Law of large numbers limits for many-server queues

By Haya Kaspi, Kavita Ramanan. 2007

128 citations recorded by the source corpus.

Abstract. This work considers a many-server queueing system in which customers with i.i.d., generally distributed service times 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 characterised 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 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.

Subfields: Management Information Systems, Management Science and Operations Research, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/de1cc6292e7b804e8bcc30523ad8f7346d689be5

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In Management Information Systems

1. Kavita Ramanan
2. Haya Kaspi
3. Weining Kang
4. Josh E. Reed
5. J. Dai
6. Tolga Tezcan
7. Shuangchi He
8. Ward Whitt

In Management Science and Operations Research

1. Kavita Ramanan
2. Haya Kaspi
3. Weining Kang
4. Josh E. Reed
5. J. Dai
6. Tolga Tezcan
7. Shuangchi He
8. Ward Whitt

In Mathematical Physics

1. Weining Kang
2. Kavita Ramanan
3. Josh E. Reed
4. Haya Kaspi
5. Daniel Paulin
6. Radosław Adamczak

Citations in the corpus

• SPDE Limits of Many Server Queues (2010, background): “This representation, which provides a Markovian description of the dynamics, was first introduced in Kaspi and Ramanan [21], where it was shown that the fluid-scaled sequence (X, ν)/N converges almost surely to a certain deterministic process (X, ν), referred to as the fluid limit.”
• The G/GI/N queue in the Halfin–Whitt regime (2009, background): “Nevertheless, with the exception of the results of Kapsi and Ramanan [12], it does not appear that any of the aforementioned approaches may be easily extended to the case of general service time distributions and so to this date there has remained no general methodology for analyzing the G/GI/N queu”
• Asymptotic approximations for stationary distributions of many-server queues with abandonment (2010, methodology): “1 of [17] or, equivalently, Proposition 4.”
• Fluid limits of many-server queues with reneging (2010, methodology): “in [14] is equivalent to the definition given in (2.”