Modeling and analysis of stochastic systems
Modeling and analysis of stochastic systems
How heavy-tailed distributions affect simulation-generated time averages
ACM Transactions on Modeling and Computer Simulation (TOMACS)
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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(&thetas;)&thetas;0 for some &thetas;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−&thetas;S). If &thetas;0−&thetas;T) is analytic in a half-plane (&thetas;1,∞), where &thetas;0≤&thetas;11 is determined by the distribution of S; then $\overline{Z}$(x)≡Pr{Tx}=o(e−sx) for any 0s1|.When &thetas;0=0, E(e−&thetas;T) is analytic in (0,∞), and now more is known about T. Inequalities on the tail $\overline{Z}$(⋅) are used to show that for any α≥1, E(Tα) is finite if and only if E(Sα) 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(S2)=∞, in which case it has Hurst index equal to [frac12](3−κ), where κ is the moment index of S.If also the tail $\overline{B}$(x)=Pr{S≥x} of the service time distribution satisfies the subexponential density condition ∫0x $\overline{B}$(x−u) $\overline{B}$(u) du/ $\overline{B}$(x)→2E(S) as x→∞, then $\overline{Z}$(x)/$\overline{B}$(x)→eλE(S), where λ is the arrival rate.