# Universality of Power-of-$d$ Load Balancing in Many-Server Systems

## Speaker:

Debankur Mukherjee

## Affiliation:

Technische Universiteit Eindhoven
Department of Mathematics and Computer Science
5612 AZ Eindhoven
Netherlands

## Time:

Wednesday, 21 December 2016, 16:00 to 17:00

• AG-80

## Organisers:

We consider a system of $N$~parallel single-server queues with unit exponential service rates and a single dispatcher where tasks arrive as a Poisson process of rate $\lambda(N)$.

When a task arrives, the dispatcher assigns it to a server with the shortest queue among $d(N)$ randomly selected servers ($1 \leq d(N) \leq N$). This load balancing strategy is referred to as a JSQ($d(N)$) scheme, marking that it subsumes the celebrated Join-the-Shortest Queue (JSQ) policy as a crucial special case for $d(N) = N$.

We construct a stochastic coupling to bound the difference in the queue length processes between the JSQ policy and a scheme with an arbitrary value of $d(N)$. We use the coupling to derive the fluid limit in the regime where $\lambda(N) / N \to \lambda < 1$ as $N \to \infty$ with $d(N) \to\infty$, along with the associated fixed point. The fluid limit turns out not to depend on the exact growth rate of $d(N)$, and in particular coincides with that for the ordinary JSQ policy. We further leverage the coupling to establish that the diffusion limit in the critical regime where $(N - \lambda(N)) / \sqrt{N} \to \beta > 0$ as $N \to \infty$ with $d(N)/(\sqrt{N} \log (N))\to\infty$ corresponds to that for the JSQ policy. These results indicate that the optimality of the JSQ policy can be preserved at the fluid-level and diffusion-level while reducing the overhead by nearly a factor O($N$) and O($\sqrt{N}/\log(N)$), respectively.