You are watching an arrival process in continuous time and you make a little game of it as follows. You are required to say the word 'now' immediately after the arrival event that you think to be the last one to occur before a particular time t > 0. You win if you succeed; otherwise you lose. If no arrival event occur before time t, you lose. Similarly, if you fail to say 'now' before time t, you lose. You may assume that the expected number of arrivals by t is greater than one. Consider the strategy in which you pick a threshold time s ∈ (0, t) and say 'now' immediately after the the first arrival that occurs after s. Answer the following questions relating to this strategy:
(a) (Poisson Arrivals) Suppose that arrivals are Poisson with rate λ > 1/t:
(i) Calculate an expression for the probability that you win using this strategy.
(ii) Which value of s maximizes the probability of winning? What is the corresponding optimal probability of winning?
(b) (Non-Memoryless Arrivals) Suppose that inter-arrival times are uniformly distributed between zero and some upper bound α < 1. Would you use the same strategy? Discuss in general terms how you would analyze this situation.