$[20$ pts$]$ Parts $($a$)$ and $($b$)$ of this problem concern an M$/$M$/1$ queueing system. Parts $($c$)$ and

$($d$)$ concern M$/$M$/1$ queueing systems that are different from the system in parts $($a$)$ and $($b$).$

$($a$)$ How many tasks are in an M$/$M$/1$ queueing system $($either waiting or being served$),$ on

average, if the average arrival rate is $8$ per second and the average service time is $\frac{1}{10}$

second $(0\mathrm{.}1$ second$)?$

$($b$)$ In part $($a$),$ how many tasks are being served, on average?

$($c$)$ What is the utilization of an $\frac{M}{M}\frac{?}{1}$ queue if the average number waiting is equal to $\frac{1}{2}?$

$($d$)$ For an M$/$M$/1$ queueing system with utilization $=\frac{1}{2},$ what is the minimum buffer length

needed to ensure that the probability of buffer overflow is less than $1\%?$