Policy Iteration: Jack's Car Rental

State: There are two locations, and maximum $20$ cars at each location:

state $=[$# of cars in first location $(0->20),$ # of cars in second location $(0->20)]$

Action: Jack moves how many cars $($up to $5$ cars per night$)$ between two locations.

We define action $=[$# of cars moving from location $1$ to location $2],$ and action $=[-5,5]$

For example, action $=3$ means moving $"3"$ cars from location $1$ to $2.$

action $=-4$ means moving $"-4"$ cars from location $1$ to $2,$ or moving $"4"$ cars from location $2$ to $1.$

Reward:

$$10$ for each car rented $($must be available$).$

$$2$ for each car moved overnight.

Transitions:

Cars returned and requested following the Poisson distribution: $P(n)=\frac{{}^n}{n!}{e}^{-}$

Location cars requested in location $1$