# On queues with service and interarrival times depending on waiting times

Onno J. Boxma, Maria Vlasiou

16 Citations (Scopus)

## Abstract

We consider an extension of the standard G/G/1 queue, described by the equation $W\stackrel{\mathcal{D}}{=}\max\{0, B-A+YW\}$, where $\mathbb{P}[Y=1]=p$ and $\mathbb{P}[Y=-1]=1-p$. 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.
Original language English 121-132 Queueing systems 56 https://doi.org/10.1007/s11134-007-9011-3 Published - 20 Mar 2007 Yes

• math.PR
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