Question: The information stand of a vendor in a trade fair is visited by customers that arrive according to a Poisson process at a rate of λ = 0.2 customer/min. Upon arrival, each customer, independently of the others and of the number of customers already in the stand, decides with probability p to talk to an employee to obtain additional information about the commercialized products, whereas with probability 1 - p she only takes a quick look at the information material made available in large exhibition area of the stand. The stand is attended by m employees and the time required by an employee to serve a customer is random, with exponential distribution of mean my = 10 minutes. If all the employees are busy, customers that wish additional information wait in an orderly manner in a single queue for the first available employee. Customers that are not interested to talk to an employee, instead, are not required to queue and can freely move in the exhibition area of the stand. In this case, a customer stays in the stand a random time, uniformly distributed in the interval U = [2, 4] min. Determine:
a) The maximum value of p such that the waiting time in the queue does not grow indefinitely if the stand is attended by a single employee, that is, m = 1.
b) The percentage of time the employee is busy serving customers, for p = 0.25 and m = 1.
c) The probability that a customer waits in queue more than 30 min, with p = 0.25 and m = 1.
d) The mean time a customer that talks to an employee stays in the stand, with p = 0.75 and m = 3.
e) The mean time any customer stays in the stand, with p = 0.75 and m = 3.