Gribaudo, Telek: Fluid models 29 2.3. Fluid Stochastic Petri Nets - Timed Transitions can be enabled , depending on the marking of the places and on the weights of the discrete and inhibitor arcs that are connected to it. - When a Timed Transition T j is enabled, it fires after an exponentially distributed time. - We denote with F ( T j , M ) the Instantaneous Firing Rate of transition T j in marking M (that is the rate parameter of the exponential distribution of the transition firing time).
Gribaudo, Telek: Fluid models 30 2.3. Fluid Stochastic Petri Nets - An enabled timed transition T j changes the marking of the discrete places to which it is connected with discrete arcs when it fires . - An enabled timed transition T j continuously changes the marking of the fluid places to which it is connected with fluid arcs as long as it is enabled .
Gribaudo, Telek: Fluid models 31 2.3. Fluid Stochastic Petri Nets t k immediate transition - Immediate Transitions represents events that happens in zero time. - They move tokens among discrete places. - They cannot change the fluid level in continuous places.
Gribaudo, Telek: Fluid models 32 2.3. Fluid Stochastic Petri Nets t k immediate transition - We call T i the set of the immediate transitions. - We address with t k ∈ T i a transition of this set.
Gribaudo, Telek: Fluid models 33 2.3. Fluid Stochastic Petri Nets - Immediate Transitions can be enabled , following the same rules as timed transitions. - Immediate transitions are characterized by their Weight , which is used to determine which transition will fire when more than one are enabled at the same time. - We denote with W ( t k , M ) the Weight of immediate transition t k in marking M .
Gribaudo, Telek: Fluid models 34 2.3. Fluid Stochastic Petri Nets - When more than one transition (timed or immediate) are enabled in a marking, a conflict arises. - The conflict resolution algorithm determines which transition actually fires.
Gribaudo, Telek: Fluid models 35 2.3. Fluid Stochastic Petri Nets - If both timed and immediate transitions are enabled in a marking, immediate transitions have priority over the timed ones (i.e. timed transitions can be ignored). - Race policy solves conflict among timed transition (whichever fires first). - Probabilistic decision, based on the transition weights, determines which fires among several immediate transition concurrently enabled.
Gribaudo, Telek: Fluid models 36 2.3. Fluid Stochastic Petri Nets discrete inhibitor arc arc - Discrete Arcs and Inhibitor Arcs connect discrete places to transitions. - They determine when a transition is enabled. - Discrete Arcs also determine what happens when a transition fires.
Gribaudo, Telek: Fluid models 37 2.3. Fluid Stochastic Petri Nets - Each Discrete Arc or Inhibitor Arcs has associated a weight. - The standard GSPNs: firing rules apply to FSPNs: • A transition is enabled if all its input places have at least as many tokens as the weight of the corresponding arc. • A transition is enabled if all the places to which it is connected with inhibitor arcs have at most as many tokens as the weight of the connecting arc, minus one. • When a transition fires, it removes from its input places and it puts into the output places as many tokens as the corresponding connecting arc.
Gribaudo, Telek: Fluid models 38 2.3. Fluid Stochastic Petri Nets fluid arc - Fluid (continuous) Arcs connect timed transitions to fluid places. - Each Fluid arc has associated a Flow Rate . - The Flow Rate is a non-negative real number.
Gribaudo, Telek: Fluid models 39 2.3. Fluid Stochastic Petri Nets fluid arc - A fluid arc directed form a timed transition to a fluid place, pumps fluid into the place at a rate equal to the arc’s Flow Rate. - A fluid arc directed form a fluid place to a timed transition, removes fluid from the place at a rate equal to the arc’s Flow Rate. - Fluid flows only when the Timed Transition at beginning or at the end of the arc is enabled.
Gribaudo, Telek: Fluid models 40 2.3. Fluid Stochastic Petri Nets - When the fluid place becomes empty (its fluid marking reaches zero), the fluid flow stop. - We denote with R ( T j , c l , M ) the flow rate of a fluid arc from timed transition T j to fluid place c l in marking M . - We use R ( c l , T j , M ) when the arc is directed in the opposite direction (from the place to the transition).
Gribaudo, Telek: Fluid models 41 2.3. Fluid Stochastic Petri Nets - Fluid Stochastic Petri Nets are analyzed by transforming them into equivalent Fluid Models. - If the FSPN has a single fluid place, then standard FM can be applied. - If the FSPN has more than one fluid place, then special FM with multiple continuous variables must be used.
Gribaudo, Telek: Fluid models 42 2.3. Fluid Stochastic Petri Nets - The Transition Rate Matrix Q and the Flow Rate Matrix R can be automatically generated starting from the FSPN Model. - The state space of the FM corresponds to the state space of the discrete part of the FSPN model. - Both the state space and the Transition Rate Matrix Q can be calculated applying standard GSPN techniques to the discrete part of FSPN the model.
