We consider a model of a large number of single server queues. When a job arrives, we assume it chooses between queues. The job then chooses to join the shortest of these queues. We show that such choice can dramatically reduce queue sizes.

## Little’s Law

Here is the long run queue length; is the expected waiting time; is the arrival rate at the queue.

## Diffusion Control Problems

- The Hamilton-Jacobi-Bellman Equation.
- Heuristic derivation of the HJB equation.

## Continuous Time Dynamic Programs

- Continuous-time dynamic programs
- The HJB equation; a heuristic derivation; and proof of optimality.

## Markov Decision Processes

- Markov Decisions Problems; Bellman’s Equation; Two examples

## Dynamic Programming

- Dynamic Programs; Bellman’s Equation; An example.

## Ito’s Formula: a heuristic derivation

- A heuristic look at the stochastic integral.
- heuristic derivation of Itô’s formula.