## Optimal control of service rates in networks of queues

### R. Weber and S. Stidham, Adv. Appl. Prob.19 202-218, 1987. Abstract

We prove a monotonicity result for the problem of optimal service rate control
in certain queueing networks.  Consider, as an illustrative example, a number pf
$\cdot/M/1$ queues which are arranged in a cycle with some number of customers
moving around the cycle.  A holding cost $h_i(x_i)$ is charged for each unit of
time that queue $i$ contains $x_i$ customers, with $h_i$ being convex.  As a
function of the queue lengths the service rate at each queue $i$ is to be chosen
in the interval $[0,\bar{\mu}]$, where the cost $c_i(\mu)$ is charged for each
unit of time that the service rate $\mu$ is in effect at queue $i$.  It is shown
that the policy which minimizes the expected total discounted cost has a
monotone structure: namely, that by moving one customer from queue $i$ to the
following queue, the optimal service rate at queue $i$ is not increased and the
optimal service rates elsewhere are not decreased.  We prove a similar result
for problems of optimal arrival rate and service rate control in general
queueing networks.  The results are extended to an average-cost measure, and an
example is included to show that in general the assumption of convex holding
costs may not be relaxed.  A further example shows that the optimal policy may
not be monotone unless the choice of possible service rates at each queue
includes $0$.