Answer to Question #182591 in Operations Research for Rashmi

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 $= \dfrac{1}{\mu-\lambda} = \dfrac{1}{\dfrac{1}{11}} = 11$