Lecture 009

Non-Preemptive Migration (NP, initial placement, remote execution): migration of newborn jobs only. (You don't migrate a job once it has started running)

Preemptive Migration (P, active process migration): migration active jobs

Job size: total CPU requirement in seconds

Job age: total CPU usage til now

Job remaining lifetime: future CPU requirement

Heavy Tails

Failure rate function:

r_X(t) = \frac{f_X(t)}{\bar{F}_X(t)}

Decreasing Failure Rate: r_X(t) strictly decrease with t. (e.g. friendship, feedback loop) Increasing Failure Rate: r_X(t) strictly increase with t. (e.g. lifetime of person, no feedback loop)

Exponential Distribution

For Exponential, Pr\{X > x + a | X > a\} = Pr\{X > x\} means the lifetime is independent of the current age by memoryless.

Pareto Distribution

For X \sim \text{Pareto}(\alpha), and 0 < \alpha < 2

\begin{align*} f_X(x) =& \alpha x^{-\alpha - 1} \tag{for $x \geq 1$}\\ F_X(x) =& Pr\{X < x\} = 1 - x^{-\alpha} \tag{for $x \geq 1$}\\ \bar{F}_X(x) =& Pr\{X > x\} = x^{-\alpha} \tag{for $x \geq 1$}\\ r_X(x) =& \frac{\alpha x^{-\alpha - 1}}{x^{-\alpha}} = \frac{\alpha}{x} \tag{for $x \geq 1$, decreasing failure rate}\\ E[X] =& \infty\\ E[X^i] =& \infty \tag{$\forall i > 1$}\\ \end{align*}

For \text{Pareto}(\alpha = 1), what is the probability a job with age a live to at least age b?

Pr\{X > b | X \geq a \geq 1\} = \frac{Pr\{X > b \cap X \geq a\}}{Pr\{X \geq a\}} = \frac{Pr\{X > b\}}{Pr\{X \geq a\}} = \frac{1/b}{1/a} = \frac{a}{b} \text{ where } b > a

Which means:

  1. of all jobs of age 1 second, half of them will live to at least 2 seconds
  2. The probability of a job of age 1 seconds use X > t seconds of CPU is \frac{1}{t}
  3. The probability of a job of age t seconds use X > 2t seconds of CPU is \frac{1}{2}

Properties of Pareto Distribution:

  1. decreasing failure rate
  2. infinite (or near infinite) variance
  3. heavy tail

Bounded Pareto

For X \sim \text{Bounded Pareto}(k, p, \alpha) for k \leq x \leq p, 0 < \alpha < 2

\begin{align*} f(x) =& \alpha x^{-\alpha - 1} \cdot \left(\frac{k^\alpha}{1 - (\frac{k}{p})^\alpha}\right)\\ \end{align*}

\frac{k^\alpha}{1 - (\frac{k}{p})^\alpha} is a normalization factor to make integral of density function between k and p come out to 1

Heavy Tails

Pareto is more heavy tail than Exponential with the same mean:

Heavy Tails:

Google's Data:

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