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]}

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