Scenario

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

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

As usual, 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 $($with a mean and standard deviation equal to each other$),"G"$ represents the general distribution $($where the mean and standard deviation do not necessarily equal each other$),$ and $"D"$ represents a deterministic$/$constant rate $($with a standard deviation equal to zero$).$ Remember that $C{V}_A^2$ 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. 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 $E[T]=12$ seconds$-$here we use "precisely" to indicate the service time is constant. 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.

Questions

Calculate the input parameters $$ and $($in vehicles per minute$).$

What is the probability of finding no vehicles in the merging lane, $P_0?$

How would you characterize the traffic light queue $,$ etc.$)?$ What is the average number of vehicles in the lane $($in line plus at the light$)$ waiting to merge into the freeway?

Using your answer from above, calculate the waiting time $($seconds$)$ for a vehicle to completely merge into the freeway.

What have you learned in this course? Would you add to $($or subtract from$)$ the material or change the format? I know this class is very different from your other courses at McNeese, and $I^{\text{'}}m$ very much open to feedback on how to present the information better.