Problem

$3$

: Queuing Theory

$($

$3$

$0$

$)$

Given: Vehicles arrive at a single EZ pass toll booth at an average rate of

$5$

$0$

$$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

$5$

$0$

$$vehicles will arrive at the toll booth within a one

$-$

minute

interval