We derive analytic expressions for the upper and lower bounds to the rate constant of a unimolecular reaction, which is treated as a competition between decay of reactive states and an arbitrary number of collisional relaxation processes all with different rate constants. The unimolecular rate is identified with the eigenvalue of smallest numerical magnitude. An analytic approximation to the corresponding eigenvector is also derived. Behaviour of the low-pressure rate constant is investigated and an analytic expression, in terms of the populations and the rates connecting them, is derived for the collision efficiency parameter βc. It is shown that there is a direct relationship between the limiting low-pressure rate and the reactive fraction of molecules only in the special case where all unreactive molecules are connected by a pure exponential collisional relaxation.