In this paper, we study Markov-dependent reflected autoregressive processes, and other related models the analysis of which results in a vector-valued fixed-point functional equation of a certain type. In queueing terms, such processes describe the workload just before a customer arrival, which makes obsolete a fraction of the work already present, and where the interarrival time and the service time depend on a common discrete time Markov chain. Our primary aim is to derive the Laplace-Stieltjes transform vector of the steady-state workload via a recursive approach. We consider the case where given the state of the underlying Markov chain, the interarrival time and the service time are conditionally independent. Moreover, we further focus on the case where there is also additional dependence based on the Farlie-Gumbel-Morgenstern copula, as well as the case where there is a dependence based on a class of multivariate matrix-exponential distributions. The transient analysis of the Markov-modulated reflected autoregressive process with a more general dependence structure is also investigated. Finally, motivated by queueing applications, we consider two other Markov-dependent models that are described by a similar stochastic recursion: the modulated shot-noise single server queue, and the single-server queue with service time randomly dependent on the waiting time.