Bounds for the Coupling Time in Queueing Networks Perfect ... · Outline 1 Queueing Networks with...
Transcript of Bounds for the Coupling Time in Queueing Networks Perfect ... · Outline 1 Queueing Networks with...
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Bounds for the Coupling Time in QueueingNetworks Perfect Simulation
J.G. Dopper2, B. Gaujal and J.-M. Vincent1
1Laboratory ID-IMAGMESCAL Project
Universities of Grenoble, France{Bruno.Gaujal,Jean-Marc.Vincent}@imag.fr
2Mathematical Institute,Leiden University, [email protected]
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 1 / 27
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Outline
1 Queueing Networks with finite capacity
2 Event modelling and monotonicity
3 Perfect simulation and coupling time
4 Acyclic networks
5 Synthesis and future works
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 2 / 27
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Outline
1 Queueing Networks with finite capacity
2 Event modelling and monotonicity
3 Perfect simulation and coupling time
4 Acyclic networks
5 Synthesis and future works
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 3 / 27
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Queueing networks with finite capacity
Network modelFinite set of resources :
servers
waiting room
Routing strategies :
state dependent
overflow strategy
blocking strategy...
Average performance :
load of the system
response time
loss rate ...
Markov model
Assumptions :- Poisson arrival,- exponential distribution for service times,- probabilistic routing with overflow
⇒ continuous time Markov chain
ProblemComputation of the stationary distribution⇒ state space explosion
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 4 / 27
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Queueing networks with finite capacity
Network modelFinite set of resources :
servers
waiting room
Routing strategies :
state dependent
overflow strategy
blocking strategy...
Average performance :
load of the system
response time
loss rate ...
Markov model
5
C
C
C
C0
1
2
3λ
λ
λ
λ
λλ
01
2
3
4
Assumptions :- Poisson arrival,- exponential distribution for service times,- probabilistic routing with overflow
⇒ continuous time Markov chain
ProblemComputation of the stationary distribution⇒ state space explosion
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 4 / 27
-
Queueing networks with finite capacity
Network modelFinite set of resources :
servers
waiting room
Routing strategies :
state dependent
overflow strategy
blocking strategy...
Average performance :
load of the system
response time
loss rate ...
Markov model
5
C
C
C
C0
1
2
3λ
λ
λ
λ
λλ
01
2
3
4
Assumptions :- Poisson arrival,- exponential distribution for service times,- probabilistic routing with overflow
⇒ continuous time Markov chain
ProblemComputation of the stationary distribution⇒ state space explosion
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 4 / 27
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Related works
Non reversible systems (reverse event)Product form solution ??Widely studied domain
- Analytical solution [Perros 94]- specific cases- numerical computation of normalization constant
- Numerical computation [Stewart 94]
- Approximation techniques [Onvural 90, Perros 94,...]
- Simulation [Banks & al. 01,...]simulation of Markov modelssimulation of event graphsdiscrete event simulationperfect simulation [Mattson 04]
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 5 / 27
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Outline
1 Queueing Networks with finite capacity
2 Event modelling and monotonicity
3 Perfect simulation and coupling time
4 Acyclic networks
5 Synthesis and future works
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 6 / 27
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Event modelling
Queueing model :
5
C
C
C
C0
1
2
3λ
λ
λ
λ
λλ
01
2
3
4
Event description :rate origin destination enabling condition routing policy
e0 λ0 Q−1 Q0 none rejection if Q0 is fulle1 λ1 Q0 Q1 s0 > 0 rejection if Q1 is fulle2 λ2 Q0 Q2 s0 > 0 rejection if Q2 is fulle3 λ3 Q1 Q3 s1 > 0 rejection if Q3 is fulle4 λ4 Q2 Q3 s2 > 0 rejection if Q3 is fulle5 λ5 Q3 Q−1 s3 > 0 none
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 7 / 27
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Event modelling
Multidimensional state spaceX = X0 × · · · × XK−1]
with Xi = {0, · · · , Ci}.Event e :; transition function Φ(., e);; Poisson process λe
Poisson driven system
Uniformization ⇒ GSMP representation
Λ =∑
e
λe and P(event e) =λeΛ
; Trajectory : {en}n∈Z i.i.d.
