Suppose that we have n jobs to distribute among m...

Suppose that we have $n$ jobs to distribute among $m$ processors. For simplicity, we assume that $m$ divides $n$. A job takes 1 step with probability $p$ and $k>1$ steps with probability $1-p$. Use Chernoff bounds to determine upper and lower bounds (that hold with high probability) on when all jobs will be completed if we randomly assign exactly $n / m$ jobs to each processor.

Key Concepts

High Probability Bounds

High probability bounds ensure that events occur with probabilities that approach one as the size of the system increases. In the context of job distribution among processors, these bounds guarantee that the completion time for all jobs will not exceed a certain threshold with overwhelming likelihood.

Parallel Processing

Parallel processing refers to performing multiple computations simultaneously across several processors. Effective analysis of parallel processing systems requires understanding the distribution of job completion times and ensuring that, with high probability, the overall system performs efficiently—a task often achieved using probabilistic methods and concentration inequalities.

Randomized Algorithms

Randomized algorithms use random choices as part of their logic, which can simplify their design and analysis while still providing strong performance guarantees. In distributed processing contexts, such algorithms are often analyzed using tools like Chernoff bounds to guarantee that performance metrics meet desired thresholds with high probability.

Chernoff Bounds

Chernoff bounds are probabilistic tools that provide exponentially decreasing bounds on the tails of sums of independent random variables. They are particularly useful in analyzing the performance of randomized algorithms and ensuring that certain events, like the completion time in distributed systems, occur with high probability.

Concentration Inequalities

Concentration inequalities quantify how a random variable deviates from its expected value. In algorithm analysis, these inequalities are employed to demonstrate that, despite random variations in processing times or job assignments, the overall behavior of the system remains tightly concentrated around its mean behavior.

Load Balancing

Load balancing involves the equitable distribution of tasks among different processors to optimize performance and prevent bottlenecks. Analyzing load balancing in a randomized setting helps in determining the overall processing time when jobs are distributed randomly, ensuring that no processor is significantly overburdened.

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