Fluid limits of many-server queues with reneging

By Weining Kang, Kavita Ramanan. 2010

120 citations recorded by the source corpus.

This work considers a many-server queueing system in which im- patient 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, essen- tially only requiring that the service and reneging distributions have densities, as the number of servers goes to innity, a law of large numbers (or uid) 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 uid 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.

Subfields: Management Information Systems, Management Science and Operations Research, Safety, Risk, Reliability and Quality

View Paper: https://www.semanticscholar.org/paper/96ae74f13aee21da4d95cbbef6a3e0669fcb8eb9

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In Management Information Systems

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

In Management Science and Operations Research

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

In Safety, Risk, Reliability and Quality

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

Citations in the corpus

• SPDE Limits of Many Server Queues (2010, methodology): “1 of [19], {(R E (t),X(t), ν t ), F (N) t , t ≥ 0} is a strong Markov process.”
• Many-server heavy-traffic limit for queues with time-varying parameters (2014, background): “Explicitly avoiding the QED regime runs counter to most of the extensive research on many-server queues, for example, as in [3, 5, 7, 8, 22].”
• Asymptotic approximations for stationary distributions of many-server queues with abandonment (2010, methodology): “This model was also considered in [15], where a functional strong law of large numbers limit for the state descriptor was established as the number of servers and the mean arrival rate both tend to infinity.”
• Many-server diffusion limits for G/Ph/n+GI queues (2010, methodology): “in Mandelbaum, Massey and Reiman (1998) for strong approximations and Mandelbaum and Pats (1998) for general state-dependent networks. In our continuous-mapping approach, we have heavily exploited some maps from DK+1 to DK+1, where K is the number of phases in the service time distribution. Variants”
• On many-server queues in heavy traffic (2010, background): “This was first accomplished in the work of Halfin and Whitt [10] for the case of exponential service time distributions and has been continued by many additional authors as well for different classes of service time distributions, see, for instance, Jelenkovic, Mandelbaum, and Momcilovic [13], Mande”