Scheduling Multi-Server Jobs is Not Easy

TL;DR

This work analyzes online scheduling of multi-server jobs on $K$ servers under worst-case inputs, focusing on flow-time minimization via competitive analysis. It proves a fundamental $\Omega(K)$ lower bound and introduces the RA algorithm, which achieves a $K+1$ competitive ratio for unit-sized jobs, while a resource-augmented RA-E with $2K$ servers matches the offline optimum with $K$ servers. For unequal sizes, RA-Size extends the approach with a bound of $\,(K+1)\log (K w_{\max})$ in the same setting, complemented by a randomized lower bound of $\Omega(\max\{K,\log w_{\max}\})$. Numerical results corroborate RA’s superior flow-time performance over ServerFilling in worst-case scenarios, underscoring implications for data-center scheduling and resource provisioning.

Abstract

The problem of online scheduling of multi-server jobs is considered, where there are a total of $K$ servers, and each job requires concurrent service from multiple servers for it to be processed. Each job on its arrival reveals its processing time, the number of servers from which it needs concurrent service and an online algorithm has to make scheduling decisions using only causal information, with the goal of minimizing the response/flow time. The worst case input model is considered and the performance metric is the competitive ratio. For the case, when all job processing time (sizes) are the same, we show that the competitive ratio of any deterministic/randomized algorithm is at least $Ω(K)$ and propose an online algorithm whose competitive ratio is at most $K+1$. With equal job sizes, we also consider the resource augmentation regime where an online algorithm has access to more servers than an optimal offline algorithm. With resource augmentation, we propose a simple algorithm and show that it has a competitive ratio of $1$ when provided with $2K$ servers with respect to an optimal offline algorithm with $K$ servers. With unequal job sizes, we propose an online algorithm whose competitive ratio is at most $2K \log (K w_{\max})$, where $w_{\max}$ is the maximum size of any job.

Figures (5)

• Figure 1: Comparison of per-job flow time for $K=16$ and ${\mathsf{s}}$ uniformly randomly among $[1 \ 2\ 4 \ 8 \ 16]$.
• Figure 2: Comparison of per-job flow time for $K=32$ and ${\mathsf{s}}$ uniformly randomly among $[1 \ 2\ 4 \ 8 \ 16 \ 32]$.
• Figure 3: Comparison of per-job flow time for $\textsf{arr}=5$ with changing $K$ with ${\mathsf{s}}$ uniformly randomly among $[1 \ 2\ 4 \ \dots \ 2^{\log K}]$.
• Figure 4: Comparison of per-job flow time for $K=8$, $\textsf{arr}=5$ with changing $p$, the probability $p$ of choosing ${\mathsf{s}}=8$.
• Figure 5: Comparison of per-job flow time with $\textsf{arr}=5$ as a function of $K$ with changing $p=1/K$.

Theorems & Definitions (45)

• Theorem 1
• proof
• Theorem 2
• Example 3
• Example 4
• Theorem 5
• Definition 6
• Definition 7
• Lemma 8
• Lemma 9