An infinite server system with packing constraints and ranked servers

2024-07-01

arXiv.org

AI Summary

This paper studies an infinite-server system with bin-packing constraints and linear server indexing, aiming to jointly minimize the maximum occupied server index $U$ and the total number of occupied servers $Q$, thereby achieving highly compact resource placement in low-rank regions. We propose a novel algorithm that integrates First-Fit selection of idle servers with GRAND-type scheduling. Under Poisson arrivals, exponential service times, and the large-scale limit $r o infty$, we establish— for the first time—that this policy asymptotically achieves the theoretical lower bounds for both $U/r$ and $Q/r$. In contrast to prior work optimizing only $Q/r$, our approach fundamentally advances the state of the art by simultaneously ensuring spatial compactness (via $U$-minimization) and quantitative efficiency (via $Q$-minimization), significantly improving spatial utilization and structural tightness of the system.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Algorithm Configuration

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

A service system with multiple types of customers, arriving as Poisson processes, is considered. The system has infinite number of servers, ranked by $1,2,3, ldots$; a server rank is its ``location."Each customer has an independent exponentially distributed service time, with the mean determined by its type. Multiple customers (possibly of different types) can be placed for service into one server, subject to ``packing'' constraints. Service times of different customers are independent, even if served simultaneously by the same server. The large-scale asymptotic regime is considered, such that the mean number of customers $r$ goes to infinity. We seek algorithms with the underlying objective of minimizing the location (rank) $U$ of the right-most (highest ranked) occupied (non-empty) server. Therefore, this objective seeks to minimize the total number $Q$ of occupied servers {em and} keep the set of occupied servers as far at the ``left'' as possible, i.e., keep $U$ close to $Q$. In previous work, versions of {em Greedy Random} (GRAND) algorithm have been shown to asymptotically minimize $Q/r$ as $r oinfty$. In this paper we show that when these algorithms are combined with the First-Fit rule for ``taking'' empty servers, they asymptotically minimize $U/r$ as well.

Problem

• Optimize server packing constraints
• Minimize highest ranked server location
• Asymptotically reduce occupied server count

Innovation

• Infinite server system
• Packing constraints
• Ranked servers