# Heavy-Traffic Limits for Queues with Many Exponential Servers

@article{Halfin1981HeavyTrafficLF, title={Heavy-Traffic Limits for Queues with Many Exponential Servers}, author={Shlomo Halfin and Ward Whitt}, journal={Oper. Res.}, year={1981}, volume={29}, pages={567-588} }

Two different kinds of heavy-traffic limit theorems have been proved for s -server queues. The first kind involves a sequence of queueing systems having a fixed number of servers with an associated sequence of traffic intensities that converges to the critical value of one from below. The second kind, which is often not thought of as heavy traffic, involves a sequence of queueing systems in which the associated sequences of arrival rates and numbers of servers go to infinity while the service…

## 754 Citations

Scheduling a multi class queue with many exponential servers: asymptotic optimality in heavy traffic

- Computer Science
- 2004

This work considers the problem of scheduling a queueing system in which many statistically identical servers cater to several classes of impatient customers, and studies the system in an asymptotic heavy-traffic regime where the number of servers n and the offered load r are simultaneously scaled up and carefully balanced.

Many-Server Heavy-Traffic Limits for Queueing Systems with Perfectly Correlated Service and Patience Times

- Mathematics
- 2020

We characterize heavy-traffic process and steady-state limits for systems staffed according to the square-root safety rule, when the service requirements of the customers are perfectly correlated…

Two-parameter heavy-traffic limits for infinite-server queues

- MathematicsQueueing Syst. Theory Appl.
- 2010

Two-parameter stochastic-process limits in the space D([0,∞),D) of D-valued functions in D draw on, and extend, previous heavy-traffic limits by Glynn and Whitt and Krichagina and Puhalskii.

Heavy-Traffic Limits for the G / H ∗ 2 / n / m Queue Ward Whitt

- Mathematics
- 2005

We establish heavy-traffic stochastic-process limits for queue-length, waiting-time and overflow stochastic processes in a class of G/GI/n/m queueing models with n servers and m extra waiting spaces.…

Heavy-traffic limits for waiting times in many-server queues with abandonment

- Mathematics
- 2009

We establish heavy-traffic stochastic-process limits for waiting times in many-server queues with customer abandonment. If the system is asymptotically critically loaded, as in the…

Heavy-Traffic Limits for the G/H2*/n/mQueue

- MathematicsMath. Oper. Res.
- 2005

To capture the impact of the service-time distribution beyond its mean within a Markovian framework, a special class of service- time distributions are considered, denoted by H2*, which are mixtures of an exponential distribution with probability p and a unit point mass at 0 with probability 1- p, which exhibit relatively high variability.

Dynamic Scheduling of a Multiclass Queue in the Halfin-Whitt Heavy Traffic Regime

- MathematicsOper. Res.
- 2004

A Markovian model of a multiclass queueing system in which a single large pool of servers attends to the various customer classes is considered, where the total arrival rate and the number of servers both become large in such a way that the system's traffic intensity parameter approaches one.

Heavy-traffic limits for many-server queues with service interruptions

- MathematicsQueueing Syst. Theory Appl.
- 2009

A stochastic-decomposition property of the steady-state distribution of the limit process in the ED regime (and for infinite-server queues) is obtained.

Departures from a Queue with Many Busy Servers

- MathematicsMath. Oper. Res.
- 1984

This paper shows that the stationary departure process is approximately Poisson when there are many busy slow servers in a large class of stationary G/GI/s congestion models having s servers, infinite waiting room, the first-come first-served discipline, and mutually independent and identically distributed service times that are independent of a stationary arrival process.

Many-Server Queues with Time-Varying Arrivals, Customer Abandonment, and non-Exponential Distributions

- Business
- 2011

Many-Server Queues with Time-Varying Arrivals, Customer A bandonment and Non-Exponential Distributions Yunan Liu This thesis develops deterministic heavy-traffic fluid appr oximations for many-server…

## References

SHOWING 1-10 OF 29 REFERENCES

On Queues in Heavy Traffic

- Mathematics
- 1962

SUMMARY We say that a single server queue is in "heavy traffic" when the traffic intensity p is less than, but very near, unity. Then the equilibrium waiting time w will be large, and it is proved…

Multiple channel queues in heavy traffic. I

- Computer ScienceAdvances in Applied Probability
- 1970

The queueing systems considered in this paper consist of r independent arrival channels and s independent service channels, where as usual the arrival and service channels are independent. Arriving…

Multiple channel queues in heavy traffic

- Computer Science
- 1971

The principal objective will be to obtain functional central limit theorems (invariance principles) for the stochastic processes characterizing these systems after appropriately scaling and translating the processes.

The diffusion approximation for tandem queues in heavy traffic

- MathematicsAdvances in Applied Probability
- 1978

Consider a pair of single server queues arranged in series. (This is the simplest example of a queuing network.) In an earlier paper [2], a limit theorem was proved to justify a heavy traffic…

Limiting diffusion approximations for the many server queue and the repairman problem

- Mathematics
- 1965

We shall consider a many server (multiple channels in parallel) queueing process in which customers arrive at the queue according to a Poisson process. The service times are assumed to be independent…

Heavy Traffic Limit Theorems for Queues: A Survey

- Mathematics
- 1974

The two principal objectives of heavy traffic research are discussed, namely, to describe unstable queueing systems and to approximate stable queues, which are each related to more general themes.

HEAVY TRAFFIC THEORY FOR QUEUES WITH SEVERAL SERVERS. II

- Mathematics
- 1974

The queues being studied here are of the type GI/Gik in statistical equilibrium (with traffic intensity less than one). The exponential limiting formula for the waiting time distribution function in…

A Diffusion Approximation to the Multi-Server Queue

- Business
- 1978

This paper presents a methodology for obtaining approximately the distribution of the number of customers in a multiserver queue for which both the customer interarnval and customer service times are drawn from arbitrary distributions.

Comparing counting processes and queues

- MathematicsAdvances in Applied Probability
- 1981

Several partial orderings of counting processes are introduced and applied to compare stochastic processes in queueing models. The conditions for the queueing comparisons involve the counting…

Approximate Stochastic Behavior of n-Server Service Systems with Large n

- Mathematics
- 1973

I. General Formulation.- 1. Introduction.- 2. Graphical Representations.- 3. Stochastic Properties.- II. Approximation Methods.- 1. Introduction.- 2. Approximations - No Customer Queue.- 3.…