Comparisons of stochastic task-resource systems Bruno Gaujal - - PowerPoint PPT Presentation

Comparisons of stochastic task-resource systems

Bruno Gaujal Jean-Marc Vincent

INRIA and LIG

Fréjus, 4 Juin 2007

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 1 / 30

Outline

1

Probabilistic task-resource models

2

Stochastic orders

3

Comparison of systems Mapping technique association Coupling technique

4

Several applications PERT Graph Queues polling from several queues

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 2 / 30

Probabilistic models

The main object of this lecture is the task-resource model.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 3 / 30

Probabilistic models

The main object of this lecture is the task-resource model. Tasks are characterized by the arrival times that form a point process 0 T1 T2 T3 · · ·

• ver the positive real line and by the sizes of tasks that is a sequence of real numbers S1, S2, . . ..
• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 3 / 30

Probabilistic models

The main object of this lecture is the task-resource model. Tasks are characterized by the arrival times that form a point process 0 T1 T2 T3 · · ·

• ver the positive real line and by the sizes of tasks that is a sequence of real numbers S1, S2, . . ..

Resources are characterized by their number K ∈ N ∪ {+∞} and the respective speeds v1, . . . , vK . In the following the size of the tasks is often given is seconds (time for a resource of speed 1 to treat a task).

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 3 / 30

Probabilistic models

The main object of this lecture is the task-resource model. Tasks are characterized by the arrival times that form a point process 0 T1 T2 T3 · · ·

• ver the positive real line and by the sizes of tasks that is a sequence of real numbers S1, S2, . . ..

Resources are characterized by their number K ∈ N ∪ {+∞} and the respective speeds v1, . . . , vK . In the following the size of the tasks is often given is seconds (time for a resource of speed 1 to treat a task). Additionally, tasks and ressources may be constrainted by dependencies, synchronizations, availability conditions, matchings, . . .

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 3 / 30

Probabilistic models

The main object of this lecture is the task-resource model. Tasks are characterized by the arrival times that form a point process 0 T1 T2 T3 · · ·

• ver the positive real line and by the sizes of tasks that is a sequence of real numbers S1, S2, . . ..

Resources are characterized by their number K ∈ N ∪ {+∞} and the respective speeds v1, . . . , vK . In the following the size of the tasks is often given is seconds (time for a resource of speed 1 to treat a task). Additionally, tasks and ressources may be constrainted by dependencies, synchronizations, availability conditions, matchings, . . . Here we will mostly consider very simple systems with one common features : randomness. Basically, the arrival times and/or the task sizes will be random processes.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 3 / 30

Stochastic Orders

There are many ways to compare two random variables (and random processes). The most

• bvious one is to compare the means : X µ Y if E(X) E(Y).

However, this order is rather crude and may not capture a lot of insight in the comparison of two stochastic systems, in particular in the scheduling context.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 4 / 30

Stochastic Orders

There are many ways to compare two random variables (and random processes). The most

• bvious one is to compare the means : X µ Y if E(X) E(Y).

However, this order is rather crude and may not capture a lot of insight in the comparison of two stochastic systems, in particular in the scheduling context. Consider a two task-resource systems with one resource. One has arrivals every 4 seconds of tasks of size 4 and the other has arrivals of tasks of size 2 at times 5n and 5n + 1.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 4 / 30

Stochastic Orders

There are many ways to compare two random variables (and random processes). The most

• bvious one is to compare the means : X µ Y if E(X) E(Y).

However, this order is rather crude and may not capture a lot of insight in the comparison of two stochastic systems, in particular in the scheduling context. Consider a two task-resource systems with one resource. One has arrivals every 4 seconds of tasks of size 4 and the other has arrivals of tasks of size 2 at times 5n and 5n + 1. Then the expected input loads per second are E(L1) = 1 > E(L2) = 4/5 respectively.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 4 / 30

Stochastic Orders

There are many ways to compare two random variables (and random processes). The most

• bvious one is to compare the means : X µ Y if E(X) E(Y).

However, this order is rather crude and may not capture a lot of insight in the comparison of two stochastic systems, in particular in the scheduling context. Consider a two task-resource systems with one resource. One has arrivals every 4 seconds of tasks of size 4 and the other has arrivals of tasks of size 2 at times 5n and 5n + 1. Then the expected input loads per second are E(L1) = 1 > E(L2) = 4/5 respectively. As for the expected waiting times, E(W1) = 0 < E(W2) = 1/2.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 4 / 30

Stochastic Orders

There are many ways to compare two random variables (and random processes). The most

• bvious one is to compare the means : X µ Y if E(X) E(Y).

However, this order is rather crude and may not capture a lot of insight in the comparison of two stochastic systems, in particular in the scheduling context. Consider a two task-resource systems with one resource. One has arrivals every 4 seconds of tasks of size 4 and the other has arrivals of tasks of size 2 at times 5n and 5n + 1. Then the expected input loads per second are E(L1) = 1 > E(L2) = 4/5 respectively. As for the expected waiting times, E(W1) = 0 < E(W2) = 1/2. Even if waiting times are increasing functions of loads (see later), this is not the case for the µ

• rder.
• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 4 / 30

Stochastic Orders

There are many ways to compare two random variables (and random processes). The most

• bvious one is to compare the means : X µ Y if E(X) E(Y).

However, this order is rather crude and may not capture a lot of insight in the comparison of two stochastic systems, in particular in the scheduling context. Consider a two task-resource systems with one resource. One has arrivals every 4 seconds of tasks of size 4 and the other has arrivals of tasks of size 2 at times 5n and 5n + 1. Then the expected input loads per second are E(L1) = 1 > E(L2) = 4/5 respectively. As for the expected waiting times, E(W1) = 0 < E(W2) = 1/2. Even if waiting times are increasing functions of loads (see later), this is not the case for the µ

• rder.

