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Bounds for the Coupling Time in Queueing Networks Perfect Simulation

J.G. Dopper2, B. Gaujal and J.-M. Vincent1

1Laboratory ID-IMAG

MESCAL Project Universities of Grenoble, France {Bruno.Gaujal,Jean-Marc.Vincent}@imag.fr

2Mathematical Institute,

Leiden University, Nederland jgdopper@math.leidenuniv.nl

J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble) Bounds for the Coupling Time in Queueing Networks Perfect Simulation MAM 2006, june12 1 / 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 Simulation MAM 2006, june12 2 / 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 Simulation MAM 2006, june12 3 / 27

Queueing networks with finite capacity

Network model

Finite set of resources : servers waiting room Routing strategies : state dependent

• verflow 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

Problem

Computation 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 Simulation MAM 2006, june12 4 / 27

Queueing networks with finite capacity

Network model

Finite set of resources : servers waiting room Routing strategies : state dependent

• verflow strategy

blocking strategy... Average performance : load of the system response time loss rate ...

Markov model

5

C C C C

1 2 3

λ λ λ λ λ λ

1 2 3 4

Assumptions :

• Poisson arrival,
• exponential distribution for service times,
• probabilistic routing with overflow

⇒ continuous time Markov chain

Problem

Computation 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 Simulation MAM 2006, june12 4 / 27

Queueing networks with finite capacity

Network model

Finite set of resources : servers waiting room Routing strategies : state dependent

• verflow strategy

blocking strategy... Average performance : load of the system response time loss rate ...

Markov model

5

C C C C

1 2 3

λ λ λ λ λ λ

1 2 3 4

Assumptions :

• Poisson arrival,
• exponential distribution for service times,
• probabilistic routing with overflow

⇒ continuous time Markov chain

Problem

Computation 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 Simulation MAM 2006, june12 4 / 27

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 models simulation of event graphs discrete event simulation perfect simulation [Mattson 04]

J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble) Bounds for the Coupling Time in Queueing Networks Perfect Simulation MAM 2006, june12 5 / 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 Simulation MAM 2006, june12 6 / 27

Event modelling

Queueing model :

5

C C C C

1 2 3

λ λ λ λ λ λ

1 2 3 4

Event description :

rate

• rigin

destination enabling condition routing policy e0 λ0 Q−1 Q0 none rejection if Q0 is full e1 λ1 Q0 Q1 s0 > 0 rejection if Q1 is full e2 λ2 Q0 Q2 s0 > 0 rejection if Q2 is full e3 λ3 Q1 Q3 s1 > 0 rejection if Q3 is full e4 λ4 Q2 Q3 s2 > 0 rejection if Q3 is full e5 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 Simulation MAM 2006, june12 7 / 27

Event modelling

Multidimensional state space

X = 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) = λe Λ ; 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 Simulation MAM 2006, june12 8 / 27

Event modelling

Multidimensional state space

X = 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) = λe Λ ; 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 Simulation MAM 2006, june12 8 / 27

Event modelling

Multidimensional state space

X = 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) = λe Λ ; 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 Simulation MAM 2006, june12 8 / 27

Monotonicity of routing strategy

(X, ≺) partially ordered set (componentwise) x = [x0, x1, · · · , xK−1] ≺ y = [y0, y1, · · · , yK−1] iff ∀i, xi 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 Simulation MAM 2006, june12 9 / 27

Monotonicity of routing strategy

(X, ≺) partially ordered set (componentwise) x = [x0, x1, · · · , xK−1] ≺ y = [y0, y1, · · · , yK−1] iff ∀i, xi 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 Simulation MAM 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 Simulation MAM 2006, june12 10 / 27

Classical forward simulation

Forward

Representation : 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 criteria return x Convergence : biased sample Sampling : Warm-up period

Trajectory Complexity

Related to the stabilization period Estimation : 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 Simulation MAM 2006, june12 11 / 27

Classical forward simulation

Forward

Representation : 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 criteria return x Convergence : biased sample Sampling : Warm-up period

