In this paper we study a MAP/M/2 queueing system with heterogeneous servers. The system operates in a Markovian random environment which modulated service rates of faster server. The allocation of customers between the servers proceeds according to a heuristic FFS-policy (Fastest Free Server) and to an optimal policy which minimizes the average number of customers in the system. It is shown that the correlated arrivals may have different impact on the average number of customers in the system and on other performance measures dependent on the value of correlation, properties of the random environment and type of the allocation policy.