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

Abate, Joseph; Whitt, Ward

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 Fc(t) ∼ αt−3/2 as t → ∞, so that the associated waiting-time ccdf’s have asymptotic form Wc(t) ∼ βt−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 hav- ing the asymptotic form Fc(t) ∼ αt−3/2e−ηt 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 Gc(t) = eηtFc(t) for Fc(t) above, which has the power tail Gc(t) ∼ αt−3/2 as t → ∞. The Boxma-Cohen long-tail service-time distribution is a special case of an undamped EMIG.


Also Published In

Operations Research Letters

More About This Work

Academic Units
Industrial Engineering and Operations Research
Published Here
September 19, 2017


Published in Operations Research Letters, vol. 25, no. 1 (August 1999), pp. 25-31.