The escape from L.A.

Probability 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 nn locations for cars x1,x2,,xnx_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 (x0x_0), a new car will enter it at the rate pinjectionp_\text{injection}.
  • If there is a car at last position on the avenue (xnx_n), the car will leave Rosecrans (and move on to the highway) at the rate premovalp_\text{removal}.
  • If there is a car at xix_i and no car at xi+1x_{i+1}, the car on xix_i will move to xi+1x_{i+1} at the rate vcarv_\text{car}.
  • If there is a car at xix_i and a car at xi+1x_{i+1}, the car at xix_i will stay put.

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

Assumptions and details

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