Question

If \(A(t)\) and \(Y(t)\) are, respectively, the age and the excess at time \(t\) of a renewal process having an interarrival distribution \(F\), calculate $$ P\\{Y(t)>x \mid A(t)=s\\} $$

Short answer

The short answer for the given problem is: $$ P\{Y(t) > x \mid A(t) = s\} = \frac{P\{X_{N(t)+1} > s + x\}}{P\{X_{N(t)+1} > s\}} $$ where \(X_{N(t)+1}\) is the remaining time till the next event in the renewal process.

Step-by-step solution

Understand the problem

We need to find $$ P\{Y(t) > x \mid A(t) = s\} $$ where \(A(t)\) and \(Y(t)\) are the age and excess of a renewal process at time \(t\), which has interarrival distribution \(F\).

Use the definition of conditional probability

By definition, the conditional probability is given by $$ P\{Y(t) > x \mid A(t) = s\} = \frac{P\{Y(t) > x \cap A(t) = s\}}{P\{A(t) = s\}} $$

Determine the joint probability of \(Y(t) > x\) and \(A(t) = s\)

\(A(t) = s\) means that the renewal process has just completed its \(s\)th arrival at time \(t\). For \(Y(t) > x\), we have the remaining time until the next arrival is greater than \(x\) with respect to that. So the joint probability is $$ P\{Y(t) > x \cap A(t) = s\} = P\{A(t) = s, N(t+s) - N(t) = 0\} $$ where \(N(t)\) is the number of arrivals in the process up to time \(t\).

Determine the probability of \(A(t) = s\)

The probability of \(A(t) = s\) can be determined by examining the time since the last event. Specifically, it is the probability of having an arrival at time \(t-s\) and no arrivals within the interval \([t-s,t)\): $$ P\{A(t) = s\} = P\{N(t-s) - N(t) = -1\} $$

Use renewal process properties

We know the renewal process has interarrival distribution \(F\). We can rewrite the joint probability as $$ P\{Y(t) > x \cap A(t) = s\} = P\{A(t) = s, N(t+s) - N(t) = 0\} = P\{X_{N(t)+1} > s+x\} $$ and we can rewrite the probability of \(A(t) = s\) as $$ P\{A(t) = s\} = P\{N(t-s) - N(t) = -1\} = P\{X_{N(t)+1} > s\} $$

Calculate the conditional probability

Now we have everything we need to calculate the conditional probability. Substitute the obtained results back into the equation obtained in Step 2: $$ P\{Y(t) > x \mid A(t) = s\} = \frac{P\{X_{N(t)+1} > s + x\}}{P\{X_{N(t)+1} > s\}} $$ Note that the conditional probability does not depend on \(t\).