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

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) similar to alpha t(-3/2) as t --> infinity, so that the associated waiting-time ccdf's have asymptotic form W-c(t) similar to beta t(-1/2) as t --> infinity. Thus the second moment of the service time and the mean of the waiting time are infinite. Our result here also extends ou; 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) similar to alpha t(-3/2)e(-eta t) as t --> infinity. 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(eta t) F-c(t) for F-c(t) above, which has the power tail G(c)(t) similar to alpha t(-3/2) as t --> infinity. The Boxma-Cohen long-tail service-time distribution is a special case of an undamped EMIG. Published by Elsevier Science B.V.