Gribaudo, Telek: Fluid models 43 2.3. Fluid Stochastic Petri Nets - The elements of the flow rate matrix can be computed from the flow rates of the fluid arcs. If we imagine to have only one single fluid place c l , then we can define: � = ( R ( T j , c l , m i ) − R ( T j , c l , m i )) r i T j ∈E ( m i ) m i is the discrete marking corresponding to state i , and E ( m i ) is the set of timed transition enabled in m i .
Gribaudo, Telek: Fluid models 44 2.3. Fluid Stochastic Petri Nets - Some important extensions have been proposed in the literature. Two of them are: • Fluid-dependent transition and flow rates, • Flush-out arcs.
Gribaudo, Telek: Fluid models 45 2.3. Fluid Stochastic Petri Nets - Both the transition rates of timed transition, and the flow rates associated to the fluid arcs can depend on the complete (discrete and continuous) marking of the process. - In this case the underlaying stochastic process should be analyzed using non-homogenous Fluid Models . - Both the transition rate matrix and the flow rate matrix depend on the fluid part of the model: Q ( x ), R ( x ).
Gribaudo, Telek: Fluid models 46 2.3. Fluid Stochastic Petri Nets flush-out arc - Flush-out arcs are special arcs that connect fluid places to timed transition (but not timed transition to fluid places). - They are drawn using thick lines. - When a transition fires, the places connected with a flush-out arc are emptied in zero time.
Gribaudo, Telek: Fluid models 47 2.3. Fluid Stochastic Petri Nets flush-out arc - Despite their simplicity, Flush-out Arcs can be exploited to obtain many interesting effects, like dropping the content of the transmission buffer. - The underlaying stochastic model is no longer a standard Fluid Model, but it can be analyzed similarly using appropriate boundary conditions.
Gribaudo, Telek: Fluid models 48 3.1 Introduction to fluid models Continuous time stochastic processes with • discrete value (state), e.g. CTMC, • continuous value, e.g. unfinished work in a queue, • hybrid (continuous and discrete) value, e.g. unfinished work and the number of customers.
Gribaudo, Telek: Fluid models 49 3.1 Introduction to fluid models General continuous and hybrid valued stochastic processes are hard to analyze. But, there are special cases: • reward models, • fluid models. A simple function of a discrete state stochastic process governs the evolution of the continuous variable. When the discrete state stochastic process is a CTMC • Markov reward models, • Markov fluid models. We focus on this Markovian case.
Gribaudo, Telek: Fluid models 50 3.1 Introduction to fluid models Reward models: unbounded (non-decreasing) evolution, Fluid models: bounded evolution. X(t) X(t) r k B r j r r r k r j k k r r i i 0 S(t) t S(t) t k k j j i i t t
Gribaudo, Telek: Fluid models 51 3.1 Introduction to fluid models Classes of fluid models: • finite buffer – infinite buffer, • first order – second order, • homogeneous – fluid level dependent, • barrier behaviour in second order case – reflecting – absorbing.
Gribaudo, Telek: Fluid models 52 3.1 Introduction to fluid models Infinite buffer: the continuous quantity is only lower bounded at zero. Finite buffer: the continuous quantity is lower bounded at zero and upper bounded at B . X(t) X(t) B r r r r r j k k r j k k r r i i 0 0 t S(t) t S(t) k k j j i i t t
Gribaudo, Telek: Fluid models 53 3.1 Introduction to fluid models First order: the continuous quantity is a deterministic function of a CTMC. Second order: the continuous quantity is a stochastic function of a CTMC. X(t) 9 B fluid level 8 state 7 r r r k j k 6 r i 0 5 S(t) t 4 k 3 j 2 i 1 0 1 1.5 2 2.5 3 3.5 4 t
Gribaudo, Telek: Fluid models 54 3.1 Introduction to fluid models Interpretation of second order fluid models. Random walk with decreasing time and fluid granularity. Fluid level CTMC state
Gribaudo, Telek: Fluid models 55 3.1 Introduction to fluid models Homogeneous: the evolution of the CTMC is independent of the fluid level. Fluid level dependent: the generator of the CTMC is a function of the fluid level. dX(t) =r S(t) dX(t) =r S(t) S(t) (X(t)) S(t) dt dt X(t) X(t) Q(X(t)) Q
Gribaudo, Telek: Fluid models 56 3.1 Introduction to fluid models Boundary behaviour of second order fluid models. Reflecting: the fluid level is immediately reflected at the boundary. Absorbing: the fluid level remains at the boundary up to a state transition of the Markov chain. X(t) X(t) B B r r >0 >0 k k σ σ =0 =0 r r i <0 i <0 j j t t S(t) S(t) k k j j i i t t
Gribaudo, Telek: Fluid models 57 3.1 Introduction to fluid models Interpretation of the boundary behaviours: Reflecting Absorbing Upper boundary Fluid level CTMC state
Gribaudo, Telek: Fluid models 58 3.2 Transient behaviour of fluid models - Transient behaviour of first order infinite buffer homogeneous Markov fluid models, - Extensions: • finite buffer, • second order, • fluid level dependency.
