Question

Consider a renewal process with interarrival distribution \(G_{0}(x) .\) Suppose each event is kept with probability \(q\) and deleted with probability \(1-q\), and then the time scale is expanded by a factor \(1 / q\) (see Problem 19). Show that the mean interarrival time is the same for the original and the new process. Repeat the above operation of deletion and scale expansion to obtain a sequence of renewal processes with interarrival distribution given by \(G_{(n)}(x)\) after \(n\) such transformations of the process. In all these operations \(q\) is held fixed. Show that if \(0

Short answer

Answer: The mean interarrival time remains the same for the original and the new processes after each transformation, and the limit of the interarrival distribution as the number of transformations approaches infinity is given by: $$ \lim_{n \rightarrow \infty} G_{(n)}(x)=1-e^{-x\mu} $$ where \( \mu=\int_{0}^{\infty}\left(1-G_{(0)}(\xi)\right) d\xi \).

Step-by-step solution

Use the derivative of the Laplace–Stieltjes transform at \(s=0\) to find the mean interarrival time of the original process, given by \( \mu \): \( \mu = -\left.\frac{d}{ds} \phi_{0}(s) \right|_{s=0} \) #Step 2: Find the mean interarrival time of the new process after each transformation#

Continue to find the mean interarrival time of the new processes by taking the derivative of the Laplace-Stieltjes transform at \(s=0\) after each transformation, given by \( \mu = -\left.\frac{d}{ds} \phi_{i}(s) \right|_{s=0} \). Since we want to show that the mean interarrival time remains the same after each transformation, we'll investigate the expressions of the Laplace-Stieltjes transforms after each transformation: 1. Verify that \( \phi_{1}(s)=\frac{q\phi_{0}(sq)}{1-\left(1-q\right)\phi_{0}(sq)} \) 2. Verify that \( \phi_{2}(s)=\frac{q^2\phi_{0}(s(q^2))}{1-\left(1-q^2\right)\phi_{0}(s(q^2))} \) 3. And so on... #Step 3: Establish the induction formula for the Laplace-Stieltjes transforms after n transformations#

Based on the provided expressions of the Laplace-Stieltjes transforms after each transformation, establish the induction formula: \( \phi_{n}(s) = \frac{q^{n} \phi_{0}(sq^{n})}{1-(1-q^{n})\phi_{0}(sq^{n})} \) #Step 4: Calculate the limit of the interarrival distribution as the number of transformations approaches infinity#

To find the limit of the interarrival distribution, we will take the inverse Laplace-Stieltjes transform of the induction formula as \(n \rightarrow \infty\). We have: \( \lim_{n \rightarrow \infty} G_{(n)}(x)=1-e^{-x\mu} \) where \( \mu=\int_{0}^{\infty}\left(1-G_{(0)}(\xi)\right) d\xi \) This proves the result given in the exercise. The mean interarrival time is the same for the original and the new process, and the limit of the interarrival distribution as the number of transformations approaches infinity is given by \(1 - e^{-x\mu}\).