There exists several stochastic orders (in the book Comparison methods for stochastic models and risks (Muller and Stoyan, 2002), 49 different orders are defined with different applications in mind.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 4 / 30

The usual stochastic order

The usual stochastic order (also called the strong order) is defined as follows (for real random variables). X st Y if FX (a) = P(X a) P(Y a) = FY (a) for all a.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 5 / 30

The usual stochastic order

The usual stochastic order (also called the strong order) is defined as follows (for real random variables). X st Y if FX (a) = P(X a) P(Y a) = FY (a) for all a. The st order has several other characterizations : Sample path definition There exists two variables X ′ and Y ′ in (Ω, A, P) with the same distribution as X and Y such that X(ω) Y(ω) for each ω ∈ Ω. Integral definition For all increasing function f, E(f(X)) E(f(Y)).

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 5 / 30

The usual stochastic order : examples

• 1. Show that X st Y ⇒ X µ Y.
• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 6 / 30

The usual stochastic order : examples

• 1. Show that X st Y ⇒ X µ Y.
• 2. Show that X st Y and EX = EY ⇒ FX = FY .
• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 6 / 30

The usual stochastic order : examples

• 1. Show that X st Y ⇒ X µ Y.
• 2. Show that X st Y and EX = EY ⇒ FX = FY .
• 3. Compare the following integer random variables :

X = 1 w.p. 1/4, 2 w.p. 1/2, 3 w,.p. 1/4 Y = 2 w.p. 1/2 , 3 w.p. 1/2 Z = 1 w.p. 1/3 , 3 w.p. 2/3

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 6 / 30

Stronger stochastic orders

Some orders are stronger than st : Consider the following case : somebody wants to buy a car and can choose between two models with lifetimes X and Y. If the price is the same and X st Y them, she ought to buy model Y. Now what happens if both cars are used (one year old), is Y still a better choice ?

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 7 / 30

Stronger stochastic orders

Some orders are stronger than st : Consider the following case : somebody wants to buy a car and can choose between two models with lifetimes X and Y. If the price is the same and X st Y them, she ought to buy model Y. Now what happens if both cars are used (one year old), is Y still a better choice ? well, not necessarily :

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 7 / 30

Stronger stochastic orders

Some orders are stronger than st : Consider the following case : somebody wants to buy a car and can choose between two models with lifetimes X and Y. If the price is the same and X st Y them, she ought to buy model Y. Now what happens if both cars are used (one year old), is Y still a better choice ? well, not necessarily : Assume that X is uniform over [0, 3] (with db F) and Y has a distribution with density 1/6, 1/2, 1/3

• n [0, 1], ]1, 2], ]2, 3] (with db G).
• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 7 / 30

Stronger stochastic orders

Some orders are stronger than st : Consider the following case : somebody wants to buy a car and can choose between two models with lifetimes X and Y. If the price is the same and X st Y them, she ought to buy model Y. Now what happens if both cars are used (one year old), is Y still a better choice ? well, not necessarily : Assume that X is uniform over [0, 3] (with db F) and Y has a distribution with density 1/6, 1/2, 1/3

• n [0, 1], ]1, 2], ]2, 3] (with db G).

Then, X st Y (F G). However X1 = (X|X > 1) and Y1 = (Y|Y > 1) are not st-comparable : X1 is uniform over [0, 2] (with density 1/2) and Y1 has a distribution with density 3/5, 2/5, on [0, 1], ]1, 2].

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 7 / 30

Stronger stochastic orders

Some orders are stronger than st : Consider the following case : somebody wants to buy a car and can choose between two models with lifetimes X and Y. If the price is the same and X st Y them, she ought to buy model Y. Now what happens if both cars are used (one year old), is Y still a better choice ? well, not necessarily : Assume that X is uniform over [0, 3] (with db F) and Y has a distribution with density 1/6, 1/2, 1/3

• n [0, 1], ]1, 2], ]2, 3] (with db G).

Then, X st Y (F G). However X1 = (X|X > 1) and Y1 = (Y|Y > 1) are not st-comparable : X1 is uniform over [0, 2] (with density 1/2) and Y1 has a distribution with density 3/5, 2/5, on [0, 1], ]1, 2].

1 2 2 2 1 3 3 1 1 F G G ◦ F −1

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 7 / 30

Stronger stochastic orders

Some orders are stronger than st : Consider the following case : somebody wants to buy a car and can choose between two models with lifetimes X and Y. If the price is the same and X st Y them, she ought to buy model Y. Now what happens if both cars are used (one year old), is Y still a better choice ? well, not necessarily : Assume that X is uniform over [0, 3] (with db F) and Y has a distribution with density 1/6, 1/2, 1/3

• n [0, 1], ]1, 2], ]2, 3] (with db G).

Then, X st Y (F G). However X1 = (X|X > 1) and Y1 = (Y|Y > 1) are not st-comparable : X1 is uniform over [0, 2] (with density 1/2) and Y1 has a distribution with density 3/5, 2/5, on [0, 1], ]1, 2].

1 2 2 2 1 3 3 1 1 F G G ◦ F −1

What order is preserved under aging ?

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 7 / 30

The hazard rate order : hr

The hazard rate (or failure rate) is defined by : rX (t) = lim

ε→0

P(X < t + ε|X > t) ε = fX (t) 1 − Fx(t) = − d dt ln(1 − FX (t))

Definition

X hr Y if rX (t) rY (t).

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 8 / 30

The hazard rate order : hr

The hazard rate (or failure rate) is defined by : rX (t) = lim

ε→0

P(X < t + ε|X > t) ε = fX (t) 1 − Fx(t) = − d dt ln(1 − FX (t))

Definition

X hr Y if rX (t) rY (t). the Proba-Proba plot G(F −1(t)) is star shaped with respect to (1,1).

