Exact sampling for some multi-dimensional queueing models with renewal input

TL;DR

This paper extends exact sampling methods to multi-dimensional queueing models with renewal input, providing new algorithms for stationary distributions in FIFO multi-server, infinite server, and Fork-Join queues, using dominated coupling from the past.

Contribution

It introduces a novel exact simulation algorithm for multi-dimensional queues with renewal arrivals, generalizing previous Poisson-based methods and covering various queueing models.

Findings

Successfully extended exact sampling to multi-dimensional queues.

Developed algorithms for FIFO GI/GI/c and GI/GI/∞ models.

Handled Fork-Join queues with multiple jobs per customer.

Abstract

Using a result of Blanchet and Wallwater (2015: Exact sampling of stationary and time-reversed queues. ACM TOMACS, 25, 26) for exactly simulating the maximum of a negative drift random walk queue endowed with independent and identically distributed (iid) increments, we extend it to a multi-dimensional setting and then we give a new algorithm for simulating exactly the stationary distribution of a first-in-first-out (FIFO) multi-server queue in which the arrival process is a general renewal process and the service times are iid; the FIFO GI/GI/c queue with 2 \le c < 1. Our method utilizes dominated coupling from the past (DCFP) as well as the Random Assignment (RA) discipline, and complements the earlier work in which Poisson arrivals were assumed, such as the recent work of Connor and Kendall (2015: Perfect simulation of M/G/c queues. Advances in Applied Probability, 47, 4). We also consider the models in continuous-time and show that with mild further assumptions, the exact simulation of those stationary distributions can also be achieved. We also give, using our FIFO algorithm, a new exact simulation algorithm for the stationary distribution of the infinite server case, the GI/GI/\infty model. Finally, we even show how to handle Fork-Join queues, in which each arriving customer brings c jobs, one for each server.