Perfect simulation of Markovian load balancing queueing networks in equilibrium

TL;DR

This paper introduces three perfect simulation algorithms for Markovian load balancing queueing networks in equilibrium, enabling accurate Monte Carlo estimation despite the infinite state space and complex network topologies.

Contribution

It develops novel perfect simulation algorithms using a preorder-based coupling, applicable to asymmetric and topologically constrained load balancing networks, with finite expected durations.

Findings

Algorithms successfully simulate equilibrium states of load balancing networks.

The first algorithm has finite duration almost surely but infinite expectation.

The other two algorithms have durations with exponential moments.

Abstract

We define a wide class of Markovian load balancing networks of identical single-server infinite-buffer queues. These networks may implement classic parallel server or work stealing load balancing policies, and may be asymmetric, for instance due to topological constraints. The invariant laws are usually not known even up to normalizing constant. We provide three perfect simulation algorithms enabling Monte Carlo estimation of quantities of interest in equilibrium. The state space is infinite, and the algorithms use a dominating process provided by the network with uniform routing, in a coupling preserving a preorder which is related to the increasing convex order. It constitutes an order up to permutation of the coordinates, strictly weaker than the product order. The use of a preorder is novel in this context. The first algorithm is in direct time and uses Palm theory and acceptance rejection. Its duration is finite, a.s., but has infinite expectation. The two other algorithms use dominated coupling from the past; one achieves coalescence by simulating the dominating process into the past until it reaches the empty state, the other, valid for exchangeable policies, is a back-off sandwiching method. Their durations have some exponential moments.