Siegel’s matrix analysis of membrane transport in the Laplace domain [J. Phys. Chem. 95, 2556 (1991)], which is restricted to zero initial distribution, has been extended to including the case of nonzero initial distribution. This extension leads to a more general transport equation with Siegel’s results as a special case. The new transport equation allows us to formulate the mean-first-passage time (t) over bar for various boundary conditions, if the initial distribution is stipulated to be of the Dirac delta-function type; and the steady-state permeability P and time lag t(L), if zero initial distribution is employed. Based on this matrix analysis we also propose an algorithm for quick and effective numerical computations of P, t(L), and ST Examples are given to demonstrate the application of this algorithm, and the numerical results are compared with the theoretical ones. The validity of the transport equation is also checked by a Green’s function.