Customers arrive according to a Poisson process with rate A outside a bank where there are two automated teller machines (ATM). The two ATMs are not identical. We estimate that 30% of the customers...

Customers arrive according to a Poisson process with rate A outside a bank where there are two automated teller machines (ATM). The two ATMs are not identical. We estimate that 30% of the customers use only ATM no. 1, while 20% of the customers use only ATM no. 2. The other customers (50%) make use of either ATM indifferently. The service times at each ATM are independent exponential random variables with parameter μ. Finally, there is space for a single waiting customer. We define the states

(a) Write the balance equations of the system. Do not solve them.

(b) Calculate, in terms of the limiting probabilities,

(i) the variance of the number of customers who are waiting to use an ATM,

(ii) the average time that an arbitrary customer, who enters the system and wishes to use ATM no. 2, will spend in the system.