DelayServiceCurve

A pure delay service model. Given delay $\tau$ , $f(t) = 0$ if $t \le \tau$ , $f(t) = +\infty$ otherwise.

public class DelayServiceCurve : SuperAdditiveCurve, Unipi.Nancy.Utility.IStableHashCode, Unipi.Nancy.MinPlusAlgebra.IToCodeString, Unipi.Nancy.MinPlusAlgebra.IToMppgString

Inheritance object → Curve → SuperAdditiveCurve → DelayServiceCurve
Implements IStableHashCode, IToCodeString, IToMppgString

Properties

Delay

Models service of a delay-only server.

public Rational Delay { get; }

Property Value

Rational

IsSuperAdditive

True if the curve is super-additive.

note

For a SuperAdditiveCurve this will always return true, without performing any checks.

public bool IsSuperAdditive { get; }

Property Value

bool

PseudoPeriodStart

Point in time after which the curve has a pseudo-periodic behavior.

note

Referred to as $T$ or Rank in [BT08] Section 4.1

public Rational PseudoPeriodStart { get; set; }

Property Value

Rational

PseudoPeriodLength

Time length of each pseudo-period.

note

Referred to as $d$ in [BT08] Section 4.1

public Rational PseudoPeriodLength { get; set; }

Property Value

Rational

PseudoPeriodHeight

Static value gain applied after each pseudo-period. If it's 0, the curve is truly periodic.

note

Referred to as $c$ in [BT08] Section 4.1

public Rational PseudoPeriodHeight { get; set; }

Property Value

Rational

PseudoPeriodSlope

Average slope of curve in pseudo-periodic behavior. If it's 0, the curve is truly periodic.

public Rational PseudoPeriodSlope { get; }

Property Value

Rational

FirstPseudoPeriodEnd

End time of the first pseudo period.

public Rational FirstPseudoPeriodEnd { get; }

Property Value

Rational

SecondPseudoPeriodEnd

End time of the second pseudo period.

public Rational SecondPseudoPeriodEnd { get; }

Property Value

Rational

BaseSequence

Sequence describing behavior of the curve in $[0, T + d[$ . Combined with the UPP property, this is also allows to derive $f(t)$ for any $t \ge T + d$ .

note

Referred to as $[t_1, ..., t_k]$ in [BT08] Section 4.1

public Sequence BaseSequence { get; set; }

Property Value

Sequence

IsFinite

True if the curve has finite value for any $t$ .

public bool IsFinite { get; }

Property Value

bool

FirstFiniteTime

The first instant around which the curve is not infinite. Does not specify whether it's inclusive or not, i.e. if $f(t)$ is finite.

public Rational FirstFiniteTime { get; }

Property Value

Rational

FirstFiniteTimeExceptOrigin

The first instant around which the curve is not infinite, excluding the origin point. Does not specify whether it's inclusive or not, i.e. if $f(t)$ is finite.

public Rational FirstFiniteTimeExceptOrigin { get; }

Property Value

Rational

FirstNonZeroTime

The first instant around which the curve is not 0. Does not specify whether it's inclusive or not, i.e. if $f(\overline{t}) = 0$ .

public Rational FirstNonZeroTime { get; }

Property Value

Rational

PseudoPeriodStartInfimum

Returns the minimum $T_L$ such that $f(t + d) = f(t) + c$ for all $t > T_L$ . It is the infimum of all valid PseudoPeriodStart, i.e. $T_L = \inf\{ T | f(t + d) = f(t) + c \forall t \ge T \}$ .

public Rational PseudoPeriodStartInfimum { get; }

Property Value

Rational

IsZero

True if the curve is 0 for all $t$ .

public bool IsZero { get; }

Property Value

bool

IsPlusInfinite

True if the curve has $+\infty$ value for any $t$ .

public bool IsPlusInfinite { get; }

Property Value

bool

IsMinusInfinite

True if the curve has $-\infty$ value for any $t$ .

public bool IsMinusInfinite { get; }

Property Value

bool

IsPassingThroughOrigin

True if $f(0) = 0$ .

public bool IsPassingThroughOrigin { get; }

Property Value

bool

IsContinuous

True if there is no infinite value or discontinuity within the curve.

public bool IsContinuous { get; }

Property Value

bool

IsContinuousExceptOrigin

True if there is no discontinuity within the curve, except at most in origin.

public bool IsContinuousExceptOrigin { get; }

Property Value

bool

IsLeftContinuous

True if there is no left-discontinuity within the curve.

public bool IsLeftContinuous { get; }

Property Value

bool

IsRightContinuous

True if there is no right-discontinuity within the curve.

public bool IsRightContinuous { get; }

Property Value

bool

IsNonNegative

True if the curve is non-negative, i.e. $f(t) \ge 0$ for any $t$ .

public bool IsNonNegative { get; }

Property Value

bool

FirstNonNegativeTime

The first instant $t^*$ around which the curve is non-negative. Does not specify whether it's inclusive or not, i.e. if $f(t^*) \ge 0$ .

note

Computes $t^* = \inf\{ t \ge 0 \mid f(t) \ge 0 \}$ .

