Capacity Provisioning Motivated Online Non-Convex Optimization Problem with Memory and Switching Cost

TL;DR

This work studies an online, memoryful capacity provisioning problem where flow-time (outstanding jobs) is minimized together with a switching-cost penalty for changing the number of active servers. The authors tackle both worst-case and stochastic inputs under linear and quadratic switching costs, deriving competitive online algorithms with memory-length independent performance. A novel primal-dual framework is developed, incorporating a work-neutrality constraint to handle memory effects and non-convexity, yielding a constant competitive ratio (≤20) for the quadratic-cost setting and, in stochastic settings, simple policies that achieve near-optimal long-run costs. The results illuminate how memory and switching-cost structure fundamentally alter online performance and scheduling strategies in data-center settings, with practical implications for SRPT-based scheduling and adaptive capacity provisioning.

Abstract

An online non-convex optimization problem is considered where the goal is to minimize the flow time (total delay) of a set of jobs by modulating the number of active servers, but with a switching cost associated with changing the number of active servers over time. Each job can be processed by at most one fixed speed server at any time. Compared to the usual online convex optimization (OCO) problem with switching cost, the objective function considered is non-convex and more importantly, at each time, it depends on all past decisions and not just the present one. Both worst-case and stochastic inputs are considered; for both cases, competitive algorithms are derived.

Figures (6)

• Figure 1: Comparison of mean flow time with different algorithms as a function of mean arrival rate per slot with linear switching cost with $\alpha=1$ .
• Figure 2: Comparison of mean flow time with different algorithms as a function of mean arrival rate per slot with linear switching cost with $\alpha=2$.
• Figure 3: Comparison of mean flow time with different algorithms as a function of mean arrival rate per slot with linear switching cost with $\alpha=4$.
• Figure 4: Comparison of mean flow time with the proposed algorithm with different values of $\beta$ and Balance $\alpha (s(t)-s(t-1))^2 = n(t)$ for quadratic switching cost with $\alpha=1$ as a function of mean arrival rate per slot.
• Figure 5: Comparison of mean flow time with the proposed algorithm with different values of $\beta$ and Balance $\alpha (s(t)-s(t-1))^2 = n(t)$ for quadratic switching cost with $\alpha=2$ as a function of mean arrival rate per slot.

Theorems & Definitions (45)

• Remark 1
• Remark 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• proof
• Remark 7
• Lemma 8
• Remark 9