This paper addresses the sojourn time asymptotics for a GI/GI/• queue operating under the
Processor Sharing (PS) discipline with stochastically varying service rate. Our focus is on the
logarithmic estimates of the tail of sojourn-time distribution, under the assumption that the jobsize
distribution has a light tail. Whereas upper bounds on the decay rate can be derived under
fairly general conditions, the establishment of the corresponding lower bounds requires that the
service process satisfies a samplepath large-deviation principle. We show that the class of
allowed service processes includes the case where the service rate is modulated by a Markov
process. Finally, we extend our results to a similar system operation under the Discriminatory
Processor Sharing (DPS) discipline. Our analysis relies predominantly on large-deviations
techniques