Trajectory Complexity

Related to the stabilization period Estimation : 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 Simulation MAM 2006, june12 11 / 27

Classical forward simulation

Forward

Representation : 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 criteria return x Convergence : biased sample Sampling : Warm-up period

Trajectory

Initial state 0001 0010 0011 0100 0101 0110 1000 1001 1010 1100 Time 1 2 3 4 5 5 6 7 8 States 0000

Complexity

Related to the stabilization period Estimation : 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 Simulation MAM 2006, june12 11 / 27

Classical forward simulation

Forward

Representation : 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 criteria return x Convergence : biased sample Sampling : Warm-up period

Trajectory

Initial state 0001 0010 0011 0100 0101 0110 1000 1001 1010 1100 Time 1 2 3 4 5 5 6 7 8 States 0000

Complexity

Related to the stabilization period Estimation : 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 Simulation MAM 2006, june12 11 / 27

Classical forward simulation

Forward

Representation : 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 criteria return x Convergence : biased sample Sampling : Warm-up period

Trajectory

Initial state 0001 0010 0011 0100 0101 0110 1000 1001 1010 1100 Time 1 2 3 4 5 5 6 7 8 States 0000

Complexity

Related to the stabilization period Estimation : 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 Simulation MAM 2006, june12 11 / 27

Classical forward simulation

Forward

Representation : 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 criteria return x Convergence : biased sample Sampling : Warm-up period

Trajectory

Initial state 0001 0010 0011 0100 0101 0110 1000 1001 1010 1100 Time 1 2 3 4 5 5 6 7 8 States 0000

Complexity

Related to the stabilization period Estimation : 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 Simulation MAM 2006, june12 11 / 27

Classical forward simulation

Forward

Representation : 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 criteria return x Convergence : biased sample Sampling : Warm-up period

Trajectory

Initial state 0001 0010 0011 0100 0101 0110 1000 1001 1010 1100 Time 1 2 3 4 5 5 6 7 8 States 0000

Complexity

Related to the stabilization period Estimation : 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 Simulation MAM 2006, june12 11 / 27

Classical forward simulation

Forward

Representation : 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 criteria return x Convergence : biased sample Sampling : Warm-up period

Trajectory

Initial state 0001 0010 0011 0100 0101 0110 1000 1001 1010 1100 Time 1 2 3 4 5 5 6 7 8 States 0000

Complexity

Related to the stabilization period Estimation : 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 Simulation MAM 2006, june12 11 / 27

Classical forward simulation

Forward

Representation : 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 criteria return x Convergence : biased sample Sampling : Warm-up period

Trajectory

6 0010 0011 0100 0101 0110 1000 1001 1010 1100 Initial state Time 1 2 3 4 5 7 8 9 States 0000 0001

Complexity

Related to the stabilization period Estimation : 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 Simulation MAM 2006, june12 11 / 27

Classical forward simulation

Forward

Representation : 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 criteria return x Convergence : biased sample Sampling : Warm-up period

Trajectory

6 0010 0011 0100 0101 0110 1000 1001 1010 1100 Initial state Time 1 2 3 4 5 7 8 9 States 0000 0001

Complexity

Related to the stabilization period Estimation : 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 Simulation MAM 2006, june12 11 / 27

Classical forward simulation

Forward

Representation : 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 criteria return x Convergence : biased sample Sampling : Warm-up period

Trajectory

6 0010 0011 0100 0101 0110 1000 1001 1010 1100 Initial state Time 1 2 3 4 5 7 8 9 States 0000 0001

Complexity

Related to the stabilization period Estimation : 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 Simulation MAM 2006, june12 11 / 27

Classical forward simulation

Forward

Representation : 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 criteria return x Convergence : biased sample Sampling : Warm-up period

Trajectory

6 0010 0011 0100 0101 0110 1000 1001 1010 1100 Initial state Time 1 2 3 4 5 7 8 9 States 0000 0001

Complexity

Related to the stabilization period Estimation : 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 Simulation MAM 2006, june12 11 / 27

