# Essay Operation Management Week 6

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

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The number of vehicles out of service is Ls = 8 / (11-8) = 8/3 = 2.667. The cost of waiting is $150 x Ls = $150 x 2.667 = $400. Server cost is $300 per day for a total of $700. (The variety of queuing models, moderate) {AACSB: Analytic Skills}

9. A crew of mechanics at the Highway Department garage repair vehicles that break down at an average of λ = 8 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 probability that the system is empty?

b. What is the probability that there is precisely one vehicle in the system?

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

d. What is the probability of 5 or more vehicles in the system?

(a) P0 = 1 – 8/10 = 0.20; (b) Pn>1 =(8/10)2 = 0.64; the probability of exactly one is .36 -.20 = .16; (c) 0.64 as previously calculated; (d) Pn>4 = (8/10)5 = 0.32768. (The variety of queuing models, moderate) {ACSB: Analytic Skills}

10. A crew of mechanics at the Highway Department garage repair