An equation for the distribution Z(.) of the duration T of the busy period in a stationary M/GI/∞ service system is constructed from first principles. Two scenarios are examined, being distinguished by the half-plane Re(θ) > θ{sub}0 for some θ{sub}0 ≤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{sup}(-θS)). If θ{sub}0 X} = o(e{sup}(-sx)) for any 0 < s <θ{sub}1. Whenθ{sub}0 = 0, E(e{sup}(-θT)) is analytic in (0, ∞), and now more is known about T. Inequalities on the tail Z{top}(-)(.) are used to show that for any α≥1, E(T{sup}α) is finite if and only if E(S{sup}α) 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{sup}2) =∞, in which case it has Hurst index equal to 1/2(3 - k), where k is the moment index of S. If also the tail B{top}(-)(x) = Pr{S ≥ X} of the service time distribution satisfies the subexponential density condition integral from x=0 to x=x of (B{top}(-)(x - u)B{top}(-)(u) )du/B{top}(-)(x) →2E(S) as x→∞, then Z{top}(-){sub} (x)/ B{top}(-){sub}(x)→e{sup}(λE(S)), where λ is the arrival rate.
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