Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal Size-Aware Dispatching Rules via Value Iteration and Some Numerical Investigations

Esa Hyytiä, Rhonda Righter

TL;DR

This work develops and refines a value-iteration framework for optimal size-aware dispatching in multi-server FCFS systems, incorporating discretization, boundary truncation, and Poisson-arrival structure. It introduces time-scale aware integration and state-space reduction to enable practical computations on grids with up to six servers, providing insights into the shape of the optimal value function and the resulting dispatching rules. Key findings show that the optimal policy tends to route short jobs to shorter queues and longer jobs to longer queues, resulting in deliberate backlog unbalancing and potential fairness tradeoffs. The methods and results have practical implications for load balancing in data centers and related systems, suggesting that near-optimal decisions can be achieved with scalable numerical techniques and that simple policies such as join-idle-queue may capture essential behavior in larger systems.

Abstract

This technical report explains how optimal size-aware dispatching policies can be determined numerically using value iteration. It also contains some numerical examples that shed light to the nature of the optimal policies itself. The report complements our ``Towards the Optimal Dynamic Size-aware Dispatching'' article that will appear in Elsevier's Performance Evaluation in 2024.

Optimal Size-Aware Dispatching Rules via Value Iteration and Some Numerical Investigations

TL;DR

This work develops and refines a value-iteration framework for optimal size-aware dispatching in multi-server FCFS systems, incorporating discretization, boundary truncation, and Poisson-arrival structure. It introduces time-scale aware integration and state-space reduction to enable practical computations on grids with up to six servers, providing insights into the shape of the optimal value function and the resulting dispatching rules. Key findings show that the optimal policy tends to route short jobs to shorter queues and longer jobs to longer queues, resulting in deliberate backlog unbalancing and potential fairness tradeoffs. The methods and results have practical implications for load balancing in data centers and related systems, suggesting that near-optimal decisions can be achieved with scalable numerical techniques and that simple policies such as join-idle-queue may capture essential behavior in larger systems.

Abstract

This technical report explains how optimal size-aware dispatching policies can be determined numerically using value iteration. It also contains some numerical examples that shed light to the nature of the optimal policies itself. The report complements our ``Towards the Optimal Dynamic Size-aware Dispatching'' article that will appear in Elsevier's Performance Evaluation in 2024.
Paper Structure (22 sections, 1 theorem, 21 equations, 11 figures, 1 table)

This paper contains 22 sections, 1 theorem, 21 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model for dispatching system
  3. Value functions and value iteration
  4. Implementing the value iteration
  5. Discretization step $\boldsymbol{\delta}$:
  6. Value iteration algorithm:
  7. Recursive algorithm for Poisson arrivals
  8. Improving value iteration under time-scale separation
  9. Tailored integration:
  10. State-space reduction
  11. Numerical experiments
  12. Optimal policy for two servers
  13. Optimal policy for three servers
  14. Convergence of the value iteration
  15. Two servers:
  16. ...and 7 more sections

Key Result

Proposition 1

The pre-assignment value function $w(\mathbf{u})$ of the optimal dispatching policy satisfies where $v(\mathbf{u})$ is the post-assignment value function, for which it holds that, where $(\cdot)^+$ denotes the component-wise maximum of zero and the argument.

Figures (11)

  • Figure 1: Dispatching system.
  • Figure 2: One decision epoch consists of the job assignment of length $X$ and the following IAT $A$. In the value iteration, the new value of state $\mathbf{u}$ depends on the current values at the shaded area.
  • Figure 3: Left: As the system size increases, the time-scales of $A$ and $X$ start to separate, which induces numerical challenges. Right: Updating $v(\mathbf{u})$ from $w(\mathbf{u})$ corresponds to taking the expectation along the depicted trajectory. With Poisson arrivals this can be achieved recursively.
  • Figure 4: State-space reduction with $m=2$ and $m=3$ servers.
  • Figure 5: The value function for two servers with $\rho=0.4$ and $\rho=0.9$ when job sizes are exponentially distributed. The black dots correspond to functions fitted to match the numerical values. For $\rho=0.4$, the value function along the diagonal ($u_1=u_2$) appears to be nearly quadratic, ${v(u,u)\approx 0.56 u^2}$. For all other cuts, we found good approximations of form $a + b u^2 /(c+u)$, where $a$, $b$, and $c$ are properly chosen.
  • ...and 6 more figures

Theorems & Definitions (1)

  • Proposition 1: Bellman equation