The various types of traffic accidents that occur in a certain tunnel, over a given time period, constitute independent stochastic processes. We suppose that accidents stopping the traffic in the northbound (respectively, southbound) direction in the tunnel occur according to a Poisson process with rate a (resp., ß). Moreover, accidents causing the complete closure of the tunnel occur at rate 7, also according to a Poisson process. Let TN (resp., Ts) be the time during which the traffic is not stopped in the northbound (resp., southbound) direction, from the moment when the tunnel has just reopened after its complete closure.
(a) Calculate the probability density function of the random variable TN
(b) Show that
(c) Check your answer in (a) with the help of the formula in (b).