## Stochastic Systems

- Stoch. Syst.
- Volume 1, Number 1 (2011), 59-108.

### An ODE for an overloaded X model involving a stochastic averaging principle

Ohad Perry and Ward Whitt

#### Abstract

We study an ordinary differential equation (ODE) arising as the many-server heavy-traffic fluid limit of a sequence of overloaded Markovian queueing models with two customer classes and two service pools. The system, known as the $X$ model in the call-center literature, operates under the fixed-queue-ratio-with-thresholds (FQR-T) control, which we proposed in a recent paper as a way for one service system to help another in face of an unanticipated overload. Each pool serves only its own class until a threshold is exceeded; then one-way sharing is activated with all customer-server assignments then driving the two queues toward a fixed ratio. For large systems, that fixed ratio is achieved approximately. The ODE describes system performance during an overload. The control is driven by a queue-difference stochastic process, which operates in a faster time scale than the queueing processes themselves, thus achieving a time-dependent steady state instantaneously in the limit. As a result, for the ODE, the driving process is replaced by its long-run average behavior at each instant of time; i.e., the ODE involves a heavy-traffic averaging principle (AP).

#### Article information

**Source**

Stoch. Syst., Volume 1, Number 1 (2011), 59-108.

**Dates**

First available in Project Euclid: 24 February 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.ssy/1393252124

**Digital Object Identifier**

doi:10.1214/10-SSY009

**Mathematical Reviews number (MathSciNet)**

MR2948918

**Zentralblatt MATH identifier**

1291.60191

**Subjects**

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

Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx] 60F17: Functional limit theorems; invariance principles 90B15: Network models, stochastic 90B22: Queues and service [See also 60K25, 68M20] 37C75: Stability theory 93D05: Lyapunov and other classical stabilities (Lagrange, Poisson, $L^p, l^p$, etc.)

**Keywords**

Many-server queues averaging principle separation of time scales heavy traffic deterministic fluid approximation quasi-birth-death processes ordinary differential equations overload control

#### Citation

Perry, Ohad; Whitt, Ward. An ODE for an overloaded X model involving a stochastic averaging principle. Stoch. Syst. 1 (2011), no. 1, 59--108. doi:10.1214/10-SSY009. https://projecteuclid.org/euclid.ssy/1393252124