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

Answer:

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%

Explanation:

As given,

Number of jobs = 5

Rate = 8 hour per day

Average hours = 1.3

Standard deviation - 1 hour

a)

Mean arrival = $\frac{No. of jobs}{rate}$

                    = $\frac{5}{8}$ = 0.6250 per hour

⇒Mean arrival rate in jobs per hour = 0.6250

b)

Mean service rate = $\frac{hour}{average hour}$

                              = $\frac{1}{1.3}$ = 0.7692 per hour

⇒Mean service rate in jobs per hour = 0.7692

c)

Average number of job waiting for service = $\frac{(0.6250)^{2} (1)^{2} + \frac{0.6250}{0.7692} }{2 ( 1 - \frac{0.6250}{0.7692} )}$

                                                                       = $\frac{1.05}{0.375}$ = 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 = $\frac{2.802}{0.6250}$

                                                                                                                  = 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 + $\frac{1}{0.7692}$

                 = 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 = $\frac{0.6250}{0.7692}$ ×100%

                                                                                = 0.8125×100%

                                                                                = 81.25% ≈ 81%

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