Bounds for the Coupling Time in Queueing Networks Perfect Simulation - PowerPoint PPT Presentation

Bounds for the Coupling Time in Queueing Networks Perfect Simulation J.G. Dopper 2 , B. Gaujal and J.-M. Vincent 1 1 Laboratory ID-IMAG MESCAL Project Universities of Grenoble, France { Bruno.Gaujal,Jean-Marc.Vincent } @imag.fr 2 Mathematical 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 Queueing Networks with finite capacity 1 Event modelling and monotonicity 2 Perfect simulation and coupling time 3 Acyclic networks 4 Synthesis and future works 5 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 Queueing Networks with finite capacity 1 Event modelling and monotonicity 2 Perfect simulation and coupling time 3 Acyclic networks 4 Synthesis and future works 5 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 Markov model Finite set of resources : Assumptions : servers - Poisson arrival, waiting room Routing strategies : - exponential distribution for service times, state dependent - probabilistic routing with overflow overflow strategy blocking strategy... ⇒ continuous time Markov chain Average performance : load of the system response time loss rate ... 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 Markov model Finite set of resources : C 1 λ 3 servers C λ C 0 3 1 λ 0 waiting room λ Routing strategies : λ 2 5 λ 4 C state dependent 2 Assumptions : overflow strategy blocking strategy... - Poisson arrival, Average performance : - exponential distribution for service times, load of the system - probabilistic routing with overflow response time loss rate ... ⇒ 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 Markov model Finite set of resources : C 1 λ 3 servers C λ C 0 3 1 λ 0 waiting room λ Routing strategies : λ 2 5 λ 4 C state dependent 2 Assumptions : overflow strategy blocking strategy... - Poisson arrival, Average performance : - exponential distribution for service times, load of the system - probabilistic routing with overflow response time loss rate ... ⇒ 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 Queueing Networks with finite capacity 1 Event modelling and monotonicity 2 Perfect simulation and coupling time 3 Acyclic networks 4 Synthesis and future works 5 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 : C 1 λ 3 C λ C 0 3 λ 1 0 λ λ 2 5 λ 4 C 2 Event description : rate origin destination enabling condition routing policy e 0 Q − 1 Q 0 none rejection if Q 0 is full λ 0 e 1 λ 1 Q 0 Q 1 s 0 > 0 rejection if Q 1 is full e 2 Q 0 Q 2 s 0 > 0 rejection if Q 2 is full λ 2 e 3 λ 3 Q 1 Q 3 s 1 > 0 rejection if Q 3 is full e 4 λ 4 Q 2 Q 3 s 2 > 0 rejection if Q 3 is full e 5 Q 3 Q − 1 s 3 > 0 none λ 5 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 Poisson driven system X = X 0 × · · · × X K − 1 ] with X i = { 0 , · · · , C i } . Event e : ❀ transition function Φ( ., e ) ; ❀ Poisson process λ e Uniformization ⇒ GSMP representation λ e and P ( event e ) = λ e � Λ = Λ ; Trajectory : { e n } n ∈ Z i.i.d . e ⇒ Homogeneous Discrete Time Markov Chain [Bremaud 99] X n + 1 = Φ( X n , e n + 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 Poisson driven system X = X 0 × · · · × X K − 1 ] States with X i = { 0 , · · · , C i } . Event e : Time Events ❀ transition function Φ( ., e ) ; e1 e2 e3 ❀ Poisson process λ e e4 Uniformization ⇒ GSMP representation λ e and P ( event e ) = λ e � Λ = Λ ; Trajectory : { e n } n ∈ Z i.i.d . e ⇒ Homogeneous Discrete Time Markov Chain [Bremaud 99] X n + 1 = Φ( X n , e n + 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 Poisson driven system X = X 0 × · · · × X K − 1 ] States with X i = { 0 , · · · , C i } . Event e : Time Events ❀ transition function Φ( ., e ) ; e1 e2 e3 ❀ Poisson process λ e e4 Uniformization ⇒ GSMP representation λ e and P ( event e ) = λ e � Λ = Λ ; Trajectory : { e n } n ∈ Z i.i.d . e ⇒ Homogeneous Discrete Time Markov Chain [Bremaud 99] X n + 1 = Φ( X n , e n + 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 = [ x 0 , x 1 , · · · , x K − 1 ] ≺ y = [ y 0 , y 1 , · · · , y K − 1 ] iff ∀ i , x i � y i . 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 = [ x 0 , x 1 , · · · , x K − 1 ] ≺ y = [ y 0 , y 1 , · · · , y K − 1 ] iff ∀ i , x i � y i . 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 Queueing Networks with finite capacity 1 Event modelling and monotonicity 2 Perfect simulation and coupling time 3 Acyclic networks 4 Synthesis and future works 5 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 X n + 1 = Φ( X n , e n + 1 ) . ← x 0 x Trajectory { 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 X n + 1 } until some empirical criteria return x Convergence : biased sample Sampling : Warm-up period 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