On queues with service and interarrival times depending on waiting times

Queueing Systems (Impact Factor: 0.84). 08/2007; 56(3):121-132. DOI: 10.1007/s11134-007-9011-3
Source: DBLP


We consider an extension of the standard G/G/1 queue, described by the equation
W= Dmaxmax{0,B-A+YW}W\stackrel{ \mathcal {D}}{=}\max\mathrm{max}\,\{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.

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Available from: Maria Vlasiou, Apr 24, 2014
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    • "This was one of the motivations of Boxma and Vlasiou [3] to investigate the distribution of W defined in (1). For the case X n = B n − A n , with B n and A n independent nonnegative random variables, the work in [3] provides explicit results for the distribution of W , assuming that either B n is phase type and A n is general or that B n is a constant and A n is exponential. It appears to be a considerable challenge to obtain the distribution of W under general assumptions on the distribution of X n . "
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