$4.(2$ pts$)$ Orders arrive at a distribution center via a Poisson process with an hourly rate of $100.$ Each

order has only one item and takes a constant processing time of $3\mathrm{.}25$ minutes.

$2|$ P a g e

$($a$)(1$ pt$)$ How many order processers are needed to ensure $90\%$ of orders are processed $($filled$)$ in

one hour? $[$Hint: Given a Poisson random variable N with $=100$ per hour, we have that the

smallest integer $$ that satisfies $\{<=\}>=0\mathrm{.}90$ is $113.$ Specifically, $\{<=113\}=0\mathrm{.}9095.]$

$($b$)(1$ pt$)$ How many order processers are needed to ensure $90\%$ of orders are processed $($filled$)$ in

three hours? $[$Hint: Given a Poisson random variable N with $=300,$ we have that the smallest

integer $$ that satisfies $\{<=\}>=0\mathrm{.}90$ is $322.$ Specifically, $\{<=322\}=0\mathrm{.}9020.]$

$5.(2$ pts$)$ Customers go through a $2-$step process at a small food pantry. The first is a check of eligibility

$($with a mean $2$ minutes$).$ The second is the collection of weekly food $($with a mean of $2$ minutes$).$

Assume a closed $($CONWIP$)$ line with $=0.$

$($a$)(0\mathrm{.}5$ pts$)$ Assume deterministic process times. What is the steady$-$state CT$,$ TH$,$ and utilization

for each station? Demonstrate line behavior with a Gantt chart through the exit of $6$ jobs from

the system. Please make sure to include jobs arriving after Job $6$ up until Job $6$ exits.

$($b$)(1$ pt$)$ Now assume a Poisson line with exponential process times. Process times for the first $6$

jobs at each station are shown below. What is CT$,$ TH and the utilization of each station given

those times? For comparative purposes with $($a$),$ do not include the first two periods of

queueing at Station $1$ in Job $2$s CT$,$ start TH calculations after Station $2$ becomes busy, and start

utilization calculations for Station $2$ once it becomes busy. These steps remove the transient

effect. $[$Hint: Demonstrate the line behavior with a Gantt chart through the exit of $6$ jobs from

the system. Please make sure to include jobs arriving after Job $6$ up until Job $6$ exits.$]$

Job S$1$ time S$2$ time

$123$

$214$

$321$

$433$

$511$

$632$

$($c$)(0\mathrm{.}5$ pts$)$ Two phenomena caused by the variability are affecting system performance in $($b$).$

Point to one example of each in the Gantt chart from $($b$).$