The rate a certain insurance company charges its policyholders alternates between r1 and r0. A new policyholder is initially charged at a rate of r1 per unit time. When a policyholder paying at rate r1 has made no claims for the most recent s time units, then the rate charged becomes r0 per unit time. The rate charged remains at r0 until a claim is made, at which time it reverts to r1.

Suppose that a given policyholder lives forever and makes claims at times chosen according to a Poisson process with rate λ, and find

(a) Pi , the proportion of time that the policyholder pays at rate ri , i = 0, 1;

(b) the long-run average amount paid per unit time.