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}
}
  • W. Whitt
  • Published 14 November 2000
  • Mathematics
  • Queueing Systems
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… 

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References

SHOWING 1-10 OF 41 REFERENCES

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

TLDR
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

TLDR
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

TLDR
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

TLDR
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

TLDR
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

TLDR
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

  • J. Cohen
  • Mathematics
    Journal 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

TLDR
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

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