We consider a two-buffer polling queueing system with a changing service rate. One of the buffers has an infinite capacity, while another is finite. Changes in service rates occur during service at random moments. Inter-change times have an exponential distribution. The arrival flow is defined by a marked Markovian arrival process. The process of the system states is defined by a multidimensional Markov chain. The generator of this chain is derived, and its steady-state distribution is computed. The expressions for the main performance characteristics of the system are obtained. Numerical results that highlight the dependence of these characteristics on the rates of the distribution of an exponentially distributed buffer visit time duration are presented. The possibility of solving an optimization problem is briefly illustrated.