In Z. Ma, P.E. Caines, and R.P. Malhamé, ``Control of Loss Network Systems: Call Admission and Routing Control", (submitted to SIAM J. Control and Optimization), the call admission control (CAC) and routing control (RC) problems for loss network systems were formulated as optimal stochastic control (OSC) problems. The HJB equation for loss network OSC problems is an array of first order PDEs which are coupled with each other by sets of integral coefficients. In the case where the value function is a set of continuously differentiable functions, V is a classical solution to the associated loss network HJB PDEs. But, in general, the smoothness of V or uniqueness of the classical solutions to the HJB PDEs may not hold. In this paper we develop the viscosity solutions to the HJB PDEs and show that, under mild conditions, the value function V is continuous and, further, subject to mild conditions any continuous value function is the unique viscosity solution to the HJB PDEs.