Asymptotic Solution to an Inverse Problem for a Shared Unbuffered Resource

We consider an unbuffered resource having capacity C, which is shared by several different services. Calls of each service arrive in a Poisson stream and request a fixed, integral amount of capacity, which depends on the service. An arriving call is blocked and lost if there is not enough free capacity. Otherwise, the capacity of the call is held for the duration of the call, and the holding period is generally distributed. 

The inverse problem of determining the traffic intensities in terms of the measured values of the carried loads for each service is investigated. It is assumed that C and the traffic intensities are commensurately large. The inverse problem is solved asymptotically in the critically loaded regime, and it involves the unique real solution of a nonlinear equation. 

An iterative solution of this equation is shown to lead to a contraction mapping, and to monotonic and geometric convergence. A separate analysis is given for the overloaded regime, and it is shown that the result matches asymptotically with that for the critically loaded regime. Numerical results are presented for two examples.