Classical forward simulation

Forward

Representation : 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 criteria return x Convergence : biased sample Sampling : Warm-up period

Trajectory

stabilization period 0101 0110 1000 1001 1010 1100 Initial state 6 Steady state ? Time 1 2 3 4 5 7 8 9 States 0000 0001 0010 0011 0100

Complexity

Related to the stabilization period Estimation : 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 Simulation MAM 2006, june12 11 / 27

Classical forward simulation

Forward

Representation : 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 criteria return x Convergence : biased sample Sampling : Warm-up period

Trajectory

stabilization period 0101 0110 1000 1001 1010 1100 Initial state 6 Steady state ? Time 1 2 3 4 5 7 8 9 States 0000 0001 0010 0011 0100

Complexity

Related to the stabilization period Estimation : 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 Simulation MAM 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

Theorem

Provided 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 Simulation MAM 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

Theorem

Provided 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 Simulation MAM 2006, june12 12 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling time: τ < +∞, properties of Φ

Trajectories Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 2006, june12 13 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling time: τ < +∞, properties of Φ

Trajectories Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 2006, june12 13 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling 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

Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 2006, june12 13 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling 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 U1

Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 2006, june12 13 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling 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 U1

Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 2006, june12 13 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling 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 U1 U2

Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 2006, june12 13 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling 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 U1 U2

Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 2006, june12 13 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling 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 U1 U2 U3

Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 2006, june12 13 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling 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 U1 U2 U3

Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 2006, june12 13 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling 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 U1 U2 U3 U4

Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 2006, june12 13 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling 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 U1 U2 U3 U4

Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 2006, june12 13 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling 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 U1 U2 U3 U4 U5

Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 2006, june12 13 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling 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 U1 U2 U3 U4 U5

Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 2006, june12 13 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling 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 U1 U2 U3 U4 U5 U6

Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 2006, june12 13 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling 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 U1 U2 U3 U4 U5 U6

Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 2006, june12 13 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling 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 U1 U2 U3 U4 U5 U6 U7

Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 2006, june12 13 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling 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 U1 U2 U3 U4 U5 U6 U7

Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 2006, june12 13 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling 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 U1 U2 U3 U4 U5 U6 U7 U8

Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 2006, june12 13 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling 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 U1 U2 U3 U4 U5 U6 U7 U8

Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 2006, june12 13 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling 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 U1 U2 U3 U4 U5 U6 U7 U8 τ∗

Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 2006, june12 13 / 27

Perfect simulation

Backward algorithm

Representation : transition fonction Xn+1 = Φ(Xn, en+1). for all x ∈ X do y(x) ← x end for repeat u ← Random; for all x ∈ X do e ← Random event(); y(x) ← y(Φ(x, e)); end for until All y(x) are equal return y(x) Convergence : If the algorithm stops, the returned value is steady state distributed Coupling 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 U1 U2 U3 U4 U5 U6 U7 U8 τ∗

Mean time complexity

cΦ mean computation cost of Φ(x, e) C 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 Simulation MAM 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 Simulation MAM 2006, june12 14 / 27

Monotonicity and perfect simulation

Monotone PS

Doubling scheme n=1;R[1]=Random event; repeat n=2.n; y(min) ← min y(Max) ← Max for i=n downto n/2+1 do R[i]=Random event; end for for i=n downto 1 do y(min) ← Φ(y(min), R[i]) y(Max) ← Φ(y(Max), R[i]) end for until y(min) = y(Max) return y(min)

Trajectories

: minimum 2 1 M −1 −2 −4 −8 −16 −32 States : : Maximum

Mean time complexity

Cm 2.(2.Eτ).cΦ. Reduction factor :

4 Card(X).

J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble) Bounds for the Coupling Time in Queueing Networks Perfect Simulation MAM 2006, june12 15 / 27

Monotonicity and perfect simulation

Monotone PS

Doubling scheme n=1;R[1]=Random event; repeat n=2.n; y(min) ← min y(Max) ← Max for i=n downto n/2+1 do R[i]=Random event; end for for i=n downto 1 do y(min) ← Φ(y(min), R[i]) y(Max) ← Φ(y(Max), R[i]) end for until y(min) = y(Max) return y(min)

