The usual stochastic order : examples 1. Show that X � st Y ⇒ X � µ Y . 2. Show that X � st Y and EX = EY ⇒ F X = F Y . 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 on [ 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 on [ 0 , 1 ] , ] 1 , 2 ] , ] 2 , 3 ] (with db G ). Then, X � st Y ( F � G ). However X 1 = ( X | X > 1 ) and Y 1 = ( Y | Y > 1 ) are not st-comparable : X 1 is uniform over [ 0 , 2 ] (with density 1/2) and Y 1 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 on [ 0 , 1 ] , ] 1 , 2 ] , ] 2 , 3 ] (with db G ). Then, X � st Y ( F � G ). However X 1 = ( X | X > 1 ) and Y 1 = ( Y | Y > 1 ) are not st-comparable : X 1 is uniform over [ 0 , 2 ] (with density 1/2) and Y 1 has a distribution with density 3 / 5 , 2 / 5, on [ 0 , 1 ] , ] 1 , 2 ] . 2 G ◦ F − 1 1 1 F G 1 2 3 1 2 3 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 on [ 0 , 1 ] , ] 1 , 2 ] , ] 2 , 3 ] (with db G ). Then, X � st Y ( F � G ). However X 1 = ( X | X > 1 ) and Y 1 = ( Y | Y > 1 ) are not st-comparable : X 1 is uniform over [ 0 , 2 ] (with density 1/2) and Y 1 has a distribution with density 3 / 5 , 2 / 5, on [ 0 , 1 ] , ] 1 , 2 ] . 2 G ◦ F − 1 1 1 F G 1 2 3 1 2 3 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 : P ( X < t + ε | X > t ) 1 − F x ( t ) = − d f X ( t ) r X ( t ) = lim = dt ln ( 1 − F X ( t )) ε ε → 0 Definition X � hr Y if r X ( t ) � r Y ( 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 : P ( X < t + ε | X > t ) 1 − F x ( t ) = − d f X ( t ) r X ( t ) = lim = dt ln ( 1 − F X ( t )) ε ε → 0 Definition X � hr Y if r X ( t ) � r Y ( 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 : P ( X < t + ε | X > t ) 1 − F x ( t ) = − d f X ( t ) r X ( t ) = lim = dt ln ( 1 − F X ( t )) ε ε → 0 Definition X � hr Y if r X ( t ) � r Y ( 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 : P ( X < t + ε | X > t ) 1 − F x ( t ) = − d f X ( t ) r X ( t ) = lim = dt ln ( 1 − F X ( t )) ε ε → 0 Definition X � hr Y if r X ( t ) � r Y ( 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 ( Z 1 , · · · , Z N ) and dynamic models X n = φ n ( X n − 1 , Z n ) , ∀ 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 ( Z 1 , · · · , Z N ) and dynamic models X n = φ n ( X n − 1 , Z n ) , ∀ n � 0 . A dynamical system is time -monotone for order F if X n � F X n − 1 . A system (static or dynamic) is F- isotone if Z k � F Z ′ k ⇒ X n � 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 ( Z 1 , · · · , Z N ) and dynamic models X n = φ n ( X n − 1 , Z n ) , ∀ n � 0 . A dynamical system is time -monotone for order F if X n � F X n − 1 . A system (static or dynamic) is F- isotone if Z k � F Z ′ k ⇒ X n � 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 ( Z 1 , . . . , Z n ) � st ( Z ′ 1 , . . . Z ′ n ) and are independent, then if Φ is increasing , then Φ( Z 1 , . . . , Z n ) � st Φ( Z ′ 1 , . . . Z ′ n ) . if ( Z 1 , . . . , Z n ) � icx ( Z ′ 1 , . . . Z ′ n ) and are independent, then If Φ is increasing and convex then Φ( Z 1 , . . . , Z n ) � icx Φ( Z ′ 1 , . . . Z ′ n ) . For a dynamic system, Theorem if ( Z 1 , . . . , Z n ) � F ( Z ′ 1 , . . . Z ′ n ) and all ϕ n are increasing, (resp. increasing and convex) then X n � 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 . X n = φ ( X n , Z n ) if φ is monotone on both variables, then if Z n are independent or associated then X n 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 . X n = φ ( X n , Z n ) if φ is monotone on both variables, then if Z n are independent or associated then X n are associated. Finally, ( X 1 , . . . X n ) associated implies that ( X 1 , . . . X n ) � o ( X ∗ 1 , . . . X ∗ n ) where ( X ∗ 1 , . . . X ∗ n ) are independent versions of ( X 1 , . . . X n ) . 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 C n � st C ′ n ? (loads at the n th arrival, T n ) 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 C n � st C ′ n ? (loads at the n th arrival, T n ) 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 C n and C ′ n are the increasingly ordered workloads at time T n 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 C n � st C ′ n ? (loads at the n th arrival, T n ) 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 C n and C ′ n are the increasingly ordered workloads at time T n in the different resources (they have s and s ′ components respectively). One has C n = R ( C n − 1 + S n e 1 − δ n 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 C n � st C ′ n ? (loads at the n th arrival, T n ) 