Multi-Timescale Online Optimization of Network Function Virtualization for Service Chaining

Network Function Virtualization (NFV) can cost-efficiently provide network services by running different virtual network functions (VNFs) at different virtual machines (VMs) in a correct order. This can result in strong couplings between the decisions of the VMs on the placement and operations of VNFs. This paper presents a new fully decentralized online approach for optimal placement and operations of VNFs. Building on a new stochastic dual gradient method, our approach decouples the real-time decisions of VMs, asymptotically minimizes the time-average cost of NFV, and stabilizes the backlogs of network services with a cost-backlog tradeoff of <inline-formula><tex-math notation="LaTeX">$[\epsilon, 1/\epsilon ]$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>[</mml:mo><mml:mi>ε</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>ε</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="chen-ieq1-2885301.gif"/></alternatives></inline-formula>, for any <inline-formula><tex-math notation="LaTeX">$\epsilon > 0$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>ε</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="chen-ieq2-2885301.gif"/></alternatives></inline-formula>. Our approach can be relaxed into multiple timescales to have VNFs (re)placed at a larger timescale and hence alleviate service interruptions. While proved to preserve the asymptotic optimality, the larger timescale can slow down the optimal placement of VNFs. A learn-and-adapt strategy is further designed to speed the placement up with an improved tradeoff <inline-formula><tex-math notation="LaTeX">$[\epsilon, \log ^2(\epsilon)/{\sqrt{\epsilon }}]$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>[</mml:mo><mml:mi>ε</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mo form="prefix">log</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>ε</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:msqrt><mml:mi>ε</mml:mi></mml:msqrt><mml:mo>]</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="chen-ieq3-2885301.gif"/></alternatives></inline-formula>. Numerical results show that the proposed method is able to reduce the time-average cost of NFV by 23 percent and reduce the queue length (or delay) by 74 percent, as compared to existing benchmarks.