Explicit M/G/1 waiting-time distributions for a class of long-tail service-time distributions

  • Authors:

  • Joseph Abate;Ward Whitt

  • Affiliations:

  • 900 Hammond Road, Ridgewood, NJ 07450-2908, USA;Room A117, AT&T Labs, Shannon Laboratory, 180 Park Avenue, Florham Park, NJ 07932-0971, USA

  • Venue:
  • Operations Research Letters
  • Year:
  • 1999

Quantified Score

Hi-index 0.00

Visualization

Abstract

O.J. Boxma and J.W. Cohen recently obtained an explicit expression for the M/G/1 steady-state waiting-time distribution for a class of service-time distributions with power tails. We extend their explicit representation from a one-parameter family of service-time distributions to a two-parameter family. The complementary cumulative distribution function (ccdf's) of the service times all have the asymptotic form F^c(t)~@at^-^3^/^2 as t-~, so that the associated waiting-time ccdf's have asymptotic form W^c(t)~@bt^-^1^/^2 as t-~. Thus the second moment of the service time and the mean of the waiting time are infinite. Our result here also extends our own earlier explicit expression for the M/G/1 steady-state waiting-time distribution when the service-time distribution is an exponential mixture of inverse Gaussian distributions (EMIG). The EMIG distributions form a two-parameter family with ccdf having the asymptotic form F^c(t)~@at^-^3^/^2e^-^@h^t as t-~. We now show that a variant of our previous argument applies when the service-time ccdf is an undamped EMIG, i.e., with ccdf G^c(t)=e^@h^tF^c(t) for F^c(t) above, which has the power tail G^c(t)~@at^-^3^/^2 as t-~. The Boxma-Cohen long-tail service-time distribution is a special case of an undamped EMIG.