Sojourn time distribution in polling systems with processor-sharing policy

We consider a polling system with a single server and multiple queues where customers arrive at the queues according to independent Poisson processes. The server visits and serves the queues in a cyclic order. The service discipline at all queues is exhaustive service. One queue uses processor-sharing as a scheduling policy, and the customers in that queue have phase-type distributed service requirements. The other queues use any work-conserving policy, and the customers in those queues have generally distributed service requirements. We derive a partial differential equation for the transform of the conditional sojourn time distribution of an arbitrary customer who arrives at the queue with processor-sharing policy, conditioned on the service requirement. We also derive a partial differential equation for the transform of the unconditional sojourn time distribution. From these equations, we obtain the first and second moments of the conditional and unconditional sojourn time distributions.