Perpetual maintenance of machines with different urgency requirements

TL;DR

The Bamboo Garden Trimming Problem (BGT) is considered, a metaphor for a collection of machines which have to be serviced, with different frequencies, by a robot which can service only one machine at a time, and two variants of BGT are considered.

Abstract

A garden $G$ is populated by $n\ge 1$ bamboos $b_1, b_2, ..., b_n$ with the respective daily growth rates $h_1 \ge h_2 \ge \dots \ge h_n$. It is assumed that the initial heights of bamboos are zero. The robotic gardener maintaining the garden regularly attends bamboos and trims them to height zero according to some schedule. The Bamboo Garden Trimming Problem (BGT) is to design a perpetual schedule of cuts to maintain the elevation of the bamboo garden as low as possible. The bamboo garden is a metaphor for a collection of machines which have to be serviced, with different frequencies, by a robot which can service only one machine at a time. The objective is to design a perpetual schedule of servicing which minimizes the maximum (weighted) waiting time for servicing. We consider two variants of BGT. In discrete BGT the robot trims only one bamboo at the end of each day. In continuous BGT the bamboos can be cut at any time, however, the robot needs time to move from one bamboo to the next. For discrete BGT, we show tighter approximation algorithms for the case when the growth rates are balanced and for the general case. The former algorithm settles one of the conjectures about the Pinwheel problem. The general approximation algorithm improves on the previous best approximation ratio. For continuous BGT, we propose approximation algorithms which achieve approximation ratios $O(\log \lceil h_1/h_n\rceil)$ and $O(\log n)$.

Figures (3)

• Figure 1: Illustration of the execution of the Main Algorithm. The top half: transformation of frequencies in layers other than the $min$ layer. The bottom half: transformation in the $min$ layer and the final powers-of-2 frequencies.
• Figure 2: Illustration of combining frequencies in the Main Algorithm. In steps 3--5 of the algorithm, frequency $f_j$ is paired twice with other frequencies by applications of Observation 1, creating a frequency $f_j/4$ (node $a$). The resulting frequency is then put into a group of $10$ frequencies $f_j/4$ and this group is replaced with one new frequency $g_i = f_j/40$ by an application of Observation 2.
• Figure 3: Example of a spiral input.

Theorems & Definitions (23)

• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• theorem thmcountertheorem
• proof
• corollary thmcountercorollary
• theorem thmcountertheorem
• proof
• theorem thmcountertheorem
• proof