Distribution of the maximum buffer content during a busy period for state-dependent M/G/L queues

We consider M/G/l queues characterized by the total unfinished work (buffer content) U(t) in the system at time t. We allow the rate of the Poisson arrivals as well as the rate at which the server works, to depend on the instantaneous value of U(t). in addition, the service density depends on the value of U(t) at the instant that the customer enters the system. We consider systems with a large arrival rate and small mean service times. We then construct asymptotic approximations to the probability that U(t) reaches the level K before completing the current busy period and we also compute the distribution of the maximum of U(t) during a busy period.