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Perfect simulation of Markovian load balancing queueing networks in equilibrium

Carl Graham

TL;DR

This work tackles the challenge of exactly sampling from the equilibrium distributions of Markovian load balancing networks with possibly asymmetric topologies. It introduces a novel preorder (up to permutation) and constructs a coupling with a UR dominating network to preserve this preorder, enabling perfect simulation on an infinite state space. Three methods are developed: a Palm-rejection-based direct-time algorithm, an empty-state DomCFTP method, and a DomCFTP-back-off sandwiching approach, the latter two providing finite exponential moments for run times. The results enable Monte Carlo estimation of equilibrium QoS metrics for a wide class of policies beyond exchangeable networks, with practical implications for performance evaluation and policy comparison. The techniques extend the DomCFTP framework to high-dimensional, non-product-ordered spaces via the new preorder, offering a principled tool for exact sampling in complex queueing networks.

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.

Perfect simulation of Markovian load balancing queueing networks in equilibrium

TL;DR

This work tackles the challenge of exactly sampling from the equilibrium distributions of Markovian load balancing networks with possibly asymmetric topologies. It introduces a novel preorder (up to permutation) and constructs a coupling with a UR dominating network to preserve this preorder, enabling perfect simulation on an infinite state space. Three methods are developed: a Palm-rejection-based direct-time algorithm, an empty-state DomCFTP method, and a DomCFTP-back-off sandwiching approach, the latter two providing finite exponential moments for run times. The results enable Monte Carlo estimation of equilibrium QoS metrics for a wide class of policies beyond exchangeable networks, with practical implications for performance evaluation and policy comparison. The techniques extend the DomCFTP framework to high-dimensional, non-product-ordered spaces via the new preorder, offering a principled tool for exact sampling in complex queueing networks.

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.
Paper Structure (20 sections, 7 theorems, 49 equations, 3 algorithms)

This paper contains 20 sections, 7 theorems, 49 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Load balancing networks
  3. Examples of parallel server load balancing policies
  4. Asymptotic performance evaluation
  5. General definitions for load balancing networks
  6. Preorder and coupling
  7. Preorder and order up to permutation of coordinates
  8. Coupling a load balancing network with the UR network
  9. Considerations for effective simulation
  10. Sampling from the invariant law of the UR network
  11. Exchangeable load balancing policies
  12. General load balancing policies
  13. Subordination and embedded Markov chain
  14. Palm theory and acceptance-rejection
  15. Dominated Coupling From The Past (DomCFTP)
  16. ...and 5 more sections

Key Result

Theorem 1

A preorder $\preccurlyeq$ between network states $x$ and $y$ in $\mathbb{N}_0^c$ is defined by The indifference relation $\sim$ defined by $x \sim y$ if and only if $x \preccurlyeq y$ and $y \preccurlyeq x$ satisfies and thus $\preccurlyeq$ defines a partial order up to permutation of the coordinates. Moreover

Theorems & Definitions (19)

  • Definition 1: Load balancing network
  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Remark 2
  • Theorem 4
  • ...and 9 more