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 functions (ccdfs) of the service times all have the asymptotic form F^c(t) ∼ αt^−3/2 as t → ∞, so that the associated waiting-time ccdfs have asymptotic form W^c(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 having the asymptotic form F^c(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 G^c(t) = e^ηtF^c(t) for F^c(t) above, which has the power tail G^c(t) ∼ αt^−3/2 as t → ∞. The Boxma–Cohen long-tail service-time distribution is a special case of an undamped EMIG.