Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Multi-stage Stochastic Fluid Models for Congestion Control Małgorzata O’Reilly * * University of Tasmania, Australia Australia New Zealand Applied Probability Workshop Brisbane 2013 1 / 57
Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control ANZAPW Auckland 2012 2 / 57
Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control ANZAPW Auckland 2012 3 / 57
Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control ANZAPW Auckland 2012 4 / 57
Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control ANZAPW Auckland 2012 5 / 57
Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Outline Introduction: Stochastic Fluid Model 1 Multi-stage SFMs with congestion control 2 Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs 6 / 57
Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Definition of a SFM Let { ( ϕ ( t ) , X ( t )) , t ≥ 0 } be a process such that: { ϕ ( t ) , t ≥ 0 } is an irreducible CTMC with a (finite) set of phases S and generator T { ϕ ( t ) , t ≥ 0 } is the driving process Level X ( t ) records some performance measure When ϕ ( t ) = i , the rate at which X ( t ) is changing is c i 7 / 57
Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control SFM with boundaries 0 and B ϕ ( t ) - phase, X ( t ) - level 8 / 57
Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Definition of a bounded SFM Let { ( ϕ ( t ) , X ( t )) , t ≥ 0 } be a process such that: { ϕ ( t ) , t ≥ 0 } is an irreducible CTMC with a (finite) set of phases S and generator T When ϕ ( t ) = i then X ( t ) = 0, c i < 0 = ⇒ dX ( t ) / dt = 0 X ( t ) = B , c i > 0 = ⇒ dX ( t ) / dt = 0 Otherwise, dX ( t ) / dt = c i 9 / 57
Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Some Notation S 1 = { i ∈ S : c i > 0 } S 2 = { i ∈ S : c i < 0 } S 0 = { i ∈ S : c i = 0 } C 1 = diag ( c i ) for all i ∈ S 1 C 2 = diag ( | c i | ) for all i ∈ S 2 T 11 = [ T ij ] for all i ∈ S 1 , j ∈ S 1 T 12 = [ T ij ] for all i ∈ S 1 , j ∈ S 2 T 10 = [ T ij ] for all i ∈ S 1 , j ∈ S 0 etc 10 / 57
Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Fluid generator Q ( s ) (Bean, O’Reilly, and Taylor 2005) Assume Re ( s ) ≥ 0 C − 1 1 [( T 11 − s I ) − T 10 ( T 00 − s I ) − 1 T 01 ] Q 11 ( s ) = C − 1 2 [( T 22 − s I ) − T 20 ( T 00 − s I ) − 1 T 02 ] Q 22 ( s ) = C − 1 1 [ T 12 − T 10 ( T 00 − s I ) − 1 T 02 ] Q 12 ( s ) = C − 1 2 [ T 21 − T 20 ( T 00 − s I ) − 1 T 01 ] Q 21 ( s ) = Definition � Q 11 ( s ) Q 12 ( s ) � Q ( s ) = Q 21 ( s ) Q 22 ( s ) Q = Q ( 0 ) 11 / 57
Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control In-Out Fluid Figure: Start in ( i , 0 ) , end in ( j , y ) at time ˆ θ ( y ) 12 / 57
Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Corresponding Laplace-Stieltjes Transform (LST) � t | Y ( t ) | = u = 0 | c ϕ ( u ) | du ˆ θ ( y ) = inf { t ≥ 0 : | Y ( t ) | = y } Definition Let ˆ ∆ y ( s ) = [ ˆ ∆ y ( s ) ij ] be such that for all i , j ∈ S 1 ∪ S 2 ∆ y ( s ) ij = E ( e − s ˆ θ ( y ) : ϕ (ˆ ˆ θ ( y )) = j | ϕ ( 0 ) = i , Y ( t ) = 0 ) Fact ˆ ∆ y ( s ) = e Q ( s ) y 13 / 57
Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Return to Level Zero Figure: Start in ( i , 0 ) , end in ( j , 0 ) at time θ ( 0 ) 14 / 57
Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Matrix Ψ ( s ) (Bean, O’Reilly, and Taylor 2005) Let θ ( 0 ) = inf { t ≥ 0 : X ( t ) = 0 } Definition For s with Re ( s ) ≥ 0 , i with c i > 0 , j with c j < 0 , let Ψ( s ) ij = E ( θ ( 0 ) < ∞ , θ ( 0 ) = i | ϕ ( 0 ) = i , X ( 0 ) = 0 ) Fact For s ≥ 0 , Ψ ( s ) is the minimum nonnegative solution of Q 12 ( s ) + Q 11 ( s ) Ψ ( s ) + Ψ ( s ) Q 22 ( s ) + Ψ ( s ) Q 21 ( s ) Ψ ( s ) = 0 15 / 57
Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control ˆ G x , y ( s ) - Draining with a Taboo 16 / 57
Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control ˆ H x , y ( s ) - Filling in with a Taboo 17 / 57
Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Draining and Filling - with taboo For i , j ∈ S 1 ∪ S 2 , 0 < x < y G x , y ( s )] ij = E [ e − s θ ( 0 ) : θ ( 0 ) < θ ( y ) , ϕ ( θ ( 0 )) = j | Y ( 0 ) = x , ϕ ( 0 ) = i ] [ˆ H x , y ( s )] ij = E [ e − s θ ( y ) : θ ( y ) < θ ( 0 ) , ϕ ( θ ( y )) = j | Y ( 0 ) = x , ϕ ( 0 ) = i ] [ˆ 18 / 57
Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control ˆ G x , y ( s ) and ˆ H x , y ( s ) (Bean, O’Reilly, and Taylor 2005) Fact � � � ˆ H y ( s ) I � � � ˆ ˆ ˆ ˆ G x , y ( s ) H x , y ( s ) G x ( s ) H y − x ( s ) = ˆ G y ( s ) I where � � � � ˆ Ψ ( s ) e ( Q 22 ( s )+ Q 22 ( s ) Ψ ( s )) x G x 0 12 ( s ) 0 G x ( s ) ˆ = = ˆ e ( Q 22 ( s )+ Q 22 ( s ) Ψ ( s )) x G x 0 22 ( s ) 0 � ˆ � � e ( Q 11 ( s )+ Q 12 ( s ) Ξ ( s )) x H x � 11 ( s ) 0 0 ˆ H x ( s ) = = ˆ H x Ξ ( s ) e ( Q 11 ( s )+ Q 12 ( s ) Ξ ( s )) x 21 ( s ) 0 0 19 / 57
Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Remark Using the above building blocks ( Q ( s ) , Ψ ( s ) , ˆ G x , y ( s ) and ˆ H x , y ( s ) ), and arguments based on appropriate partitioning of sample paths, the (transient and stationary) analysis of (different classes of) SFMs follows. We use these building blocks in the analysis of the multi-stage SFMs with congestion control, which is discussed below. 20 / 57
Two-stage SFMs Transient Analysis Introduction: Stochastic Fluid Model Stationary Analysis Multi-stage SFMs with congestion control Additional measures Multi-stage SFMs Outline Introduction: Stochastic Fluid Model 1 Multi-stage SFMs with congestion control 2 Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs 21 / 57
Two-stage SFMs Transient Analysis Introduction: Stochastic Fluid Model Stationary Analysis Multi-stage SFMs with congestion control Additional measures Multi-stage SFMs Two-stage buffer 22 / 57
Two-stage SFMs Transient Analysis Introduction: Stochastic Fluid Model Stationary Analysis Multi-stage SFMs with congestion control Additional measures Multi-stage SFMs Two-stage SFM (with lower boundary 0) Thresholds b 1 , b 2 , 0 < b 1 < b 2 , for controlling congestion. The process starts from Stage 1 in level 0 Stage 1 → Stage 2 when reaching b 2 from below Stage 2 → Stage 1 when reaching b 1 from above Matrices P ( b 2 ) , P ( b 1 ) record the probabilities of these transitions While in Stage ℓ ∈ { 1 , 2 } , the process evolves according to a traditional SFM with a set of phases S ℓ , generator T ℓ and fluid rates c ℓ i 23 / 57
Two-stage SFMs Transient Analysis Introduction: Stochastic Fluid Model Stationary Analysis Multi-stage SFMs with congestion control Additional measures Multi-stage SFMs Mutli-stage SFMs This class of models contains a model introduced by Malhotra, Mandjes, Scheinhardt and van den Berg (2009). Generalizations here: Any real fluid change rates c ( ℓ ) (including zero), where i i ∈ S ( ℓ ) , and ℓ = 1 , 2 is the current stage. The transition between the stages may involve not only the change in T ( ℓ ) , but also in S ( ℓ ) . The change in c ( ℓ ) at the moment of the transition between i the stages allows all possible types of changes of sign (from + or − to + , − or 0). We treat the model with an upper boundary B > b 2 . We consider a generalization to multi-stage SFMs. 24 / 57
Two-stage SFMs Transient Analysis Introduction: Stochastic Fluid Model Stationary Analysis Multi-stage SFMs with congestion control Additional measures Multi-stage SFMs Mutli-stage SFMs This class of models contains a model introduced by Malhotra, Mandjes, Scheinhardt and van den Berg (2009). Generalizations here: Any real fluid change rates c ( ℓ ) (including zero), where i i ∈ S ( ℓ ) , and ℓ = 1 , 2 is the current stage. The transition between the stages may involve not only the change in T ( ℓ ) , but also in S ( ℓ ) . The change in c ( ℓ ) at the moment of the transition between i the stages allows all possible types of changes of sign (from + or − to + , − or 0). We treat the model with an upper boundary B > b 2 . We consider a generalization to multi-stage SFMs. 25 / 57
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