On queues with service and interarrival times depending on waiting times

We consider an extension of the standard G/G/1 queue, described by the equation W D=\,\0,B-A+YW\ , where ℙ[Y=1]=p and ℙ[Y=−1]=1−p. For p=1 this model reduces to the classical Lindley equation for the waiting time in the G/G/1 queue, whereas for p=0 it describes the waiting time of the server in an alternating service model. For all other values of p, this model describes a FCFS queue in which the service times and interarrival times depend linearly and randomly on the waiting times. We derive the distribution of W when A is generally distributed and B follows a phase-type distribution, and when A is exponentially distributed and B deterministic.