Problem $3$: Queuing Theory $(30)$

Given: Vehicles arrive at a single EZ pass toll booth at an average rate of $50$ vehicles per

minute. On average, a single vehicle requires $1$ second to be serviced. Assume that the vehicle

arrival pattern can be described by the Poisson distribution and that service time is exponentially

distributed.

Find: Calculate the following performance characteristics of the queuing system considering

steady state condition.

$($a$)$ percent of the time that the booth will be idle

$($b$)$ probability that at any given time there will be three vehicles in the system

$($c$)$ average number of vehicles in the system.

$($d$)$ average queue length.

$($e$)$ average time each vehicle spends in the queuing system. seconds.

$($f$)$ probability that more than $50$ vehicles will arrive at the toll booth within a one$-$minute

interval