This thesis is naturally broken down into two main problems, one concerning
optimal routing control and the other optimal service control. In the routing control
problem the arriving customers must be allocated to one of the 'K' possible service
stations. We assume that the customers arrive in a single Poisson stream. We take
the service at each of the stations to be exponentially distributed, but perhaps with
different parameters. The system cost rate is additive across the queues formed at
each station. We also have that at each station the holding cost function is
increasing convex. Following Whittle's approach to a class of restless bandit
problems, we develop a Lagrangian relaxation of the routing control problem which
serves to motivate the development of index heuristics. The index by a particular
station is characterised as a fair charge for rejecting the arriving customer at that
station. We also consider a policy improvement index for comparison to the
heuristic. We develop these indices and report an extensive numerical investigation
which exhibits strong performance of the index heuristic for both discounted and
average costs.The second problem concerns the optimal service control of a multi-class M/G/l
queueing system in which customers are served non preemptively. The system cost
rate is additive across classes and increasing convex in the numbers present within
each class. We again follow the method prescribed by Whittle when considering a
class of restless bandits. Hence we develop a Lagrangian relaxation of the service
control problem which motivates the development of a class of index heuristics. For
a particular customer class the index is characterised as a fair charge for service of
that class. These indices are developed and we again report representative results
from an extensive numerical study which again implies a strong performance of the
index heuristic for both discounted and average costs
