A bank wants to determine how many automated teller machines (ATMs) to install at a very busy site. It is willing to model the time between arrivals of customers as exponentially distributed with mean...

A bank wants to determine how many automated teller machines (ATMs) to install at a very busy site. It is willing to model the time between arrivals of customers as exponentially distributed with mean 3 minutes. The time a person spends at the ATM is uncertain, but can be modeled as an exponentially distributed random variable with mean 2 minutes. The bank’s performance measure is the number of customers that must wait more than 5 minutes to begin using an ATM. It wants no more than 15% of all customers who must wait to have to wait more than 5 minutes. Of course, the bank does not want to install more ATMs than necessary. How many should it install? (Hint: You will need the result from Exercise 17.)

Exercise 17

Let W_q
be a random variable representing the steady-state delay in queue of a customer arriving to an M/M/s queue (therefore (therefore wq = E[W_q]). Show that

for a > 0. (Hint: Try the M/M/1 case first. Let L be a random variable representing the number of customers in the system when a customer arrives. Therefore Pr{L = j} = πj. Derive an expression for Pr{Wq > a | L = j} for j = 0, 1, 2, … by recalling that the sum of independent, exponentially distributed random variables is Erlang. Then apply the law of total probability to obtain

Finally, use the definition of conditional probability directly.)