# The escape from L.A.

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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