_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

Uz U1 Uo } where Uo1 and Uk, fork E {0, 1, 2}, are square matrices of orders J 0 = c(K +2) and J 1 = c(K +3), respectively, and U oo and U 12 are matrices of respective dimensions Jo x J 1 and J 1 x Jo. Expressions for these blocks are readily derived from Table 1. For later use, we note (see [3, Theorem 7.2.4]) that X is positive recurrent, null recurrent or transient if p < 1, p = 1 or p > 1, respectively, where

. vK+2_ K+2 . the traffic load pIS defined asp=>.+(>,- +p,vvK+3_pR+3)-I, If !L =F v, and p = >.+(.>,- +p,~!D-\ if p, = v. Let x be the steady-state probability vector of Q. By partitioning x according to the levelled state space into sub-vectors x(n), we have that x(n) = x(O)SRn-1, for n 2 1, where S = Uoo( -U1-RU2)- 1 and R is the minimal non-negative solution to Uo + RU1 + R 2U 2 = 0 J, x J,. We may observe that R does not have necessarily any special block structure since its major property seems to be the property of being a non-negative matrix. Thus, to compute R, we can use the Latouche-Ramaswami algorithm [3]. Once R is numerically computed, the subvector x(O) is evaluated by solving the boundary equations given by x(O) (Uol + SU 12) = 0}0 and x(O) ( eJ0 + S(IJ, - R)- 1eh) = 1.