We consider single-server queueing systems that are modulated by a discrete-time Markov chain on a countable state space. The underlying stochastic process is a Markov random walk (MRW) whose increments can be expressed as differences between service times and interarrival times. We derive the joint distributions of the waiting and idle times in the presence of the modulating Markov chain. Our approach is based on properties of the ladder sets associated with this MRW and its time-reversed counterpart. The special case of a Markov-modulated M/M/1 queueing system is then analysed and results analogous to the classical case are obtained.
