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Perfect simulation of monotone queueing networks application to index based routing policies Jean-Marc Vincent 1 1 Laboratory ID-IMAG MESCAL Project Universities of Grenoble Jean-Marc.Vincent@imag.fr Jean-Marc Vincent (Universities of

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SLIDE 1

Perfect simulation of monotone queueing networks

application to index based routing policies Jean-Marc Vincent1

1Laboratory ID-IMAG

MESCAL Project Universities of Grenoble Jean-Marc.Vincent@imag.fr

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 1 / 33

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SLIDE 2

Outline

1

Queueing Networks with finite capacity

2

Perfect simulation

3

Events in queueing networks

4

Application

5

Conclusion and future works

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 2 / 33

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SLIDE 3

Outline

1

Queueing Networks with finite capacity

2

Perfect simulation

3

Events in queueing networks

4

Application

5

Conclusion and future works

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 3 / 33

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SLIDE 4

Interconnexion networks

Delta network

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 1 2 3 4 5 6 7 10 9 8

Input rates Service rates Homogeneous routing Overflow strategy

Problem

Loss probability at each level Analysis of hot spot ...

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 4 / 33

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SLIDE 5

Interconnexion networks

Delta network

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 1 2 3 4 5 6 7 10 9 8

Input rates Service rates Homogeneous routing Overflow strategy

Problem

Loss probability at each level Analysis of hot spot ...

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 4 / 33

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SLIDE 6

Call centers

Multilevel Erlang model

Traffic 2 Overflow traffic 2 Overflow traffic 1 Type I servers Type II servers Type III servers Traffic 1

Types of requests Input rates Different service rates Overflow strategy

Problem

Optimization of resources Quality of service (waiting time, rejection probability,...) ...

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 5 / 33

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SLIDE 7

Call centers

Multilevel Erlang model

Traffic 2 Overflow traffic 2 Overflow traffic 1 Type I servers Type II servers Type III servers Traffic 1

Types of requests Input rates Different service rates Overflow strategy

Problem

Optimization of resources Quality of service (waiting time, rejection probability,...) ...

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 5 / 33

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SLIDE 8

Resource broker

Grid model

Q9 Resource Broker λ µ2 µ1 µ3 µ4 µ5 µ6 µ7 µ8 µ9 Overflow Q2 Q3 Q1 Q10 Q11 Q12 Q4 Q6 Q5 Q7 Q8

Input rates Allocation strategy State dependent allocation Index based routing : destination minimize a criteria

Problem

Optimization of throughput, response time,... Comparison of policies, analysis of heuristics ...

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 6 / 33

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SLIDE 9

Resource broker

Grid model

Q9 Resource Broker λ µ2 µ1 µ3 µ4 µ5 µ6 µ7 µ8 µ9 Overflow Q2 Q3 Q1 Q10 Q11 Q12 Q4 Q6 Q5 Q7 Q8

Input rates Allocation strategy State dependent allocation Index based routing : destination minimize a criteria

Problem

Optimization of throughput, response time,... Comparison of policies, analysis of heuristics ...

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 6 / 33

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SLIDE 10

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

Poisson arrival, exponential services distribution, probabilistic routing ⇒ continuous time Markov chain

Problem

Computation of steady state distribution ⇒ state space explosion

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 7 / 33

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SLIDE 11

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

1−p λ µ ν C1 C2 Rejection Blocking p

Poisson arrival, exponential services distribution, probabilistic routing ⇒ continuous time Markov chain

C1 1 2 C2−1 C2 Queue 2 2 Queue 1 1 C1−1

Problem

Computation of steady state distribution ⇒ state space explosion

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 7 / 33

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SLIDE 12

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

1−p λ µ ν C1 C2 Rejection Blocking p

Poisson arrival, exponential services distribution, probabilistic routing ⇒ continuous time Markov chain

C1 1 2 C2−1 C2 Queue 2 2 Queue 1 1 C1−1

Problem

Computation of steady state distribution ⇒ state space explosion

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 7 / 33

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SLIDE 13

Modeling discrete event systems

SYSTEM STATE + TRANSITIONS Markov model ALGEBRAIC RESOLUTION piQ=0 COST DISTRIBUTION Cost(pi)

Difficulties:

  • complex structure
  • large state space
  • analytical/numerical method
  • approximation/bounding techniques

⇒ reduction of the state space

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 8 / 33

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SLIDE 14

Markov chain : Two states automaton

q 1 1−p 1−q p

P = 1 − p p q 1 − q

  • P(Xn = 1|X0 = 0)

