# Telephone calls arrives at telephone booth following Poisson distribution at an average of 5 minutes between one and next. The length of phone call is assumed to be experientially distributed with an average of 4 minute What is the probability that a person arriving at the booth will have to wait? What is the average length of queue that forms time to time?

Solution:

λ = 1 call per minute

= 1 / 5 per minute

μ = 1 call per 4 minute

= 1 / 4 per minute

i) Probability that server is busy (ρ) = $$\frac{λ}{μ}$$ = $$\frac{\frac{1}{5}}{\frac{1}{4}}$$ = 0.8

ii) Average length of queue (Lq) = $$\frac{ρ^2}{1-ρ}$$ = $$\frac{0.64}{1-0.8}$$ = 3.2