You're driving from Caltech, in Pasadena, to San Diego after a long day in the lab. You're cruising along and everything is going great until you hit Rosecrans Avenue, a horrible street in horrible Los Angeles. Here, everything grinds to a halt and you sit in mind numbing gridlock for a long time. Bored to tears, you start thinking about your situation and come up with the following simple model.
Let's approximate Rosecrans Ave as a 1d lattice with locations for cars each of which can hold only one car. At each moment in time, traffic flow on Rosecrans works as follows:
What is the flow rate (on average) of cars leaving Rosecrans?
Assumptions and details
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Let p i represent the long-run probability that a car is present in space x i . Considering the long run net flow in and and out of x n we have p n − 1 ( 1 − p n ) = 0 . 3 p n .
Since the road is long we can assume p n − 1 = p n and this value is clearly non-zero! Simple algebra gives p n = 0 . 7 , and the flow out is 0 . 3 × 0 . 7 = 0 . 2 1 .
Notice that the injection rate does not influence the net flow out at Rosecrans which is intuitively obvious- but it will influence the values of p i when i is small. If the injection rate were to satisfy p i n j e c t i o n = 1 − p r e m o v a l then p i would be constant throughout the road (by symmetry).
Here because 0.6<0.7 ( p i n j e c t i o n < p r e m o v a l ) the traffic will get denser as we approach Rosecrans. If p i n j e c t i o n were bigger than 0.7 then the jam would be heaviest closest to the Caltech.