Can you help me with a possible methodology?

Approximately 380 student & parent-driven drop-off vehicles arrive within a 25 minute period to access parking at the school. (Approximately 1/5 of the vehicles are parent driven). The in-Queue is a 1 mile stretch of county road that runs from a traffic light to the school. Prior to that, students arrive from any one of three directions at a variable rate. The traffic light throws a dimension into this that we really do not need to consider because the queue is consistently solid for the whole mile for at least 15 of the 25 minutes. Suffice it to say that everyday (M-F), a line forms from the light to the school entrance… bumper to bumper. The dead-end county road has light commuter traffic – mostly out to the light… maybe 30-35 vehicles (including school arrivals) within that same 25 minute period. Buses have another entrance.

The students have devised a “brilliant” way to circumvent interference by the out-traffic, without considering how much they are slowing down the in-traffic by doing so. Periodically, a student will drive past the school entrance (there is a double lane to facilitate thru in-traffic). These students then make a u-turn at the next available community entrance. Each student who does this then helps create a 4-way stop situation (thereby causing a queue to form for the out-traffic) by letting alternating in-traffic make the turn into the school. In doing so, they are probably doubling the wait time for those students in the in-traffic queue.

How best to show the difference between this method and allowing the thru out-traffic to flow at 30 mph with no stops so they are not “in the way” of the students waiting to turn? Obviously those students approaching the turn into the school when there are no outbound vehicles would not have to come to a complete stop.

I would like to demonstrate for a math class how much time is added to each student’s wait time by their “brilliant” 4-way stop approach. Just the hesitation alone while everyone ensures the out-bound vehicle is going to allow the in-bound vehicle to turn can be 3-5 seconds… not to mention everyone having to come to a full stop at this point.

I am trying to discover if there is a reasonably simple way to predict the probability of a Queue Wait time exceeding a certain length for a multi-server queueing system with a poisson arrival rate and a constant service rate. For example, for a call centre with a call arrival rate of 50 / Hour, and a service time of 5 minutes, how can I calculate how many servers I will need in order that 80% of calls have a wait time of > 20 seconds.

I would be very grateful if you could let me know whether this calculation is possible, and if so, what the equation is. Many thanks

James Nutting