Meeting of Vehicles. Vehicles enter a one-mile stretch of a twoway highway at both ends by independent Poisson processes and move at 60 miles per hour to the opposite end. Let λ and μ denote the rate...

Meeting of Vehicles. Vehicles enter a one-mile stretch of a twoway highway at both ends by independent Poisson processes and move at 60 miles per hour to the opposite end. Let λ and μ denote the rate of Poisson arrivals at the two ends labeled 0 and 1. Assuming the highway is empty at time 0, show that the probability is (λe^−μ
+ μe^−λ)/(λ + μ) that the first vehicle that enters at either end does not encounter another vehicle coming from the other direction during the one-mile stretch.

Show that this probability is (λe−μ + μe−(λ+2μ))/(λ + μ) for such an encounter avoidance for the first vehicle to arrive from end 0. Hint: For this second problem, let N1(t) denote the Poisson process of arrivals at end 1, and let T denote the time of the first arrival at end 0. Then N1(T ) denotes the number of arrivals at 1 before the first arrival at 0. Use its distribution from Exercise 6.Exercise 6From Theorem 22, we know that the sum N = N1+···+Nn of independent Poisson processes is Poisson. Prove this statement by verifying that N satisfies the defining properties of a Poisson process.