Answer to Question #173568 in Operations Research for ANJU JAYACHANDRAN

Question

3(b) 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

Solution:

Arrival rate of each custom officer, $\lambda=\frac{30}{8\times3}=1.25$ per hour

Service rate, $\mu=\frac1{32}\times 60=1.875$ per hour

(1). The probability that all the custom officers are idle $p=\frac{\lambda}{\mu}=\frac{1.25}{1.875}=\frac23$

(2). The expected number of passengers in the queues $N=\dfrac{\lambda}{\mu-\lambda}=\dfrac{1.25}{1.875-1.25}=2$

(3). Average waiting time in the system $\mathrm{W}=\dfrac{\lambda}{\mu(\mu-\lambda)}$

$=\dfrac{1.25}{1.875(1.875-1.25)}=\dfrac{16}{15}$ hours