⇒ Homogeneous Discrete Time Markov Chain [Bremaud 99]
Xn+1 = Φ(Xn, en+1).J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 8 / 27
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Event modelling
Multidimensional state spaceX = X0 × · · · × XK−1]
with Xi = {0, · · · , Ci}.Event e :; transition function Φ(., e);; Poisson process λe
Poisson driven system
Time
States
Events
e1
e2
e3
e4
Uniformization ⇒ GSMP representation
Λ =∑
e
λe and P(event e) =λeΛ
; Trajectory : {en}n∈Z i.i.d.
⇒ Homogeneous Discrete Time Markov Chain [Bremaud 99]
Xn+1 = Φ(Xn, en+1).J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 8 / 27
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Event modelling
Multidimensional state spaceX = X0 × · · · × XK−1]
with Xi = {0, · · · , Ci}.Event e :; transition function Φ(., e);; Poisson process λe
Poisson driven system
Time
States
Events
e1
e2
e3
e4
Uniformization ⇒ GSMP representation
Λ =∑
e
λe and P(event e) =λeΛ
; Trajectory : {en}n∈Z i.i.d.
⇒ Homogeneous Discrete Time Markov Chain [Bremaud 99]
Xn+1 = Φ(Xn, en+1).J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 8 / 27
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Monotonicity of routing strategy
(X ,≺) partially ordered set (componentwise)
x = [x0, x1, · · · , xK−1] ≺ y = [y0, y1, · · · , yK−1] iff ∀i , xi 6 yi .
An event e is said to be monotone if
x ≺ y ⇒ Φ(x , e) ≺ Φ(y , e).
Examples [Glasserman and Yao]All of these routing events are monotone:- external arrival with overflow and rejection- routing with overflow and rejection or blocking- routing to the shortest available queue- routing to the shortest mean available response time- general index policies [Palmer-Mitrani]- rerouting inside queues...
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 9 / 27
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Monotonicity of routing strategy
(X ,≺) partially ordered set (componentwise)
x = [x0, x1, · · · , xK−1] ≺ y = [y0, y1, · · · , yK−1] iff ∀i , xi 6 yi .
An event e is said to be monotone if
x ≺ y ⇒ Φ(x , e) ≺ Φ(y , e).
Examples [Glasserman and Yao]All of these routing events are monotone:- external arrival with overflow and rejection- routing with overflow and rejection or blocking- routing to the shortest available queue- routing to the shortest mean available response time- general index policies [Palmer-Mitrani]- rerouting inside queues...
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 9 / 27
-
Outline
1 Queueing Networks with finite capacity
2 Event modelling and monotonicity
3 Perfect simulation and coupling time
4 Acyclic networks
5 Synthesis and future works
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 10 / 27
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Classical forward simulation
ForwardRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
x ← x0{choice of the initial state at time =0}n = 0;repeat
n ← n + 1;e ← Random event();x ← Φ(x, e);{computation of the next state Xn+1}
until some empirical criteriareturn x
Convergence : biased sampleSampling : Warm-up period
Trajectory
ComplexityRelated to the stabilization periodEstimation : replication or ergodic estimation
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 11 / 27
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Classical forward simulation
ForwardRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
x ← x0{choice of the initial state at time =0}n = 0;repeat
n ← n + 1;e ← Random event();x ← Φ(x, e);{computation of the next state Xn+1}
until some empirical criteriareturn x
Convergence : biased sampleSampling : Warm-up period
Trajectory
ComplexityRelated to the stabilization periodEstimation : replication or ergodic estimation
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 11 / 27
-
Classical forward simulation
ForwardRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
x ← x0{choice of the initial state at time =0}n = 0;repeat
n ← n + 1;e ← Random event();x ← Φ(x, e);{computation of the next state Xn+1}
