1. 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