DeltaQ.PiecewisePolynomial

DQ is a probability distribution of completion time using the numeric type Rational. This type represents a mixed discrete / continuous probability distribution where the continuous part is represented in terms of piecewise polynomials.

Type

data DQ

Probability distribution of durations.

Instances

Show DQ

Defined in DeltaQ.PiecewisePolynomial

Methods

showsPrec :: Int -> DQ -> ShowS

show :: DQ -> String

showList :: [DQ] -> ShowS

NFData DQ

Defined in DeltaQ.PiecewisePolynomial

Methods

rnf :: DQ -> ()

DeltaQ DQ

Defined in DeltaQ.PiecewisePolynomial

Associated Types

type Probability DQ

Methods

choice :: Probability DQ -> DQ -> DQ -> DQ

choices :: [(Probability DQ, DQ)] -> DQ

uniform :: Duration DQ -> Duration DQ -> DQ

failure :: DQ -> Probability DQ

successWithin :: DQ -> Duration DQ -> Probability DQ

quantile :: DQ -> Probability DQ -> Eventually (Duration DQ)

earliest :: DQ -> Eventually (Duration DQ)

deadline :: DQ -> Eventually (Duration DQ)

Outcome DQ

Defined in DeltaQ.PiecewisePolynomial

Associated Types

type Duration DQ

Methods

never :: DQ

wait :: Duration DQ -> DQ

sequentially :: DQ -> DQ -> DQ

(.>>.) :: DQ -> DQ -> DQ

firstToFinish :: DQ -> DQ -> DQ

(.\/.) :: DQ -> DQ -> DQ

lastToFinish :: DQ -> DQ -> DQ

(./\.) :: DQ -> DQ -> DQ

Eq DQ

Defined in DeltaQ.PiecewisePolynomial

Methods

(==) :: DQ -> DQ -> Bool

(/=) :: DQ -> DQ -> Bool

PartialOrd DQ

Partial order of cumulative distribution functions.

leq x y holds if and only if for all completion times t, the probability to succeed within the time t is always larger (or equal) for x compared to y. In other words, x has a higher probability of completing faster.

x `leq` y  <=>  ∀ t. successWithin x t >= successWithin y t

Defined in DeltaQ.PiecewisePolynomial

Methods

leq :: DQ -> DQ -> Bool

comparable :: DQ -> DQ -> Bool

type Duration DQ

Defined in DeltaQ.PiecewisePolynomial

type Duration DQ = Rational

type Probability DQ

Defined in DeltaQ.PiecewisePolynomial

type Probability DQ = Rational

distribution :: DQ -> Piecewise (Poly Rational)

Get the distribution function as piecewise function of polynomials.

fromPositiveMeasure :: Measure Rational -> Maybe DQ

Interpret a finite, signed Measure as a probability distribution.

In order to admit an interpretation as probability, the measure needs to be positive. This condition is checked, and if it does not hold, the function returns Nothing.

unsafeFromPositiveMeasure :: Measure Rational -> DQ

Interpret a finite, positive Measure as a probability distribution.

The precondition that the measure is positive is not checked!

Operations

meetsQTA :: DQ -> DQ -> Bool

Test whether the given probability distribution of completion times is equal to or better than a given quantitative timeliness agreement (QTA).

Synonym for leq of the partial order,

p `meetsQTA` qta  =  p `leq` qta

data Moments a

The first four commonly used moments of a probability distribution.

Constructors

Moments

Fields

• mean :: a

  Mean or Expected Value \( \mu \). Defined as \( \mu = E[X] \).

• variance :: a

  Variance \( \sigma^2 \). Defined as \( \sigma^2 = E[(X - \mu)^2] \). Equal to \( \sigma^2 = E[X^2] - \mu^2 \).

• skewness :: a

  Skewness \( \gamma_1 \). Defined as \( \gamma_1 = E\left[\left(\frac{(X - \mu)}{\sigma}\right)^3 \right] \).

• kurtosis :: a

  Kurtosis \( \kappa \). Defined as \( \kappa = E\left[\left(\frac{(X - \mu)}{\sigma}\right)^4 \right] \).

  The kurtosis is bounded below: \( \kappa \geq \gamma_1^2 + 1 \).

Instances

Show a => Show (Moments a)

Defined in Numeric.Probability.Moments

Methods

showsPrec :: Int -> Moments a -> ShowS

show :: Moments a -> String

showList :: [Moments a] -> ShowS

Eq a => Eq (Moments a)

Defined in Numeric.Probability.Moments

Methods

(==) :: Moments a -> Moments a -> Bool

(/=) :: Moments a -> Moments a -> Bool

moments :: DQ -> (Rational, Moments Rational)

Compute the success probability of a DQ, and the first commonly used Moments of the probability distribution conditioned on success.

Internal

complexity :: DQ -> Int

Size of the representation of a probability distribution, i.e. number of pieces of the piecewise function and degrees of the polynomials.

This quantity is relevant to stating and analyzing the asymptotic time complexity of operations.