until some empirical criteriareturn x
Convergence : biased sampleSampling : Warm-up period
Trajectory
Initial state
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
0
Time
1 2 3 4 5 5 6 7 8
States
0000
ComplexityRelated to the stabilization periodEstimation : replication or ergodic estimation
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 11 / 27
-
Classical forward simulation
ForwardRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
x ← x0{choice of the initial state at time =0}n = 0;repeat
n ← n + 1;e ← Random event();x ← Φ(x, e);{computation of the next state Xn+1}
until some empirical criteriareturn x
Convergence : biased sampleSampling : Warm-up period
Trajectory
Initial state
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
0
Time
1 2 3 4 5 5 6 7 8
States
0000
ComplexityRelated to the stabilization periodEstimation : replication or ergodic estimation
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 11 / 27
-
Classical forward simulation
ForwardRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
x ← x0{choice of the initial state at time =0}n = 0;repeat
n ← n + 1;e ← Random event();x ← Φ(x, e);{computation of the next state Xn+1}
until some empirical criteriareturn x
Convergence : biased sampleSampling : Warm-up period
Trajectory
Initial state
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
0
Time
1 2 3 4 5 5 6 7 8
States
0000
ComplexityRelated to the stabilization periodEstimation : replication or ergodic estimation
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 11 / 27
-
Classical forward simulation
ForwardRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
x ← x0{choice of the initial state at time =0}n = 0;repeat
n ← n + 1;e ← Random event();x ← Φ(x, e);{computation of the next state Xn+1}
until some empirical criteriareturn x
Convergence : biased sampleSampling : Warm-up period
Trajectory
Initial state
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
0
Time
1 2 3 4 5 5 6 7 8
States
0000
ComplexityRelated to the stabilization periodEstimation : replication or ergodic estimation
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 11 / 27
-
Classical forward simulation
ForwardRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
x ← x0{choice of the initial state at time =0}n = 0;repeat
n ← n + 1;e ← Random event();x ← Φ(x, e);{computation of the next state Xn+1}
until some empirical criteriareturn x
Convergence : biased sampleSampling : Warm-up period
Trajectory
Initial state
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
0
Time
1 2 3 4 5 5 6 7 8
States
0000
ComplexityRelated to the stabilization periodEstimation : replication or ergodic estimation
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 11 / 27
-
Classical forward simulation
ForwardRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
x ← x0{choice of the initial state at time =0}n = 0;repeat
n ← n + 1;e ← Random event();x ← Φ(x, e);{computation of the next state Xn+1}
until some empirical criteriareturn x
Convergence : biased sampleSampling : Warm-up period
Trajectory
Initial state
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
0
Time
1 2 3 4 5 5 6 7 8
States
0000
ComplexityRelated to the stabilization periodEstimation : replication or ergodic estimation
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 11 / 27
-
Classical forward simulation
ForwardRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
x ← x0{choice of the initial state at time =0}n = 0;repeat
n ← n + 1;e ← Random event();x ← Φ(x, e);{computation of the next state Xn+1}
until some empirical criteriareturn x
Convergence : biased sampleSampling : Warm-up period
Trajectory
6
0010
0011
0100
0101
0110
1000
1001
1010
1100
0
Initial state
Time
1 2 3 4 5 7 8 9
States
0000
0001
ComplexityRelated to the stabilization periodEstimation : replication or ergodic estimation
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 11 / 27
-
Classical forward simulation
ForwardRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
