The applicability of queueing network models to real-world systems is often enhanced by characterizing service times using phase-type (PH) distributions. Unfortunately, the combinatorial explosion of the underlying state space makes it difficult to evaluate efficiently the resulting models. We propose to tackle this issue by a solution algorithm inspired by the Reversed Compound Agent Theorem (RCAT) [2]. Let the QBD process for the uphill queue be specified by the forward transition matrix F , by the local transition matrix L, and by the backward transition matrix B. Also, let us indicate with αn the vector of state probability when the queue size is n. Further, let (β,T ) be the representation of the PH distribution at the uphill queue. Then a RCAT product-form exists if there exist a scalar q > 0 such that αnL = αn+1B, n ≥ 1, where L = diag(qδ1, qδ2, . . . , qδK) and δk is 1 if the kth column of −Teβ has at least one nonzero element, 0 otherwise [1]. An interesting observation arising in [1] from the above formula is that, for models with a RCAT product-form, there exist a matrix H = −F (L+ L) such that αn = αn−1H , n ≥ 2. In general, this matrix differs from the rate matrix R, even though they must share the same dominating eigenvalue η and αn = αn−1R. Observe now that αn+1 = αnR = αnH, n ≥ 1, thus αn(R −H) = 0, which implies αn(L − RB) = 0, n ≥ 1. Since the last equation must hold for all n ≥ 1, it suggests that the general solution admits the form αn+k = cn+kαn+k−1, for some sequence of coefficients ci, i > 1. Applying the Cayley-Hamilton theorem to R, it can be shown that choosing cn+k ≡ η such that αn+1 = ηαn provides always a solution that is consistent with the matrix geometric one for all n ≥ 1. In particular, if H and R share only the eigenvalue η, the relation can be proved exactly. Based on the above observations, one may then approximate the steady state of a queue satisfying RCAT’s conditions by the relation αn = ηαn−1 which, upon imposing