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) ¸ fft \Gamma3=2 as t ! 1, so that the associated waiting-time ccdf's have asymptotic form W c (t) ¸ fit \Gamma1=2 as t ! 1. 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) ¸ fft \Gamma3=2 e \Gammajt as ..

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