Highway Model. Vehicles enter an infinite highway denoted by R at times that form a Poisson process N on the time axis R+ with intensity measure μ. For simplicity, assume the highway is empty at time...

Highway Model. Vehicles enter an infinite highway denoted by R at times that form a Poisson process N on the time axis R+ with intensity measure μ. For simplicity, assume the highway is empty at time 0. The vehicle arriving at time Tn enters at a location Xn on the highway R and moves on it with a velocity Vn for a time τn and then exits the highway. The velocity may be negative, denoting a movement in the negative direction and vehicles may automatically pass one another on the highway with no change in velocity. The Xn are i.i.d. with distribution F and are independent of N. The pairs (Vn, τn) are independent of N and, they are conditionally independent given the Xn with G_x(v, t) = P{Vn ≤ v, τ_n
≤ t|X_k, k ≥ 1, Xn = x}, a non-random distribution independent of n. (a) Justify that M = n δ(Tn,Xn,Vn,τn), is a Poisson process on R+ ×R2×R+ and describe its intensity. (b) Consider the departure process D on R × R+ where D(A × (a, b]) is the number of departures from A in the time interval (a, b]. Justify that D is a Poisson process and specify its intensity. Find the expected number of departures in (0, t]. (c) For a fixed t, let Nt(A) denote the number of vehicles in A ⊆ R at time t. Justify that Nt is a Poisson process on R and specify its intensity. (d) Suppose a vehicle is at the location x ∈ R at time t and let X(t) denote the distance to the nearest vehicle. Specify assumptions on μ, F and Gx(v, t) that would guarantee that there is at most one point at any location on the highway. Under these assumptions, find the distribution of X(t).