We consider an extension of the standard G/G/1 queue, described by the equation W = max{0, B - A + YW}, where PY = 1 = p and PY = -1 = 1 - p. For p = 1 this model reduces to the classical Lindley equation for the waiting time in the G/G/l 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.
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