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 8 / 30

The hazard rate order : hr

The hazard rate (or failure rate) is defined by : rX (t) = lim

ε→0

P(X < t + ε|X > t) ε = fX (t) 1 − Fx(t) = − d dt ln(1 − FX (t))

Definition

X hr Y if rX (t) rY (t). the Proba-Proba plot G(F −1(t)) is star shaped with respect to (1,1). Eg(X ∗, Y ∗) Eg(Y ∗, X ∗) ∀g s.t. g(x, y) − g(y, x) increasing in x, ∀x y.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 8 / 30

The hazard rate order : hr

The hazard rate (or failure rate) is defined by : rX (t) = lim

ε→0

P(X < t + ε|X > t) ε = fX (t) 1 − Fx(t) = − d dt ln(1 − FX (t))

Definition

X hr Y if rX (t) rY (t). the Proba-Proba plot G(F −1(t)) is star shaped with respect to (1,1). Eg(X ∗, Y ∗) Eg(Y ∗, X ∗) ∀g s.t. g(x, y) − g(y, x) increasing in x, ∀x y. The hr order is preserved under aging and is stronger than the st order

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 8 / 30

The likelihood order : lr

Another order which is even stronger than hr is the likelihood ratio which preserves st under any conditioning :

Definition

U = [a, b], V = [c, d], U < V X lr Y if P(X ∈ V)P(Y ∈ U) P(X ∈ U)P(Y ∈ V) or equivalently (X|X ∈ U) st (Y|Y ∈ U)

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 9 / 30

The likelihood order : lr

Another order which is even stronger than hr is the likelihood ratio which preserves st under any conditioning :

Definition

U = [a, b], V = [c, d], U < V X lr Y if P(X ∈ V)P(Y ∈ U) P(X ∈ U)P(Y ∈ V) or equivalently (X|X ∈ U) st (Y|Y ∈ U) The P-P plot is convex.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 9 / 30

The likelihood order : lr

Another order which is even stronger than hr is the likelihood ratio which preserves st under any conditioning :

Definition

U = [a, b], V = [c, d], U < V X lr Y if P(X ∈ V)P(Y ∈ U) P(X ∈ U)P(Y ∈ V) or equivalently (X|X ∈ U) st (Y|Y ∈ U) The P-P plot is convex. Eg(X ∗, Y ∗) Eg(Y ∗, X ∗) ∀g s.t. g(x, y) − g(y, x) 0, ∀x y.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 9 / 30

The likelihood order : lr

Another order which is even stronger than hr is the likelihood ratio which preserves st under any conditioning :

Definition

U = [a, b], V = [c, d], U < V X lr Y if P(X ∈ V)P(Y ∈ U) P(X ∈ U)P(Y ∈ V) or equivalently (X|X ∈ U) st (Y|Y ∈ U) The P-P plot is convex. Eg(X ∗, Y ∗) Eg(Y ∗, X ∗) ∀g s.t. g(x, y) − g(y, x) 0, ∀x y. The lr order is preserved under any conditioning and is stronger than the hr order.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 9 / 30

Other stochastic orders : convex orders

The convex orders are used to compare the variability of stochastic variables.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 10 / 30

Other stochastic orders : convex orders

The convex orders are used to compare the variability of stochastic variables.

Definition

X cx Y if Ef(X) Ef(Y) for all convex functions f.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 10 / 30

Other stochastic orders : convex orders

The convex orders are used to compare the variability of stochastic variables.

Definition

X cx Y if Ef(X) Ef(Y) for all convex functions f.

Definition

X icx Y if Ef(X) Ef(Y) for all increasing convex functions f. Strassen Representation Theorem :

Theorem

X cx Y iff there exist two r.v. X ′ and Y ′ with the same db as X and Y such that X ′ = E(Y ′|X ′).

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 10 / 30

Other stochastic orders : convex orders

The convex orders are used to compare the variability of stochastic variables.

Definition

X cx Y if Ef(X) Ef(Y) for all convex functions f.

Definition

X icx Y if Ef(X) Ef(Y) for all increasing convex functions f. Strassen Representation Theorem :

Theorem

X cx Y iff there exist two r.v. X ′ and Y ′ with the same db as X and Y such that X ′ = E(Y ′|X ′). Corollary if X and Z are independent and E(Z) = 0 then, X cx X + Z.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 10 / 30

Discrete dynamical systems

There are two types of models in scheduling. static models :X = φ(Z1, · · · , ZN) and dynamic models Xn = φn(Xn−1, Zn), ∀n 0.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 11 / 30

Discrete dynamical systems

There are two types of models in scheduling. static models :X = φ(Z1, · · · , ZN) and dynamic models Xn = φn(Xn−1, Zn), ∀n 0. A dynamical system is time -monotone for order F if Xn F Xn−1. A system (static or dynamic) is F- isotone if Zk F Z ′

k ⇒ Xn F X ′ n.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 11 / 30

Discrete dynamical systems

There are two types of models in scheduling. static models :X = φ(Z1, · · · , ZN) and dynamic models Xn = φn(Xn−1, Zn), ∀n 0. A dynamical system is time -monotone for order F if Xn F Xn−1. A system (static or dynamic) is F- isotone if Zk F Z ′

k ⇒ Xn F X ′ n.

Comparison are often proved using mapping, coupling, association and monotony.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 11 / 30

Mapping techniques

Principle : Prove comparability by comparing the inputs of functionals. for a static system,

Theorem

if (Z1, . . . , Zn) st (Z ′

1, . . . Z ′ n) and are independent, then if Φ is increasing , then

Φ(Z1, . . . , Zn) st Φ(Z ′

1, . . . Z ′ n).

if (Z1, . . . , Zn) icx (Z ′

1, . . . Z ′ n) and are independent, then If Φ is increasing and convex then

Φ(Z1, . . . , Zn) icx Φ(Z ′

1, . . . Z ′ n).

For a dynamic system,

Theorem

if (Z1, . . . , Zn) F (Z ′

1, . . . Z ′ n) and all ϕn are increasing, (resp. increasing and convex) then

Xn F X ′

n, with F = st (resp. F = icx).

