The escape from L.A.

Discrete Mathematics Level 5

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 \(n\) locations for cars \(x_1,x_2,\ldots,x_n\) each of which can hold only one car. At each moment in time, traffic flow on Rosecrans works as follows:

  • If there is no car on the first position on Rosecrans (\(x_0\)), a new car will enter it at the rate \(p_\text{injection}\).
  • If there is a car at last position on the avenue (\(x_n\)), the car will leave Rosecrans (and move on to the highway) at the rate \(p_\text{removal}\).
  • If there is a car at \(x_i\) and no car at \(x_{i+1}\), the car on \(x_i\) will move to \(x_{i+1}\) at the rate \(v_\text{car}\).
  • If there is a car at \(x_i\) and a car at \(x_{i+1}\), the car at \(x_i\) will stay put.

What is the flow rate (on average) of cars leaving Rosecrans?

Assumptions and details

  • \(p_\text{injection} = 0.6\)
  • \(p_\text{removal} = 0.3\)
  • \(v_\text{car} = 1\)
  • Assume the street is very long (\(n\rightarrow\infty\))
  • Give your answer in \(\text{cars}/\text{unit time}\)

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