd ed. Applications of Mathematics (New York) 51. Springer, New York.
  • [7] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [8] Bramson, M. (2011). Stability of join the shortest queue networks. Ann. Appl. Probab. 21 1568–1625.
  • [9] Bramson, M., Lu, Y. and Prabhakar, B. (2010). Randomized load balancing with general service time distributions. SIGMETRICS Perform. Eval. Rev. 38 275–286.
  • [10] Bramson, M., Lu, Y. and Prabhakar, B. (2012). Asymptotic independence of queues under randomized load balancing. Queueing Syst. 71 247–292.
  • [11] 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.
  • [12] Brémaud, P. (1981). Point Processes and Queues: Martingale Dynamics. Springer Series in Statistics. Springer, New York.
  • [13] Brown, L., Gans, N., Mandelbaum, A., Sakov, A., Shen, H., Zeltyn, S. and Zhao, L. (2005). Statistical analysis of a telephone call center: A queueing-science perspective. J. Amer. Statist. Assoc. 100 36–50.
  • [14] Chen, H. and Yao, D. D. (2001). Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization, Stochastic Modelling and Applied Probability. Applications of Mathematics (New York) 46. Springer, New York.
  • [15] Chen, S., Sun, Y., Kozat, U. C., Huang, L., Sinha, P., Liang, G., Liu, X. and Shroff, N. B. (2014). When queueing meets coding: Optimal-latency data retrieving scheme in storage clouds. In INFOCOM 2014—IEEE Conference on Computer Communications 1042–1050.
  • [16] Daley, D. J. and Mohan, N. R. (1978). Asymptotic behaviour of the variance of renewal processes and random walks. Ann. Probab. 6 516–521.
  • [17] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York.
  • [18] Farias, V. F., Moallemi, C. C. and Prabhakar, B. (2005). Load balancing with migration penalties. In Proceedings of the IEEE International Symposium on Information Theory 558–562.
  • [19] Ganesh, A., Lilienthal, S., Manjunath, D., Proutiere, A. and Simatos, F. (2012). Load balancing via random local search in closed and open systems. Queueing Syst. 71 321–345.
  • [20] Graham, C. (2000). Chaoticity on path space for a queueing network with selection of the shortest queue among several. J. Appl. Probab. 37 198–211.
  • [21] Jakubowski, A. (1986). On the Skorokhod topology. Ann. Inst. Henri Poincaré Probab. Stat. 22 263–285.
  • [22] Kang, W. and Ramanan, K. (2010). Fluid limits of many-server queues with reneging. Ann. Appl. Probab. 20 2204–2260.
  • [23] Kardassakis, K. (2014). Load balancing in stochastic networks: Algorithms, analysis, and game theory. Undergraduate Honors Thesis, Brown Univ., Providence, RI.
  • [24] Kaspi, H. and Ramanan, K. (2011). Law of large numbers limits for many-server queues. Ann. Appl. Probab. 21 33–114.
  • [25] Klebaner, F. C. (2005). Introduction to Stochastic Calculus with Applications, 2nd ed. Imperial College Press, London.
  • [26] Kolesar, P. (1984). Stalking the endangered CAT: A queueing analysis of congestion at Automatic Teller Machines. Interfaces 14 16–26.
  • [27] Liang, G. and Kozat, U. C. (2014). TOFEC: achieving optimal throughput-delay trade-off of cloud storage using erasure codes. In INFOCOM 2014—IEEE Conference on Computer Communications 826–834.
  • [28] Luczak, M. J. and Norris, J. (2005). Strong approximation for the supermarket model. Ann. Appl. Probab. 15 2038–2061.
  • [29] Mitzenmacher, M. (2001). Analyses of load stealing models based on families of differential equations. Theory Comput. Syst. 34 77–98.
  • [30] Mitzenmacher, M. (2001). The power of two choices in randomized load balancing. IEEE Trans. Parallel Distrib. Syst. 12 1094–1104.
  • [31] Mukhopadhyay, A. and Mazumdar, R. R. (2016). Analysis of randomized join-the-shortest-queue (JSQ) schemes in large heterogeneous processor-sharing systems. IEEE Trans. Control Netw. Syst. 3 116–126.
  • [32] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Vol. 2. Itô Calculus. Cambridge Mathematical Library. Cambridge Univ. Press, Cambridge. Reprint of the second (1994) edition.
  • [33] 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.