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 C n and C ′ n are the increasingly ordered workloads at time T n in the different resources (they have s and s ′ components respectively). One has C n = R ( C n − 1 + S n e 1 − δ n 1 ) + Then, it should be clear that the s first components of C n 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 C n � st C ′ n ? (loads at the n th arrival, T n ) 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 C n and C ′ n are the increasingly ordered workloads at time T n in the different resources (they have s and s ′ components respectively). One has C n = R ( C n − 1 + S n e 1 − δ n 1 ) + Then, it should be clear that the s first components of C n and C ′ n are comparable, by induction. Therefore, C n = st C n ( 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 C n � st C ′ n ? (loads at the n th arrival, T n ) 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 C n and C ′ n are the increasingly ordered workloads at time T n in the different resources (they have s and s ′ components respectively). One has C n = R ( C n − 1 + S n e 1 − δ n 1 ) + Then, it should be clear that the s first components of C n and C ′ n are comparable, by induction. Therefore, C n = st C n ( 1 ) � st C ′ n ( 1 ) = st C ′ n . G / G / 2 G / G / 1 T n 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 C n � st C ′ n ? (loads at the n th arrival, T n ) 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 C n and C ′ n are the increasingly ordered workloads at time T n in the different resources (they have s and s ′ components respectively). One has C n = R ( C n − 1 + S n e 1 − δ n 1 ) + Then, it should be clear that the s first components of C n and C ′ n are comparable, by induction. Therefore, C n = st C n ( 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 C n � st C ′ n ? (loads at the n th arrival, T n ) 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 C n and C ′ n are the increasingly ordered workloads at time T n in the different resources (they have s and s ′ components respectively). One has C n = R ( C n − 1 + S n e 1 − δ n 1 ) + Then, it should be clear that the s first components of C n and C ′ n are comparable, by induction. Therefore, C n = st C n ( 1 ) � st C ′ n ( 1 ) = st C ′ n . G / G / 2 G / G / 1 T n + 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 C n � st C ′ n ? (loads at the n th arrival, T n ) 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 C n and C ′ n are the increasingly ordered workloads at time T n in the different resources (they have s and s ′ components respectively). One has C n = R ( C n − 1 + S n e 1 − δ n 1 ) + Then, it should be clear that the s first components of C n and C ′ n are comparable, by induction. Therefore, C n = st C n ( 1 ) � st C ′ n ( 1 ) = st C ′ n . G / G / 2 G / G / 1 T n + 1 B. Gaujal, J.-M. Vincent (UJF , INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 14 / 30
The problem 1 || � C i 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 S 1 , . . . , S N . B. Gaujal, J.-M. Vincent (UJF , INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 15 / 30
The problem 1 || � C i 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 S 1 , . . . , S N . For a given schedule (or permutation) σ , the objective function is T σ = P N i = 1 C i = P N i = 1 ( N − i + 1 ) S σ ( i ) . B. Gaujal, J.-M. Vincent (UJF , INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 15 / 30
The problem 1 || � C i 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 S 1 , . . . , S N . For a given schedule (or permutation) σ , the objective function is T σ = P N i = 1 C i = P N i = 1 ( N − i + 1 ) S σ ( 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 , ET SEPT � ET σ � ET LEPT . Indeed, ET σ = E P N i = 1 C i = P N i = 1 ( N − i + 1 ) ES σ ( i ) . B. Gaujal, J.-M. Vincent (UJF , INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 15 / 30
