A convenience store has only one checkout counter. Customers arrive at the counter accord$-$

ing a Poisson process with rate $20$ per hour and wait in a queue if the counter is busy. The

time to checkout of a customer is i$.$i$.$d$.$ with mean $2$ minutes.

$($a$)$ What is the long$-$run fraction of customers who will find the counter free?

For the following problems, assume that the checkout times have exponential distribution.

$($b$)$ What is the long$-$run average number of customers waiting in the system $($at the queue

or at the counter$)?$

$[$ You may need to use the identity P $1$

n$=0$ nx n $=$ x

$(1$x$)2.]$

$($c$)$ What is the long$-$run average waiting time of a customer at the queue