Exact Sampling of Stationary and Time-Reversed Queues

TL;DR

This paper introduces the first exact algorithm for simulating the stationary waiting-time sequence of a single-server queue backwards in time, useful for bias-free steady-state analysis in applications like DCFTP, with minimal assumptions on input distributions.

Contribution

The paper presents a novel, finite-time algorithm for simulating stationary queue processes backwards in time under minimal assumptions, advancing steady-state simulation methods.

Findings

Algorithm terminates in finite time with finite mean inter-arrival and service times.

Finite variance is necessary for finite expected idle periods.

Applicable in bias-free steady-state simulation methods like DCFTP.

Abstract

We provide the first algorithm that under minimal assumptions allows to simulate the stationary waiting-time sequence of a single-server queue backwards in time, jointly with the input processes of the queue (inter-arrival and service times). The single-server queue is useful in applications of DCFTP (Dominated Coupling From The Past), which is a well known protocol for simulation without bias from steady-state distributions. Our algorithm terminates in finite time assuming only finite mean of the inter-arrival and service times. In order to simulate the single-server queue in stationarity until the first idle period in finite expected termination time we require the existence of finite variance. This requirement is also necessary for such idle time (which is a natural coalescence time in DCFTP applications) to have finite mean. Thus, in this sense, our algorithm is applicable under minimal assumptions.