The city security officer $("($is tasked with designing a new security checkpost

for traffic entering the city of Ariel. The checkpost is expected to handle an average

arrival rate of $288$ vehicles per hour during peak times. A security guard takes an

average of $20$ seconds to inspect a vehicle, with negligible time between vehicles.

Assume the arrival rate of vehicles follows a Poisson distribution and the inspection

times by the guards follow an exponential distribution.

a$)(6$ pts$)$ What is the minimum number of security guards required to handle the traffic

entering the system? Justify your answer.

b$)$ The security officer is considering two operating options for the system:

In the first option $($Option $1),$ the arriving vehicles can join two separate queues,

each to be inspected by one guard, and the vehicles cannot switch from one queue to

another. $($We assume the two queues are mathematically identical, and that $50\%$ of

the arriving vehicles randomly join each queue.$)$

In the second option $($Option $2),$ the vehicles wait in one queue. Two guards inspect

the vehicles, where the first car in the queue proceeds to the first guard who

becomes available.

For each of the two options, calculate:

$1.(5$ pts$)$ The probability that the guards are idle, i$.$e$.$ they don$$t inspect any

vehicles.

$2.(8$ pts$)$ The average number of vehicles in the system.

$3.(8$ pts$)$ The average time a vehicle is waiting in queue.

$[$Table of Po for multiple$-$channel queues is attached.$]$

c$)(6$ pts$)$ Discuss which option $($Option $1$ or Option $2)$ would be more suitable for this

system based on your calculations. Explain why the system you chose is more

effective.