## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 21, Number 1 (2011), 33-114.

### Law of large numbers limits for many-server queues

#### Abstract

This work considers a many-server queueing system in which customers with independent and identically distributed service times, chosen from a general distribution, enter service in the order of arrival. The dynamics of the system are represented in terms of a process that describes the total number of customers in the system, as well as a measure-valued process that keeps track of the ages of customers in service. Under mild assumptions on the service time distribution, as the number of servers goes to infinity, a law of large numbers (or fluid) limit is established for this pair of processes. The limit is characterized as the unique solution to a coupled pair of integral equations which admits a fairly explicit representation. As a corollary, the fluid limits of several other functionals of interest, such as the waiting time, are also obtained. Furthermore, when the arrival process is time-homogeneous, the measure-valued component of the fluid limit is shown to converge to its equilibrium. Along the way, some results of independent interest are obtained, including a continuous mapping result and a maximality property of the fluid limit. A motivation for studying these systems is that they arise as models of computer data systems and call centers.

#### Article information

**Source**

Ann. Appl. Probab., Volume 21, Number 1 (2011), 33-114.

**Dates**

First available in Project Euclid: 17 December 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1292598028

**Digital Object Identifier**

doi:10.1214/09-AAP662

**Mathematical Reviews number (MathSciNet)**

MR2759196

**Zentralblatt MATH identifier**

1208.60095

**Subjects**

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

Secondary: 60H99: None of the above, but in this section 35D99: None of the above, but in this section

**Keywords**

Multi-server queues GI∕G∕N queue fluid limits mean-field limits strong law of large numbers measure-valued processes call centers

#### Citation

Kaspi, Haya; Ramanan, Kavita. Law of large numbers limits for many-server queues. Ann. Appl. Probab. 21 (2011), no. 1, 33--114. doi:10.1214/09-AAP662. https://projecteuclid.org/euclid.aoap/1292598028