Traffic Engineering (TE) is important for improving QoS in forwarding paths by efficient use of network resources. In fact, MPLS allows several detour paths to be (pre-)established for some source-destination pair as well as its primary path of minimum hops. Thus, we focus on a two-phase path management scheme using these two kinds of paths. In the first phase, each primary path is allocated to a flow on a specific source-destination pair if the path is not congested, i.e., if its utilization is less than some predetermined threshold; otherwise, as the second phase, one of the detour paths is allocated randomly if the path is available. Therefore, in this paper, we analytically evaluate this path management scheme by extending the M/M/c/c queueing system, and through some numerical results we investigate the impact of a threshold on the flow-blocking probability. Through some numerical results, we discuss the adequacy of the path management scheme for MPLS-TE.