Analytical Approximation: Reservation Model

We estimate $\lambda=(\lambda_1,...,\lambda_l)$ by minimizing the square deviations $\Delta (\mu,\lambda, r)$ between the stationary probabilities that there are $k_s$ calls in the $s$-th server $P_{k_s}(\mu_s),\lambda_s, r_s),\ s=1,...,l$ defined by expressions similar to (2.3) assuming that there are $r_s$ waiting places reserved for the calls $\lambda_s$ Here $r=(r_1,...,r_l)$ is the reservation vector and $\sum_s r_s=r(c)$.

$$P_{k_s}(\mu_s,\lambda_s, r_s) = \cases{\frac {\rho^k_s}{k_s!} P_0(\mu_s,\lambda_s, r_s),$$

and their estimates $P_{k_s}^0$

$$\Delta_s (\mu,\lambda, r_s)=\sum_{k_s=0}^{m_s+r_s} (P_{k_s}(\mu_s,\lambda_s, r_s)-P_{k_s}^0)^2 .$$ (96)

The estimates $P_{k_s}^0$ are obtained by counting the numbers of waiting $s$-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-vector $\lambda$ is as follows

$$\lambda^o= \arg \min_{\lambda} \min_r (\sum_s \Delta (\mu_s,\lambda_s, r_s)).$$ (97)

The reservation model is simple and clear. However one must test this model to define when the reservation assumption is reasonable one. A way to do this is by using a Monte Carlo model.