Answer to Question #259391 in Operations Research for DEEPAK KUMAR

Question

Three custom officers check the luggage of the passengers of an airport. The passengers are found to arrive at an average rate of 30 per 8 hours a day. The amount of time a custom officer spends with the passenger is found to have an exponential distribution with mean service time 32 minutes. (5) (i) Find the probability that all the custom officers are idle. (ii) Find the expected number of passengers in the queues. (iii) Find the expected waiting time of passenger in the system

Answer 1

Average arrival rate $= \lambda = \dfrac{30}{8\times 24} = \dfrac{5}{32}$

Average service time $= \mu = \dfrac{60}{32} = \dfrac{15}{8}$

Hence, utilisation factor, $\rho = \dfrac{\lambda}{\mu} = \dfrac{5}{60}$

a.) The probability that all the custom officers are idle $= \rho = \dfrac{5}{60}$

b.) The expected number of passengers in the queues $= \dfrac{\lambda}{\mu-\lambda} = \dfrac{55}{32} = 1.71$

c.) The expected waiting time of passenger in the system $= \dfrac{1}{\mu-\lambda} = \dfrac{1}{\dfrac{1}{11}} = 11$