Trajectories

: minimum 2 1 M −1 −2 −4 −8 −16 −32 States : : Maximum

Mean time complexity

Cm 2.(2.Eτ).cΦ. Reduction factor :

4 Card(X).

J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble) Bounds for the Coupling Time in Queueing Networks Perfect Simulation MAM 2006, june12 15 / 27

Monotonicity and perfect simulation

Monotone PS

Doubling scheme n=1;R[1]=Random event; repeat n=2.n; y(min) ← min y(Max) ← Max for i=n downto n/2+1 do R[i]=Random event; end for for i=n downto 1 do y(min) ← Φ(y(min), R[i]) y(Max) ← Φ(y(Max), R[i]) end for until y(min) = y(Max) return y(min)

Trajectories

: minimum 2 1 M −1 −2 −4 −8 −16 −32 States : : Maximum

Mean time complexity

Cm 2.(2.Eτ).cΦ. Reduction factor :

4 Card(X).

J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble) Bounds for the Coupling Time in Queueing Networks Perfect Simulation MAM 2006, june12 15 / 27

Monotonicity and perfect simulation

Monotone PS

Doubling scheme n=1;R[1]=Random event; repeat n=2.n; y(min) ← min y(Max) ← Max for i=n downto n/2+1 do R[i]=Random event; end for for i=n downto 1 do y(min) ← Φ(y(min), R[i]) y(Max) ← Φ(y(Max), R[i]) end for until y(min) = y(Max) return y(min)

Trajectories

: minimum 2 1 M −1 −2 −4 −8 −16 −32 States : : Maximum

Mean time complexity

Cm 2.(2.Eτ).cΦ. Reduction factor :

4 Card(X).

J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble) Bounds for the Coupling Time in Queueing Networks Perfect Simulation MAM 2006, june12 15 / 27

Monotonicity and perfect simulation

Monotone PS

Doubling scheme n=1;R[1]=Random event; repeat n=2.n; y(min) ← min y(Max) ← Max for i=n downto n/2+1 do R[i]=Random event; end for for i=n downto 1 do y(min) ← Φ(y(min), R[i]) y(Max) ← Φ(y(Max), R[i]) end for until y(min) = y(Max) return y(min)

Trajectories

: minimum 2 1 M −1 −2 −4 −8 −16 −32 States : : Maximum

Mean time complexity

Cm 2.(2.Eτ).cΦ. Reduction factor :

4 Card(X).

J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble) Bounds for the Coupling Time in Queueing Networks Perfect Simulation MAM 2006, june12 15 / 27

Monotonicity and perfect simulation

Monotone PS

Doubling scheme n=1;R[1]=Random event; repeat n=2.n; y(min) ← min y(Max) ← Max for i=n downto n/2+1 do R[i]=Random event; end for for i=n downto 1 do y(min) ← Φ(y(min), R[i]) y(Max) ← Φ(y(Max), R[i]) end for until y(min) = y(Max) return y(min)

Trajectories

: minimum 2 1 M −1 −2 −4 −8 −16 −32 States : : Maximum

Mean time complexity

Cm 2.(2.Eτ).cΦ. Reduction factor :

4 Card(X).

J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble) Bounds for the Coupling Time in Queueing Networks Perfect Simulation MAM 2006, june12 15 / 27

Monotonicity and perfect simulation

Monotone PS

Doubling scheme n=1;R[1]=Random event; repeat n=2.n; y(min) ← min y(Max) ← Max for i=n downto n/2+1 do R[i]=Random event; end for for i=n downto 1 do y(min) ← Φ(y(min), R[i]) y(Max) ← Φ(y(Max), R[i]) end for until y(min) = y(Max) return y(min)

Trajectories

: minimum 2 1 M −1 −2 −4 −8 −16 −32 States : : Maximum

Mean time complexity

Cm 2.(2.Eτ).cΦ. Reduction factor :