x ← x0{choice of the initial state at time =0}n = 0;repeat
n ← n + 1;e ← Random event();x ← Φ(x, e);{computation of the next state Xn+1}
until some empirical criteriareturn x
Convergence : biased sampleSampling : Warm-up period
Trajectory
6
0010
0011
0100
0101
0110
1000
1001
1010
1100
0
Initial state
Time
1 2 3 4 5 7 8 9
States
0000
0001
ComplexityRelated to the stabilization periodEstimation : replication or ergodic estimation
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 11 / 27
-
Classical forward simulation
ForwardRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
x ← x0{choice of the initial state at time =0}n = 0;repeat
n ← n + 1;e ← Random event();x ← Φ(x, e);{computation of the next state Xn+1}
until some empirical criteriareturn x
Convergence : biased sampleSampling : Warm-up period
Trajectory
6
0010
0011
0100
0101
0110
1000
1001
1010
1100
0
Initial state
Time
1 2 3 4 5 7 8 9
States
0000
0001
ComplexityRelated to the stabilization periodEstimation : replication or ergodic estimation
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 11 / 27
-
Classical forward simulation
ForwardRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
x ← x0{choice of the initial state at time =0}n = 0;repeat
n ← n + 1;e ← Random event();x ← Φ(x, e);{computation of the next state Xn+1}
until some empirical criteriareturn x
Convergence : biased sampleSampling : Warm-up period
Trajectory
6
0010
0011
0100
0101
0110
1000
1001
1010
1100
0
Initial state
Time
1 2 3 4 5 7 8 9
States
0000
0001
ComplexityRelated to the stabilization periodEstimation : replication or ergodic estimation
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 11 / 27
-
Classical forward simulation
ForwardRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
x ← x0{choice of the initial state at time =0}n = 0;repeat
n ← n + 1;e ← Random event();x ← Φ(x, e);{computation of the next state Xn+1}
until some empirical criteriareturn x
Convergence : biased sampleSampling : Warm-up period
Trajectory
stabilization period
0101
0110
1000
1001
1010
1100
0
Initial state
6
Steady
state ?
Time
1 2 3 4 5 7 8 9
States
0000
0001
0010
0011
0100
ComplexityRelated to the stabilization periodEstimation : replication or ergodic estimation
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 11 / 27
-
Classical forward simulation
ForwardRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
x ← x0{choice of the initial state at time =0}n = 0;repeat
n ← n + 1;e ← Random event();x ← Φ(x, e);{computation of the next state Xn+1}
until some empirical criteriareturn x
Convergence : biased sampleSampling : Warm-up period
Trajectory
stabilization period
0101
0110
1000
1001
1010
1100
0
Initial state
6
Steady
state ?
Time
1 2 3 4 5 7 8 9
States
0000
0001
0010
0011
0100
ComplexityRelated to the stabilization periodEstimation : replication or ergodic estimation
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 11 / 27
-
Perfect simulation : backward idea
Representation : transition fonction
Xn+1 = Φ(Xn, en+1), {en}n∈Z i.i.d. sequence.In what state could I be at time n = 0 ?
X0 ∈ X = Z0∈ Φ(X , e0) = Z1∈ Φ(Φ(X , e−1), e0) = Z2· · ·∈ Φ(Φ(· · ·Φ(X , e−n+1), · · · ), e0) = Zn
TheoremProvided some condition on the sequence of events, the sequence of sets
Z0 ⊇ Z1 ⊇ Z2 ⊇ · · · ⊇ Zn ⊇ · · · is decreasing to a single state.
The generated state is stationary distributed (steady state sample).
τb = inf{n ∈ N; Card(Zn) = 1}.
backward coupling time
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 12 / 27
-
Perfect simulation : backward idea
Representation : transition fonction
Xn+1 = Φ(Xn, en+1), {en}n∈Z i.i.d. sequence.In what state could I be at time n = 0 ?
X0 ∈ X = Z0∈ Φ(X , e0) = Z1∈ Φ(Φ(X , e−1), e0) = Z2· · ·∈ Φ(Φ(· · ·Φ(X , e−n+1), · · · ), e0) = Zn
TheoremProvided some condition on the sequence of events, the sequence of sets
Z0 ⊇ Z1 ⊇ Z2 ⊇ · · · ⊇ Zn ⊇ · · · is decreasing to a single state.
The generated state is stationary distributed (steady state sample).
