In this paper a model of a heterogeneous resource queueing system with a Markovian arrival process is considered. The customer accepted for servicing occupies random amount of resource with a given distribution function depending on the class of the customer and on the type of service it needs. At the end of the service, the customer leaves the system and releases the occupied resource. In this work, asymptotic formulas for calculating the main probability characteristics of the model, including the joint distribution functions of the customers number and the total resource amounts occupied by them, are obtained. Finally, the accuracy of the approximation is verified by using simulation. © Springer Nature Switzerland AG 2019.