On the moment distance of Poisson processes

Consider the distance between two i.i.d. Poisson processes with arrival rate λ > 0 and respective arrival times X1, X2, . . . and Y1, Y2, . . . on a line. We give a closed analytical formula for the E [|Xk+r − Yk|] , for any integer k ≥ 1, r ≥ 0, when a is natural number. The expected distance to the power a between two i.i.d. Poisson processes we represent as the combination of the Pochhammer polynomials. Especially, for r = 0, the following identity is valid E [|Xk − Yk|] = a! λa Γ ( a 2 + k ) Γ(k)Γ ( a 2 + 1 ) , where Γ(z) is Gamma function. As an application to sensor networks, we derive that the expected transportation cost to the power b of the bicolored matching with edges {Xk, Yk} between two i.i.d. Poisson processes with arrival rate λ = n and respective arrival times X1, X2, . . . and Y1, Y2, . . . is in Θ ( n b 2 ) , when b ≥ 1, and in O ( n b 2 ) , when 0 < b < 1.