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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 Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 1 / 33

Outline Queueing Networks with finite capacity 1 Perfect simulation 2 Events in queueing networks 3 Application 4 5 Conclusion and future works Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 2 / 33

Outline Queueing Networks with finite capacity 1 Perfect simulation 2 Events in queueing networks 3 Application 4 5 Conclusion and future works Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 3 / 33

Interconnexion networks Delta network 0 8 16 24 1 9 17 25 Input rates 2 10 18 26 Service rates 3 11 19 27 Homogeneous routing 4 12 20 28 Overflow strategy 5 13 21 29 6 14 22 30 7 15 23 31 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

Interconnexion networks Delta network 0 8 16 24 1 9 17 25 Input rates 2 10 18 26 Service rates 3 11 19 27 Homogeneous routing 4 12 20 28 Overflow strategy 5 13 21 29 6 14 22 30 7 15 23 31 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

Call centers Multilevel Erlang model Traffic 1 Traffic 2 Overflow traffic 1 Overflow traffic 2 Types of requests Input rates Different service rates Overflow strategy Type III servers Type I servers Type II servers 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

Call centers Multilevel Erlang model Traffic 1 Traffic 2 Overflow traffic 1 Overflow traffic 2 Types of requests Input rates Different service rates Overflow strategy Type III servers Type I servers Type II servers 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

Resource broker Grid model Q1 µ1 Q2 Input rates µ2 Q10 Q3 Allocation strategy µ3 Q4 µ4 State dependent allocation Index based routing : Resource Q5 Broker Q11 µ5 λ destination minimize a Q6 µ6 criteria Q7 µ7 Overflow Q12 Q8 µ8 Q9 µ9 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

Resource broker Grid model Q1 µ1 Q2 Input rates µ2 Q10 Q3 Allocation strategy µ3 Q4 µ4 State dependent allocation Index based routing : Resource Q5 Broker Q11 µ5 λ destination minimize a Q6 µ6 criteria Q7 µ7 Overflow Q12 Q8 µ8 Q9 µ9 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

Queueing networks with finite capacity Network model Finite set of resources : servers waiting room Routing strategies : state dependent Markov model overflow strategy blocking strategy Poisson arrival, exponential services distribution, probabilistic routing ⇒ continuous time Markov chain ... Average performance : load of the system response time loss rate ... 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

Queueing networks with finite capacity Network model Markov model Finite set of resources : C1 µ λ p servers 1−p waiting room Routing strategies : Blocking ν C2 Rejection Poisson arrival, exponential services distribution, probabilistic routing state dependent ⇒ continuous time Markov chain overflow strategy Queue 2 C2 blocking strategy C2−1 ... Average performance : load of the system 2 response time 1 loss rate ... 0 0 1 2 C1−1 C1 Queue 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

Queueing networks with finite capacity Network model Markov model Finite set of resources : C1 µ λ p servers 1−p waiting room Routing strategies : Blocking ν C2 Rejection Poisson arrival, exponential services distribution, probabilistic routing state dependent ⇒ continuous time Markov chain overflow strategy Queue 2 C2 blocking strategy C2−1 ... Average performance : load of the system 2 response time 1 loss rate ... 0 0 1 2 C1−1 C1 Queue 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

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

Markov chain : Two states automaton 1−p 1−q p 0 1 q � 1 − p � p P = 1 − q q P ( X n = 1 | X 0 = 0 ) = ( 1 − q ) .π n − 1 + p . ( 1 − π n ) = p + ( 1 − p − q ) π n q � q � ( 1 − p − q ) n . = p + q + π 0 − p + q Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 9 / 33

Markov chain : Two states automaton 1−p 1−q p 0 1 q � 1 − p � p P = 1 − q q P ( X n = 1 | X 0 = 0 ) = ( 1 − q ) .π n − 1 + p . ( 1 − π n ) = p + ( 1 − p − q ) π n q � q � ( 1 − p − q ) n . = p + q + π 0 − p + q Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 9 / 33

Markov chain general discrete state space { X n } n ∈ N Stochastic process Markov memoryless property : P ( X n = j | X n − 1 = i , X n − 2 = i n − 2 , · · · , X 0 = i 0 ) = P ( X n = j | X n − 1 = i ) = P ( X 1 = j | X 0 = i ) = p i , j P = (( p i , j )) : transition matrix (stochastic matrix) Iteration ⇐ ⇒ matrix product Steady state convergence lim n P ( X n = j | X 0 = i ) = π ∞ ( j ) Ergodic theorem N 1 � lim 1 1 ( X n = j ) = π ∞ ( j ) N n i = 1 Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 10 / 33

Markov chain general discrete state space { X n } n ∈ N Stochastic process Markov memoryless property : P ( X n = j | X n − 1 = i , X n − 2 = i n − 2 , · · · , X 0 = i 0 ) = P ( X n = j | X n − 1 = i ) = P ( X 1 = j | X 0 = i ) = p i , j P = (( p i , j )) : transition matrix (stochastic matrix) Iteration ⇐ ⇒ matrix product Steady state convergence lim n P ( X n = j | X 0 = i ) = π ∞ ( j ) Ergodic theorem N 1 � lim 1 1 ( X n = j ) = π ∞ ( j ) N n i = 1 Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 10 / 33

Markov chain general discrete state space { X n } n ∈ N Stochastic process Markov memoryless property : P ( X n = j | X n − 1 = i , X n − 2 = i n − 2 , · · · , X 0 = i 0 ) = P ( X n = j | X n − 1 = i ) = P ( X 1 = j | X 0 = i ) = p i , j P = (( p i , j )) : transition matrix (stochastic matrix) Iteration ⇐ ⇒ matrix product Steady state convergence lim n P ( X n = j | X 0 = i ) = π ∞ ( j ) Ergodic theorem N 1 � lim 1 1 ( X n = j ) = π ∞ ( j ) N n i = 1 Jean-Marc Vincent (Universities of Grenoble)Perfect simulation of monotone queueing networks ENSL Oct. 2005 10 / 33

Modeling discrete event systems STATE DIRECT STATE COST + SYSTEM TRANSITIONS SIMULATION COMPUTATION Cost(x) state x Markov model Difficulties: - stopping criteria : burn in time - simulation biases || π n − 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

Modeling discrete event systems STATE BACKWARD COST + SYSTEM SIMULATION TRANSITIONS cost c Markov model 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