A single-server queue with work conserving FIFO discipline is considered. The input process is a multiple marked. Markovian arrival process governed by a continuous-time finite state Markovian chain. The service time distributions of customers may be different for different arrival streams. The virtual waiting time is considered under a heavy traffic. The probability distributions of virtual waiting time and state of the random environment are asymptotically independent. The virtual waiting time is asymptotically exponential with the mean depending on the characteristics of the modulated process.