Online Job Scheduling with Redundancy and Opportunistic Checkpointing: A Speedup-Function-Based Analysis

In a large-scale computing cluster, the job completions can be substantially delayed due to two sources of variability, namely, variability in the job size and that in the machine service capacity. To tackle this issue, existing works have proposed various scheduling algorithms which exploit redundancy wherein a job runs on multiple servers until the first completes. In this paper, we explore the impact of variability in the machine service capacity and adopt a rigorous analytical approach to design scheduling algorithms using redundancy and checkpointing. We design several online algorithms which can dynamically vary the number of redundant copies for jobs. We also provide new theoretical performance bounds for these algorithms in terms of the overall job flowtime by introducing the notion of a speedup function, based on which a novel potential function can be defined to enable the corresponding competitive ratio analysis. In particular, by adopting the online primal-dual fitting approach, we prove that our SRPT+R Algorithm in a non-multitasking cluster is <inline-formula><tex-math notation="LaTeX">$(1+\epsilon)$</tex-math><alternatives><mml:math><mml:mrow><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="xu-ieq1-2871135.gif"/></alternatives><alternatives><mml:math><mml:mrow><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="xu-ieq1-2871135.gif"/></alternatives></inline-formula>-speed, <inline-formula><tex-math notation="LaTeX">$\ O(\frac{1}{\epsilon })$</tex-math><alternatives><mml:math><mml:mrow><mml:mspace width="4pt"/><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>ε</mml:mi></mml:mfrac><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="xu-ieq2-2871135.gif"/></alternatives><alternatives><mml:math><mml:mrow><mml:mspace width="4pt"/><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>ε</mml:mi></mml:mfrac><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="xu-ieq2-2871135.gif"/></alternatives></inline-formula>-competitive. We also show that our proposed Fair+R and LAPS+R(<inline-formula><tex-math notation="LaTeX">$\beta$</tex-math><alternatives><mml:math><mml:mi>β</mml:mi></mml:math><inline-graphic xlink:href="xu-ieq3-2871135.gif"/></alternatives><alternatives><mml:math><mml:mi>β</mml:mi></mml:math><inline-graphic xlink:href="xu-ieq3-2871135.gif"/></alternatives></inline-formula>) Algorithms for a multitasking cluster are <inline-formula><tex-math notation="LaTeX">$(4+\epsilon)$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>(</mml:mo><mml:mn>4</mml:mn><mml:mo>+</mml:mo><mml:mi>ε</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="xu-ieq4-2871135.gif"/></alternatives><alternatives><mml:math><mml:mrow><mml:mo>(</mml:mo><mml:mn>4</mml:mn><mml:mo>+</mml:mo><mml:mi>ε</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="xu-ieq4-2871135.gif"/></alternatives></inline-formula>-speed, <inline-formula><tex-math notation="LaTeX">$\ O(\frac{1}{\epsilon })$</tex-math><alternatives><mml:math><mml:mrow><mml:mspace width="4pt"/><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>ε</mml:mi></mml:mfrac><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="xu-ieq5-2871135.gif"/></alternatives><alternatives><mml:math><mml:mrow><mml:mspace width="4pt"/><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>ε</mml:mi></mml:mfrac><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="xu-ieq5-2871135.gif"/></alternatives></inline-formula>-competitive and (<inline-formula><tex-math notation="LaTeX">$2 + 2\beta + 2\epsilon)$</tex-math><alternatives><mml:math><mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>β</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>ε</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="xu-ieq6-2871135.gif"/></alternatives><alternatives><mml:math><mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>β</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>ε</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="xu-ieq6-2871135.gif"/></alternatives></inline-formula>-speed <inline-formula><tex-math notation="LaTeX">$O(\frac{1}{\beta \epsilon })$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>β</mml:mi><mml:mi>ε</mml:mi></mml:mrow></mml:mfrac><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="xu-ieq7-2871135.gif"/></alternatives><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>β</mml:mi><mml:mi>ε</mml:mi></mml:mrow></mml:mfrac><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="xu-ieq7-2871135.gif"/></alternatives></inline-formula>-competitive respectively. We demonstrate via extensive simulations that our proposed algorithms can significantly reduce job flowtime under both the non-multitasking and multitasking modes.