Renewal Theory

Summary of Chapter 7 from Introduction to Probability Models (11th ed) by Sheldon M. Ross.

1 - Introduction

• Definition of inter-arrival/holding times, jump times, renewal process.
• Proof that N(t) is finite for all t.
• Proof that N(∞) = lim_{t → ∞} N(t) = ∞.

2 - Distribution of N(t)

• P(N(t) = n) in terms of cumulative distribution functions of jump times.
• P(N(t) = n) as an integral.
• m(t) = E(N(t)) in terms of cumulative distribution functions of jump times.
• Claim (without proof) that m(t) uniquely determines F.
• Claim (without proof) that m(t) is finite.
• Renewal equation

3 - Limit Theorems and their Applications

• N(t)/t converges to 1/μ when t → ∞.
• Example 7.7: analyzing dice games using renewal theory
• Elementary renewal theorem (without proof).
• Stopping time for a sequence of random variables.
• Wald's equation for stopping time.
• Proof of the elementary renewal theorem.
• Central limit theorem for renewal processes.

4 - Renewal Reward Processes

• Notation for rewards.
• R(t)/t converges to E(R)/E(X) when t → ∞.
• Elementary reward renewal theorem (without proof).
• Example 7.16
• Example 7.18: average age of renewal process.
• Example 7.20: average no. of people waiting at bus stop.

5 - Regenerative Process

TODO

6 - Semi-Markov Process

• Definition of semi-markov process.
• Long-run proportion of semi-markov process.

7 - Inspection Paradox

TODO

8 - Computing the Renewal Function

TODO

9 - Application to Patters

TODO

10 - The Insurance Ruin Problem

TODO