= (1 − q).πn−1 + p.(1 − πn) = p + (1 − p − q)πn = q p + q +

  • π0 −

q p + q

  • (1 − p − q)n.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 9 / 33

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SLIDE 15

Markov chain : Two states automaton

q 1 1−p 1−q p

P = 1 − p p q 1 − q

  • P(Xn = 1|X0 = 0)

= (1 − q).πn−1 + p.(1 − πn) = p + (1 − p − q)πn = q p + q +

  • π0 −

q p + q

  • (1 − p − q)n.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 9 / 33

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SLIDE 16

Markov chain general discrete state space

{Xn}n∈N Stochastic process Markov memoryless property : P(Xn = j|Xn−1 = i, Xn−2 = in−2, · · · , X0 = i0) = P(Xn = j|Xn−1 = i) = P(X1 = j|X0 = i) = pi,j P = ((pi,j)) : transition matrix (stochastic matrix) Iteration ⇐ ⇒ matrix product

Steady state convergence

lim

n P(Xn = j|X0 = i) = π∞(j)

Ergodic theorem

lim

n

1 N

N

  • i=1

1 1(Xn=j) = π∞(j)

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 10 / 33

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SLIDE 17

Markov chain general discrete state space

{Xn}n∈N Stochastic process Markov memoryless property : P(Xn = j|Xn−1 = i, Xn−2 = in−2, · · · , X0 = i0) = P(Xn = j|Xn−1 = i) = P(X1 = j|X0 = i) = pi,j P = ((pi,j)) : transition matrix (stochastic matrix) Iteration ⇐ ⇒ matrix product

Steady state convergence

lim

n P(Xn = j|X0 = i) = π∞(j)

Ergodic theorem

lim

n

1 N

N

  • i=1

1 1(Xn=j) = π∞(j)

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 10 / 33

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SLIDE 18

Markov chain general discrete state space

{Xn}n∈N Stochastic process Markov memoryless property : P(Xn = j|Xn−1 = i, Xn−2 = in−2, · · · , X0 = i0) = P(Xn = j|Xn−1 = i) = P(X1 = j|X0 = i) = pi,j P = ((pi,j)) : transition matrix (stochastic matrix) Iteration ⇐ ⇒ matrix product

Steady state convergence

lim

n P(Xn = j|X0 = i) = π∞(j)

Ergodic theorem

lim

n

1 N

N

  • i=1

1 1(Xn=j) = π∞(j)

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 10 / 33

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SLIDE 19

Modeling discrete event systems

SYSTEM STATE + TRANSITIONS Markov model DIRECT STATE SIMULATION state x COST COMPUTATION Cost(x)

Difficulties:

  • stopping criteria : burn in time
  • simulation biases ||πn − π∞||
  • estimation biases : confidence intervals O( 1

√n)

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 11 / 33

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SLIDE 20

Modeling discrete event systems

SYSTEM STATE + TRANSITIONS Markov model BACKWARD COST SIMULATION cost c

Properties:

  • Exact stopping criteria

⇒ no simulation bias Constraints:

  • N parallel trajectories

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 12 / 33

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SLIDE 21

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]

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 13 / 33

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SLIDE 22

Main results

1

Adaptation of perfect simulation method (Propp & al, Vincent & al.) to finite capacity queueing networks

2

Monotonicity property of events : index routing (blocking and

  • verflow)

3

PSI2 : a simulation framework

4

Validation and examples

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 14 / 33

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SLIDE 23

Outline

1

Queueing Networks with finite capacity

2

Perfect simulation

3

Events in queueing networks

4

Application

5

Conclusion and future works

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 15 / 33

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SLIDE 24

Event modelling

Multidimensional state space : X = X1 × · · · × XK with Xi = {0, · · · , Ci }. Event e : ❀ transition function Φ(., e); (skip rule) ❀ Poisson process λe

Uniformization

Λ = X

e

λe and P(event e) = λe Λ ; Trajectory : {en}n∈Z i.i.d. sequence. ⇒ Homogeneous Discrete Time Markov Chain [Bremaud 99] Xn+1 = Φ(Xn, en+1). Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 16 / 33

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SLIDE 25

Event modelling

Multidimensional state space : X = X1 × · · · × XK with Xi = {0, · · · , Ci }. Event e : ❀ transition function Φ(., e); (skip rule) ❀ Poisson process λe

Time States Events e1 e2 e3 e4

Uniformization

Λ = X

e

λe and P(event e) = λe Λ ; Trajectory : {en}n∈Z i.i.d. sequence. ⇒ Homogeneous Discrete Time Markov Chain [Bremaud 99] Xn+1 = Φ(Xn, en+1). Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 16 / 33

