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Expanding a Truck Service Facility

You manage the *Big-Rig Service Center*, which runs a
thriving business performing repairs on large trucks. The number of trucks arriving at the facility in need of
repair each morning is well-modeled by a Poisson random variable with a mean
value of 3.8. If you have a free
service bay available for an arriving truck, you start servicing it immediately.
Trucks that you cannot immediately accommodate with a free service bay do
not wait, but go to a competitor’s facility instead.
You currently have 5 service bays, but since you frequently have to turn
away business, you are considering expanding to 6, 7, or possibly 8 bays.

Including debt service, insurance, and taxes, your costs
for these expansion options, converted to a daily payment, would be:

__Total Bays__ |
__Incremental Daily Cost__ |

6 (add 1 bay) |
$175 |

7 (add 2 bays) |
$325 |

8 (add 3 bays) |
$475 |

Repairing each truck can take either 1, 2, or 3 full days; and you cannot tell
in advance how long the work will take. There
is a 30% chance that a truck can be completed by the end of one day, in which
case you charge $800. Given that service cannot be completed the first day, there
is 75% chance it will be finished at the end of the next day, in which case your
total charge is $1,500. Otherwise,
completing the service takes three full days or work, for which you charge
$2,100. Your operating costs are
$425 per occupied service bay per day.

Should you expand your facility, and if so, how
many service bays should you add? With this number of bays, what is the
average number of service requests denied per day?