Question

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 five jobs per 8-hour day. The time required to complete the jobs follows a normal probability distribution, with a mean time of 1.3 hours and a standard deviation of 1 hour. Answer the following questions, assuming that Gubser uses one welder to complete all jobs:

What is the mean arrival rate in jobs per hour? Round your answer to four decimal places.
jobs per hour _________
What is the mean service rate in jobs per hour? Round your answer to four decimal places.
jobs per hour _________
What is the average number of jobs waiting for service? Round your answer to three decimal places.
__________
What is the average time a job waits before the welder can begin working on it? Round your answer to one decimal place.
_________ hours
What is the average number of hours between when a job is received and when it is completed? Round your answer to one decimal place.
_________ hours
What percentage of the time is Gubser's welder busy? Round your answer to the nearest whole number.
_________ % of the time the welder is busy.

Answer 1

a) Mean arrival rate in jobs per hour = 0.6250

b) Mean service rate in jobs per hour = 0.7692

c) The average number of jobs waiting for service = 2.802

d) Average time a job waits before the welder can begin working on it = 4.5 hours

e) Average number of hours between when a job is received and when it is completed = 5.8 hours

f) Percentage of the time is Gubser's welder busy = 81%

Step-by-step explanation:

As given,

Number of jobs = 5

Rate = 8 hour per day

Average hours = 1.3

Standard deviation - 1 hour

a)

Mean arrival =
`(No. of jobs)/(rate)`

=
`(5)/(8)` = 0.6250 per hour

⇒Mean arrival rate in jobs per hour = 0.6250

b)

Mean service rate =
`(hour)/(average hour)`

=
`(1)/(1.3)` = 0.7692 per hour

⇒Mean service rate in jobs per hour = 0.7692

c)

Average number of job waiting for service =
`((0.6250)^(2) (1)^(2) + (0.6250)/(0.7692) )/(2 ( 1 - (0.6250)/(0.7692) ))`

=
`(1.05)/(0.375)` = 2.802

⇒The average number of jobs waiting for service = 2.802

d)

Average time a job waits before the welder can begin working on it =
`(2.802)/(0.6250)`

= 4.5 hr

⇒Average time a job waits before the welder can begin working on it = 4.5 hours

e)

Average number of hours between when a job is received and when it is completed = 4.5 +
`(1)/(0.7692)`

= 4.5 + 1.3

= 5.8 hours

⇒Average number of hours between when a job is received and when it is completed = 5.8 hours

f)

Percentage of the time is Gubser's welder busy =
`(0.6250)/(0.7692)` ×100%

= 0.8125×100%

= 81.25% ≈ 81%

⇒Percentage of the time is Gubser's welder busy = 81%