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SLIDE 26

Event modelling

Multidimensional state space : X = X1 × · · · × XK with Xi = {0, · · · , Ci }. Event e : ❀ transition function Φ(., e); (skip rule) ❀ Poisson process λe

Time States Events e1 e2 e3 e4

Uniformization

Λ = X

e

λe and P(event e) = λe Λ ; Trajectory : {en}n∈Z i.i.d. sequence. ⇒ Homogeneous Discrete Time Markov Chain [Bremaud 99] Xn+1 = Φ(Xn, en+1). Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 16 / 33

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SLIDE 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

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 17 / 33

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SLIDE 28

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

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 17 / 33

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SLIDE 29

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

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 17 / 33

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SLIDE 30

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

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 17 / 33

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SLIDE 31

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

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 17 / 33

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SLIDE 32

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

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 17 / 33

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SLIDE 33

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

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 17 / 33

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SLIDE 34

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

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 17 / 33

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SLIDE 35

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

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 17 / 33

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SLIDE 36

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

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 17 / 33

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SLIDE 37

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

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 17 / 33

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SLIDE 38

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

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 17 / 33

slide-39
SLIDE 39

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

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 17 / 33

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SLIDE 40

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

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 17 / 33

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SLIDE 41

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, e−1) = Z1 ∈ Φ(Φ(X, e−2), e−1) = Z2 . . . . . . ∈ Φ(Φ(· · · Φ(X, e−n), · · · ), e−2), e−1) = Zn

Theorem

Provided some condition on the events the sequence of sets {Zn}n∈N is decreasing to a single state, steady state distributed. τ∗ = inf{n ∈ N; Card(Zn) = 1}. backward coupling time Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 18 / 33

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SLIDE 42

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, e−1) = Z1 ∈ Φ(Φ(X, e−2), e−1) = Z2 . . . . . . ∈ Φ(Φ(· · · Φ(X, e−n), · · · ), e−2), e−1) = Zn

Theorem

Provided some condition on the events the sequence of sets {Zn}n∈N is decreasing to a single state, steady state distributed. τ∗ = inf{n ∈ N; Card(Zn) = 1}. backward coupling time Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 18 / 33

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SLIDE 43

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

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SLIDE 44

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

slide-45
SLIDE 45

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

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SLIDE 46

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

slide-47
SLIDE 47

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

slide-48
SLIDE 48

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

slide-49
SLIDE 49

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

slide-50
SLIDE 50

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

slide-51
SLIDE 51

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

slide-52
SLIDE 52

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

slide-53
SLIDE 53

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

slide-54
SLIDE 54

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

slide-55
SLIDE 55

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

slide-56
SLIDE 56

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

slide-57
SLIDE 57

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

slide-58
SLIDE 58

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

slide-59
SLIDE 59

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

slide-60
SLIDE 60

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

slide-61
SLIDE 61

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

slide-62
SLIDE 62

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

slide-63
SLIDE 63

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.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 19 / 33

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SLIDE 64

Monotonicity and perfect simulation : idea

(X, ≺) partially ordered set (componentwise) min = (0, · · · , 0) and Max = (C1, · · · , Cn). An event e is monotone if Φ(., e) is monotone on X If all events are monotone then

X0 ∈ Zn ⊂ ˆ Φ(Φ(· · · Φ(min, e−n), · · · ), e−2), e−1), Φ(Φ(· · · Φ(Max, e−n), · · · ), e−2), e−1) ˜ ⇒ 2 trajectories Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 20 / 33

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SLIDE 65

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).

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 21 / 33

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SLIDE 66

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).

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 21 / 33

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SLIDE 67

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).

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 21 / 33

slide-68
SLIDE 68

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).

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 21 / 33

slide-69
SLIDE 69

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).

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 21 / 33

slide-70
SLIDE 70

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).

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 21 / 33

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SLIDE 71

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).

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 21 / 33

slide-72
SLIDE 72

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).

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 21 / 33

slide-73
SLIDE 73

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).

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 21 / 33

slide-74
SLIDE 74

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).

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 21 / 33

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SLIDE 75

Outline

1

Queueing Networks with finite capacity

2

Perfect simulation

3

Events in queueing networks

4

Application

5

Conclusion and future works

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 22 / 33

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SLIDE 76

Index routing in queueing networks

Index functions for event e

For queue i Ie

i : {0, · · · , Ci} −

→ O (totally ordered set). Property : ∀xi, xj Ie

i (xi) = Ie j (xj).

ex: inverse of a priority,...

