Erlang capacity and uniform approximations for shared unbuffered resources

We consider a basic teletraffic model, which has applications to integrated multirate services on ATM and wireless systems. In the “finite-sources” version of the model an unbuffered resource with C channels is shared by heterogeneous sources which alternate between arbitrarily distributed random periods in the on and off states, and in the on state require a fixed number of channels. If a source does not find enough free channels when it turns on, then it is blocked and the burst is lost. In the “infinite-sources” version of the model requests for connections form Poisson streams and connections hold fixed numbers of channels for random periods. The stationary distribution of the system has the product-form and the insensitivity property. Our main results for the finite-sources model are for the asymptotic scaling in which C and the number of sources of each type are large. The central result is a uniform asymptotic approximation (UAA) for the blocking probabilities. It is uniformly effective for the complete range of loadings, simple to calculate and gives accurate results even for relatively small systems. The UAA is also specialized to the overloaded, critical and underloaded regimes. For the admission control of the system we calculate its Erlang capacity, i.e., the set of combinations of sources of various types such that the blocking probabilities for all types do not exceed specified values. For the first two regimes we obtain the boundaries of the admissible sets in the form of hyperplanes, and thus the effective bandwidths of sources of each type. For the underloaded regime the boundary is nonlinear and we obtain a convenient parameterized characterization. Finally, various numerical results are presented