# An Introduction to Stochastic-Process Limits and their Application to Queues

@inproceedings{Whitt2002AnIT, title={An Introduction to Stochastic-Process Limits and their Application to Queues}, author={Ward Whitt}, year={2002} }

Experiencing Statistical Regularity.- Random Walks in Applications.- The Framework for Stochastic-Process Limits.- A Panorama of Stochastic-Process Limits.- Heavy-Traffic Limits for Fluid Queues.- Unmatched Jumps in the Limit Process.- More Stochastic-Process Limits.- Fluid Queues with On-Off Sources.- Single-Server Queues.- Multi-Server Queues.- More on the Mathematical Framework.- The Space D Useful Functions.- Queueing Networks.- The Spaces E and F.- Appendices.

## 815 Citations

Heavy-traffic limits for queues with periodic arrival processes

- MathematicsOper. Res. Lett.
- 2014

HEAVY-TRAFFIC LIMITS FOR THE STATIONARY FLOWS IN GENERALIZED JACKSON NETWORKS

- Mathematics
- 2018

Motivated by interest in approximations for the steady-state performance of non-Markovian open queueing networks based on the index of dispersion for counts of each stationary arrival process, we…

A large deviations principle for infinite-server queues in a random environment

- MathematicsQueueing Syst. Theory Appl.
- 2016

This paper studies an infinite-server queue in a random environment, meaning that the arrival rate, the service requirements, and the server work rate are modulated by a general càdlàg stochastic background process and a large deviations principle for the number of jobs in the system is derived using attainable parameters.

Limit Theorems for Queuing Systems with Regenerative Doubly Stochastic Input Flow*

- Mathematics
- 2016

This article focuses on queuing systems with doubly stochastic Poisson regenerative input flow and an infinite number of servers. Service times have the heavy-tailed distribution. The analogs of the…

Diffusion limits for networks of Markov-modulated infinite-server queues

- MathematicsPerform. Evaluation
- 2019

Join the Shortest Queue with Many Servers. The Heavy-Traffic Asymptotics

- MathematicsMath. Oper. Res.
- 2018

The martingale method is used to prove that a scaled process counting the number of idle servers and queues of length exactly two weakly converges to a two-dimensional reflected Ornstein-Uhlenbeck process, while processes counting longer queues converge to a deterministic system decaying to zero in constant time.

2 The Stationary Flows in an Open Queueing Network

- Mathematics
- 2019

This paper studies the stationary customer flows in open queueing network. The flows are the processes counting customers flowing from one queue to another or out of the network. We establish the…

An approximation algorithm for the queue length distributions of time-varying many-server queues

- Mathematics, Business
- 2015

This paper proposes an approximation algorithm for estimating the queue length (the number of customers in the system) distributions of time-varying non-Markovian many-server queues (e.g., Gt/Gt/nt…

Diffusion Parameters of Flows in Stable Queueing Networks

- Mathematics
- 2013

We consider open multi-class queueing networks with general arrival processes, general processing time sequences and Bernoulli routing. The network is assumed to be operating under an arbitrary…