τb = inf{n ∈ N; Card(Zn) = 1}.
backward coupling time
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 12 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Time
States
0000
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
−4 −3 −2 −1−5−6−7−8−9−10 0
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Time
States
0000
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
−4 −3 −2 −1−5−6−7−8−9−10 0U1
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Time
States
0000
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
−4 −3 −2 −1−5−6−7−8−9−10 0U1
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Time
States
0000
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
−4 −3 −2 −1−5−6−7−8−9−10 0U1U2
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Time
States
0000
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
−4 −3 −2 −1−5−6−7−8−9−10 0U1U2
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Time
States
0000
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
−4 −3 −2 −1−5−6−7−8−9−10 0U1U2U3
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Time
States
0000
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
−4 −3 −2 −1−5−6−7−8−9−10 0U1U2U3
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Time
States
0000
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
−4 −3 −2 −1−5−6−7−8−9−10 0U1U2U3U4
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Time
States
0000
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
−4 −3 −2 −1−5−6−7−8−9−10 0U1U2U3U4
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Time
States
0000
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
−4 −3 −2 −1−5−6−7−8−9−10 0U1U2U3U4U5
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Time
States
0000
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
−4 −3 −2 −1−5−6−7−8−9−10 0U1U2U3U4U5
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Time
States
0000
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
−4 −3 −2 −1−5−6−7−8−9−10 0U1U2U3U4U5U6
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Time
States
0000
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
−4 −3 −2 −1−5−6−7−8−9−10 0U1U2U3U4U5U6
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Time
States
0000
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
−4 −3 −2 −1−5−6−7−8−9−10 0U1U2U3U4U5U6U7
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Time
States
0000
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
−4 −3 −2 −1−5−6−7−8−9−10 0U1U2U3U4U5U6U7
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Time
States
0000
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
−4 −3 −2 −1−5−6−7−8−9−10 0U1U2U3U4U5U6U7U8
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Time
States
0000
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
−4 −3 −2 −1−5−6−7−8−9−10 0U1U2U3U4U5U6U7U8
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Time
States
0000
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
−4 −3 −2 −1−5−6−7−8−9−10 0U1U2U3U4U5U6U7U8
τ∗
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Perfect simulation
Backward algorithmRepresentation : transition fonction
Xn+1 = Φ(Xn, en+1).
for all x ∈ X doy(x) ← x
end forrepeat
u ← Random;for all x ∈ X do
e ← Random event();y(x) ← y(Φ(x, e));
end foruntil All y(x) are equalreturn y(x)
Convergence : If the algorithm stops, thereturned value is steady state distributedCoupling time: τ < +∞, properties of Φ
Trajectories
Time
States
0000
0001
0010
0011
0100
0101
0110
1000
1001
1010
1100
−4 −3 −2 −1−5−6−7−8−9−10 0U1U2U3U4U5U6U7U8
τ∗
Mean time complexitycΦ mean computation cost of Φ(x , e)
C 6 Card(X ).Eτ.cΦ.J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 13 / 27
-
Monotonicity and perfect simulation : idea
min = (0, · · · , 0) and Max = (C1, · · · , Cn).
If all events are monotone then
X0 ∈ Zn ⊂ [Φ(min, e−n→0),Φ(Max , e−n→0)]
⇒ 2 trajectories
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 14 / 27
-
Monotonicity and perfect simulation
Monotone PSDoubling scheme
n=1;R[1]=Random event;repeat
n=2.n;y(min) ← miny(Max) ← Maxfor i=n downto n/2+1 do
R[i]=Random event;end forfor i=n downto 1 do
y(min) ← Φ(y(min), R[i])y(Max) ← Φ(y(Max), R[i])
end foruntil y(min) = y(Max)return y(min)
Trajectories
: minimum
2
1
M
−1−2−4−8−16−32 0
States
: : Maximum
0
Mean time complexity
Cm 6 2.(2.Eτ).cΦ. Reduction factor : 4Card(X ) .
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 15 / 27
-
Monotonicity and perfect simulation
Monotone PSDoubling scheme
n=1;R[1]=Random event;repeat
n=2.n;y(min) ← miny(Max) ← Maxfor i=n downto n/2+1 do
R[i]=Random event;end forfor i=n downto 1 do
y(min) ← Φ(y(min), R[i])y(Max) ← Φ(y(Max), R[i])
end foruntil y(min) = y(Max)return y(min)
Trajectories
: minimum
2
1
M
−1−2−4−8−16−32 0
States
: : Maximum
0
Mean time complexity
Cm 6 2.(2.Eτ).cΦ. Reduction factor : 4Card(X ) .
