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

@article{Whitt2000TheIO, title={The impact of a heavy-tailed service-time distribution upon the M/GI/s waiting-time distribution}, author={Ward Whitt}, journal={Queueing Systems}, year={2000}, volume={36}, pages={71-87} }

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 first-come first-served discipline are bounded below by tails of Poisson distributions. As a consequence, the tail of the steady-state waiting-time distribution is bounded below by a constant times the sth power of the tail of the service-time stationary-excess distribution. We apply that bound to show…

## 67 Citations

### 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…

### How Big Queues Occur in Multi-Server System with Heavy Tails

- Mathematics
- 2011

We present upper and lower bounds for the tail distribution of the stationary waiting time D in the stable GI/GI/s FCFS queue. These bounds depend on the value of the traffic loadwhich is the ratio…

### 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…

### Waiting-Time Asymptotics for the M/G/2 Queue with Heterogeneous Servers

- MathematicsQueueing Syst. Theory Appl.
- 2002

An exact analysis of the queue length and waiting time distribution in case B(⋅) has a rational Laplace–Stieltjes transform is presented.

### Heavy-tailed queues in the Halfin-Whitt regime

- Mathematics, Computer Science
- 2017

This work considers the FCFS G/G/n queue in the Halfin-Whitt regime, in the presence of heavy-tailed distributions, and proves that under minimal assumptions, the sequence of stationary queue length distributions, normalized by $n^{\frac{1}{2}}$, is tight.

### Heavy tails in multi-server queues

- Mathematics
- 2013

In this paper, the asymptotic behaviour of the distribution tail of the stationary waiting time $W$ in the $GI/GI/2$ FCFS queue is studied. Under subexponential-type assumptions on the service time…

### The consequences of heavy-tailed service time distribution on a basic queuing model and its performance indicators

- Computer Science
- 2010

The behavior of the average length of the queue and the average waiting-time were analyzed through simulation, varying system capacity, average service utilization factor and the number of servers in the systems as parameters and showed more sensitive variations of Lq and Wq for heavy-tailed service times than for Poisson-based service times.

### Heavy Tails in Multi-Server Queue

- MathematicsQueueing Syst. Theory Appl.
- 2006

Under subexponential-type assumptions on the service time distribution, bounds and sharp asymptotics are given for the distribution tail of the stationary waiting time W in the GI/GI/2 FCFS queue and of a stationary queue length.

### A new optimal cost model of queue systems with Heavy-Tailed distribution

- Computer ScienceThe 2nd International Conference on Information Science and Engineering
- 2010

It is proved that the efficiency of multiple servers is equal to the single server when all of servers work with full load and the optimal number of servers for the M/Heavy-Tailed/K queue system is less than that of system with the average response time as its objective function.

### Steady-State Analysis for Multiserver Queues Under Size Interval Task Assignment in the Quality-Driven Regime

- MathematicsMath. Oper. Res.
- 2013

If the job arrival rate and the number of servers increase to infinity with the traffic intensity held fixed, the SITA policy parameterized by α minimizes in a large deviation sense the steady-state probability that the total number of jobs in the system is greater than or equal to thenumber of servers.

## References

SHOWING 1-10 OF 41 REFERENCES

### Waiting-time tail probabilities in queues with long-tail service-time distributions

- MathematicsQueueing Syst. Theory Appl.
- 1994

Algorithms for computing the waiting-time distribution by Laplace transform inversion when the Laplace transforms of the interarrival-time and service-time distributions are known are developed and a convenient two-parameter family of long-tail distributions on the positive half line with explicit Laplace transformations is introduced.

### Asymptotics for M/G/1 low-priority waiting-time tail probabilities

- MathematicsQueueing Syst. Theory Appl.
- 1997

It is shown that the low-priority steady-state waiting-time can be expressed as a geometric random sum of i.i.d. random variables, just like the M/G/1 FIFO waiting- time distribution, and asymptotic results for cases with long-tail service-time distributions are established.

### Waiting-Time Asymptotics for the M/G/2 Queue with Heterogeneous Servers

- MathematicsQueueing Syst. Theory Appl.
- 2002

An exact analysis of the queue length and waiting time distribution in case B(⋅) has a rational Laplace–Stieltjes transform is presented.

### The Physics of the Mt/G/∞ Queue

- MathematicsOper. Res.
- 1993

It is significant that the well known insensitivity property of the stationary M/G/∞ model does not hold for the nonstationary Mt/G/, and the time-dependent mean function m depends on the service-time distribution beyond its mean.

### A Light-Traffic Theorem for Multi-Server Queues

- Mathematics, Computer ScienceMath. Oper. Res.
- 1983

It is shown that as the traffic goes to zero, the probability of delay depends only on the mean of the service-time distributions and that the delay when positive converges in distribution to the minimum of c independent equilibrium-excess service-times is zero.

### Peak congestion in multi-server service systems with slowly varying arrival rates

- MathematicsQueueing Syst. Theory Appl.
- 1997

The value and lag in peak congestion predicted by the MOL approximation are compared with exact values for Mt/M/s delay models with sinusoidal arrival-rate functions obtained by numerically solving the Chapman–Kolmogorov forward equations.

### Some results on regular variation for distributions in queueing and fluctuation theory

- MathematicsJournal of Applied Probability
- 1973

For the distribution functions of the stationary actual waiting time and of the stationary virtual waiting time of the GI/G/l queueing system it is shown that the tails vary regularly at infinity if…

### Control and recovery from rare congestion events in a large multi-server system

- MathematicsQueueing Syst. Theory Appl.
- 1997

Deterministic fluid approximations are developed to describe the recovery from rare congestion events in a large multi-server system in which customer holding times have a general distribution and it is proved that, under regularity conditions, the fluid approxIMations are asymptotically correct as the arrival rate increases.

### On Stochastic Bounds for the Delay Distribution in the GI/G/s Queue

- MathematicsOper. Res.
- 1981

A counterexample is constructed to show that the steady-state delay distribution in the GI/G/s queue with the FIFO discipline need not be stochastically less (in the sense of first-order stochastic…

### Long-Tail Buffer-Content Distributions in Broadband Networks

- MathematicsPerform. Evaluation
- 1997