Question

Each airline passenger and his or her luggage must be checked to determine whether he or she is carrying weapons on to the airplane. Suppose that at O’hare International Airport, an average of 10 passengers per minute arrive (interarrival times are exponential). To check passengers for weapons, the airport must have a checkpoint consisting of a metal detector and baggage X-ray machine. Whenever a checkpoint is in operation, two employees are required. A checkpoint can check an average of 12 passengers per minute (the time to check a passenger is exponential). Under the assumption that the airport has only one checkpoint, answer the following questions:
a. What is the probability that a passenger will have to wait before being checked for weapons?
b. On the average, how many passengers are waiting in line to enter the checkpoint?
c. On the average, how long will a passenger spend at the checkpoint?

Answer 1

Answer:

a

$P_k = 0.83$

b

 $N_{\mu} \approx 4 \ passengers$

c

$T_{\lambda} = 0.5 \ minutes$

Step-by-step explanation:

From the question we are told that

The average number of passengers that arrive per minute is $\lambda = 10$

The average number of check that can be carried out in one minute is $\mu= 12$

Generally the probability that a passenger will have to wait before being checked for weapons is mathematically represented as

        $P_k = \frac{\lambda }{ \mu }$

=>    $P_k = \frac{10 }{ 12}$

=>    $P_k = 0.83$

Generally the number of passengers are waiting in line to enter the checkpoint is mathematically represented as

     $N_{\mu} = \frac{\lambda^2}{\mu (\mu -\lambda) }$

=>  $N_{\mu} = \frac{10^2}{12 (12 -10) }$

=>  $N_{\mu} \approx 4 \ passengers$

Generally the average time a passenger spend at the checkpoint is mathematically represented as

      $T_{\lambda} = \frac{ \frac{\lambda}{(\mu - \lambda)} }{ \lambda}$

=>   $T_{\lambda} = \frac{ \frac{ 10}{(12 - 10)} }{10}$

=>   $T_{\lambda} = 0.5 \ minutes$