The Busy Period of the M/GI/∞ Queue

Abstract

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)>θ₀ for some θ₀≤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^−θS). If θ₀<0 then E(e^−θT) is analytic in a half-plane (θ₁,∞), where θ₀≤θ₁<0 and θ₁ is determined by the distribution of S; then $\bar Z(x) \equiv \Pr \{ T >x\} = {\text{o(e}}^{--sx} )$ for any 0<s<|θ₁|.

When θ₀=0, E(e^−θT) is analytic in (0,∞), and now more is known about T. Inequalities on the tail $\bar Z$(⋅) are used to show that for any α≥1, E(T ^α) is finite if and only if E(S ^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 ²)=∞, in which case it has Hurst index equal to [frac12](3−κ), where κ is the moment index of S.

If also the tail $\bar B$(x)=Pr{S≥x} of the service time distribution satisfies the subexponential density condition ∫₀ ^x $\bar B$(x−u) $\bar B$(u) du/ $\bar B$(x)→2E(S) as x→∞, then $\bar Z$(x)/$\bar B$(x)→e^λE(S), where λ is the arrival rate.