## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 24, Number 1 (2014), 378-421.

### Many-server heavy-traffic limit for queues with time-varying parameters

Yunan Liu and Ward Whitt

#### Abstract

A many-server heavy-traffic FCLT is proved for the $G_{t}/M/s_{t}+\mathit{GI} $ queueing model, having time-varying arrival rate and staffing, a general arrival process satisfying a FCLT, exponential service times and customer abandonment according to a general probability distribution. The FCLT provides theoretical support for the approximating deterministic fluid model the authors analyzed in a previous paper and a refined Gaussian process approximation, using variance formulas given here. The model is assumed to alternate between underloaded and overloaded intervals, with critical loading only at the isolated switching points. The proof is based on a recursive analysis of the system over these successive intervals, drawing heavily on previous results for infinite-server models. The FCLT requires careful treatment of the initial conditions for each interval.

#### Article information

**Source**

Ann. Appl. Probab., Volume 24, Number 1 (2014), 378-421.

**Dates**

First available in Project Euclid: 9 January 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1389278729

**Digital Object Identifier**

doi:10.1214/13-AAP927

**Mathematical Reviews number (MathSciNet)**

MR3161651

**Zentralblatt MATH identifier**

1290.60092

**Subjects**

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

Secondary: 60F17: Functional limit theorems; invariance principles 90B22: Queues and service [See also 60K25, 68M20]

**Keywords**

Many-server queues queues with time-varying arrivals nonstationary queues customer abandonment nonexponential patience distribution heavy traffic functional central limit theorem Gaussian approximation deterministic fluid approximation

#### Citation

Liu, Yunan; Whitt, Ward. Many-server heavy-traffic limit for queues with time-varying parameters. Ann. Appl. Probab. 24 (2014), no. 1, 378--421. doi:10.1214/13-AAP927. https://projecteuclid.org/euclid.aoap/1389278729