# Heavy-tailed queues in the Halfin-Whitt regime

@article{Goldberg2017HeavytailedQI, title={Heavy-tailed queues in the Halfin-Whitt regime}, author={David Alan Goldberg and Yuan Li}, journal={arXiv: Probability}, year={2017} }

We consider the FCFS G/G/n queue in the Halfin-Whitt regime, in the presence of heavy-tailed distributions (i.e. infinite variance). We prove that under minimal assumptions, i.e. only that processing times have finite 1 + epsilon moment and inter-arrival times have finite second moment, the sequence of stationary queue length distributions, normalized by $n^{\frac{1}{2}}$, is tight. All previous tightness results for the stationary queue length required that processing times have finite 2…

## 4 Citations

### Large deviations analysis for the $M/H_2/n + M$ queue in the Halfin-Whitt regime

- Mathematics
- 2018

We consider the FCFS $M/H_2/n + M$ queue in the Halfin-Whitt heavy traffic regime. It is known that the normalized sequence of steady-state queue length distributions is tight and converges weakly to…

### Simple and explicit bounds for multi-server queues with $1/(1 - \rho)$ (and sometimes better) scaling

- Mathematics
- 2017

We consider the FCFS GI/GI/n queue, and prove the first simple and explicit bounds that scale as 1 1−ρ (and sometimes better). Here ρ denotes the corresponding traffic intensity. Conceptually, our…

### Economies-of-Scale in Many-Server Queueing Systems: Tutorial and Partial Review of the QED Halfin-Whitt Heavy-Traffic Regime

- Computer ScienceSIAM Rev.
- 2019

The mathematics behind the quality- and efficiency-driven (QED) regime is reviewed, which lets the system operate close to full utilization, while the number of servers grows simultaneously large and delays remain manageable.

### Economies-of-scale in resource sharing systems: tutorial and partial review of the QED heavy-traffic regime

- Computer Science
- 2017

The mathematics behind the Quality-and-Efficiency Driven (QED) regime, which lets the system operate close to full utilization, while the number of servers grows simultaneously large and delays remain manageable, is reviewed.

## References

SHOWING 1-10 OF 54 REFERENCES

### The G/GI/N queue in the Halfin–Whitt regime

- Mathematics, Computer Science
- 2007

The first result is to obtain a deterministic fluid limit for the properly centered and scaled number of customers in the system which may be used to provide a first-order approximation to the queue length process.

### Steady-state analysis of a multiserver queue in the Halfin-Whitt regime

- Mathematics, Computer ScienceAdvances in Applied Probability
- 2008

This work characterize the limiting scaled stationary queue length distribution in terms of the stationary distribution of an explicitly constructed Markov chain, and obtains an explicit expression for the critical exponent for the moment generating function of a limiting stationary queuelength.

### On Large Delays in Multi-Server Queues with Heavy Tails

- MathematicsMath. Oper. Res.
- 2012

We present upper and lower bounds for the tail distribution of the stationary waiting time D in the stable GI/GI/s first-come first-served (FCFS) queue. These bounds depend on the value of the…

### The impact of a heavy-tailed service-time distribution upon the M/GI/s waiting-time distribution

- MathematicsQueueing Syst. Theory Appl.
- 2000

By exploiting an infinite-server-model lower bound, we show that the tails of the steady-state and transient waiting-time distributions in the M/GI/s queue with unlimited waiting room and the…

### Multiclass multiserver queueing system in the Halfin–Whitt heavy traffic regime: asymptotics of the stationary distribution

- Mathematics, Computer ScienceQueueing Syst. Theory Appl.
- 2012

This paper is the first to address the case of heterogeneity in the steady state regime of a heterogeneous queueing system consisting of one large pool of O(r) identical servers, and shows that every weak limit of $r^{-{1\over2}}Q^{r}(\infty)$ has a sub-Gaussian tail.

### Tail asymptotics for delay in a half-loaded GI/GI/2 queue with heavy-tailed job sizes

- MathematicsQueueing Syst. Theory Appl.
- 2015

We obtain asymptotic bounds for the tail distribution of steady-state waiting time in a two-server queue where each server processes incoming jobs at a rate equal to the rate of their arrivals (that…

### The M/G/1 queue with heavy-tailed service time distribution

- MathematicsIEEE J. Sel. Areas Commun.
- 1998

An approximation for W(t), which is based on a heavy-traffic limit theorem for the M/G/1 queue with heavy-tailed service time distribution (with infinite variance), is presented and shown to yield excellent results for values of t which are not too small, even when the load is not heavy.

### On the transition from heavy traffic to heavy tails for the M/G/1 queue: The regularly varying case.

- Mathematics
- 2011

Two of the most popular approximations for the distribution of the steady-state waiting time, W∞, of the M/G/1 queue are the socalled heavy-traffic approximation and heavy-tailed asymptotic,…

### Simple and explicit bounds for multi-server queues with universal 1 / (1 - rho) scaling

- Mathematics, Computer Science
- 2017

We consider the FCFS GI/GI/n queue, and prove the first simple and explicit bounds that scale gracefully and universally as 1 / (1 - rho) (and better), with rho the corresponding traffic intensity.…

### Excursion-Based Universal Approximations for the Erlang-A Queue in Steady-State

- MathematicsMath. Oper. Res.
- 2014

This work revisits many-server approximations for the well-studied Erlang-A queue and proposes a diffusion approximation that applies simultaneously to all existing many- server heavy-traffic regimes: quality and efficiencydriven, efficiency driven, quality driven, and nondegenerate slowdown.