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 15 / 27
-
Monotonicity and perfect simulation
Monotone PSDoubling scheme
n=1;R[1]=Random event;repeat
n=2.n;y(min) ← miny(Max) ← Maxfor i=n downto n/2+1 do
R[i]=Random event;end forfor i=n downto 1 do
y(min) ← Φ(y(min), R[i])y(Max) ← Φ(y(Max), R[i])
end foruntil y(min) = y(Max)return y(min)
Trajectories
: minimum
2
1
M
−1−2−4−8−16−32 0
States
: : Maximum
0
Mean time complexity
Cm 6 2.(2.Eτ).cΦ. Reduction factor : 4Card(X ) .
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 15 / 27
-
Monotonicity and perfect simulation
Monotone PSDoubling scheme
n=1;R[1]=Random event;repeat
n=2.n;y(min) ← miny(Max) ← Maxfor i=n downto n/2+1 do
R[i]=Random event;end forfor i=n downto 1 do
y(min) ← Φ(y(min), R[i])y(Max) ← Φ(y(Max), R[i])
end foruntil y(min) = y(Max)return y(min)
Trajectories
: minimum
2
1
M
−1−2−4−8−16−32 0
States
: : Maximum
0
Mean time complexity
Cm 6 2.(2.Eτ).cΦ. Reduction factor : 4Card(X ) .
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 15 / 27
-
Monotonicity and perfect simulation
Monotone PSDoubling scheme
n=1;R[1]=Random event;repeat
n=2.n;y(min) ← miny(Max) ← Maxfor i=n downto n/2+1 do
R[i]=Random event;end forfor i=n downto 1 do
y(min) ← Φ(y(min), R[i])y(Max) ← Φ(y(Max), R[i])
end foruntil y(min) = y(Max)return y(min)
Trajectories
: minimum
2
1
M
−1−2−4−8−16−32 0
States
: : Maximum
0
Mean time complexity
Cm 6 2.(2.Eτ).cΦ. Reduction factor : 4Card(X ) .
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 15 / 27
-
Monotonicity and perfect simulation
Monotone PSDoubling scheme
n=1;R[1]=Random event;repeat
n=2.n;y(min) ← miny(Max) ← Maxfor i=n downto n/2+1 do
R[i]=Random event;end forfor i=n downto 1 do
y(min) ← Φ(y(min), R[i])y(Max) ← Φ(y(Max), R[i])
end foruntil y(min) = y(Max)return y(min)
Trajectories
: minimum
2
1
M
−1−2−4−8−16−32 0
States
: : Maximum
0
Mean time complexity
Cm 6 2.(2.Eτ).cΦ. Reduction factor : 4Card(X ) .
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 15 / 27
-
Monotonicity and perfect simulation
Monotone PSDoubling scheme
n=1;R[1]=Random event;repeat
n=2.n;y(min) ← miny(Max) ← Maxfor i=n downto n/2+1 do
R[i]=Random event;end forfor i=n downto 1 do
y(min) ← Φ(y(min), R[i])y(Max) ← Φ(y(Max), R[i])
end foruntil y(min) = y(Max)return y(min)
Trajectories
: minimum
2
1
M
−1−2−4−8−16−32 0
States
: : Maximum
0
Mean time complexity
Cm 6 2.(2.Eτ).cΦ. Reduction factor : 4Card(X ) .
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 15 / 27
-
Monotonicity and perfect simulation
Monotone PSDoubling scheme
n=1;R[1]=Random event;repeat
n=2.n;y(min) ← miny(Max) ← Maxfor i=n downto n/2+1 do
R[i]=Random event;end forfor i=n downto 1 do
y(min) ← Φ(y(min), R[i])y(Max) ← Φ(y(Max), R[i])
end foruntil y(min) = y(Max)return y(min)
Trajectories
: minimum
2
1
M
−1−2−4−8−16−32 0
States
: : Maximum
0
Mean time complexity
Cm 6 2.(2.Eτ).cΦ. Reduction factor : 4Card(X ) .
