Question

At a production machine, parts arrive according to a Poisson process at the rate of 0.35 parts per minute. Processing time for parts has an exponential distribution with a mean of 2 minutes. What is the probability that a random part arrival finds that there are already 8 parts in the system (in machine + in the queue)

1. 0.0247
2. 0.0576
3. 0.0173
4. 0.082

Answer 1

Correct Answer - Option 3 : 0.0173

Concept:

 

The probability that there is 'n' number of parts are in the system is:

P_n = ρⁿP_o

where ρ = traffic intensity, P_o represents that the system is idle i.e. (1 - ρ) and

\(ρ=\frac{λ}{μ}\)

where λ = arrival rate and μ = service rate.

Calculation:

Given:

λ = 0.35 parts/min, μ = 1 parts in 2 min ⇒ 0.5 parts/min

\(\rho=\frac{\lambda}{\mu}⇒\frac{0.35}{0.5}=0.7\)

The probability that there is '8' number of parts are in the system is

P₈ = ρ⁸(1 - ρ)

∴ P₈ = (0.7)⁸(1 - 0.7) ⇒ 0.0173