4 Card(X).

J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble) Bounds for the Coupling Time in Queueing Networks Perfect Simulation MAM 2006, june12 15 / 27

Monotonicity and perfect simulation

Monotone PS

Doubling scheme n=1;R[1]=Random event; repeat n=2.n; y(min) ← min y(Max) ← Max for i=n downto n/2+1 do R[i]=Random event; end for for i=n downto 1 do y(min) ← Φ(y(min), R[i]) y(Max) ← Φ(y(Max), R[i]) end for until y(min) = y(Max) return y(min)

Trajectories

: minimum 2 1 M −1 −2 −4 −8 −16 −32 States : : Maximum

Mean time complexity

Cm 2.(2.Eτ).cΦ. Reduction factor :

4 Card(X).

J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble) Bounds for the Coupling Time in Queueing Networks Perfect Simulation MAM 2006, june12 15 / 27

Monotonicity and perfect simulation

Monotone PS

Doubling scheme n=1;R[1]=Random event; repeat n=2.n; y(min) ← min y(Max) ← Max for i=n downto n/2+1 do R[i]=Random event; end for for i=n downto 1 do y(min) ← Φ(y(min), R[i]) y(Max) ← Φ(y(Max), R[i]) end for until y(min) = y(Max) return y(min)

Trajectories

State 2 1 M −1 −2 −4 −8 −16 −32 States : : Maximum : minimum Generated

Mean time complexity

Cm 2.(2.Eτ).cΦ. Reduction factor :

4 Card(X).

J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble) Bounds for the Coupling Time in Queueing Networks Perfect Simulation MAM 2006, june12 15 / 27

Monotonicity and perfect simulation

Monotone PS

Doubling scheme n=1;R[1]=Random event; repeat n=2.n; y(min) ← min y(Max) ← Max for i=n downto n/2+1 do R[i]=Random event; end for for i=n downto 1 do y(min) ← Φ(y(min), R[i]) y(Max) ← Φ(y(Max), R[i]) end for until y(min) = y(Max) return y(min)

Trajectories

State 2 1 M −1 −2 −4 −8 −16 −32 States : : Maximum : minimum Generated

Mean time complexity

Cm 2.(2.Eτ).cΦ. Reduction factor :

4 Card(X).

J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble) Bounds for the Coupling Time in Queueing Networks Perfect Simulation MAM 2006, june12 15 / 27

Coupling time

definition

τ b = min{n ∈ N; Card(Zn) = 1}; = min{n ∈ N; |Φ(X, e−n→0| = 1}.

Properties

• Backward τ b and forward τ f coupling times have the same

probability distribution;

• Marginal coupling : denote by τ b

i the backward coupling time for

Qi τ b = max τ b

i .

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 Simulation MAM 2006, june12 16 / 27

Coupling time

definition

τ b = min{n ∈ N; Card(Zn) = 1}; = min{n ∈ N; |Φ(X, e−n→0| = 1}.

Properties

• Backward τ b and forward τ f coupling times have the same

probability distribution;

• Marginal coupling : denote by τ b

i the backward coupling time for

Qi τ b = max τ b

i .

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 Simulation MAM 2006, june12 16 / 27

Coupling time

definition

τ b = min{n ∈ N; Card(Zn) = 1}; = min{n ∈ N; |Φ(X, e−n→0| = 1}.

Properties

• Backward τ b and forward τ f coupling times have the same

probability distribution;

• Marginal coupling : denote by τ b

i the backward coupling time for

Qi τ b = max τ b

i .

