Time-sharing service systems. II

The paper shows how to find the service order that minimizes an additive loss functional.
Consider a characteristical result (see example 3). Independent Poisson inputs come to a service system. The service time of the $i$-th item of the input has distribution function $B_i(x)$. Interruption of service is possible. Let $c_i(t)$ be the cost of waiting in the time unit for the $(i,t)$-item, i. e. for the $i$-th item of the input which has been served for the period of time equal to $t\ge 0$. Let each $(i,t)$-item have now a priority index.
$$ R_i(t)=\sup_{x>t}\biggl\{[c_i(t)(1-B_i(t))-c_i(x)(1-B_i(x))]\biggl[\int_t^x(1-B_i(u))\,du\biggr]^{-1}\biggr\}. $$
Then the optimal service order (that minimizes the mean loss in the unit time in stationary regime) is the following: those items should have priority which have the maximum priority index. In particular, if $\gamma_i(t)$ is the mean time necessary to complete the service of the $(i,t)$-item and, for each $i$, the function $c_i(t)/\gamma_i(t)$ is non-decreasing as a function of $t$, it means that $R_i(t)=c_i(t)/\gamma_i(t)$.