Operation Management Week 6

Waiting Line Questions Render with A
PROBLEMS

1. A waiting line meeting the M/M/1 assumptions has an arrival rate of 4 per hour and a service rate of 12 per hour. What is the probability that the waiting line is empty?
Po = 1 - λ/μ = 1 - 4/12 = 8/12 or 0.667. (The variety of queuing models, easy) {AACSB: Analytic Skills}

2. A waiting line meeting the M/M/1 assumptions has an arrival rate of 4 per hour and a service rate of 12 per hour. What is the average time a unit spends in the system and the average time a unit spends waiting?
Ws = 1 / (μ - λ) = 1 / (12 – 4) = 1/8 or 0.125; Wq = λ / (μ*(μ-λ)) = 4 / (12*8) = 1/24 or 0.0417. (The variety of queuing models, easy) {AACSB: Analytic Skills}

3. A waiting line meeting the M/M/1 assumptions has an arrival rate of 10 per hour and a service rate of 12 per hour. What is the average time a unit spends in the system and the average time a unit spends waiting?
Ws = 1 / (μ - λ) = 1 / (12 – 10) = 1/2 or 0.5; Wq = λ / (μ*(μ-λ)) = 10 / (12*2) = 10 / 24 or 0.4167. (The variety of queuing models, easy) {AACSB: Analytic Skills}

4. A waiting line meeting the M/M/1 assumptions has an arrival rate of 10 per hour and a service rate of 12 per hour. What is the probability that the waiting line is empty?
Po = 1 - λ/μ = 1 - 10/12 = 2/12 or 0.1667. (The variety of queuing models, easy) {AACSB: Analytic Skills}

5. A crew of mechanics at the Highway Department garage repair vehicles that break down at an average of λ = 7.5 vehicles per day (approximately Poisson in nature). The mechanic crew can service an average of μ = 10 vehicles per day with a repair time distribution that approximates an exponential distribution.
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 in the system at any one time? (a) Utilization is ρ = 7.5 / 10 =