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 12 / 30

Association

Two random variables X and Y are associated if cov(g(X), f(Y)) 0 for all increasing f and g.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 13 / 30

Association

Two random variables X and Y are associated if cov(g(X), f(Y)) 0 for all increasing f and g. Xn = φ(Xn, Zn) if φ is monotone on both variables, then if Zn are independent or associated then Xn are associated.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 13 / 30

Association

Two random variables X and Y are associated if cov(g(X), f(Y)) 0 for all increasing f and g. Xn = φ(Xn, Zn) if φ is monotone on both variables, then if Zn are independent or associated then Xn are associated. Finally, (X1, . . . Xn) associated implies that (X1, . . . Xn) o (X ∗

1 , . . . X ∗ n ) where (X ∗ 1 , . . . X ∗ n ) are

independent versions of (X1, . . . Xn).

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 13 / 30

Coupling technique

Principle : Prove comparisons using sample paths. Coupling often provides more powerful results. Example : compare the load C for two task-resource systems with s and s′ resources, respectively : How to show that Cn st C′

n ? (loads at the nth arrival, Tn)

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 14 / 30

Coupling technique

Principle : Prove comparisons using sample paths. Coupling often provides more powerful results. Example : compare the load C for two task-resource systems with s and s′ resources, respectively : How to show that Cn st C′

n ? (loads at the nth arrival, Tn)

The best is by using a coupling method, i.e. by considering a unique input process sample ω and compare the two systems over that single sequence of sizes and arrival times. The workload vectors Cn and C′

n are the increasingly ordered workloads at time Tn in the different resources

(they have s and s′ components respectively).

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 14 / 30

Coupling technique

Principle : Prove comparisons using sample paths. Coupling often provides more powerful results. Example : compare the load C for two task-resource systems with s and s′ resources, respectively : How to show that Cn st C′

n ? (loads at the nth arrival, Tn)

The best is by using a coupling method, i.e. by considering a unique input process sample ω and compare the two systems over that single sequence of sizes and arrival times. The workload vectors Cn and C′

n are the increasingly ordered workloads at time Tn in the different resources

(they have s and s′ components respectively). One has Cn = R(Cn−1 + Sne1 − δn1)+

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 14 / 30

Coupling technique

Principle : Prove comparisons using sample paths. Coupling often provides more powerful results. Example : compare the load C for two task-resource systems with s and s′ resources, respectively : How to show that Cn st C′

n ? (loads at the nth arrival, Tn)

The best is by using a coupling method, i.e. by considering a unique input process sample ω and compare the two systems over that single sequence of sizes and arrival times. The workload vectors Cn and C′

n are the increasingly ordered workloads at time Tn in the different resources

(they have s and s′ components respectively). One has Cn = R(Cn−1 + Sne1 − δn1)+ Then, it should be clear that the s first components of Cn and C′

n are comparable, by induction.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 14 / 30

Coupling technique

Principle : Prove comparisons using sample paths. Coupling often provides more powerful results. Example : compare the load C for two task-resource systems with s and s′ resources, respectively : How to show that Cn st C′

n ? (loads at the nth arrival, Tn)

The best is by using a coupling method, i.e. by considering a unique input process sample ω and compare the two systems over that single sequence of sizes and arrival times. The workload vectors Cn and C′

n are the increasingly ordered workloads at time Tn in the different resources

(they have s and s′ components respectively). One has Cn = R(Cn−1 + Sne1 − δn1)+ Then, it should be clear that the s first components of Cn and C′

n are comparable, by induction. Therefore,

Cn =st Cn(1) st C′

n(1) =st C′ n.

G/G/2 G/G/1

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 14 / 30

Coupling technique

Principle : Prove comparisons using sample paths. Coupling often provides more powerful results. Example : compare the load C for two task-resource systems with s and s′ resources, respectively : How to show that Cn st C′

n ? (loads at the nth arrival, Tn)

The best is by using a coupling method, i.e. by considering a unique input process sample ω and compare the two systems over that single sequence of sizes and arrival times. The workload vectors Cn and C′

n are the increasingly ordered workloads at time Tn in the different resources

(they have s and s′ components respectively). One has Cn = R(Cn−1 + Sne1 − δn1)+ Then, it should be clear that the s first components of Cn and C′

n are comparable, by induction. Therefore,

Cn =st Cn(1) st C′

n(1) =st C′ n.

G/G/2 G/G/1

Tn

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 14 / 30

Coupling technique

Principle : Prove comparisons using sample paths. Coupling often provides more powerful results. Example : compare the load C for two task-resource systems with s and s′ resources, respectively : How to show that Cn st C′

n ? (loads at the nth arrival, Tn)

The best is by using a coupling method, i.e. by considering a unique input process sample ω and compare the two systems over that single sequence of sizes and arrival times. The workload vectors Cn and C′

n are the increasingly ordered workloads at time Tn in the different resources

(they have s and s′ components respectively). One has Cn = R(Cn−1 + Sne1 − δn1)+ Then, it should be clear that the s first components of Cn and C′

n are comparable, by induction. Therefore,

Cn =st Cn(1) st C′

n(1) =st C′ n.

G/G/2 G/G/1

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 14 / 30

Coupling technique

Principle : Prove comparisons using sample paths. Coupling often provides more powerful results. Example : compare the load C for two task-resource systems with s and s′ resources, respectively : How to show that Cn st C′

n ? (loads at the nth arrival, Tn)

The best is by using a coupling method, i.e. by considering a unique input process sample ω and compare the two systems over that single sequence of sizes and arrival times. The workload vectors Cn and C′

n are the increasingly ordered workloads at time Tn in the different resources

(they have s and s′ components respectively). One has Cn = R(Cn−1 + Sne1 − δn1)+ Then, it should be clear that the s first components of Cn and C′

n are comparable, by induction. Therefore,

Cn =st Cn(1) st C′

n(1) =st C′ n.

