Markov Chain: Long-Run Average Costs

This lecture covers the Markov Law of Large Numbers (LLN) applied to an irreducible Markov chain with transition matrix P and invariant distribution π. It discusses the first return time to a state, reward functions, and the long-run average costs in an (S, S) inventory model. The proof of the Markov LLN implies that the expectation of the reward converges to a limit. The lecture also explores the dynamics of the inventory level based on demand, the cost of lost customers, and the unique invariant distribution. It concludes with the implications of the Markov LLN on the long-run average holding cost and the cost of lost customers.