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

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)∼ αt −3/2 as t→∞, so that the associated waiting-time ccdf's 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 ηt F 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.