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Expected Cost Analysis of Online Facility Assignment on Regular Polygons

Md Rawha Siddiqi Riad, Md Manzurul Hasan

TL;DR

The paper addresses the online facility assignment problem in a geometric setting where facilities at the vertices of a regular polygon each have unit capacity and customers arrive on edges uniformly at random. It develops a recursive integral framework for the expected total cost, $V(S)$, and provides exact solutions for small $n$ via dynamic programming, complemented by Monte Carlo methods for larger instances. Key contributions include the rigorous recurrence, a Monte Carlo interpretation, and structural insights, accompanied by computational results up to $n=12$ and a detailed square ($n=4$) worked example that yields $V_4=71/32$. The work delivers a foundational probabilistic approach for online geometric assignment, offering practical algorithms for moderate-sized polygons and a basis for extending to more complex geometries and arrival models.

Abstract

This paper analyzes the online facility assignment problem in a geometric setting where facilities with unit capacity are positioned at the vertices of a regular $n$-gon. Customers arrive sequentially at uniformly random positions along the edges. They must be assigned immediately to the nearest available facility, with ties broken by coin toss. The sequential nature and unknown future arrivals require a probabilistic analysis of the expected assignment cost. Our main contribution is a recursive characterization of the expected cost: for any occupancy state $S$, the expected remaining cost $V(S)$ equals the average over all edge positions of the immediate assignment cost plus the expected future cost after assignment. We prove that this integral equation can calculate a solution and provide the expected value for small $n$ ($n = 3, 4, 5$). For larger values of $n$ and expected cost, we develop efficient numerical methods, including a discretized dynamic programming approach and Monte Carlo simulation. The work establishes a fundamental probabilistic approach for online assignment in polygonal environments.

Expected Cost Analysis of Online Facility Assignment on Regular Polygons

TL;DR

The paper addresses the online facility assignment problem in a geometric setting where facilities at the vertices of a regular polygon each have unit capacity and customers arrive on edges uniformly at random. It develops a recursive integral framework for the expected total cost, , and provides exact solutions for small via dynamic programming, complemented by Monte Carlo methods for larger instances. Key contributions include the rigorous recurrence, a Monte Carlo interpretation, and structural insights, accompanied by computational results up to and a detailed square () worked example that yields . The work delivers a foundational probabilistic approach for online geometric assignment, offering practical algorithms for moderate-sized polygons and a basis for extending to more complex geometries and arrival models.

Abstract

This paper analyzes the online facility assignment problem in a geometric setting where facilities with unit capacity are positioned at the vertices of a regular -gon. Customers arrive sequentially at uniformly random positions along the edges. They must be assigned immediately to the nearest available facility, with ties broken by coin toss. The sequential nature and unknown future arrivals require a probabilistic analysis of the expected assignment cost. Our main contribution is a recursive characterization of the expected cost: for any occupancy state , the expected remaining cost equals the average over all edge positions of the immediate assignment cost plus the expected future cost after assignment. We prove that this integral equation can calculate a solution and provide the expected value for small (). For larger values of and expected cost, we develop efficient numerical methods, including a discretized dynamic programming approach and Monte Carlo simulation. The work establishes a fundamental probabilistic approach for online assignment in polygonal environments.

Paper Structure

This paper contains 34 sections, 4 theorems, 20 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Motivation
  3. Problem Definition
  4. Contributions
  5. Related Work
  6. Problem Formulation
  7. Online Assignment Process
  8. Theoretical Analysis
  9. Expected Cost Recurrence
  10. Monte Carlo Interpretation of the Recurrence
  11. Structural Properties
  12. Special Case Analysis
  13. Computational Results
  14. Worked example: Exact expected cost for a square ($n=4$)
  15. First customer.
  16. ...and 19 more sections

Key Result

theorem thmcountertheorem

For any state $S \subset V$ with $|S| < n$, the expected total cost satisfies the recurrence: with the boundary condition $V(V) = 0$. Here:

Figures (1)

  • Figure 1: Distance Calculation Method

Theorems & Definitions (7)

  • theorem thmcountertheorem
  • proof
  • lemma thmcounterlemma: Probability Normalization
  • proof
  • corollary thmcountercorollary: Both Endpoints Available
  • proof
  • corollary thmcountercorollary: One Endpoint Available