A Markovian single server working vacation queue with server state dependent arrival rate and with randomly varying environment

An M/M/1 queue with working vacation state has been considered. If the system is in busy state, it functions as a single server Markovian queue. When it is on vacation, again it functions as a single server Markovian queue but with different arrival and service rates. In addition, during busy period, the arrival and service are generated by K distinct randomly varying environments. The server takes vacation whenever there are no one in the system. But, the vacationing server serves at a lower rate as an arrival occur. The vacation policy is multiple vacation policy and the vacation period follows negative exponential. For this model using Matrix-Geometric method the probability distribution of number of customers in the queue in steady state has been obtained. Some illustrative examples are also provided.