# M/M/1 Transient Queue length distribution

Computing steady state probabilities for a queueing system is somewhat easier than computing the transient distributions. The latter typically comprises a differential-difference equation and the usual trick of recursive substitution fails as there is a derivative in the equation. The tools employed in solving a differential-difference equation are Generating functions, Laplace transforms and PDE solving tricks. Only for simple systems such as M/M/1 can one slog out and find a closed form solution. For a generic system, the algebraic manipulations get extremely tedious and one usually resorts to numerical methods. Having said that, it pays to know how the tools work for a simple case.

Link : Detailed derivation of M/M/1 transient queue length distribution