Routing algorithm:

if xorigin >0 then { a client is available in the origin queue} xorigin = xorigin − 1; { the client is removed from the origin queue} j = argmini Ie

i (xi); { computation of the destination}

if j = -1 then xj = xj+1; { arrival of the client in queue j } { in the other case, the client goes out of the network} end if end if

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 23 / 33

slide-77
SLIDE 77

Index routing in queueing networks

Index functions for event e

For queue i Ie

i : {0, · · · , Ci} −

→ O (totally ordered set). Property : ∀xi, xj Ie

i (xi) = Ie j (xj).

ex: inverse of a priority,...

Routing algorithm:

if xorigin >0 then { a client is available in the origin queue} xorigin = xorigin − 1; { the client is removed from the origin queue} j = argmini Ie

i (xi); { computation of the destination}

if j = -1 then xj = xj+1; { arrival of the client in queue j } { in the other case, the client goes out of the network} end if end if

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 23 / 33

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SLIDE 78

Monotonicity of index routing policies

Proposition

If all index functions Ie

i are monotone then event e is monotone.

Proof : Let x ≺ y two states and let be an index routing event. Let i be the origin queue for the event. jx = argminjIe

j (xj) and jy = argminjIe j (yj)

Case 1 xi = yi = 0 nothing happens and Φ(x, e) = x ≺ y = Φ(y, e) Case 2 xi = 0, yi > 0 then Φ(x, e) = x ≺ y − ei + ejy = Φ(y, e) Case 3 xi > 0, yi > 0 then Ie

jx(xjx) < Ie jy(xjy) Ie jy(yjy) < Ie jx(yjx);

then xjx < yjx and Φ(x, e) = x − ei + ejx y − ei y − ei + ejy = Φ(y, e)

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 24 / 33

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SLIDE 79

Monotonicity of index routing policies

Proposition

If all index functions Ie

i are monotone then event e is monotone.

Proof : Let x ≺ y two states and let be an index routing event. Let i be the origin queue for the event. jx = argminjIe

j (xj) and jy = argminjIe j (yj)

Case 1 xi = yi = 0 nothing happens and Φ(x, e) = x ≺ y = Φ(y, e) Case 2 xi = 0, yi > 0 then Φ(x, e) = x ≺ y − ei + ejy = Φ(y, e) Case 3 xi > 0, yi > 0 then Ie

jx(xjx) < Ie jy(xjy) Ie jy(yjy) < Ie jx(yjx);

then xjx < yjx and Φ(x, e) = x − ei + ejx y − ei y − ei + ejy = Φ(y, e)

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 24 / 33

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SLIDE 80

Monotonicity of routing

Examples [Glasserman and Yao]

All of these events could be expressed as index based routing policies :

  • 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

...

Join the shortest response time

Ie

i (xi) =

  • xi+1

µi

if xi < Ci; +∞ elsewhere; Ie

−1 = max i

Ci.

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 25 / 33

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SLIDE 81

Outline

1

Queueing Networks with finite capacity

2

Perfect simulation

3

Events in queueing networks

4

Application

5

Conclusion and future works

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 26 / 33

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SLIDE 82

Priority servers

Erlang model

Output Arrivals Servers Overflow

  • n next free server

Rejection if all servers are buzy

X = {0, 1}3 E = {e0, e1, e2, e3} Card(X) = 2K

Events

Event type Rate Origin Destination list Arrival λ −1 Q1 ; Q2 ; Q3 ; −1 Departure µ1 Q1 −1 Departure µ2 Q2 −1 Departure µ3 Q3 −1

Results

Validation χ2 test K = 30 µi decreasing Saturation probability 0.0579 ± 4.710−4 Simulation time 0.4ms τ = 577

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 27 / 33

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SLIDE 83

Priority servers

Erlang model

Output Arrivals Servers Overflow

  • n next free server

Rejection if all servers are buzy

X = {0, 1}3 E = {e0, e1, e2, e3} Card(X) = 2K

Events

Event type Rate Origin Destination list Arrival λ −1 Q1 ; Q2 ; Q3 ; −1 Departure µ1 Q1 −1 Departure µ2 Q2 −1 Departure µ3 Q3 −1

Results

Validation χ2 test K = 30 µi decreasing Saturation probability 0.0579 ± 4.710−4 Simulation time 0.4ms τ = 577

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 27 / 33

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SLIDE 84

Priority servers

Erlang model

Output Arrivals Servers Overflow

  • n next free server

Rejection if all servers are buzy

X = {0, 1}40 µ1 = 1, µ2 = 0.8, µ3 = 0.5 Sample size 5.106 Card(X) = 2K

Saturation probability

0.1 0.2 0.3 0.4 0.5 0.6 10 20 30 40 50 60 70 λ Prob Proba 1e−06 1e−05 1e−04 0.001 8 8.5 9 9.5 10 10.5 11 11.5

