I remember doing this problem in a stochastic models class (although I don't think it was a hardware store and I don't think the customers had names).

Here's what I did:

Let = the total amount of time Bob spends in the hardware store

Let = the amount of time Bob spends waiting in line for Al to get his goods from server 1

Let = the amount of time it takes for Bob to get his goods from server 1

Let = the amount of time Bob spends waiting in line for Al to pay server 2 for his goods

and Let = the amount of time it takes for Bob to pay server 2 for his goods

Then

and

because of memoryless property of the exponential distribution

To find , condition on the availability of server 2 after Bob is done with server 1

(I used the probability than one exponential random variable is smaller than another exponential random variable.)

and

so E[T] = 6+6+1+3= 16 minutes