Call Rate Estimation

It is difficult to estimate the call rate $\lambda$ directly if $r$ is limited, because the lost calls are not registered as usual. In such cases the indirect estimate can be used by minimizing the square deviations $\Delta (\mu,\lambda)$ between the stationary probabilities that there are $k$ calls in the system $P_{k}(\mu,\lambda)$ defined by expression (2.3)

$$P_{k}(\mu,\lambda) = \cases{\frac{\rho^k}{k!} P_0(\mu,\lambda),$$ (75)

and their estimates $P_{k}^0$

$$\Delta (\mu,\lambda)=\sum_{k=0}^{m+r} (P_{k}(\mu,\lambda)-P_{k}^0)^2 .$$ (76)

The estimates $P_{k}^0$ are obtained by counting the numbers of waiting calls at different time moments. More moments we consider the better estimation will be. Using the average numbers of waiting calls in a time interval instead of moment numbers some additional errors are expected that are increasing with the length of the time interval. The least square estimation of the call rate is as follows

$$\lambda^o= \arg \min_{\lambda} \Delta (\mu,\lambda).$$ (77)