λ

Coupling time

Coupling time Reward coupling time s

50 100 150 200 250 300 350 10 20 30 40 50 60 70 λ µ 400

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 28 / 33

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SLIDE 85

Priority servers

Erlang model

Output Arrivals Servers Overflow

  • n next free server

Rejection if all servers are buzy

X = {0, 1}40 µ1 = 1, µ2 = 0.8, µ3 = 0.5 Sample size 5.106 Card(X) = 2K

Saturation probability

0.1 0.2 0.3 0.4 0.5 0.6 10 20 30 40 50 60 70 λ Prob Proba 1e−06 1e−05 1e−04 0.001 8 8.5 9 9.5 10 10.5 11 11.5

λ

Coupling time

Coupling time Reward coupling time s

50 100 150 200 250 300 350 10 20 30 40 50 60 70 λ µ 400

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 28 / 33

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SLIDE 86

Line of servers

Tandem queues

reject µ2 C2 C3 µ3 C4 µ4 C5 µ5 C1 µ1 λ

X = {0, · · · , 100}5 E = {e0, · · · , e5} Card(X) = CK

Events

Event type Rate Origin

  • Dest. list

Arrival λ −1 Q1 ; −1 Routing/block µ1 Q1 Q2 ; Q1 Routing/block µ2 Q2 Q3 ; Q2 · · · · · · · · · · · · Departure µ5 Q5 −1

Results

C = 100 λ = 0.9; µ = 1 p = 1

2

Blocking probability b1 = 0.34, b2 = 0.02 b3 = 0.02, b4, = 0.02. Simulation time < 1ms

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 29 / 33

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SLIDE 87

Line of servers

Tandem queues

reject µ2 C2 C3 µ3 C4 µ4 C5 µ5 C1 µ1 λ

X = {0, · · · , 100}5 E = {e0, · · · , e5} Card(X) = CK

Events

Event type Rate Origin

  • Dest. list

Arrival λ −1 Q1 ; −1 Routing/block µ1 Q1 Q2 ; Q1 Routing/block µ2 Q2 Q3 ; Q2 · · · · · · · · · · · · Departure µ5 Q5 −1

Results

C = 100 λ = 0.9; µ = 1 p = 1

2

Blocking probability b1 = 0.34, b2 = 0.02 b3 = 0.02, b4, = 0.02. Simulation time < 1ms

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 29 / 33

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SLIDE 88

Multistage network

Delta network

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 1 2 3 4 5 6 7 10 9 8

X = {0, · · · , 100}32 E = {e0, · · · , e64} Card(X) = CK

Events

Event type Rate Origin

  • Dest. list

Arrival λ −1 Qi ; −1 Routing/rejection

1 2µ

Qi Qj ; −1 · · · · · · · · · · · · Departure µ Qk −1

Results

C = 100 λ = 0.9; µ = 1 Loss rate Simulation time 135ms τ ≃ 400000

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 30 / 33

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SLIDE 89

Multistage network

Delta network

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 1 2 3 4 5 6 7 10 9 8

X = {0, · · · , 100}32 E = {e0, · · · , e64} Card(X) = CK

Events

Event type Rate Origin

  • Dest. list

Arrival λ −1 Qi ; −1 Routing/rejection

1 2µ

Qi Qj ; −1 · · · · · · · · · · · · Departure µ Qk −1

Results

C = 100 λ = 0.9; µ = 1 Loss rate Simulation time 135ms τ ≃ 400000

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 30 / 33

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SLIDE 90

Multistage network

Delta network

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 1 2 3 4 5 6 7 10 9 8

X = {0, · · · , 100}32 E = {e0, · · · , e64} Card(X) = CK Sample size 100000

Queue length and saturation proba at level 3

1 2 3 4 5 6 7 E(N_34) 0.2 0.4 0.6 0.8

λ

0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.58 0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78

λ

Coupling time

Reward queue 31 saturated Reward at least 1 queue saturated (3rd level) Global coupling

s

2000 4000 6000 8000 10000 12000 14000 16000 18000 0.2 0.4 0.6 0.8

λ µ Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 31 / 33

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SLIDE 91

Outline

1

Queueing Networks with finite capacity

2

Perfect simulation

3

Events in queueing networks

4

Application

5

Conclusion and future works

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 32 / 33

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SLIDE 92

Conclusion and future works

Summary

monotone structure of queueing networks perfect simulation ⇒ direct sample of steady-state avoid burn in time practically efficient

Future works

Event model extensions Acceleration techniques : control variables Software environment PSI2

Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 33 / 33