Study the following queuing models, assuming Poisson arrivals and exponential service time distributions, and that mean service time $=1$ in all cases:

$(1)$ M$/$M$/1$ queuing model with $=0\mathrm{.}8$;

$(2)$ M$/$M$/1$ queuing model with $=1\mathrm{.}2$;

$(3)$ M$/$M$/3/10$ queuing model with $=1\mathrm{.}2$;

Determine whether the steady state will be reached in each case. If yes, determine

$($a$)$ the average number of jobs in the system $($either waiting in queue or in service$)$;

$($b$)$ the average time that a job spends waiting in the queue $($not including service time$).$

Use both analytical method and simulation to demonstrate your results.