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 15 / 27
-
Monotonicity and perfect simulation
Monotone PSDoubling scheme
n=1;R[1]=Random event;repeat
n=2.n;y(min) ← miny(Max) ← Maxfor i=n downto n/2+1 do
R[i]=Random event;end forfor i=n downto 1 do
y(min) ← Φ(y(min), R[i])y(Max) ← Φ(y(Max), R[i])
end foruntil y(min) = y(Max)return y(min)
Trajectories
State
2
1
M
−1−2−4−8−16−32 0
States
: : Maximum
: minimum
Generated
0
Mean time complexity
Cm 6 2.(2.Eτ).cΦ. Reduction factor : 4Card(X ) .
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 15 / 27
-
Monotonicity and perfect simulation
Monotone PSDoubling scheme
n=1;R[1]=Random event;repeat
n=2.n;y(min) ← miny(Max) ← Maxfor i=n downto n/2+1 do
R[i]=Random event;end forfor i=n downto 1 do
y(min) ← Φ(y(min), R[i])y(Max) ← Φ(y(Max), R[i])
end foruntil y(min) = y(Max)return y(min)
Trajectories
State
2
1
M
−1−2−4−8−16−32 0
States
: : Maximum
: minimum
Generated
0
Mean time complexity
Cm 6 2.(2.Eτ).cΦ. Reduction factor : 4Card(X ) .
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 15 / 27
-
Coupling time
definition
τb = min{n ∈ N; Card(Zn) = 1};= min{n ∈ N; |Φ(X , e−n→0| = 1}.
Properties
- Backward τb and forward τ f coupling times have the sameprobability distribution;
- Marginal coupling : denote by τbi the backward coupling time forQi
τb = max τbi .
Problem : compute the mean coupling time
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 16 / 27
-
Coupling time
definition
τb = min{n ∈ N; Card(Zn) = 1};= min{n ∈ N; |Φ(X , e−n→0| = 1}.
Properties
- Backward τb and forward τ f coupling times have the sameprobability distribution;
- Marginal coupling : denote by τbi the backward coupling time forQi
τb = max τbi .
Problem : compute the mean coupling time
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 16 / 27
-
Coupling time
definition
τb = min{n ∈ N; Card(Zn) = 1};= min{n ∈ N; |Φ(X , e−n→0| = 1}.
Properties
- Backward τb and forward τ f coupling times have the sameprobability distribution;
- Marginal coupling : denote by τbi the backward coupling time forQi
τb = max τbi .
Problem : compute the mean coupling time
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 16 / 27
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Outline
1 Queueing Networks with finite capacity
2 Event modelling and monotonicity
3 Perfect simulation and coupling time
4 Acyclic networks
5 Synthesis and future works
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 17 / 27
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Coupling experiment
Queueing model :
5
C
C
C
C0
1
2
3λ
λ
λ
λ
λλ
01
2
3
4
Estimation of Eτ :
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 18 / 27
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Coupling experiment
Queueing model :
5
C
C
C
C0
1
2
3λ
λ
λ
λ
λλ
01
2
3
4
Estimation of Eτ :
0
50
100
150
200
250
300
350
400
τ
0 1 2 3 4 λ
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 18 / 27
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Main result
Theorem (Bound on coupling time)
Eτ 6K∑
i=1
Λ
Λi
Ci + C2i2
,
- Λ : global event rate in the network,
- Λi the rate of events affecting Qi- Ci is the capacity of Queue i.
Sketch of the proof- Explicit computation for the M/M/1/C
- Computable bounds for the M/M/1/C
- Bound with isolated queues
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 19 / 27
-
Main result
Theorem (Bound on coupling time)
Eτ 6K∑
i=1
Λ
Λi
Ci + C2i2
,
- Λ : global event rate in the network,
- Λi the rate of events affecting Qi- Ci is the capacity of Queue i.
