4.5.1 Solving the Balance Equations

       We can now proceed to solve the balance equations expressing all steadystate probabilities P_n, 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 K_n, 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   P_n, 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,