Estimating Call Rates

The statistical model can be used estimating the call rates $\lambda$ in the same way as the analytical one. We estimate $\lambda=(\lambda_1,...,\lambda_l)$ by minimizing the square deviations $\Delta (\mu,\lambda, r)$ between the probabilitiesthat there are $k_s$ calls in the $s$-th serverand their estimates $P_{k_s}^0$ by the actual observations

$$\Delta (\mu,\lambda,r)=\sum_s \sum_{k_s=0}^{m_s+r} (P_{k_s}(\mu,\lambda, r)-P_{k_s}^0)^2 .$$ (98)

The estimates $P_{k_s}^0$ are obtained by counting the numbers of waiting $s$-calls at different time moments. The least square estimation of the call-vector $\lambda$ is as follows

$$\lambda^o= \arg \min_{\lambda} \Delta (\mu,\lambda, r)).$$ (99)

However, using the statistical model this way one needs considerable computing power.