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$q,$ CAMMIMS$1611.$E$\mathrm{.}017.$

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The Fore and Aft Marina is located on the Ohio River near Madison, Indiana. Fore and Aft currently operates a docking facility where only one boat at a time can stop for gas and servicing.

The manager of the Fore and Aft Marina wants to investigate the possibility of enlarging the docking facility so that two boats can stop for gas and servicing simultaneously. Assume that the arrival rate is $4$ boats per hour and that the service rate for each server is $9$ boats per hour. $($Round your answers to four decimal places.$)$

$($a$)$ What is the probability that the boat dock will be idle?

$($b$)$ What is the average number of boats that will be waiting for service?

$($c$)$ What is the average time $($in hours$)$ a boat will spend waiting for service?

$h$

$($d$)$ What is the average time $($in hours$)$ a boat will spend at the dock?

$h$

$($e$)$ If customers expect to be able to be in and out of the marina within $10$ minutes on average, does this system provide acceptable service?

$,$ since the average time a boat will spend at the dock is $---$ Select$---$ hat$(v)10$ minutes.

$($f$)$ The manager of Fore and Aft Marina is interested in balancing good customer service with the cost of providing this service. To achieve this, the manager would like the customer's average time in the system to be as close to $10$ minutes as possible, but not exceeding $10$ minutes. Is enlarging the capacity of the dock to handle two boats at a time a good way of achieving this?

Both channels will be empty approximately $---$Select$---$ hat$(v)\%$ of the time, and when customers do show up$,$ they $---$Select$---$ hat$(v)$ likely to have to wait for $10$ minutes. On the whole, the expansion $---$Select$---$ hat$(v)$ be best way of achieving their goal.

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$3.$ Points$]$

CAMMIMS$1611.$E$\mathrm{.}027.$

Gubser Welding, Inc., operates a welding service for construction and automotive repair jobs. Assume that the arrival of jobs at the company's office can be described by a Poisson probability distribution with an arrival rate of three jobs per $8-$hour day. The time required to complete the jobs follows a normal probability distribution, with a mean time of $2\mathrm{.}2$ hours and a standard deviation of $1\mathrm{.}4$ hours. Answer the following questions, assuming that Gubser uses one welder to complete all jobs.

$($a$)$ What is the mean arrival rate in jobs per hour? $($Round your answer to four decimal places.$)$ per hour

$($b$)$ What is the mean service rate in jobs per hour? $($Round your answer to four decimal places.$)$ per hour

$($c$)$ What is the average number of jobs waiting for service? $($Round your answer to four decimal places.$)$

$($d$)$ What is the average time $($in hours$)$ a job waits before the welder can begin working on it$?($Round your answer to two decimal places.$)$

$h$

$($e$)$ What is the average number of hours between when a job is received and when it is completed? $($Round your answer to two decimal places.$)$

$h$

$($f$)$ What percentage of the time is Gubser's welder busy? $($Round your answer to the nearest whole number.$)$

$\%$

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