The busy period of the M/GI/infinity queue

An equation for the distribution Z((.)) of the duration T of the busy period in a stationary M/GI/infinity service system is constructed from first principles. Two scenarios are examined, being distinguished by the half-plane Re(theta) > theta (0) for some theta (0) less than or equal to 0 in which the generic service time random variable S, always assumed to have a finite mean E(S), has an analytic Laplace-Stieltjes transform E(e(-thetaS)). If theta (0) < 0 then E(e(-T)) is analytic in a half-plane (theta (1), infinity), where theta (0) less than or equal to theta (1) < 0 and (1) is determined by the distribution of S; then (Z) over bar (x) = Pr{T > x} = o(e(-sx)) for any 0 < s < \theta (1)\. When theta (0) = 0, E(e(-thetaT)) is analytic in (theta, infinity), and now more is known about T. Inequalities on the tail (Z) over bar((.)) are used to show that for any alpha greater than or equal to 1, E(T-alpha) is finite if and only if E(S-alpha) is finite. It follows that the point process consisting of the starting epochs of busy periods is long range dependent if and only if E(S-2) = infinity, in which case it has Hurst index equal to 1/2(3 - kappa), where kappa is the moment index of S. If also the tail (B) over bar (x) = Pr{S greater than or equal to x} of the service time distribution satisfies the subexponential density condition f(0)(x) (B) over bar (x - u)(B) over bar (u) du/(B) over bar (x) --> 2E(S) as x --> infinity, then (Z) over bar (x)/(B) over bar (x) --> e(lambdaE(S)), where lambda is the arrival rate.