## The Annals of Applied Probability

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

### Supermarket model on graphs

Amarjit Budhiraja, Debankur Mukherjee, and Ruoyu Wu

#### Abstract

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

**Source**

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

**Dates**

Received: December 2017

Revised: September 2018

First available in Project Euclid: 19 February 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1550566841

**Digital Object Identifier**

doi:10.1214/18-AAP1437

**Mathematical Reviews number (MathSciNet)**

MR3914555

**Zentralblatt MATH identifier**

07057465

**Subjects**

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]

**Keywords**

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

#### Citation

Budhiraja, Amarjit; Mukherjee, Debankur; Wu, Ruoyu. Supermarket model on graphs. Ann. Appl. Probab. 29 (2019), no. 3, 1740--1777. doi:10.1214/18-AAP1437. https://projecteuclid.org/euclid.aoap/1550566841