## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 20, Number 6 (2010), 2204-2260.

### Fluid limits of many-server queues with reneging

Weining Kang and Kavita Ramanan

#### Abstract

This work considers a many-server queueing system in which impatient customers with i.i.d., generally distributed service times and i.i.d., generally distributed patience times enter service in the order of arrival and abandon the queue if the time before possible entry into service exceeds the patience time. The dynamics of the system is represented in terms of a pair of measure-valued processes, one that keeps track of the waiting times of the customers in queue and the other that keeps track of the amounts of time each customer being served has been in service. Under mild assumptions, essentially only requiring that the service and reneging distributions have densities, as both the arrival rate and the number of servers go to infinity, a law of large numbers (or fluid) limit is established for this pair of processes. The limit is shown to be the unique solution of a coupled pair of deterministic integral equations that admits an explicit representation. In addition, a fluid limit for the virtual waiting time process is also established. This paper extends previous work by Kaspi and Ramanan, which analyzed the model in the absence of reneging. A strong motivation for understanding performance in the presence of reneging arises from models of call centers.

#### Article information

**Source**

Ann. Appl. Probab., Volume 20, Number 6 (2010), 2204-2260.

**Dates**

First available in Project Euclid: 19 October 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1287494559

**Digital Object Identifier**

doi:10.1214/10-AAP683

**Mathematical Reviews number (MathSciNet)**

MR2759733

**Zentralblatt MATH identifier**

1208.60094

**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**

Many-server queues GI/G/N queue fluid limits reneging abandonment strong law of large numbers measure-valued processes call centers

#### Citation

Kang, Weining; Ramanan, Kavita. Fluid limits of many-server queues with reneging. Ann. Appl. Probab. 20 (2010), no. 6, 2204--2260. doi:10.1214/10-AAP683. https://projecteuclid.org/euclid.aoap/1287494559