The problem 1 || � C i 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 S 1 , . . . , S N . For a given schedule (or permutation) σ , the objective function is T σ = P N i = 1 C i = P N i = 1 ( N − i + 1 ) S σ ( 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 , ET SEPT � ET σ � ET LEPT . Indeed, ET σ = E P N i = 1 C i = P N i = 1 ( N − i + 1 ) ES σ ( 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 || � C i revisited Theorem (Shanthikumar, Yao, 1993) If S i � lr S i + 1 for all i, then T SEPT � st T σ � st T LEPT . If S i � hr S i + 1 for all i, then T SEPT � icx T σ � icx T LEPT . 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 || � C i revisited Theorem (Shanthikumar, Yao, 1993) If S i � lr S i + 1 for all i, then T SEPT � st T σ � st T LEPT . If S i � hr S i + 1 for all i, then T SEPT � icx T σ � icx T LEPT . 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 || � C i revisited Theorem (Shanthikumar, Yao, 1993) If S i � lr S i + 1 for all i, then T SEPT � st T σ � st T LEPT . If S i � hr S i + 1 for all i, then T SEPT � icx T σ � icx T LEPT . 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, T σ = X j + k ( X i + X j ) + Y and T µ = X i + k ( X i + X j ) + Y where Y is the contribution of the other jobs, independent of S i and S j . B. Gaujal, J.-M. Vincent (UJF , INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 16 / 30
The problem 1 || � C i revisited Theorem (Shanthikumar, Yao, 1993) If S i � lr S i + 1 for all i, then T SEPT � st T σ � st T LEPT . If S i � hr S i + 1 for all i, then T SEPT � icx T σ � icx T LEPT . 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, T σ = X j + k ( X i + X j ) + Y and T µ = X i + k ( X i + X j ) + Y where Y is the contribution of the other jobs, independent of S i and S j . 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, S i � hr S j implies Ef ( X j + k ( X i + X j )) � Ef ( X i + k ( X i + X j )) 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 || � C i revisited Theorem (Shanthikumar, Yao, 1993) If S i � lr S i + 1 for all i, then T SEPT � st T σ � st T LEPT . If S i � hr S i + 1 for all i, then T SEPT � icx T σ � icx T LEPT . 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, T σ = X j + k ( X i + X j ) + Y and T µ = X i + k ( X i + X j ) + Y where Y is the contribution of the other jobs, independent of S i and S j . 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, S i � hr S j implies Ef ( X j + k ( X i + X j )) � Ef ( X i + k ( X i + X j )) for all increasing convex f . Finally, X i + k ( X i + X j ) � icx X j + k ( X i + X j ) implies T µ � icx T σ . 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 of 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 of 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( X 1 , . . . X N ) = max c ∈P ( G ) P i ∈ c X i . 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( X 1 , . . . X N ) = max c ∈P ( G ) P i ∈ c X i . Note that Φ is convex and increasing. This implies the following first result. Z i � 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( X 1 , . . . X N ) = max c ∈P ( G ) P i ∈ c X i . Note that Φ is convex and increasing. This implies the following first result. Z i � 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( X 1 , . . . X N ) = max c ∈P ( G ) P i ∈ c X i . Note that Φ is convex and increasing. This implies the following first result. Z i � 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 : c and C � st C ∗ = max c ∈P ( G ) S ∗ for all c , S c � st 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( X 1 , . . . X N ) = max c ∈P ( G ) P i ∈ c X i . Note that Φ is convex and increasing. This implies the following first result. Z i � 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 : c and C � st C ∗ = max c ∈P ( G ) S ∗ for all c , S c � st S ∗ c 3- Next, if X i is NBUE (New Better than Used in Expectation : E ( X − t | X > t ) � EX ) then X i � cx exp ( E ( X i )) . 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 X X max EX i � icx C � icx max exp ( E ( X i )) c ∈P ( G ) c ∈P ( G ) i ∈ c i ∈ c 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 X X max EX i � icx C � icx max exp ( E ( X i )) c ∈P ( G ) c ∈P ( G ) i ∈ c i ∈ c and X Y X C � icx c ∈P ( G ) exp ( max E ( X i )) = db 1 − exp ( − t E ( X i )) , i ∈ c c ∈P ( G ) i ∈ c as soon as X i 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 a a a a a a 6 1 2 3 4 5 δ δ δ δ δ δ 1 2 3 4 5 6 W(t) σ 4 σ 2 σ 3 σ 5 σ 1 σ 6 t 0 0 d1 d d d d 2 3 4 5 W n is the waiting time of the n -th task. It is a dynamical system of the form W n = ϕ ( W n − 1 , X n ) with X n = S n − 1 − δ n and ϕ defined by the Lindley’s equation : W n = max ( W n − 1 + X n , 0 ) . B. Gaujal, J.-M. Vincent (UJF , INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 21 / 30
Loynes’ scheme Theorem W n � st W n + 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 W n � st W n + 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 : S 1 i − n ( ω ) = S 2 i − n − 1 ( ω ) with distribution S i and T 1 i − n ( ω ) = T 2 i − n − 1 ( ω ) , with distribution T i − T n + 1 for all 0 � i � n + 1 and S 1 − n − 1 ( ω ) = 0. B. Gaujal, J.-M. Vincent (UJF , INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 22 / 30
