Some Basic Relationships At(z) = e–λt(1–z) z-transform of number of Poisson arrivals at rate λ in time t AS(z) = S(λ(1–z)) z-transform of number of Poisson arrivals at rate λ in time with c.d.f. S() N(z) = T(λ–λz) = AT(z) z-transform of number in system is Laplace transform of system’s response time evaluated at (λ–λz) Bx(s) = e–x(s+λ–λB(s)) Laplace transform of busy period started by job of size x BW(s) = W(s+λ–λB(s)) Laplace transform of busy period started by initial work with c.d.f. W() Hence, B(s) = S(s+λ–λB(s)) where S() is the c.d.f. of service times