Airplanes enter a rectangular land as shown in the following figure. The sector length is $50 \text{ nm}$. The spacing between airplane as they enter the land is $20 \text{ nm}$ plus an exponentially distributed random variable with a mean of $1 \text{ nm}$. Suppose that airplane travels at $300 \text{ nm}$ per hour. What is the average number of airplanes in a land?
My thought: Even though I spent many hours thinking of the way to use Little's Law to link the relationship of the distances between airplanes and the sector length, I don't really see the relationship between those two. Furthermore, why do we need the information about the width of the land $= 20 \text{ nm}$? In short, I'm completely stumbled upon this problem. If anyone could give some hints on how to approach this problem, I would really appreciate it.
First let us suppose the spacing is $20$ nm, without the added random variable. If we measure distance from the left edge of the box we will have three planes in the box when the leftmost is between $0$ and $10$ nm and two planes in the box when the leftmost is between $10$ and $20$ nm. This leads to an average of $2.5$ airplanes in the box. The width of the box is immaterial.
Adding in the random variable to the spacing makes it possible that there are less planes in the box. When the leftmost plane is at $0$, it will be the only one in the box if the random variable of the spacing in front of it exceeds $30$, which is rather unlikely but possible. There will be only two if the sum of two random variables exceeds $10$, still unlikely. For each position from $0$ to $20$ for the leftmost plane you need to compute the chance that there is one or two more planes in the box. It is now possible that the leftmost plane is at a position greater than $20$, so you need to compute the chance of that and the expected number of planes in front of it.