Consider an M/G/∞ queue, i.e., a queue with Poisson arrivals of rate λ in which each arrival i, independent of other arrivals, remains in the system for a time Xi, where {Xi; i ≥ 1} is a set of IID rv s with some given CDF F(x).
You may assume that the number of arrivals in any interval (t, t + E) that are still in the system at some later time τ ≥ t + E is statistically independent of the number of arrivals in that same interval (t, t + E) that have departed from the system by time τ .
(a) Let N(τ ) be the number of customers in the system at time τ . Find the mean, m(τ ), of N(τ ) and find Pr{N(τ ) = n}.
(b) Let D(τ ) be the number of customers that have departed from the system by time τ . Find the mean, E [D(τ )], and find Pr{D(τ ) = d}.
(c) Find Pr{N(τ ) = n, D(τ ) = d}.
(d) Let A(τ ) be the total number of arrivals up to time τ . Find Pr{N(τ ) = n | A(τ ) = a}.
(e) Find Pr{D(τ + E) - D(τ ) = d}.
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.