ASYMPTOTICS FOR M G/1 LOW-PRIORITY WAITING-TIME TAIL PROBABILITIES/

We consider the classical M/G/1 queue with two priority classes and th e nonpreemptive and preemptive-resume disciplines. We show that the lo w-priority steady-state waiting-time can be expressed as a geometric r andom sum of i.i.d. random variables, just like the M/G/1 FIFO waiting -time distribution. We exploit this structures to determine the asympt otic behavior of the tail probabilities. Unlike the FIFO case, there i s routinely a region of the parameters such that the tail probabilitie s have non exponential asymptotics. This phenomenon even occurs when b oth service-time distributions are exponential. When non-exponential a symptotics holds, the asymptotic form tends to be determined by the no n-exponential asymptotics for the high-priority busy-period distributi on. We obtain asymptotic expansions for the low-priority waiting-time distribution by obtaining an asymptotic expansion for the busy-period transform from Kendall's functional equation. We identify the boundary between the exponential and non-exponential asymptotic regions. For t he special cases of an exponential high-priority service-time distribu tion and of common general service-time distributions, we obtain conve nient explicit forms for the low-priority waiting-time transform. We a lso establish asymptotic results for cases with long-tail service-time distributions. As with FIFO, the exponential asymptotics tend to prov ide excellent approximations, while the non-exponential asymptotics do not, but the asymptotic relations indicate the general form. In all c ases, exact results can be obtained by numerically inverting the waiti ng-time transform.