[Paper Review] Perfect Sampling of GI/GI/c Queues

José Blanchet, Jing Dong+1 more|arXiv (Cornell University)|Aug 10, 2015

Advanced Queuing Theory Analysis3 references5 citations

TL;DR

This paper presents the first perfect sampling algorithm for GI/GI/c queues with general interarrival and service time distributions, using a dominated coupling from the past (DCFTP) framework with a coupled multi-server vacation system as a stationary upper bound. The algorithm ensures finite expected termination time under mild moment conditions (2+ε moments), enabling exact sampling from the steady-state distribution without bias.

ABSTRACT

We introduce the first class of perfect sampling algorithms for the steady-state distribution of multi-server queues with general interarrival time and service time distributions. Our algorithm is built on the classical dominated coupling from the past protocol. In particular, we use a coupled multi-server vacation system as the upper bound process and develop an algorithm to simulate the vacation system backwards in time from stationarity at time zero. The algorithm has finite expected termination time with mild moment assumptions on the interarrival time and service time distributions.

Motivation & Objective

• To develop a perfect sampling algorithm for multi-server queues with general interarrival and service time distributions, where prior methods required Poisson arrivals.
• To ensure finite expected running time for the sampling algorithm under mild moment assumptions on the input distributions.
• To construct a stationary upper bound process (a coupled multi-server vacation system) that enables backward simulation and coalescence detection.
• To overcome the challenge of simulating general renewal processes backwards in time by introducing a novel coupling construction.
• To validate the algorithm's correctness and efficiency through numerical experiments on M/M/c queues and scalability analysis under varying system load and scale.

Proposed method

• Uses Dominated Coupling From The Past (DCFTP) protocol to sample exactly from the steady-state distribution of GI/GI/c queues.
• Employs a coupled multi-server vacation system as a stationary upper bound process, which is reversible and amenable to backward simulation.
• Applies common random numbers in reverse time to simulate both the upper and lower bound processes, leveraging monotonicity to detect coalescence.
• Implements a recursive doubling search strategy to efficiently locate the coalescence time by doubling the time interval until the bounds meet.
• Uses renewal process simulation for arrivals and service/vacation times, with backward sampling via inverse transform and equilibrium distribution sampling.
• Reconstructs the target process from the coalescence time forward to produce a stationary draw at time zero.

Experimental results

Research questions

• RQ1 Can perfect sampling be achieved for GI/GI/c queues with general interarrival and service time distributions, without assuming Poisson arrivals?
• RQ2 What conditions on the moments of interarrival and service time distributions ensure finite expected running time for the perfect sampling algorithm?
• RQ3 How can a stationary upper bound process be constructed for general-service multi-server queues to enable backward simulation?
• RQ4 Can monotonicity and common random numbers be leveraged to improve coalescence detection in the DCFTP framework for non-Markovian queues?
• RQ5 How does the computational complexity of the algorithm scale with system size and traffic intensity?

Key findings

• The algorithm achieves finite expected termination time under the assumption that interarrival and service time distributions have finite 2+ε moments for some ε>0.
• For M/M/c queues, the empirical distribution from 5,000 perfect simulation draws closely matches the theoretical steady-state distribution, validating correctness.
• In the quality-driven (QD) regime, mean coalescence time increases only slightly with system scale: from 6.42 at s=100 to 7.75 at s=1000.
• In the quality-and-efficiency-driven (QED) regime, mean coalescence time increases more noticeably but remains moderate: from 6.51 at s=100 to 9.47 at s=1000.
• Computational complexity scales approximately as (1−ρ)⁻², with the product of total renewals sampled and (1−ρ)² remaining nearly constant across ρ values, confirming the theoretical scaling behavior.
• The algorithm’s performance is robust under varying traffic intensity, with complexity increasing significantly as ρ→1, but normalized complexity (multiplied by (1−ρ)²) stabilizes around a constant factor proportional to λ.