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Time-sharing service systems. II
G. P. Klimov Moscow
Abstract:
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
$R_i(t)=c_i(t)/\gamma_i(t)$.
Received: 10.05.1976
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