
exponentials
In a hardware store you must first go to server 1 to get your goods and then to server 2 to pay for them. Suppose that the times for the times for the two activities are exponentially distributed with means 6 min and 3 min. Compute the average amount of time it takes Bob to get his goods and pay if when he comes in there is a customer named Al with server 1 and no one at server 2.

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 $\displaystyle T$ = the total amount of time Bob spends in the hardware store
Let $\displaystyle T_{1} $ = the amount of time Bob spends waiting in line for Al to get his goods from server 1
Let $\displaystyle T_{2}$ = the amount of time it takes for Bob to get his goods from server 1
Let $\displaystyle T_{3} $ = the amount of time Bob spends waiting in line for Al to pay server 2 for his goods
and Let $\displaystyle T_{4} $ = the amount of time it takes for Bob to pay server 2 for his goods
Then $\displaystyle T= T_{1}+T_{2}+T_{3}+T_{4} $
and $\displaystyle E[T] = E[T_{1}]+E[T_{2}]+E[T_{3}]+E[T_{4}] $
$\displaystyle E[T_{1}] = 6 $ because of memoryless property of the exponential distribution
$\displaystyle E[T_{2}] = 6 $
To find $\displaystyle E[T_{3}] $, condition on the availability of server 2 after Bob is done with server 1
$\displaystyle E[T_{3}] = E[T_{3}\text{server 2 is free}]P(\text{server 2 is free}) + E[T_{3}\text{server 2 is not free}]$$\displaystyle P(\text{server 3 is not free}) $
$\displaystyle = 0 + 3 \cdot P(\text{Bob is done with server 1 before Al is done with server 2}) $
$\displaystyle = 3 \cdot \frac{1/6}{1/6+1/3} = 1 $ (I used the probability than one exponential random variable is smaller than another exponential random variable.)
and $\displaystyle E[T_{4}] = 3 $
so E[T] = 6+6+1+3= 16 minutes

Thanks so much. I tried to construct something similar, but got lost around E(T3). And your final answer matches with the back of my book btw.