Gribaudo, Telek: Fluid models 59 3.2 Transient behaviour of fluid models First order, infinite buffer, homogeneous Markov fluid models During a sojourn of the CTMC in state i ( S ( t ) = i ) the fluid level ( X ( t )) increases at rate r i when X ( t ) > 0: d X ( t +∆) − X ( t ) = r i ∆ → dtX ( t ) = r i if S ( t ) = i, X ( t ) > 0 . When X ( t ) = 0 the fluid level can not decrease: d dtX ( t ) = max( r i , 0) if S ( t ) = i, X ( t ) = 0 . That is r S ( t ) if X ( t ) > 0 , d dtX ( t ) = max( r S ( t ) , 0) if X ( t ) = 0 .
Gribaudo, Telek: Fluid models 60 3.2 Transient behaviour of fluid models First order, finite buffer, homogeneous Markov fluid models When X ( t ) = B the fluid level can not increase: d dtX ( t ) = min( r i , 0) , if S ( t ) = i, X ( t ) = B. That is r S ( t ) , if X ( t ) > 0 , d dtX ( t ) = max( r S ( t ) , 0) , if X ( t ) = 0 , min( r S ( t ) , 0) , if X ( t ) = B.
Gribaudo, Telek: Fluid models 61 3.2 Transient behaviour of fluid models Second order, infinite buffer, homogeneous Markov fluid models with reflecting barrier During a sojourn of the CTMC in state i ( S ( t ) = i ) in the sufficiently small ( t, t + ∆) interval the distribution of the fluid increment ( X ( t + ∆) − X ( t )) is normal distributed with mean r i ∆ and variance σ 2 i ∆: X ( t + ∆) − X ( t ) = N ( r i ∆ , σ 2 i ∆) , if S ( u ) = i, u ∈ ( t, t + ∆) , X ( t ) > 0 . At X ( t ) = 0 the fluid process is reflected immediately, − → Pr ( X ( t ) = 0) = 0.
Gribaudo, Telek: Fluid models 62 3.2 Transient behaviour of fluid models Second order, infinite buffer, homogeneous Markov fluid models with absorbing barrier Between the boundaries the evolution of the process is the same as before. First time when the fluid level decreases to zero the fluid process stops, − → Pr ( X ( t ) = 0) > 0. Due to the absorbing property of the boundary the probability that the fluid level is close to it is very low, P r (0 <X ( t ) < ∆) − → lim ∆ → 0 = 0. ∆
Gribaudo, Telek: Fluid models 63 3.2 Transient behaviour of fluid models Inhomogeneous (fluid level dependent), first order, infinite buffer Markov fluid models The evolution of the fluid level is the same: r S ( t ) ( X ( t )) , if X ( t ) > 0 , d dtX ( t ) = max( r S ( t ) ( X ( t )) , 0) , if X ( t ) = 0 . But the evolution of the CTMC depends on the fluid level: Pr ( S ( t + ∆) = j | S ( t ) = i ) lim = q ij ( X ( t )) . ∆ ∆ → 0 The generator of the CTMC is Q ( X ( t )) and the rate matrix is R ( X ( t )).
Gribaudo, Telek: Fluid models 64 3.3 Transient description of fluid models Notations: π i ( t ) = Pr ( S ( t ) = i ) – state probability, u i ( t ) = Pr ( X ( t ) = B, S ( t ) = i ) – buffer full probability, ℓ i ( t ) = Pr ( X ( t ) = 0 , S ( t ) = i ) – buffer empty probability, 1 p i ( t, x ) = lim ∆ Pr ( x < X ( t ) < x + ∆ , S ( t ) = i ) ∆ → 0 – fluid density. � = ⇒ π i ( t ) = ℓ i ( t ) + u i ( t ) + x p i ( t, x ) dx .
Gribaudo, Telek: Fluid models 65 3.3 Transient description of fluid models First order, infinite buffer, homogeneous behaviour. Forward argument: If S ( t + δ ) = i , then between t and t + ∆ the CTMC • stays in i with probability 1 + q ii ∆, • moves from k to i with probability q ki ∆, • has more than 1 state transition with probability σ (∆).
Gribaudo, Telek: Fluid models 66 3.3 Transient description of fluid models Fluid density: (1 + q ii ∆) p i ( t, x − r i ∆)+ p i ( t + ∆ , x ) = � q ki ∆ p k ( t, x − O (∆))+ k ∈S ,k � = i σ (∆) , where lim ∆ → 0 σ (∆) / ∆ = 0 and lim ∆ → 0 O (∆) = 0.