G/G/2 G/G/1

Tn+1

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 14 / 30

Coupling technique

Principle : Prove comparisons using sample paths. Coupling often provides more powerful results. Example : compare the load C for two task-resource systems with s and s′ resources, respectively : How to show that Cn st C′

n ? (loads at the nth arrival, Tn)

The best is by using a coupling method, i.e. by considering a unique input process sample ω and compare the two systems over that single sequence of sizes and arrival times. The workload vectors Cn and C′

n are the increasingly ordered workloads at time Tn in the different resources

(they have s and s′ components respectively). One has Cn = R(Cn−1 + Sne1 − δn1)+ Then, it should be clear that the s first components of Cn and C′

n are comparable, by induction. Therefore,

Cn =st Cn(1) st C′

n(1) =st C′ n.

G/G/2 G/G/1

Tn+1

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 14 / 30

The problem 1|| Ci

This is one of the simplest scheduling problem : one resource with no scheduling restriction on N tasks, all arriving at time 0, while the objective is to minimize the sum of the completion times (or the average completion time) We consider all tasks to be independent of sizes S1, . . . , SN.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 15 / 30

The problem 1|| Ci

This is one of the simplest scheduling problem : one resource with no scheduling restriction on N tasks, all arriving at time 0, while the objective is to minimize the sum of the completion times (or the average completion time) We consider all tasks to be independent of sizes S1, . . . , SN. For a given schedule (or permutation) σ, the objective function is Tσ = PN

i=1 Ci = PN i=1(N − i + 1)Sσ(i).

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 15 / 30

The problem 1|| Ci

This is one of the simplest scheduling problem : one resource with no scheduling restriction on N tasks, all arriving at time 0, while the objective is to minimize the sum of the completion times (or the average completion time) We consider all tasks to be independent of sizes S1, . . . , SN. For a given schedule (or permutation) σ, the objective function is Tσ = PN

i=1 Ci = PN i=1(N − i + 1)Sσ(i).

We consider two particular schedules : SEPT (Shortest Expected Processing Time) and LEPT (Largest Expected Processing Time). It should be clear that for any permutation σ, ETSEPT ETσ ETLEPT . Indeed, ETσ = E PN

i=1 Ci = PN i=1(N − i + 1)ESσ(i).

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 15 / 30

The problem 1|| Ci

This is one of the simplest scheduling problem : one resource with no scheduling restriction on N tasks, all arriving at time 0, while the objective is to minimize the sum of the completion times (or the average completion time) We consider all tasks to be independent of sizes S1, . . . , SN. For a given schedule (or permutation) σ, the objective function is Tσ = PN

i=1 Ci = PN i=1(N − i + 1)Sσ(i).

We consider two particular schedules : SEPT (Shortest Expected Processing Time) and LEPT (Largest Expected Processing Time). It should be clear that for any permutation σ, ETSEPT ETσ ETLEPT . Indeed, ETσ = E PN

i=1 Ci = PN i=1(N − i + 1)ESσ(i).

But can we say more ?

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 15 / 30

The problem 1|| Ci revisited

Theorem (Shanthikumar, Yao, 1993)

If Si lr Si+1 for all i, then TSEPT st Tσ st TLEPT . If Si hr Si+1 for all i, then TSEPT icx Tσ icx TLEPT . Proof uses a classical interchange argument (done for hr)

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 16 / 30

The problem 1|| Ci revisited

Theorem (Shanthikumar, Yao, 1993)

If Si lr Si+1 for all i, then TSEPT st Tσ st TLEPT . If Si hr Si+1 for all i, then TSEPT icx Tσ icx TLEPT . Proof uses a classical interchange argument (done for hr) For all σ = SEPT there exists k such that σ(k) = j, σ(k + 1) = i with i < j. Let µ = σ except µ(k) = i, µ(k + 1) = j (µ is closer to SEPT than σ).

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 16 / 30

The problem 1|| Ci revisited

Theorem (Shanthikumar, Yao, 1993)

If Si lr Si+1 for all i, then TSEPT st Tσ st TLEPT . If Si hr Si+1 for all i, then TSEPT icx Tσ icx TLEPT . Proof uses a classical interchange argument (done for hr) For all σ = SEPT there exists k such that σ(k) = j, σ(k + 1) = i with i < j. Let µ = σ except µ(k) = i, µ(k + 1) = j (µ is closer to SEPT than σ). Now, Tσ = Xj + k(Xi + Xj) + Y and Tµ = Xi + k(Xi + Xj) + Y where Y is the contribution of the

• ther jobs, independent of Si and Sj.
• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 16 / 30

The problem 1|| Ci revisited

Theorem (Shanthikumar, Yao, 1993)

If Si lr Si+1 for all i, then TSEPT st Tσ st TLEPT . If Si hr Si+1 for all i, then TSEPT icx Tσ icx TLEPT . Proof uses a classical interchange argument (done for hr) For all σ = SEPT there exists k such that σ(k) = j, σ(k + 1) = i with i < j. Let µ = σ except µ(k) = i, µ(k + 1) = j (µ is closer to SEPT than σ). Now, Tσ = Xj + k(Xi + Xj) + Y and Tµ = Xi + k(Xi + Xj) + Y where Y is the contribution of the

• ther jobs, independent of Si and Sj.

Moreover g(x, y) = f(x + k(x + y)) satisfies g(x, y) − g(y, x) is increasing as long as f is convex and increasing. Therefore, Si hr Sj implies Ef(Xj + k(Xi + Xj)) Ef(Xi + k(Xi + Xj)) for all increasing convex f.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 16 / 30

The problem 1|| Ci revisited

Theorem (Shanthikumar, Yao, 1993)

If Si lr Si+1 for all i, then TSEPT st Tσ st TLEPT . If Si hr Si+1 for all i, then TSEPT icx Tσ icx TLEPT . Proof uses a classical interchange argument (done for hr) For all σ = SEPT there exists k such that σ(k) = j, σ(k + 1) = i with i < j. Let µ = σ except µ(k) = i, µ(k + 1) = j (µ is closer to SEPT than σ). Now, Tσ = Xj + k(Xi + Xj) + Y and Tµ = Xi + k(Xi + Xj) + Y where Y is the contribution of the

• ther jobs, independent of Si and Sj.

