$4$ Queuing at a Traffic Light

To begin this problem, let us first introduce the queue length of a basic G$/$G$/1$ queuing system:

$E[L_q]=\frac{{}^2}{1-}\frac{C{V}_A^2+C{V}_T^2}{2}$

Notice the right$-$hand side of the multiplication sign is the conversion factor to adjust for the

interarrival and service time distributions$-$where $"$M$"$ represents the exponential distribution,

$"$G$"$ represents the general distribution, and $"$D$"$ represents a deterministic$/$constant rate. Here,

the $C{V}_A^2$ is the squared coefficient of variation for the interarrival times:

$C{V}_A^2=\frac{{}_A^2}{E[A{]}^2}$

Of course the $C{V}_T^2$ term is defined analogously for service time in the form $\frac{{}_T^2}{E}[T{]}^2.$ Now for

our problem. To regulate merging traffic onto a freeway, the transportation department installed

a traffic light signal on an on$-$ramp. When vehicles are present, the signal lets one vehicle into

the expressway precisely every $12$ seconds $($service time$).$ Vehicles wait in a single queue in the

merging lane. Assume the times between consecutive vehicle arrivals to the on$-$ramp lane are

exponentially distributed with a mean of $E[A]=15$ seconds.

What is the probability of finding no vehicles in the merging lane? Note we can express $$

and $$ in vehicles per minute to keep things simple.

How would you characterize the traffic light queue $(\frac{M}{M}\frac{?}{1},\frac{G}{G}\frac{?}{1},$ etc$)?$ What is the

average number of vehicles in the lane $($in the line plus at the light$)$ waiting to merge into the

freeway? $($Hint: think about the "exponentially distributed" interarrival times and "precise"

service times.$)$