  This paper deals with a single-server queue with modulated arrivals, service
requirements and service capacity. In our first result, we derive the mean of
the total workload assuming generally distributed service requirements and any
service discipline which does not depend on the modulating environment. We then
show that the workload is exponentially distributed under heavy-traffic
scaling. In our second result, we focus on the discriminatory processor sharing
(DPS) discipline. Assuming exponential, class-dependent service requirements,
we show that the joint distribution of the queue lengths of different customer
classes under DPS undergoes a state-space collapse when subject to
heavy-traffic scaling. That is, the limiting distribution of the queue length
vector is shown to be exponential, times a deterministic vector. The
distribution of the scaled workload, as derived for general service
disciplines, is a key quantity in the proof of the state-space collapse.
