Starting with P1 = l/uP0, the next few equations unroll into
| P2 | = | ((l + u)/u)P1 - (l/u)P0 |
| = | ((l + u)/u)(l/uP0) - (l/u)P0 | |
| = | (l/u)2P0 | |
| P3 | = | ((l + u)/u)P2 - (l/u)P1 |
| = | ((l + u)/u)(l/u2P0) - (l/u)(l/u)P0 | |
| = | (l/u)3P0 |
An inductive argument following this pattern leads to the equivalence
Pj is the probability that there are j customers in the system; the definition of probabilities (and independent outcomes) leads to the equations
which can be rewritten to
sum(i = 0 to inf, (l/u)i) is a geometric series and converges if and only if l/u = r < 1; when converges, it converges to the value 1/(1 - r).
This page last modified on 2 March 2005.