Question: (M/M/∞ model.) Consider a trade fair where visitors arrive according to a Poisson process at a rate of λ = 240 visitors per hour. The time a visitor spends in the trade fair can be modeled as an exponential rv with mean my hours, independent of the number of visitors. Let x(t) be the number of customers in fair at time t.
a) Prove that x(t) is a BDP and determines the birth and death rates.
b) Find the stationary PMD and mean of x(t) and the system stability conditions.
c) Find the probability that the fair is visited by more than M customers in the first 30 minutes after its opening.
d) Find the stationary probability of having more than M customers in the fair at any given time.