On queues with service and interarrival times depending on waiting times

Authors:
O. J. Boxma;M. Vlasiou
Affiliations:
EURANDOM and Department of Mathematics & Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands 5600 MB;H. Milton Stewart School of Industrial & Systems Engineering, Georgia Institute of Technology, Atlanta, USA 30332-0205
Venue:
Queueing Systems: Theory and Applications
Year:
2007

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Abstract

We consider an extension of the standard G/G/1 queue, described by the equation $W\stackrel{ \mathcal {D}}{=}\max\mathrm{max}\,\{0,B-A+YW\}$ , where 驴[Y=1]=p and 驴[Y=驴1]=1驴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.