Limiting Behavior of M/G/∞ System. Consider the M/G/∞ system in Section 3.12 with arrival rate λ and service distribution G, which has a mean α. Show that the limiting distribution of the quantity of...

Limiting Behavior of M/G/∞ System. Consider the M/G/∞ system in Section 3.12 with arrival rate λ and service distribution G, which has a mean α. Show that the limiting distribution of the quantity of items in the system Q(t) is Poisson with mean λα as t → ∞. Use the fact that α = ! ∞ 0 [1 − G(u)]du. Turning to the departures, consider the point process Dt(B) on R+ that records the numbers of departures in a time set B after time t. In particular, show that the number of departures Dt(0, b] in the interval (t, t + b] has a Poisson distribution with