The rockwell electronics corporation retains a service crew to repair machine breakdowns that occur on an average of 3 per day (approximately poisson in nature). the crew can service an average of 8 machines per day, with a repair time distribution that resembles the exponential distribution. a. what is the utilization rate of this service system? b. what is the average downtime for a machine that is broken? c. how many machines are waiting to be serviced at any given time? d. what is the probability that more than one machine is in the system? e. what is the probability that more than two are broken and waiting to be repaired or being serviced? f. what is the probability that more than three are in the system? g. what is the probability that more than four are in the system?

Consider the following calculation

Explanation:

Here Machine break down rate = arrival rate = λ = 3

Service rate = μ = 8

a) utilization rate of this service system = ρ = λ/μ = 3/8 = 0.375

b) Average downtime for a machine that

is broken = W = E[T] = 1/(μ-λ)= 1/5 = 0.2 day

c) No of machines are waiting to be serviced at any given time = Lq = = 0.225

d) probability that more than one machine is in the system = P(N >=1) = ρ = 0.375

Probability that more than two are broken and waiting to be repaired or being serviced = Conditional Probability that queue is not empty but have more than 2 machines in the system

= Pn/ρ2 = ρ(n-2)*(1- ρ)

= 1 - ρ = 0.625

Probability that more than 3 are broken and waiting to be repaired or being serviced = = ρ(n-2)*(1- ρ) = ρ(1-ρ) = 0.2343

Probability that more than 4 are broken and waiting to be repaired or being serviced = ρ*ρ*(1-ρ) = 0.087

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