On queues with service and interarrival times depending on waiting times

TL;DR

This paper extends the classical G/G/1 queue model to include waiting times that influence service and interarrival times, deriving the distribution of waiting times for specific cases.

Contribution

It introduces a generalized queue model where service and interarrival times depend on waiting times, providing explicit distribution results for certain distributions.

Findings

Derived the distribution of waiting times for general and specific cases.

Connected the model to classical queues and alternating service models.

Provided analytical solutions for phase-type and exponential distributions.

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.