Optimal priority assignment with hard constraint

We consider a discrete-time queueing system consisting of
K + 1
classes of customers competing for a single server at an infinite capacity queue. For each customer class the arrival sequence forms a renewal sequence but is otherwise arbitrary. The service requirements are geometric with class-dependent parameter. The optimization criterion is to minimize a linear combination of the average line lengths for classes 1 through
K
, while simultaneously subjecting the average line length of class-0 customers to a hard constraint. The optimal policy is shown to be a randomized modification of a static-priority policy. The optimization problem is thereby reduced to a problem of finding the optimal randomization factor. This is done in a particular case, when the arrival processes are independent and geometrically distributed.