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1/ passengers arrive at a train station according to a poisson process of rate lambda per minute and trains depart station according to a renewal process with inter-departure times uniformly distributed between a and b minutes. Find the long run fraction of time when the station is empty.

2/ An item with exponential life time distribution of rate lambda is installed in a system. It is inspected periodically, and is replaced immediately by a new item of same life time distribution if found defective at an inspection. The inspection is performed every h units of time, so they occur at times t=h,2h,3h..., and the time to perform an inspection may be ignored. Suppose each inspection costs a, a failed item in system incurs a continuous cost at rate of b per unit time, and there is no replacement cost. Find the long run cost per unit time.

Thank you so much for your time.