## Stochastic Systems

- Stoch. Syst.
- Volume 2, Number 2 (2012), 277-321.

### Diffusion approximation for an input-queued switch operating under a maximum weight matching policy

#### Abstract

For $N\geq 2$, we consider an $N\times N$ input-queued switch operating under a maximum weight matching policy. We establish a diffusion approximation for a $(2N-1)$-dimensional workload process associated with this switch when all input ports and output ports are heavily loaded. The diffusion process is a semimartingale reflecting Brownian motion living in a polyhedral cone with $N^{2}$ boundary faces, each of which has an associated constant direction of reflection. Our proof builds on our own prior work [13] on an invariance principle for semimartingale reflecting Brownian motions in piecewise smooth domains and on a multiplicative state space collapse result for switched networks established by Shah and Wischik in [19].

#### Article information

**Source**

Stoch. Syst., Volume 2, Number 2 (2012), 277-321.

**Dates**

First available in Project Euclid: 24 February 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.ssy/1393252025

**Digital Object Identifier**

doi:10.1214/12-SSY061

**Mathematical Reviews number (MathSciNet)**

MR3354769

**Zentralblatt MATH identifier**

1296.60212

**Subjects**

Primary: 60J60: Diffusion processes [See also 58J65] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx] 90B36: Scheduling theory, stochastic [See also 68M20]

**Keywords**

Input-queued switch maximum weight matching policy diffusion approximation heavy traffic semimartingale reflecting Brownian motion (SRBM)

#### Citation

Kang, W. N.; Williams, R. J. Diffusion approximation for an input-queued switch operating under a maximum weight matching policy. Stoch. Syst. 2 (2012), no. 2, 277--321. doi:10.1214/12-SSY061. https://projecteuclid.org/euclid.ssy/1393252025