First passage processes in Queuing system MX/Gr/1 with service delay discipline

This article deals with a general single-server bulk queueing system with a server waiting until the queue will reach level before it starts processing customers. If at least r customers are available the server takes a batch of the fixed size of units for service. The input stream is assumed to be a compound Poisson process modulated by a semi-Markov process and with a multilevel control of service time.