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 Simulation MAM 2006, june12 16 / 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 Simulation MAM 2006, june12 17 / 27

Coupling experiment

Queueing model :

5

C C C C

1 2 3

λ λ λ λ λ λ

1 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 Simulation MAM 2006, june12 18 / 27

Coupling experiment

Queueing model :

5

C C C C

1 2 3

λ λ λ λ λ λ

1 2 3 4

Estimation of Eτ :

50 100 150 200 250 300 350 400 τ 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 Simulation MAM 2006, june12 18 / 27

Main result

Theorem (Bound on coupling time)

Eτ

K

• i=1

Λ Λi Ci + C2

i

2 ,

• 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 Simulation MAM 2006, june12 19 / 27

Main result

Theorem (Bound on coupling time)

Eτ

K

• i=1

Λ Λi Ci + C2

i

2 ,

• 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 Simulation MAM 2006, june12 19 / 27

Explicit computation for the M/M/1/C

Eτ b = E min(h0→C, hC→0) Absorbing time in a finite Markov chain; p =

λ λ+µ = 1 − q

1,C 1,C−1 0,C−2 1,C−2 2,C−1 2,C 3,C C−2,C−1 C−1,C C,C 0,0 0,1 1,2 0,C 0,C−1 0,C−3 p p p p p p p p p p p p p p p p q q q q q q q q q q q q q q q q Level 3 Level 4 Level 5 Level C+1 Level C+2 Level 2

Explicit recurrence equations Case λ = µ Eτ b = C+C2

2

.

J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble) Bounds for the Coupling Time in Queueing Networks Perfect Simulation MAM 2006, june12 20 / 27

Computable bounds for M/M/1/C

If the stationary distribution is concentrated on 0 (λ < µ), Eτ b Eh0→C is an accurate bound.

Theorem

The mean coupling time Eτ b of a M/M/1/C queue with arrival rate λ and service rate µ is bounded using p = λ/(λ + µ) = 1 − q. Critical bound: ∀p ∈ [0, 1], Eτ b C2+C

2

. Heavy traffic Bound: if p > 1

2,

Eτ b

C p−q − q(1− “

q p

”C ) (p−q)2

. Light traffic bound: if p < 1

2,

Eτ b

C q−p − p(1− “

p q

”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 Simulation MAM 2006, june12 21 / 27

Computable bounds for M/M/1/C

Example with C = 10

20 40 60 80 100 120 0.2 0.4 0.6 0.8 1

Eτ b p heavy traffic Light traffic bound

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 Simulation MAM 2006, june12 22 / 27

Example for tandem queues

Coupling of Queue 0 Time X 0

0 = 5

5 4 2 1 3 = C1 6 = C0 −τ b Coupling of queue 1 conditionned by state of queue 0 4 3 = C1 1 −τ b

1 (s0 = 2)

Time X 1

1 = 2

X 1

0 = 3

5 = X 0 6 2 X 1

1 = 2

5 4 2 1 3 = C1 6 = C0 −τ b

0 − τ b 1 (s0 = 5)

X 0

0 = 5

τ b

1 (s0 = 5)

Time X 1

0 = 3

Then τ b st

∞τ b 1 + τ b 0 , normalized

J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble) Bounds for the Coupling Time in Queueing Networks Perfect Simulation MAM 2006, june12 23 / 27

Bound with isolated queues

Theorem

In an acyclic stable network of K M/M/1/Ci queues with Bernoulli routing and losses in case of overflow, the coupling time from the past satisfies in expectation, E[τ b]

• K−1
• i=0

Λ ℓi + µi    Ci qi − pi − pi(1 −

• pi

qi

Ci) (qi − pi)2   

• K−1
• i=0

Λ ℓi + µi (Ci + C2

i ).

J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble) Bounds for the Coupling Time in Queueing Networks Perfect Simulation MAM 2006, june12 24 / 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 Simulation MAM 2006, june12 25 / 27

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 stationary

distribution.

J.G. Dopper, B. Gaujal and J.-M. Vincent (Universities of Grenoble) Bounds for the Coupling Time in Queueing Networks Perfect Simulation MAM 2006, june12 26 / 27

Future works

Conjecture for general networks.

700 800 0.5 1 1.5 2 2.5 3 3.5 4 500 400 300 200 100 600

λ5

Eτ b B1 (proven) B1 ∧ B2 ∧ B3 B3 (conjecture) B2 (conjecture)

Extension to cyclic networks, Generalization to several types of events Application : 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 Simulation MAM 2006, june12 27 / 27