# A NOTE ON BIAS OPTIMALITY IN CONTROLLED QUEUEING SYSTEMS

## Mark E. Lewis and Martin L. Puterman

Faculty of Commerce and
Business

The University of British Columbia

2053 Main Mall

Vancouver, BC V6T 1Z2 Canada

The use of * bias optimality*
to distinguish among gain optimal policies was recently studied by Haviv
and Puterman and extended in Lewis, et al. In Haviv and Puterman's model,
upon arrival to an $M/M/1$ queue, customers offer the gatekeeper a reward
$R$. If accepted, the gatekeeper immediately receives the reward, but is
charged a holding cost, $c(s)$, depending on, the number of customers in
the system. The gatekeeper, whose objective is to ``maximize'' rewards,
must decide whether to admit the customer. If the customer is accepted,
the customer joins the queue and awaits service. Haviv and Puterman showed
there can only be two Markovian, stationary, deterministic gain optimal
policies and that only the policy which uses the * larger * control
limit is bias optimal. This showed the usefulness of bias optimality to
distinguish between gain optimal policies. In the same paper, they
conjectured that if the gatekeeper receives the reward upon * completion
* of a job instead of upon entry, the bias optimal policy will be the
lower control limit. This note confirms that conjecture.