## 4.5.1 Solving the Balance Equations

We can now proceed to solve the balance equations expressing all steadystate probabilities Pn, n = 0, 1, 2, . . . in terms of one of them and then taking advantage of the fact that

It is common practice in queueing theory to express P1, P2, P3, . . . in terms of P0, the steady-state probability of an empty system. Working with (4.22) and (4.23) or, equivalently (and preferably) with (4.24), we have

or, defining the coefficient of P0, in (4.28) as the quantity Kn, we have

Going now back to (4.25),

It follows that the system can reach steady state only if . For, otherwise, P0 = P1 = P2 = ... = 0 (i.e., the number of users in the system never "stabilizes").

Assuming that the system does reach steady state, the probabilities   Pn, n = 0, 1, 2,. . . , are now given by the fundamental expressions (4.30) and (4.28), and other quantities of interest can be computed from these probabilities. For instance,

and q, , and q, can then be obtained from (4.10), (4.11), and (4.13).

Exercise 4.2 Argue that the value of that should be used with Little's equations, (4.10) and (4.11) is in this case,