TITLE: Durrett's Probability: Example 5.5.14 (M/G/1 queue)
QUESTION [1 upvotes]: I am reading over the stationary distribution section Durrett's textbook Probability: Theory and Example, and I got stuck by a statement in an example he gave about the M/G/1 queue.
Here's the problem: Let $\mu = \Sigma k a_k$ be the mean number of customers that arrive during one service time where $a_k$ is the probability of the event that k customers arrive. We would like to show that the chain is positive recurrent if and only if $\mu < 1$. (The full example and proof are given in the attached screenshot)
Summary: In the forward direction, Durrett first proved that $E_x N \leq \frac{x}{1-\mu}$. He then showed that this $\leq$ is actually $=$.
To prove the equality, Durrett brought up several observations including $X_n$ decreases by at most one each time and for $x \geq 1, E_x T_{x-1} = E_1 T_0$. These are fine.
My question is: Durrett then claimed that based on the given observations, $E_x N = cx$, where $N = \{n \geq 0: X_n = 0\}$. This is not very obvious to me. May you show me how he got this result?
Thank you very much in advance.
The attached screenshot is the full example and proof, and my question is highlighted in blue.

REPLY [0 votes]: We have
$$
   E_x N = E_x T_{x-1} + E_{x-1} T_{x-2} + E_{x-2} T_{x-3} + \dots + E_1 T_0
$$
since in order to get to $0$, we must first get to $x-1$, then to $x-2$, then to $x-3$, and so on. This simplifies to $x E_1 T_0$.


TITLE: Algebraic Curves and Phase Diagrams of Physical Systems
QUESTION [10 upvotes]: Lots of low degree curves arise naturally as the phase spaces of physical systems (that is, the curve parameterized by $(q,p)$ where $q$ is a generalized position variable and $p$ is a generalized momentum variable (such that $p=\dot{q}$, etc).
For instance, if degree is 1, then we can construct the curve as the phase space of a particle moving with constant velocity, and it is parameterized by $(x,0)$, so is given by the equation $y=0$, up to a choice of coordinates.
For $d=2$, a conic, we can use, for instance, a spring, whose position is $\cos(x)$, and so $(\cos x,\sin x)$ parameterizes a circle (ellipses are also fairly easy to see how to do this way, and parabolas are linearly accelerated particles, though I don't see hyperbolas immediately)
For $d=3$, it is well known that a simple pendulum has phase space an elliptic curve with roots the initial height, the length of the pendulum, and minus the length.
Are there other nice examples like this? Is there a natural physical system that realizes hyperbolas? Does every real elliptic curve arise in this way? How about quartic or higher curves? I assume there are physical systems that work, as every ODE is modeling SOME system, but are there well-known examples that arise naturally in physics?

REPLY [2 votes]: I have a nice example concerning the electric field in two dimensions, which I'm not sure if physicists know about. One can easily phrase it in terms of ODE.
The electric field lines of a system of point charges in two dimensions form a pencil of (real) algebraic curves, the degree of which equals the sum of (the absolute values of) the charges (which have to be integers to begin with, and the system only has finite number of charges). You get a lot of singular ones in particular, and irreducible singular ones.
I'd be happy to see a reference of it. It's so natural that one would expect that the 19th century physicists/mathematicians know about them.