Average residual service Time (with vacations) As t-> M(t)t→>λandL(t)t→= vacation rate Now, let I= 1 if system is on vacation and I=0 if system is busy By Little's Theorem we have E[=Efvacations]=P(system idle)=1-P=A EM >my=(1-P)EMV Hence, remember W=R/(1-P) E ElV R=λ EⅨ21,(1-p)E 2 2EV] 2(1-P)2E[V]Average Residual Service Time (with vacations) • As t-> ∞ , M(t)/t -> λ and L(t)/t -> λv = vacation rate • Now, let I = 1 if system is on vacation and I = 0 if system is busy • By Little’s Theorem w e have, – E[I] =E[#vacations] = P(system idle) = 1- ρ = λ v E[V] – => λv = (1- ρ)/E[V] • Hence, remember W = R/(1- ρ) R λ E[X 2] 2 + (1- ρ )E[V 2] 2 E[V] = W λ E[X 2] 2(1- ρ ) + E[V 2 ] 2 E[V] = Eytan Modiano Slide 9