Moreover g(x, y) = f(x + k(x + y)) satisfies g(x, y) − g(y, x) is increasing as long as f is convex and increasing. Therefore, Si hr Sj implies Ef(Xj + k(Xi + Xj)) Ef(Xi + k(Xi + Xj)) for all increasing convex f. Finally, Xi + k(Xi + Xj) icx Xj + k(Xi + Xj) implies Tµ icx Tσ.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 16 / 30

PERT Graph

A PERT graph is a more general static model : N tasks are to be executed over an infinite number

• f resources and are constrained by an acyclic graph.
• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 17 / 30

PERT Graph

A PERT graph is a more general static model : N tasks are to be executed over an infinite number

• f resources and are constrained by an acyclic graph.

PERT graphs are impossible to solve (compute the makespan) analytically in general (Kamburowski, 1992). However, one can use comparisons to prove several results.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 17 / 30

PERT Graph, continued

Here are the ingredients used to compare (and compute bounds) for PERT graphs.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 18 / 30

PERT Graph, continued

Here are the ingredients used to compare (and compute bounds) for PERT graphs. 1- Using the mapping technique : C = Φ(X1, . . . XN) = maxc∈P(G) P

i∈c Xi.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 18 / 30

PERT Graph, continued

Here are the ingredients used to compare (and compute bounds) for PERT graphs. 1- Using the mapping technique : C = Φ(X1, . . . XN) = maxc∈P(G) P

i∈c Xi.

Note that Φ is convex and increasing. This implies the following first result. Zi F Z ′

i implies

C F C′ with F = st or icx.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 18 / 30

PERT Graph, continued

Here are the ingredients used to compare (and compute bounds) for PERT graphs. 1- Using the mapping technique : C = Φ(X1, . . . XN) = maxc∈P(G) P

i∈c Xi.

Note that Φ is convex and increasing. This implies the following first result. Zi F Z ′

i implies

C F C′ with F = st or icx. 2- Next, if tasks are independent (or associated), then the paths are all associated and are therefore bounded by independent versions :

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 18 / 30

PERT Graph, continued

Here are the ingredients used to compare (and compute bounds) for PERT graphs. 1- Using the mapping technique : C = Φ(X1, . . . XN) = maxc∈P(G) P

i∈c Xi.

Note that Φ is convex and increasing. This implies the following first result. Zi F Z ′

i implies

C F C′ with F = st or icx. 2- Next, if tasks are independent (or associated), then the paths are all associated and are therefore bounded by independent versions : for all c, Sc st S∗

c and C st C∗ = maxc∈P(G) S∗ c

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 18 / 30

PERT Graph, continued

Here are the ingredients used to compare (and compute bounds) for PERT graphs. 1- Using the mapping technique : C = Φ(X1, . . . XN) = maxc∈P(G) P

i∈c Xi.

Note that Φ is convex and increasing. This implies the following first result. Zi F Z ′

i implies

C F C′ with F = st or icx. 2- Next, if tasks are independent (or associated), then the paths are all associated and are therefore bounded by independent versions : for all c, Sc st S∗

c and C st C∗ = maxc∈P(G) S∗ c

3- Next, if Xi is NBUE (New Better than Used in Expectation : E(X − t|X > t) EX) then Xi cx exp(E(Xi)).

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 18 / 30

PERT Graph, continued

This allows us to show that max

c∈P(G)

X

i∈c

EXi icx C icx max

c∈P(G)

X

i∈c

exp(E(Xi))

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 19 / 30

PERT Graph, continued

This allows us to show that max

c∈P(G)

X

i∈c

EXi icx C icx max

c∈P(G)

X

i∈c

exp(E(Xi)) and C icx max

c∈P(G) exp(

X

i∈c

E(Xi)) =db Y

c∈P(G)

1 − exp(−t X

i∈c

E(Xi)), as soon as Xi are all NBUE and associated.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 19 / 30

Queues

Queues are among simplest dynamic systems, but are still the source of many open problems. Tasks do not have any constraints, sizes and arrival times are often independent.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 20 / 30

Queues

Queues are among simplest dynamic systems, but are still the source of many open problems. Tasks do not have any constraints, sizes and arrival times are often independent.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 20 / 30

Queues

Queues are among simplest dynamic systems, but are still the source of many open problems. Tasks do not have any constraints, sizes and arrival times are often independent.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 20 / 30

Queues

Queues are among simplest dynamic systems, but are still the source of many open problems. Tasks do not have any constraints, sizes and arrival times are often independent.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 20 / 30

Queues

Queues are among simplest dynamic systems, but are still the source of many open problems. Tasks do not have any constraints, sizes and arrival times are often independent.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 20 / 30

Queues

Queues are among simplest dynamic systems, but are still the source of many open problems. Tasks do not have any constraints, sizes and arrival times are often independent.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 20 / 30

Lindley’s formula

σ 4

5

σ σ 3 σ 2 σ 1 δ δ δ δ δ σ6 δ

d d d d 2 3 4 5 d1

a a a a a

1 2 3 4 5

a6 W(t)

3 2 4 5

t

6 1

Wn is the waiting time of the n-th task. It is a dynamical system of the form Wn = ϕ(Wn−1, Xn) with Xn = Sn−1 − δn and ϕ defined by the Lindley’s equation : Wn = max (Wn−1 + Xn, 0) .

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 21 / 30

Loynes’ scheme

Theorem

Wn st Wn+1 in a G/G/1 queue, initialy empty.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 22 / 30

Loynes’ scheme

Theorem

Wn st Wn+1 in a G/G/1 queue, initialy empty.

