## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 23, Number 5 (2013), 2099-2138.

### Systems with large flexible server pools: Instability of “natural” load balancing

Alexander L. Stolyar and Elena Yudovina

#### Abstract

We consider general large-scale service systems with multiple customer classes and multiple server (agent) pools, mean service times depend both on the customer class and server pool. It is assumed that the allowed activities (routing choices) form a tree (in the graph with vertices being both customer classes and server pools). We study the behavior of the system under a natural (load balancing) routing/scheduling rule, *Longest-Queue Freest-Server* (LQFS-LB), in the many-server asymptotic regime, such that the exogenous arrival rates of the customer classes, as well as the number of agents in each pool, grow to infinity in proportion to some scaling parameter $r$. *Equilibrium point* of the system under LQBS-LB is the desired operating point, with server pool loads minimized and perfectly balanced.

Our main results are as follows. (a) We show that, quite surprisingly (given the tree assumption), for certain parameter ranges, the *fluid limit* of the system may be *unstable* in the vicinity of the equilibrium point; such instability may occur if the activity graph is not “too small.” (b) Using (a), we demonstrate that the sequence of stationary distributions of *diffusion-scaled* processes [measuring $O(\sqrt{r})$ deviations from the equilibrium point] may be nontight, and in fact may escape to infinity. (c) In one special case of interest, however, we show that the sequence of stationary distributions of diffusion-scaled processes is tight, and the limit of stationary distributions is the stationary distribution of the limiting diffusion process.

#### Article information

**Source**

Ann. Appl. Probab., Volume 23, Number 5 (2013), 2099-2138.

**Dates**

First available in Project Euclid: 28 August 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1377696307

**Digital Object Identifier**

doi:10.1214/12-AAP895

**Mathematical Reviews number (MathSciNet)**

MR3134731

**Zentralblatt MATH identifier**

1290.60093

**Subjects**

Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60F17: Functional limit theorems; invariance principles

**Keywords**

Many server models fluid limit diffusion limit load balancing instability tightness of invariant distributions

#### Citation

Stolyar, Alexander L.; Yudovina, Elena. Systems with large flexible server pools: Instability of “natural” load balancing. Ann. Appl. Probab. 23 (2013), no. 5, 2099--2138. doi:10.1214/12-AAP895. https://projecteuclid.org/euclid.aoap/1377696307