## The Annals of Applied Probability

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

### The hydrodynamic limit of a randomized load balancing network

Reza Aghajani and Kavita Ramanan

#### Abstract

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

**Source**

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

**Dates**

Received: October 2017

Revised: July 2018

First available in Project Euclid: 23 July 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1563869039

**Digital Object Identifier**

doi:10.1214/18-AAP1444

**Mathematical Reviews number (MathSciNet)**

MR3984253

**Subjects**

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

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aoap/1563869039