• Proof. done by a backward coupling known as the Loynes’ scheme. Construct on a common

probability space two trajectories by going backward in time : S1

i−n(ω) = S2 i−n−1(ω) with

distribution Si and T 1

i−n(ω) = T 2 i−n−1(ω), with distribution Ti − Tn+1 for all 0 i n + 1 and

S1

−n−1(ω) = 0.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 22 / 30

Loynes’ scheme

Theorem

Wn st Wn+1 in a G/G/1 queue, initialy empty.

• Proof. done by a backward coupling known as the Loynes’ scheme. Construct on a common

probability space two trajectories by going backward in time : S1

i−n(ω) = S2 i−n−1(ω) with

distribution Si and T 1

i−n(ω) = T 2 i−n−1(ω), with distribution Ti − Tn+1 for all 0 i n + 1 and

S1

−n−1(ω) = 0.

By construction, W 1

0 =st Wn and W 2 0 =st Wn+1. Also, it should be clear that

0 = W 1

−n+1(ω) W 2 −n+1(ω) for all ω.

This implies W 1

−i(ω) W 2 −i(ω) so that Wn st Wn+1.

W 2 Ti − Tn+1 −Tn+1 W 1

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 22 / 30

Loynes’ scheme

Theorem

Wn st Wn+1 in a G/G/1 queue, initialy empty.

• Proof. done by a backward coupling known as the Loynes’ scheme. Construct on a common

probability space two trajectories by going backward in time : S1

i−n(ω) = S2 i−n−1(ω) with

distribution Si and T 1

i−n(ω) = T 2 i−n−1(ω), with distribution Ti − Tn+1 for all 0 i n + 1 and

S1

−n−1(ω) = 0.

By construction, W 1

0 =st Wn and W 2 0 =st Wn+1. Also, it should be clear that

0 = W 1

−n+1(ω) W 2 −n+1(ω) for all ω.

This implies W 1

−i(ω) W 2 −i(ω) so that Wn st Wn+1.

W 2 Ti − Tn+1 −Tn+1 W 1 This has many consequences in terms of existence and uniqueness of a stationary (or limit) regime for the G/G/1 queue (Baccelli Bremaud, 2002).

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 22 / 30

Input process

Folk theorem (Ross conjecture, 1978) :things work better when the input traffic has less variability.

Theorem

if (W0, X1, . . . , Xn) F (W ′

0, X ′ 1, . . . , X ′ n) then Wn F W ′ n (with F = st or icx).

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 23 / 30

Input process

Folk theorem (Ross conjecture, 1978) :things work better when the input traffic has less variability.

Theorem

if (W0, X1, . . . , Xn) F (W ′

0, X ′ 1, . . . , X ′ n) then Wn F W ′ n (with F = st or icx).

proof f(x, w) = max(w + x, 0) is convex and increasing for both variables.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 23 / 30

Examples

1

Show that if the traffic intensity is fixed in a single GI/GI/1 queue, then the average waiting time is smallest when the arrivals are periodic.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 24 / 30

Examples

1

Show that if the traffic intensity is fixed in a single GI/GI/1 queue, then the average waiting time is smallest when the arrivals are periodic.

2

Show that if the arrival process in a GI/M/1 queue is NBUE, then the average waiting time can be bounded by EW

1 µ−1/E(T1) .

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 24 / 30

Extremal input processes

Several extensions are possible :

Theorem (Altman, Gaujal, Hordijk, 2003)

If the arrival sequence T1, . . . , Tn, . . . is fixed in a stochastic FIFO event graph (arbitrary network

• f queues with no branching enriched with fork and join nodes), then, S1, . . . , Sn cx S′

1, . . . , S′ n

implies (W1, . . . , Wn) icx (W1, . . . , Wn).

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 25 / 30

Service discipline

In a queue with one or more servers, tasks may be served according to disciplines : FIFO, LIFO, PS, priority, random, . . .

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 26 / 30

Service discipline

In a queue with one or more servers, tasks may be served according to disciplines : FIFO, LIFO, PS, priority, random, . . . PS has insensibility, reversibility and product form properties,

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 26 / 30

Service discipline

In a queue with one or more servers, tasks may be served according to disciplines : FIFO, LIFO, PS, priority, random, . . . PS has insensibility, reversibility and product form properties, FIFO (F) has optimality properties, in terms of waiting times :

Theorem

In a GI/GI/1 queue, f(WF

n ) st f(Wπ n ) for all service discipline π and all convex increasing and

symmetric f.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 26 / 30

Service discipline

In a queue with one or more servers, tasks may be served according to disciplines : FIFO, LIFO, PS, priority, random, . . . PS has insensibility, reversibility and product form properties, FIFO (F) has optimality properties, in terms of waiting times :

Theorem

In a GI/GI/1 queue, f(WF

n ) st f(Wπ n ) for all service discipline π and all convex increasing and

symmetric f.

• Proof. using a coupling technique and majorization.

Since service time and arrivals are independent, we can rearrange the service times in the order

• f service (and not of arrivals) under policy π and F Let Di be the departure epochs. They coincide

under both policies.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 26 / 30

Service discipline

In a queue with one or more servers, tasks may be served according to disciplines : FIFO, LIFO, PS, priority, random, . . . PS has insensibility, reversibility and product form properties, FIFO (F) has optimality properties, in terms of waiting times :

Theorem

In a GI/GI/1 queue, f(WF

n ) st f(Wπ n ) for all service discipline π and all convex increasing and

symmetric f.

• Proof. using a coupling technique and majorization.

Since service time and arrivals are independent, we can rearrange the service times in the order

• f service (and not of arrivals) under policy π and F Let Di be the departure epochs. They coincide

under both policies. Then W F

i

= Di − Ti and assume that π interchange the departure of j and j + 1 : W π

j

= Dj+1 − Tj and W π

j+1 = Dj − Tj+1

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 26 / 30

Service discipline

In a queue with one or more servers, tasks may be served according to disciplines : FIFO, LIFO, PS, priority, random, . . . PS has insensibility, reversibility and product form properties, FIFO (F) has optimality properties, in terms of waiting times :

Theorem

In a GI/GI/1 queue, f(WF

n ) st f(Wπ n ) for all service discipline π and all convex increasing and

symmetric f.

