Virginia's Ron McPherson Electronics Corporation retains a service crew to repair machine breakdowns that occur on an average of λ = 4 per 8-hour workday (approximately Poisson in nature). The crew can service an average of µ = 10 machines per workday, with a repair time distribution that resembles the negative exponential distribution.

a. The utilization rate of this service system =.
b. The average downtime for a broken machine =
c. The average number of machines waiting to receive service at any given time =
d. The probability that there is more than one machine that is in the system =

(a) The utilization rate = 50%

(b) The average down time = 2 hours

(c) Average number of machines waiting = 0.5 Machines

(d) Pn>1   = 0.25

Explanation:

Given that:

Arrival Rate= λ = 4 per day

Service Rate = μ = 8 hour per day

Now,

a)

The utilization rate of this service system  is given as;

The utilization rate =  λ / μ

                               = 4/8

                                = 0.5 = 50%

b)

The average downtime for a broken machine  is calculated as;

The average down time = 1/( μ- λ)

                                         = 1/(8-4)

                                         = 1/4

                                         = 0.25 days

It is given that work day = 8 hours

Therefore, 0.25*8 = 2 hours

c)

Average number of machines waiting to be serviced at any given time  is calculated as;

Average number of machines waiting = λ²/(μ*(μ-λ))

                                                               = 4²/(8*(8-4))

                                                               = 16/32

                                                                = 0.5 Machines

d) The probability that more than 1 machine is in the system  is calculated as;

Pn>k = (λ/μ)^k+1

where k number of machine = 1 machine

Pn>1 = (4/8)¹⁺¹

         =(4/8)²

         = 0.5²

         = 0.25

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