Customers arrive at the ATM at the rate of 12 customers per hour and spend 2 minutes, on average, on all the transactions. This system is modeled by the single-server Bernoulli queuing process with 1-minute frames.
a. Compute the transition probability matrix for the number of customers at the ATM at the end of each frame.
b. Currently the ATM is idle. Find the probabilities that, in 3 minutes, the ATM will be idle; will be serving one customer with no one in the queue; will be serving one customer with one customer waiting in the queue; and will be serving one customer with two customers waiting in the queue.
c. Use your answer in part b to compute the expected number of customers in the system 3 minutes after the ATM was idle.
d. Use your answers in parts b and c to compute the expected number customers waiting in the queue 3 minutes after the ATM was idle.