Problem $1$:Problem $2$:

$(20$ points$)$ Consider a queueing system a Poisson arrival process, with the mean interarrival

time $7$ minutes. The customers observe the queue. If they observe $k$ customers in the queue, they

do not join the queue with probability $l_k.$

$l_k=\{\begin{array}{ccc}\frac{k}{4},& if& k<mn\; id="EditorElement\_582036">4</mn>\\ 0,& 1\text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

The service time is exponentially distributed with mean time of $6$ minutes.

$($a$)$ Determine the average number of customers in the system

$($b$)$ Determine the number of customers served in $100$ seconds.

$(15$ points$)$Ture or False

Consider a single$-$server queueing system with a Poisson arrival process and exponentially

distributed service times. The interarrival times have a mean value of $20$ mins and the mean

service time is $25$ mins.

$($a$)$ The server will be always busy after the first customer arrives $($T$/$F$).$

$($b$)$ The queue will grow and explode $(\frac{T}{F}.$

$($c$)$ If a second server with the same service time distribution is added to the service facility, the

system can reach a steady$-$state condition $($T$/$F$).$