Question

A mechanic at the Highway Department Garage repairs vehicles that break down at an average   of ? = 3 vehicles per hour. The mechanic can service a vehicle on an average of 15 minutes.  (20 Points)

a. What is the utilization rate for this service system?

b. What is the average time before the facility can return a breakdown to service?

c. How much of that time is spent waiting for service?

d. How many vehicles are likely to be waiting for service at any one time?

e. What is the probability that the system is empty?

f. What is the probability that there is more than one vehicle in the system?

g. What will be the waiting time if another mechanic is added?

Answer 1

Arrival rate = 3 per hr

Sevrice time is 15 min per vehicle which means 60 min per 4 vehicle or 1 hr for 4 vehicle

Service rate   = 4 per hr

a) Utilization rate = / = 3/4 =.75 =75%

b) average time before the facility can return a breakdown to service is the average waiting time in system

Ws = 1 /( - ) = 1 / (4-3) = 1 hr

c) time is spent waiting for service is waiting time in queue

Wq = / ( ( - ) ) = 3 / (4*(4-3) ) = 0.75 Hr

d) vehicles likely to be waiting for service at any one time is length of queue

Lq =  ² / ( ( - ) ) = 3² / (4*(4-3) ) = 2.25 vehicles

e) Probability that system is empty

P0 = 1 - = 1-0.75 = 0.25 =25%

f) P n>1 = ( / ) ⁿ⁺¹

P n>1 = (3/4)² = 0.5625 =56.25%

g) If another mechanic is added, service rate would double

so = 8 per hr

Wq = / ( ( - ) ) = 3 / (8*(8-3) ) = 0.075 Hr = 4.5 min