help plz, Traffic Engineering question !

[kintoki]

after observing arrival & departures at toll booth over 60 min period, it was noted that the arrival and service rate are deterministic
but instead of being uniform, they change over time according to a known function

- the arrival rate is given by :

- the departure rate is given by :

what is minutes after the beginning of the observation period. determine the total vehicular-delay at the toll booth & longest queue assuming D/D/1 regime?

 

[kintoki]

Can someone help solve the problem?

 

[IlPadrino]

As deterministic queuing goes, this one makes you work the math (integrals and derivatives!) to get the answer. Do you understand the D/D/1 notation?

If you use wolframalpha.com, you can more easily visualize what's going on...

Plot the arrival function. Plot the departure function. Where they cross over is where the queue dissipates (after which, there is no wait because the departure rate is greater than the arrival rate and you've cleaned out the queue. A quick calculation (which might have some mistakes!) gives me 39.2 minutes.

The total vehicle delay is the area between the two curves. A quick calculation (same disclaimer!) gives me me 52 veh-mins.

The longest queue is the max of the difference between the two functions. You need to find where the derivative of the difference between the two curves is 0 (a quick calculation gives me 15.6) and evaluate the difference between the two curves at that point (a quick calculation gives me 1.8).

Frankly, this doesn't seem like a very good problem.

Again, I did some quick calcs so I might have made a mistake. If you have the answer, I can easily verify. Do you?

 

[kintoki]

OMG

ur genius, I asked everyone i could but they didn't have any idea.

//

unfortunately,i don't have the answer .. actually i resolved one example about this thing

and read some ebook but it was different style.

but it look right to me lol

give me a sec to try and i'll reply to u

thx man, u r the best.

 

[kintoki]

finally hhh

u r surely a genius,i got the same answer.

thx again ^^

 

[IlPadrino]

Yikes... don't celebrate just yet! I went back and thought about this again. Because you're given arrival and departure rates that vary with time (rather than being constant), you need to take the integrals of both the arrival and departure rates and set them equal to each other in order to find out how long it takes the queue to dissipate. Does this make sense to you? The integral of the arrival rate is the total number of vehicles that have arrived. And the integral of the departure rate is the total number of vehicles that have departed. When the number of vehicles that departed equal the number that have arrived (and you departure rate is greater than your arrival rate), the queue is empty.

So some more quick math gets me 61.5 minutes until queue dissipation.

The total vehicle delay is now the integral of the difference between the two integrals evaluated at t=61.5 minutes... quick math is about 1900 vehicles.

The queue length is the difference between the two integrals and the max occurs where the derivative of that equals 0... which is the 39.2 minutes I calculated in my first post. Evaluating the queue length at t=39.2 gives you about 50 vehicles.

Sorry for the confusion! The principles are the same, but in order to graph the number of vehicles that have arrived and departed you need to integrate first. Everything else is the same concept.