## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 11, Number 3 (2001), 569-607.

### Join the shortest queue: stability and exact asymptotics

R. D. Foley and D. R. McDonald

#### Abstract

We consider the stability of a network serving a patchwork of
overlapping regions where customers from a local region are assigned to a
collection of local servers.These customers join the queue of the local server
with the shortest queue of waiting customers.We then describe how the backlog
in the network overloads.We do this in the simple case of two servers each of
which receives a *dedicated* stream of customers in addition to customers
from a stream of *smart* customers who join the shorter queue. There are
three distinct ways the backlog can overload. If one server is very fast, then
that server takes all the smart customers along with its dedicated customers
and keeps its queue small while the dedicated customers at the other server
cause the overload.We call this the unpooled case. If the proportion of smart
customers is large, then the two servers overload in tandem.We call this the
strongly pooled case. Finally, there is the weakly pooled case where both
queues overload but in different proportions. The fact that strong pooling can
be attained based on a local protocol for overlapping regions may have
engineering significance. In addition, this paper extends the methodology
developed in McDonald (to appear *The Annals of Applied Probability*) to
cover periodicities. The emphasis here is on sharp asymptotics, not rough
asymptotics as in large deviation theory. Moreover, the limiting distributions
are for the unscaled process, not for the fluid limit as in large deviation
theory. In the strongly pooled case, for instance, we give the limiting
distribution of the difference between the two queues as the backlog grows.We
also give the exact asymptotics of the mean time until overload.

#### Article information

**Source**

Ann. Appl. Probab., Volume 11, Number 3 (2001), 569-607.

**Dates**

First available in Project Euclid: 5 March 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1015345342

**Digital Object Identifier**

doi:10.1214/aoap/1015345342

**Mathematical Reviews number (MathSciNet)**

MR1865017

**Zentralblatt MATH identifier**

1016.60078

**Subjects**

Primary: 60K25: Queueing theory [See also 68M20, 90B22]

Secondary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) [See also 90Bxx]

**Keywords**

Join the shortest queue rare events change of measure h-transform

#### Citation

Foley, R. D.; McDonald, D. R. Join the shortest queue: stability and exact asymptotics. Ann. Appl. Probab. 11 (2001), no. 3, 569--607. doi:10.1214/aoap/1015345342. https://projecteuclid.org/euclid.aoap/1015345342