Efficient Fault-Tolerant Consensus for Collaborative Services in Edge Computing

In many edge computing applications, edge devices are required to reach fault-tolerant consensus in order to provide collaborative services in outdoor environments. In this paper, we study a comprehensive <inline-formula><tex-math notation="LaTeX">$(a,b)$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>(</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="zou-ieq1-3238138.gif"/></alternatives></inline-formula>-majority consensus problem based on a novel failure model, which takes <inline-formula><tex-math notation="LaTeX">$a$</tex-math><alternatives><mml:math><mml:mi>a</mml:mi></mml:math><inline-graphic xlink:href="zou-ieq2-3238138.gif"/></alternatives></inline-formula> distinct opinions as inputs and outputs a <inline-formula><tex-math notation="LaTeX">$b$</tex-math><alternatives><mml:math><mml:mi>b</mml:mi></mml:math><inline-graphic xlink:href="zou-ieq3-3238138.gif"/></alternatives></inline-formula>-majority opinion as the final agreement. This problem formulation is drastically different from traditional ones, which usually require a majority consensus from the binary opinions of multiple supporters. It is more practical and flexible as it can accommodate more than 2 input opinions and output one that satisfies the application requirement defined by parameter <inline-formula><tex-math notation="LaTeX">$b$</tex-math><alternatives><mml:math><mml:mi>b</mml:mi></mml:math><inline-graphic xlink:href="zou-ieq4-3238138.gif"/></alternatives></inline-formula>. We also consider physical layer in our failure model while previous models mainly focus on faults occurred in protocol layer and data layer. Based on this more realistic failure model and a more practical consensus problem definition, we present a distributed protocol for <inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="zou-ieq5-3238138.gif"/></alternatives></inline-formula> edge devices to reach an <inline-formula><tex-math notation="LaTeX">$(a,b)$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>(</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="zou-ieq6-3238138.gif"/></alternatives></inline-formula>-majority consensus within <inline-formula><tex-math notation="LaTeX">$\Theta (n)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>Θ</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="zou-ieq7-3238138.gif"/></alternatives></inline-formula> time steps with high probability. Empirical results from our simulation studies validate the fault tolerance property and efficiency of our work in achieving the <inline-formula><tex-math notation="LaTeX">$(a,b)$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>(</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="zou-ieq8-3238138.gif"/></alternatives></inline-formula>-majority consensus.