Analysis of queues under batch service in an M/G/1 system in a random environment

A queue system with batch service is investigated. The batch size depends on both the state of the system and the state of some random environment. The server state changes under the action of the random environment. When the system is empty, a birth–death process acts as the random environment, whereas when the system is busy the random environment is disregarded and affects only the initial service probabilistic characteristics (in this sense, the random environment in the busy state can be regarded as arbitrary). An analytical model of the system is constructed and studied in terms of Laplace (Laplace–Stieltjes) images. The generating function of the queue length is derived.