public Rational FirstNonNegativeTime { get; }

Property Value

Rational

IsNonDecreasing

True if for any $t > s$ , $f(t) \ge f(s)$ .

public bool IsNonDecreasing { get; }

Property Value

bool

IsIncreasing

True if for any $t > s$ , $f(t) > f(s)$ .

public bool IsIncreasing { get; }

Property Value

bool

IsUltimatelyFinite

True if for all $t \ge$ PseudoPeriodStart the curve is finite.

note

This property does not check if $f(t), t <$ PseudoPeriodStart is either infinite, finite or both.

public bool IsUltimatelyFinite { get; }

Property Value

bool

IsUltimatelyPlusInfinite

True if, for some $T$ , $f(t) = +\infty$ for all $t \ge T$ .

public bool IsUltimatelyPlusInfinite { get; }

Property Value

bool

IsUltimatelyMinusInfinite

True if, for some $T$ , $f(t) = -\infty$ for all $t \ge T$ .

public bool IsUltimatelyMinusInfinite { get; }

Property Value

bool

IsUltimatelyInfinite

True if, for $b \in \{-\infty, +\infty\}$ and some $T$ , $f(t) = b$ for all $t \ge T$ .

public bool IsUltimatelyInfinite { get; }

Property Value

bool

IsUltimatelyPlain

True if for all $t >$ PseudoPeriodStart the curve is either always finite or always infinite.

note

Defined in [BT08], Definition 1.

public bool IsUltimatelyPlain { get; }

Property Value

bool

IsPlain

True if $f$ is plain, i.e., it is either a) always finite, b) always plus or minus infinite (without changing sign), or c) finite up to a $T$ , then always plus or minus infinite (without changing sign)

note

Formally defined in [BT08], Definition 1.

public bool IsPlain { get; }

Property Value

bool

IsUltimatelyAffine

True if for all $t \ge$ PseudoPeriodStart the curve is affine.

public bool IsUltimatelyAffine { get; }

Property Value

bool

IsUltimatelyConstant

True if for $t \ge$ PseudoPeriodStart the curve is constant.

public bool IsUltimatelyConstant { get; }

Property Value

bool

IsSubAdditive

True if the curve is subadditive, i.e. $f(t+s) \le f(t) + f(s)$ .

note

Based on [Zippo23] Lemma 9.3: $f(0) \ge 0, f$ is subadditive $\iff f^\circ = f^\circ \otimes f^\circ$ , where $f^\circ$ is defined in WithZeroOrigin(). Can be computationally expensive the first time it is invoked, the result is cached afterwards.

public bool IsSubAdditive { get; }

Property Value

bool

IsRegularSubAdditive

True if the curve is subadditive with $f(0) = 0$ .

public bool IsRegularSubAdditive { get; }

Property Value

bool

IsRegularSuperAdditive

True if the curve is super-additive with $f(0) = 0$ .

public bool IsRegularSuperAdditive { get; }

Property Value

bool

IsConcave

Tests if the curve is concave, i.e. for any two points $(t, f(t))$ the straight line joining them is below $f$ .

note

The property is checked via the following property: $f$ is concave $\iff$ a) $f$ is continuous, or it is continuous for $t > 0$ and $f(0) \le f(0^+)$ , and b) $f$ is composed of segments with decreasing slopes.

public bool IsConcave { get; }

Property Value

bool

IsRegularConcave

Tests if the curve is concave with $f(0) = 0$ .

public bool IsRegularConcave { get; }

Property Value

bool

IsConvex

Tests if the curve is convex, i.e. for any two points $(t, f(t))$ the straight line joining them is above $f$ .

note

The property is checked via the following property: $f$ is convex $\iff$ a) $f$ is continuous, or it is continuous for $t > 0$ and $f(0) \ge f(0^+)$ , and b) $f$ is composed of segments with increasing slopes.

public bool IsConvex { get; }

Property Value

bool

IsRegularConvex

Tests if the curve is convex with $f(0) = 0$ .

public bool IsRegularConvex { get; }

Property Value

bool

HasTransient

True if pseudo-periodic behavior starts at $T > 0$ .

public bool HasTransient { get; }

Property Value

bool

TransientSequence

Sequence describing the curve in $[0, T[$ , before pseudo-periodic behavior.

public Sequence TransientSequence { get; }

Property Value

Sequence

TransientElements

Elements describing the curve from $[0, T[$ , before pseudo-periodic behavior.

note

Referred to as $[t_1, ..., t_{i_0 - 1}]$ in [BT08] Section 4.1

public IEnumerable<Element> TransientElements { get; }

Property Value

IEnumerable<Element>

PseudoPeriodicSequence

Sequence describing the pseudo-periodic behavior of the curve in $[T, T + d[$ .

public Sequence PseudoPeriodicSequence { get; }

Property Value

Sequence

PseudoPeriodicElements

Elements describing the pseudo-periodic behavior of the curve in $[T, T + d[$ .

note

Referred to as $[t_{i_0}, ..., t_k]$ in [BT08] Section 4.1

public IEnumerable<Element> PseudoPeriodicElements { get; }

Property Value

IEnumerable<Element>

Constructors