• Proof. using a coupling technique and majorization.

Since service time and arrivals are independent, we can rearrange the service times in the order

• f service (and not of arrivals) under policy π and F Let Di be the departure epochs. They coincide

under both policies. Then W F

i

= Di − Ti and assume that π interchange the departure of j and j + 1 : W π

j

= Dj+1 − Tj and W π

j+1 = Dj − Tj+1

Now, it should be obvious that W π

j

+ W π

j+1 = W F j

+ W F

j+1 and if f is increasing convex and

symmetric (or Schur convex) f(W π

j , W π j+1) f(W F j , W F j+1).

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 26 / 30

Service discipline

In a queue with one or more servers, tasks may be served according to disciplines : FIFO, LIFO, PS, priority, random, . . . PS has insensibility, reversibility and product form properties, FIFO (F) has optimality properties, in terms of waiting times :

Theorem

In a GI/GI/1 queue, f(WF

n ) st f(Wπ n ) for all service discipline π and all convex increasing and

symmetric f.

• Proof. using a coupling technique and majorization.

Since service time and arrivals are independent, we can rearrange the service times in the order

• f service (and not of arrivals) under policy π and F Let Di be the departure epochs. They coincide

under both policies. Then W F

i

= Di − Ti and assume that π interchange the departure of j and j + 1 : W π

j

= Dj+1 − Tj and W π

j+1 = Dj − Tj+1

Now, it should be obvious that W π

j

+ W π

j+1 = W F j

+ W F

j+1 and if f is increasing convex and

symmetric (or Schur convex) f(W π

j , W π j+1) f(W F j , W F j+1).

In general, consider all tasks (n) within a busy period of the system, then, W π

1 + · · · + W π j+1 = W F 1 + · · · + W F n and interchanging a pair of customers out of order under π,

reduces the value of f(W π

1 , . . . , W π n ) down to the value of f(W F 1 , . . . , W F n ) for any Schur convex

function f.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 26 / 30

Service discipline

In a queue with one or more servers, tasks may be served according to disciplines : FIFO, LIFO, PS, priority, random, . . . PS has insensibility, reversibility and product form properties, FIFO (F) has optimality properties, in terms of waiting times :

Theorem

In a GI/GI/1 queue, f(WF

n ) st f(Wπ n ) for all service discipline π and all convex increasing and

symmetric f.

• Proof. using a coupling technique and majorization.

Since service time and arrivals are independent, we can rearrange the service times in the order

• f service (and not of arrivals) under policy π and F Let Di be the departure epochs. They coincide

under both policies. Then W F

i

= Di − Ti and assume that π interchange the departure of j and j + 1 : W π

j

= Dj+1 − Tj and W π

j+1 = Dj − Tj+1

Now, it should be obvious that W π

j

+ W π

j+1 = W F j

+ W F

j+1 and if f is increasing convex and

symmetric (or Schur convex) f(W π

j , W π j+1) f(W F j , W F j+1).

In general, consider all tasks (n) within a busy period of the system, then, W π

1 + · · · + W π j+1 = W F 1 + · · · + W F n and interchanging a pair of customers out of order under π,

reduces the value of f(W π

1 , . . . , W π n ) down to the value of f(W F 1 , . . . , W F n ) for any Schur convex

function f. If the first n tasks do not form a busy period, then W π

1 + · · · + W π j+1 W F 1 + · · · + W F n and again

f(W π

1 , . . . , W π n ) f(W F 1 , . . . , W F n ) for any Schur convex function f

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 26 / 30

Polling systems

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 27 / 30

Polling systems

Choosing the best open loop schedule for the server corresponds to choose the most regular service in each queue. (Gaujal, Hordijk, Van der Laan, 2007)

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 27 / 30

Polling systems

Choosing the best open loop schedule for the server corresponds to choose the most regular service in each queue. (Gaujal, Hordijk, Van der Laan, 2007) Example : for two queues (1 and 2) 12121212. . . is a better schedule than 1122112211. . .

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 27 / 30

Polling systems

Choosing the best open loop schedule for the server corresponds to choose the most regular service in each queue. (Gaujal, Hordijk, Van der Laan, 2007) Example : for two queues (1 and 2) 12121212. . . is a better schedule than 1122112211. . . The main difficulty is to compute the frequency of the visits to each queue.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 27 / 30

Polling systems, continued

λ1 λ2

αopt = 1/2 Instability

FIG.: The frequency of the server allocations w.r.t input intensities

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 28 / 30

Polling systems, continued

0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.75 0.8 0.85 0.9 0.95 1

ρ αopt

FIG.: The frequency of the server allocations w.r.t the total load, the ratio of input intensities being fixed.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 29 / 30

Conclusion

There exists a systematic framework to deal with task-resource systems involving randomness through the theory of stochastic comparisons.

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 30 / 30

Conclusion

There exists a systematic framework to deal with task-resource systems involving randomness through the theory of stochastic comparisons. Main actors in that field :

• R. Righter, Z. Liu, J . Shanthikumar, C. Cassandras, T. Rolski. . .
• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 30 / 30

Conclusion

There exists a systematic framework to deal with task-resource systems involving randomness through the theory of stochastic comparisons. Main actors in that field :

• R. Righter, Z. Liu, J . Shanthikumar, C. Cassandras, T. Rolski. . .

Main bibliography for that talk : Comparison Methods for Stochastic Models and Risks (A. Muller and D. Stoyan, 2002). Stochastic Modeling and the Theory of Queues (R. W. Wolff, 1989). Discrete-Event Control of Stochastic Networks : Multimodularity and Regularity (E. Altman, B. Gaujal and A. Hordijk, 2003).

• B. Gaujal, J.-M. Vincent (UJF

, INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 30 / 30