## Stochastic Systems

### Control of parallel non-observable queues: Asymptotic equivalence and optimality of periodic policies

#### Abstract

We consider a queueing system composed of a dispatcher that routes jobs to a set of non-observable queues working in parallel. In this setting, the fundamental problem is which policy should the dispatcher implement to minimize the stationary mean waiting time of the incoming jobs. We present a structural property that holds in the classic scaling of the system where the network demand (arrival rate of jobs) grows proportionally with the number of queues. Assuming that each queue of type $r$ is replicated $k$ times, we consider a set of policies that are periodic with period $k\sum_{r}p_{r}$ and such that exactly $p_{r}$ jobs are sent in a period to each queue of type $r$. When $k\to\infty$, our main result shows that all the policies in this set are equivalent, in the sense that they yield the same mean stationary waiting time, and optimal, in the sense that no other policy having the same aggregate arrival rate to all queues of a given type can do better in minimizing the stationary mean waiting time. This property holds in a strong probabilistic sense. Furthermore, the limiting mean waiting time achieved by our policies is a convex function of the arrival rate in each queue, which facilitates the development of a further optimization aimed at solving the fundamental problem above for large systems.

#### Article information

Source
Stoch. Syst., Volume 5, Number 1 (2015), 120-145.

Dates
First available in Project Euclid: 23 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.ssy/1450879283

Digital Object Identifier
doi:10.1214/14-SSY146

Mathematical Reviews number (MathSciNet)
MR3442391

Zentralblatt MATH identifier
1336.60175

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

#### Citation

Anselmi, Jonatha; Gaujal, Bruno; Nesti, Tommaso. Control of parallel non-observable queues: Asymptotic equivalence and optimality of periodic policies. Stoch. Syst. 5 (2015), no. 1, 120--145. doi:10.1214/14-SSY146. https://projecteuclid.org/euclid.ssy/1450879283

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