Let X be a finite set, P be a stochastic matrix on X, and P = lim ,,_,(l/n)X$LAPk.

We consider a multi-class M/GI/1 system, in which an average response time objective is associated with each class. The performance of each class is measured by the ratio of the average response time over the corresponding value of the objective. To achieve fairness in service allocation it is required to Þnd a policy that lexicographically minimizes the vector of performance ratios arranged in non-increasing order. We provide such a policy that is adaptive, uses only knowledge of arrival and departure instants and is thus easy to implement. We also consider a variant of this policy which adapts faster to changes in the statistical parameters of the model. Both policies are analyzed via associated stochastic recursions using techniques of stochastic approximation.

Interchange arguments are applied to establish the optimality of priority list policies in three problems. First, we prove that in a multiclass tandem of two ·/M/1 queues it is always optimal in the second node to serve according to the cµ rule. The result holds more generally if the first node is replaced by a multiclass network consisting of ·/M/1 queues with Bernoulli routing. Next, for scheduling a single server in a multiclass node with feedback, a simplified proof of Klimov's result is given. From it follows the optimality of the index rule among idling policies for general service time distributions, and among pre-emptive policies when the service time distributions are exponential. Lastly, we consider the problem of minimizing the blocking in a communication link with lossy channels and exponential holding times.

Adaptive algorithms are obtained for the solution of separable optimization problems in multiclass M/GI/1 queues with Bernoulli feedback. Optimality of the algorithms is established by modifying and extending methods of stochastic approximation. These algorithms, can be used as a basis for designing policies for semi-separable and approximate lexicographic optimization problems and in the case of M/GI/1 queues without feedback, they also provide a simple policy for lexicographic optimization. The results obtained on stochastic approximation imply convergence of classical recursions such as Robbins-Monroe in cases where the conditional second moment of their increments is not Þnite.

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