points$]$ Transit service is to be provided along a circular bus route of length

$D=20km$ that visits $n=24$ stops. Buses travel at a cruising speed $v_{\text{m}\text{a}\text{x}}=20k\frac{m}{h}r$ and

experience a fixed loss and dwell time of $t_s=1$min at each stop $($for all activities, including

passenger boarding and alighting, door opening and closing, etc.$).$ Demand is steady with

the same rate $(q=0\mathrm{.}25pa\frac{x}{h}r)$ between every origin$-$destination pair. The transit agency's

operating contract stipulates that the agency must serve this route with an average service

time across all passengers equal to $T=0\mathrm{.}80hrs.$ This time is the sum of the average waiting

time at an origin and the average riding time.

$($a$)$ Write an expression for the cycle time for a bus to complete service on the loop in terms

of the variables provided if there is not terminal time.

$($b$)$ How many buses $($at a minimum$)$ does the agency need to run if it decides to operate the

a single clockwise route? Provide a numerical answer using the values provided. $($Hint:

You should use the largest possible headway that will fulfill the contract.$)$

$($c$)$ How many buses $($at a minimum$)$ would it need to run both clockwise and counter$-$

clockwise routes? Assume that passengers use the direction with the least riding time.

Provide a numerical answer using the values provided.

$($d$)$ Compare the number of buses required to operate only one direction of service $($b$)$ with

the number of buses required to operate in both directions $($c$).$ Determine the general

conditions in terms of $D,n,v_{\text{m}\text{a}\text{x}},t_s,q$ and $T,$ which favor two directions over one.

$($e$)$ The agency now needs to decide what size buses to purchase. Suppose that the de$-$

mand rate for each origin$-$destination pair, $q=0\mathrm{.}25pa\frac{x}{h}r,$ is the mean of a Poisson

distribution. What is the smallest bus capacity that can be used for the one$-$directional

service $($part b$)$ if the agency wants to ensure that there is enough space onboard for

passengers $95\%$ of the time? What is the smallest bus capacity that can be used for the

two$-$directional service $($part c$)?$