Average Age of Renewal Process

1. Definition

• Let $S_n$ be the n-th renewal
• Then $S_{N(t)}$ = time of the last renewal (prior to time t)
• Let $A(t) = t - S_{N(t)}$
• $A(t)$ is called the age of the renewal process
• Interpretation: A(t) is the time since last renewal 

2. Average Age of Renewal Process

• What is $R_n$? $R_n = \int^{S_n}_{S_{n-1}} A(t) dt$
• $R_n$ is the area under the curve for one cycle
• so $E(R_n)) = E[(X_n)^2/2] = E[X^2_n]/2$
• Thus the long-run average of the process is 
• $\frac{E[R_n]}{E[X_n]} = \frac{E[X^2_n]}{2E[X_n]}$