Answer:
a
[tex]P_k = 0.83[/tex]
b
[tex]N_{\mu} \approx 4 \ passengers[/tex]
c
[tex]T_{\lambda} = 0.5 \ minutes[/tex]
Step-by-step explanation:
From the question we are told that
The average number of passengers that arrive per minute is [tex]\lambda = 10[/tex]
The average number of check that can be carried out in one minute is [tex]\mu= 12[/tex]
Generally the probability that a passenger will have to wait before being checked for weapons is mathematically represented as
[tex]P_k = \frac{\lambda }{ \mu }[/tex]
=> [tex]P_k = \frac{10 }{ 12}[/tex]
=> [tex]P_k = 0.83[/tex]
Generally the number of passengers are waiting in line to enter the checkpoint is mathematically represented as
[tex]N_{\mu} = \frac{\lambda^2}{\mu (\mu -\lambda) }[/tex]
=> [tex]N_{\mu} = \frac{10^2}{12 (12 -10) }[/tex]
=> [tex]N_{\mu} \approx 4 \ passengers[/tex]
Generally the average time a passenger spend at the checkpoint is mathematically represented as
[tex]T_{\lambda} = \frac{ \frac{\lambda}{(\mu - \lambda)} }{ \lambda}[/tex]
=> [tex]T_{\lambda} = \frac{ \frac{ 10}{(12 - 10)} }{10}[/tex]
=> [tex]T_{\lambda} = 0.5 \ minutes[/tex]