Loynes’ scheme Theorem W n � st W n + 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 : S 1 i − n ( ω ) = S 2 i − n − 1 ( ω ) with distribution S i and T 1 i − n ( ω ) = T 2 i − n − 1 ( ω ) , with distribution T i − T n + 1 for all 0 � i � n + 1 and S 1 − n − 1 ( ω ) = 0. By construction, W 1 0 = st W n and W 2 0 = st W n + 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 W n � st W n + 1 . W 2 0 W 1 0 T i − T n + 1 − T n + 1 0 B. Gaujal, J.-M. Vincent (UJF , INRIA ) Stochastic task-resource systems Fréjus, 4 Juin 2007 22 / 30
Loynes’ scheme Theorem W n � st W n + 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 : S 1 i − n ( ω ) = S 2 i − n − 1 ( ω ) with distribution S i and T 1 i − n ( ω ) = T 2 i − n − 1 ( ω ) , with distribution T i − T n + 1 for all 0 � i � n + 1 and S 1 − n − 1 ( ω ) = 0. By construction, W 1 0 = st W n and W 2 0 = st W n + 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 W n � st W n + 1 . W 2 0 W 1 0 T i − T n + 1 − T n + 1 0 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 ( W 0 , X 1 , . . . , X n ) � F ( W ′ 0 , X ′ 1 , . . . , X ′ n ) then W n � 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 ( W 0 , X 1 , . . . , X n ) � F ( W ′ 0 , X ′ 1 , . . . , X ′ n ) then W n � 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 Show that if the traffic intensity is fixed in a single GI/GI/1 queue, then the average waiting 1 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 Show that if the traffic intensity is fixed in a single GI/GI/1 queue, then the average waiting 1 time is smallest when the arrivals are periodic. Show that if the arrival process in a GI/M/1 queue is NBUE, then the average waiting time can 2 1 be bounded by EW � µ − 1 / E ( T 1 ) . 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 T 1 , . . . , T n , . . . is fixed in a stochastic FIFO event graph (arbitrary network of queues with no branching enriched with fork and join nodes), then, S 1 , . . . , S n � cx S ′ 1 , . . . , S ′ n implies ( W 1 , . . . , W n ) � icx ( W 1 , . . . , W n ) . 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 ( W F n ) � st f ( W π 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 ( W F n ) � st f ( W π 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 of service (and not of arrivals) under policy π and F Let D i 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 ( W F n ) � st f ( W π 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 of service (and not of arrivals) under policy π and F Let D i be the departure epochs. They coincide under both policies. Then W F = D i − T i and assume that π interchange the departure of j and j + 1 : W π = D j + 1 − T j i j and W π j + 1 = D j − T 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 ( W F n ) � st f ( W π 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 of service (and not of arrivals) under policy π and F Let D i be the departure epochs. They coincide under both policies. Then W F = D i − T i and assume that π interchange the departure of j and j + 1 : W π = D j + 1 − T j i j and W π j + 1 = D j − T j + 1 Now, it should be obvious that W π + W π j + 1 = W F + W F j + 1 and if f is increasing convex and j j 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 ( W F n ) � st f ( W π 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 of service (and not of arrivals) under policy π and F Let D i be the departure epochs. They coincide under both policies. Then W F = D i − T i and assume that π interchange the departure of j and j + 1 : W π = D j + 1 − T j i j and W π j + 1 = D j − T j + 1 Now, it should be obvious that W π + W π j + 1 = W F + W F j + 1 and if f is increasing convex and j j 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 ( W F n ) � st f ( W π 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 of service (and not of arrivals) under policy π and F Let D i be the departure epochs. They coincide under both policies. Then W F = D i − T i and assume that π interchange the departure of j and j + 1 : W π = D j + 1 − T j i j and W π j + 1 = D j − T j + 1 Now, it should be obvious that W π + W π j + 1 = W F + W F j + 1 and if f is increasing convex and j j 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 λ 2 Instability α opt = 1 / 2 λ 1 F IG .: 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
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