Sketch of the proof- Explicit computation for the M/M/1/C
- Computable bounds for the M/M/1/C
- Bound with isolated queues
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 19 / 27
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Explicit computation for the M/M/1/C
Eτb = E min(h0→C , hC→0)Absorbing time in a finite Markov chain; p = λλ+µ = 1− q
1,C
1,C−10,C−2
1,C−2 2,C−1
2,C
3,C
C−2,C−1 C−1,C
C,C0,0
0,1 1,2
0,C
0,C−1
0,C−3
p
p
p
p
p
pp p p p
ppp
p p
pq
q
q
q
q
q q q q q
qqq
q q
qLevel 3
Level 4
Level 5
Level C+1
Level C+2
Level 2
Explicit recurrence equationsCase λ = µ Eτb = C+C22 .
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 20 / 27
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Computable bounds for M/M/1/C
If the stationary distribution is concentrated on 0 (λ < µ),
Eτb 6 Eh0→C is an accurate bound.
Theorem
The mean coupling time Eτb of a M/M/1/C queue with arrival rate λand service rate µ is bounded using p = λ/(λ + µ) = 1− q.
Critical bound: ∀p ∈ [0, 1], Eτb 6 C2+C2 .
Heavy traffic Bound: if p > 12 , Eτb 6 Cp−q −
q(1−“
qp
”C)
(p−q)2.
Light traffic bound: if p < 12 , Eτb 6 Cq−p −
p(1−“
pq
”C)
(q−p)2.
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 21 / 27
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Computable bounds for M/M/1/C
Example with C = 10
0
20
40
60
80
100
120
0 0.2 0.4 0.6 0.8 1
Eτb
p
heavy trafficLight trafficbound
C+C2
2
C + C2
bound
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 22 / 27
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Example for tandem queues
Coupling of Queue 0
Time
0
X 00 = 55
4
2
1
3 = C1
6 = C0
0
−τ b0Coupling of queue 1 conditionned by state of queue 0
4
3 = C1
1
0
−τ b1 (s0 = 2) 0Time
X 11 = 2
X 10 = 3
5 = X 00
6
2
X 11 = 2
5
4
2
1
3 = C1
6 = C0
0
−τ b0 − τ b1 (s0 = 5)
X 00 = 5
τ b1 (s0 = 5) 0Time
X 10 = 3
Then τb 6st ∞τb1 + τb0 , normalized
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 23 / 27
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Bound with isolated queues
TheoremIn an acyclic stable network of K M/M/1/Ci queues with Bernoullirouting and losses in case of overflow, the coupling time from the pastsatisfies in expectation,
E[τb] 6K−1∑i=0
Λ
`i + µi
Ciqi − pi
−pi(1−
(piqi
)Ci)
(qi − pi)2
6
K−1∑i=0
Λ
`i + µi(Ci + C
2i ).
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 24 / 27
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Outline
1 Queueing Networks with finite capacity
2 Event modelling and monotonicity
3 Perfect simulation and coupling time
4 Acyclic networks
5 Synthesis and future works
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 25 / 27
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Synthesis
Computable bound for the mean coupling time :
- linear in the number of component of the model;
- at most quadratic in queues sizes;
- large capacity queues ( bound is accurate).
Practical impact
- Accurate bounds, dimensionning of trajectories length;
- Simulation useful even for low probability events;
- Coupling time is explained by the spread of the stationarydistribution.
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 26 / 27
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Future works
Conjecture for general networks.
0
700
800
0 0.5 1 1.5 2 2.5 3 3.5 4
500
400
300
200
100
600
λ5
Eτb
B1 (proven)
B1 ∧ B2 ∧ B3
B3 (conjecture)
B2 (conjecture)
Extension to cyclic networks,Generalization to several types of eventsApplication : Grid and call centers
J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble)Bounds for the Coupling Time in Queueing Networks Perfect SimulationMAM 2006, june12 27 / 27
Queueing Networks with finite capacityEvent modelling and monotonicityPerfect simulation and coupling timeAcyclic networksSynthesis and future works