## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 8, Number 3 (1998), 822-848.

### Heavy traffic analysis of a system with parallel servers: asymptotic optimality of discrete-review policies

#### Abstract

This paper is concerned with dynamic scheduling in a queueing system that has two independent Poisson input streams, two servers, deterministic service times and linear holding costs. One server can process both classes of incoming jobs, but the other can process only one class, and the service time for the shared job class is different depending on which server is involved. A bound on system performance is developed in terms of a single pooled resource, or super-server, whose capabilities combine those of the original two servers. Thereafter, attention is focused on the heavy traffic regime, where the combined capacity of the two servers is approximately equal to the total input rate. We construct a discrete-review control policy and show that if its parameters are chosen correctly as one approaches the heavy traffic limit, then its cost performance approaches the bound associated with a single pooled resource. Thus the discrete-review policy is proved to be asymptotically optimal in the heavy traffic limit. Although resource pooling in heavy traffic has been observed to occur in other network scheduling problems, there have been very few studies that rigorously proved the pooling phenomenon, or that proved the asymptotic optimality of a specific policy. Our discrete-review policy is obtained by applying a general method, called the BIGSTEP method in an earlier paper, to the parallel-server model.

#### Article information

**Source**

Ann. Appl. Probab., Volume 8, Number 3 (1998), 822-848.

**Dates**

First available in Project Euclid: 9 August 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1028903452

**Digital Object Identifier**

doi:10.1214/aoap/1028903452

**Mathematical Reviews number (MathSciNet)**

MR1627791

**Zentralblatt MATH identifier**

0938.60094

**Subjects**

Primary: 60K25: Queueing theory [See also 68M20, 90B22] 90B15: Network models, stochastic 90B22: Queues and service [See also 60K25, 68M20]

**Keywords**

Queueing theory heavy traffic BIGSTEP method resource pooling

#### Citation

Harrison, J. Michael. Heavy traffic analysis of a system with parallel servers: asymptotic optimality of discrete-review policies. Ann. Appl. Probab. 8 (1998), no. 3, 822--848. doi:10.1214/aoap/1028903452. https://projecteuclid.org/euclid.aoap/1028903452