Event-dependent arrivals in the M/G/1 queue

The paper investigates a single-server queueing system with an unlimited waiting buffer and an arbitrary service-time distribution. A distinctive feature of the system is that the intensity of the Poisson input flow depends on the most recent event in the system — either the arrival of a request or the completion of service. Input flows of this type make it possible to model situations in which the behavior of the incoming flow depends on the operation of the system itself. The supplementary variable method is used as the main analytical tool, allowing the derivation of the distribution of the number of requests in the system in the transient regime, as well as the corresponding stationary distribution.