A college-owned van is used until it will not run anymore, and then it is immediately replaced by a similar new one whose lifetime \(Z\) has discrete distribution \(p_{1}=P[Z=1], p_{2}=P[Z=2], \ldots\), where the times are in months. The process continues through successive van replacements. Let \(\left(X_{n}\right)\) be the chain defined by \(X_{n}=\) remaining life of the van in use at month \(n\). Notice that a new van has a remaining life with the distribution of \(Z\); otherwise the remaining life of a van next month is one month less than the remaining life this month. Find the transition matrix of this Markov chain. Show that if the mean van lifetime \(m\) is finite, then the limiting distribution of \(X_{n}\) exists. Find that limiting distribution.