In this paper, we analyse a queueing model with a Markov arrival process and the possibility of simultaneous service of a finite number of customers defined by the multidimensional Markov chain. Service in this system is implemented in cooperation with another, infinite-server, queueing system. The variants when service in the infinite-server system is implemented before, after, and in parallel with service in the main system are considered. An explicit form of the generator of the multidimensional Markov chains describing the analysed variants of the queueing system is obtained. It is shown that in all these variants, the system is stable. The way for solving the problem of the stationary distribution computation in these variants is outlined as well. The results can be used for optimization of the choice of the system parameters.