Problem $3.$ Managing and Estimating Delays. AUzon$$s customer call center wants you to analyze its operation for the time between $1$pm and $2$pm$.$ On average, it takes $15$ minutes to handle a call with a standard deviation of $12$ minutes. Calls come in every $2$ minutes on average with a standard deviation of $2\mathrm{.}5$ minutes during this period. DO NOT use the Queuing Models and the Excel spreadsheet to answer the questions below. Instead, use the table below to answer the questions.

a$)(5$ points$)$ What is the capacity of each call agent in calls per hour? Show your work!

Capacity$=60$min$/$ avg call time

$60$min$/15$min$=4$ calls per hr

b$)(5$ points$)$ What is the minimum number of call agents to strictly handle the demand? Show your work!

Arrival rate$=60$min$/$ avg time between calls

$60$min$/2$min$=30$min

Number of agents needed$=$ arrival rate$/$ capacity per agent

$30$ calls per hr$/4$ calls per agent$=7\mathrm{.}5$ agents

You need a minimum of $8$ agents

c$)(3$ points$)$ What is the capacity of the call center, if $9$ agents are scheduled to answer calls, in calls per hour? Show your work!

Calculate capacity with $9$ agents:

Number of agents x capacity per agent

$9$ agents x $4$ calls per hr$=36$ calls per hr

d$)(6$ points$)$ If $9$ agents are scheduled to answer calls, what is the average time a customer spends in the system in minutes? Show your work!

e$)(6$ points$)$ If $9$ agents are scheduled to answer calls, what is the average total number of customers at the call center either waiting or talking to call agent? Show your work!

f$)(6$ points$)$ If $9$ agents are scheduled to answer calls, what is the percentage of time $($probability$)$ that the customer call center is idle? Show your work!

g$)(8$ points$)$ At most, how many call agents should be scheduled for this period to keep the utilization of the call center at least at $70\%?$ Show your work!