Asymptotic Shape of the Erlang Capacity Region of a Multi-Service Shared Resource

We consider a loss model of an unbuffered resource having C channels, which are shared by several different types of service connections. Connections of each type arrive in a Poisson stream and request a number of channels, which depends on the type. An arriving connection is blocked and lost if there are not enough free channels. 

Otherwise, the channels are held for the duration of the connection, and the holding period is generally distributed. It is assumed that C and the traffic intensities are proportionately large. The admission control problem is considered for specified upper bounds on the blocking probabilities, and the boundary of the admissible set is investigated asymptotically. 

The results are derived by investigating the local behavior with respect to the tangent hyperplane at a point on the boundary of the admissible set. The lowest order results that hold in the asymptotic limit C -> inf are given first. Importantly, the boundary is linear for the key critically loaded and also for the overloaded regimes, and weakly convex for the underloaded regime. Next, refined results that hold for C >> 1 are given, which indicate that the boundary is not convex, although only slighty so.