<p>This paper investigates a semi-open queueing network with a restricted capacity for concurrent client processing. The network nodes are modeled as multi-server systems with finite buffers, operating within a fluctuating random environment controlled by a continuous-time Markov chain. A change in the environment state triggers immediate transitions in key system parameters, including arrival process matrices (<i>MMAP</i>), server counts, service rates, reneging rates, and routing probabilities. The system’s dynamics are characterized by a multidimensional continuous-time Markov chain. We derive an explicit generator for this chain, enabling the calculation of the steady-state distribution. Furthermore, we provide analytical formulas for key performance measures of both the network and its individual nodes. The study includes numerical examples demonstrating how performance metrics depend on concurrent processing limits and environment transition rates, followed by a brief discussion on optimization applications.</p>