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Asymptotic optimality of dynamic first-fit packing on the half-axis

Philip Ernst, Alexander Stolyar

TL;DR

The paper proves the asymptotic optimality of the dynamic first-fit packing on the half-axis for a non-degenerate two-type item size distribution $(1,2)$ under Poisson arrivals and exponential service. Using hydrodynamic scaling and generator-based Lyapunov drift arguments, it shows that the stationary configuration, scaled by $r$, converges in probability to the optimal left-to-right packing with no gaps, i.e., $(1/r)(F_1(p_1 r; ty), F_2(p_1 r+2 p_2 r; ty)) o (p_1,p_2)$. This implies that, in the limit, almost all 1-items occupy $[0,p_1 r)$ and 2-items occupy $[p_1 r,(p_1+2p_2)r)$ with vanishing empty space in the occupied span. The results extend the classical unit-length asymptotic optimality to a non-degenerate setting and are proposed to generalize to arbitrary integer item sizes, impacting the understanding of online packing and dynamic storage efficiency.

Abstract

We revisit a classical problem in dynamic storage allocation. Items arrive in a linear storage medium, modeled as a half-axis, at a Poisson rate $r$ and depart after an independent exponentially distributed unit mean service time. The arriving item sizes are assumed to be independent and identically distributed (i.i.d.) from a common distribution $H$. A widely employed algorithm for allocating the items is the ``first-fit'' discipline, namely, each arriving item is placed in the the left-most vacant interval large enough to accommodate it. In a seminal 1985 paper, Coffman, Kadota, and Shepp [6] proved that in the special case of unit length items (i.e. degenerate $H$), the first-fit algorithm is asymptotically optimal in the following sense: the ratio of expected empty space to expected occupied space tends towards $0$ as the occupied space tends towards infinity. In a sequel to [6], the authors of [5] conjectured that the first-fit discipline is also asymptotically optimal for non-degenerate $H$. In this paper we provide the first proof of first-fit asymptotic optimality for a non-degenerate distribution $H$, namely the case when items can be of sizes 1 and 2. Specifically, we prove that, under first-fit, the steady-state packing configuration, scaled down by $r$, converges in distribution to the optimal limiting packing configuration, i.e. the one with smaller items on the left, larger items on the right, and with no gaps between.

Asymptotic optimality of dynamic first-fit packing on the half-axis

TL;DR

The paper proves the asymptotic optimality of the dynamic first-fit packing on the half-axis for a non-degenerate two-type item size distribution under Poisson arrivals and exponential service. Using hydrodynamic scaling and generator-based Lyapunov drift arguments, it shows that the stationary configuration, scaled by , converges in probability to the optimal left-to-right packing with no gaps, i.e., . This implies that, in the limit, almost all 1-items occupy and 2-items occupy with vanishing empty space in the occupied span. The results extend the classical unit-length asymptotic optimality to a non-degenerate setting and are proposed to generalize to arbitrary integer item sizes, impacting the understanding of online packing and dynamic storage efficiency.

Abstract

We revisit a classical problem in dynamic storage allocation. Items arrive in a linear storage medium, modeled as a half-axis, at a Poisson rate and depart after an independent exponentially distributed unit mean service time. The arriving item sizes are assumed to be independent and identically distributed (i.i.d.) from a common distribution . A widely employed algorithm for allocating the items is the ``first-fit'' discipline, namely, each arriving item is placed in the the left-most vacant interval large enough to accommodate it. In a seminal 1985 paper, Coffman, Kadota, and Shepp [6] proved that in the special case of unit length items (i.e. degenerate ), the first-fit algorithm is asymptotically optimal in the following sense: the ratio of expected empty space to expected occupied space tends towards as the occupied space tends towards infinity. In a sequel to [6], the authors of [5] conjectured that the first-fit discipline is also asymptotically optimal for non-degenerate . In this paper we provide the first proof of first-fit asymptotic optimality for a non-degenerate distribution , namely the case when items can be of sizes 1 and 2. Specifically, we prove that, under first-fit, the steady-state packing configuration, scaled down by , converges in distribution to the optimal limiting packing configuration, i.e. the one with smaller items on the left, larger items on the right, and with no gaps between.
Paper Structure (9 sections, 7 theorems, 26 equations, 1 figure)

This paper contains 9 sections, 7 theorems, 26 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model and Main Result
  3. Proof of Theorem \ref{['conj1']}
  4. Outline and basic intuition
  5. Preliminaries
  6. Proof of Proposition \ref{['prop1']}
  7. Proof of Proposition \ref{['prop2']}
  8. Proof of Proposition \ref{['prop4']}
  9. Discussion

Key Result

Theorem 2.1

As $r\to\infty$, $(1/r)(F_1(p_1 r;\infty), F_2(p_1 r + 2 p_2 r;\infty)) \stackrel{P}{\rightarrow} (p_1,p_2)$.

Figures (1)

  • Figure 1: A simulation of the system state evolution for $r=5000$, $p_1=p_2=1/2$. The 1-items appear in blue and the 2-items appear in red. Grey represents empty space.

Theorems & Definitions (10)

  • Theorem 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof
  • proof