Consider a miniBison with only three stations $($pasta$,$ burritos, deli$)$ and a single server at each

station. An average of $9$ customers per hour arrive to the pasta station from outside of the miniBison,

an average of $16$ customers per hour arrive to the burrito station from outside of the miniBison,

and an average of $3$ customers per hour arrive to the deli station from outside of the miniBison, all

with exponential interarrival times. The pasta station can serve $40$ customers per hour, the burrito

station can serve $50$ customers per hour, and the deli can serve $30$ customers per hour, all with

exponential service times. After completing service at a station, a customer will go to a different

station according to the probabilities in the table given below. $($A customer may visit the same

station more than once before leaving the miniBison.$)$

$($a$)$ Suppose that Jen is currently being served at the burrito station, where she has patiently been

waiting for $7$ minutes. What is the probability that a customer arrives to the burrito station

from outside of the miniBison before Jen$$s service is completed?

$($b$)$ Calculate the total arrival rate at each of the stations.

$($c$)$ Find the steady state distributions of the number of customers at each station. Then give the

product form of the joint distribution of the number of customers at the three stations.

$($d$)$ What is the probability that there are less than $3$ total customers in the miniBison?

$($e$)$ Find the expected total number of customers in the miniBison.

$($f$)$ Find the expected total amount of time that a customer spends in the miniBison.

$($g$)$ To promote social distancing, suppose that Bucknell imposes a new rule saying that all students

must enter the miniBison at the pasta station, then go directly to the burrito station, then go

directly to the deli station, and then leave. Assume that this new rule does not change the

overall rate of students arriving to the miniBison, nor does it change the service rates. How

would this new rule change $($if at all$)$ your answers to questions $($e$)$ and $($f$)?$ Be as specific as

you can.