An independence result concerning the arrival rate of and the provision of transport to tourists

What is the relationship between the arrival rate of tourists at a transport providing firm's depot and the long run expected wait time experienced by individual tourists in this depot? We use a renewal theoretic framework in this note to answer this question. Our analysis yields two results. First, we show that the long run expected wait time is independent of the arrival rate of the tourists. Second, we explain why this long run expected wait time is equal to the average residual life of a particular renewal process.