he Chinese take$-$out restaurant serves two dishes chow mein and spareribs. There are two

separate windows one for chow mein and one for spareribs. Customers arrive according to a

Poisson process of rate $20/$hr$.$ Sixty percent go to the chow Mein window and forty percent

to the spareribs window. Twenty percent of those who come from the chow mein window

with their order go next to the rib window; the other eighty percent leave the restaurant.

Ten percent of those who purcahse ribs then go to the chow mein window, and the the other

$90$ percent leave the restaurant. It takes on average $4$ minute to fill a chow$-$mein order and

$5$ minutes to fill a sparerib order and the service times are exponentially distributed. $($Recall

that for a stable M$/$M$/1$ queue with arival rate $\backslash$lambda and service rate $\backslash$mu the long run probability

of having n customers is $(1\backslash$lambda

$\backslash$mu $)\backslash$lambda n

$\backslash$mu n $)$

$($a$)$ What is the expected number of customers in the restaurant in the long run?

$($b$)$ What is the average amount of time that an arbitrary customer spends in the restaurant

in the long run?