From moment explosions to the asymptotic behavior of the cumulative distribution
We study the Tauberlan relations between the moment generating function of a random variable and the complementary cumulative distribution function of this variable. We show that if the moment generating function is finite only on part of the real line, then the behavior of the MGF near the critical moment gives the asymptotic behavior of the complementary cumulative distribution function of this variable. We apply our results to an arbitrary superposition of a CIR process and the time-integral of this process.
Abstract The aim of the present paper is to give the main characteristics of the finite-source G / M /r queue in equilibrium. Here unit i stays in the source for a random time having general distribution function F i ( x ) with density f i ( x ). The service times of all units are assumed to be identically and exponentially distributed random variables with means 1/μ. It is shown that the solution to this G / M /r model is similar in most important respects to that for the M/M/ r model.