Gribaudo, Telek: Fluid models 67 3.3 Transient description of fluid models p i ( t + ∆ , x ) − p i ( t, x − r i ∆) = � q ki ∆ p k ( t, x − O (∆)) + σ (∆) , k ∈S p i ( t + ∆ , x ) − p i ( t, x ) + r i p i ( t, x ) − p i ( t, x − r i ∆) = ∆ r i ∆ q ki p k ( t, x − O (∆)) + σ (∆) � , ∆ k ∈S ∂tp i ( t, x ) + r i ∂ ∂ � ∂xp i ( t, x ) = q ki p k ( t, x ) . k ∈S
Gribaudo, Telek: Fluid models 68 3.3 Transient description of fluid models Empty buffer probability: If r i > 0, − → the fluid level increases in state i , − → ℓ i ( t ) = Pr ( X ( t ) = 0 , S ( t ) = i ) = 0.
Gribaudo, Telek: Fluid models 69 3.3 Transient description of fluid models If r i ≤ 0: ℓ i ( t + ∆) = � − r i ∆ (1 + q ii ∆) ℓ i ( t ) + p i ( t, x ) dx + 0 � �� � ∗ � O (∆) � q ki ∆ ℓ k ( t ) + p k ( t, x ) dx + 0 k ∈S ,k � = i � �� � O (∆) σ (∆) .
Gribaudo, Telek: Fluid models 70 3.3 Transient description of fluid models When x ≤ − r i ∆, then p i ( t, x ) = p i ( t, 0) + xp ′ i ( t, 0) + σ (∆) , and � − r i ∆ ∗ = p i ( t, x ) dx 0 � − r i ∆ � − r i ∆ � − r i ∆ xp ′ = p i ( t, 0) dx + i ( t, 0) dx + σ (∆) dx 0 0 0 + ( − r i ∆) 2 p ′ = − r i ∆ p i ( t, 0) + ( − r i ∆) σ (∆) i ( t, 0) . 2 � �� � � �� � σ (∆) σ (∆)
Gribaudo, Telek: Fluid models 71 3.3 Transient description of fluid models From which the empty buffer probability: � � ℓ i ( t + ∆) = (1 + q ii ∆) ℓ i ( t ) − r i ∆ p i ( t, 0) + σ (∆) + � q ki ∆ ( ℓ k ( t ) + O (∆)) + σ (∆) , k ∈S ,k � = i ℓ i ( t + ∆) − ℓ i ( t ) = q ii ∆ ℓ i ( t ) − r i ∆ p i ( t, 0)+ � q ki ∆ ( ℓ k ( t ) + O (∆)) + σ (∆) , k ∈S ,k � = i
Gribaudo, Telek: Fluid models 72 3.3 Transient description of fluid models and ℓ i ( t + ∆) − ℓ i ( t ) = ∆ q ki ( ℓ k ( t ) + O (∆)) + σ (∆) � − r i p i ( t, 0) + , ∆ k ∈S d � − r i p i ( t, 0) + dtℓ i ( t ) = q ki ℓ k ( t ) . k ∈S
Gribaudo, Telek: Fluid models 73 3.3 Transient description of fluid models Set of governing equations: Fluid density: ∂tp i ( t, x ) + r i ∂ ∂ � ∂xp i ( t, x ) = q ki p k ( t, x ) , k ∈S Empty buffer probability: if r i < = 0: d � − r i p i ( t, 0) + dtℓ i ( t ) = q ki ℓ k ( t ) , k ∈S if r i > 0: ℓ i ( t ) = 0 .
Gribaudo, Telek: Fluid models 74 3.3 Transient description of fluid models By the definition of fluid density and empty buffer probability: � ∞ p i ( t, x ) dx + ℓ i ( t ) = π i ( t ) . 0 In the homogeneous case: d � π i ( t ) = π i (0) e Qt . dtπ i ( t ) = q ki π k ( t ) , − → k ∈S
Gribaudo, Telek: Fluid models 75 3.3 Transient description of fluid models First order, finite buffer , homogeneous behaviour. If there is also an upper boundary: if r i < 0: u i ( t ) = 0 , if r i ≥ 0: d � dtu i ( t ) = r i p i ( t, B ) + q ki u k ( t ) . k ∈S
Gribaudo, Telek: Fluid models 76 3.3 Transient description of fluid models Second order , infinite buffer, homogeneous behaviour. Fluid density: p i ( t + ∆ , x ) = � ∞ p i ( t, x − u ) f N (∆ r i , ∆ σ 2 (1 + q ii ∆) i ) ( u ) du + −∞ � �� � ∗∗ � q ki ∆ p k ( t, x − O (∆))+ k ∈S ,k � = i σ (∆)
Gribaudo, Telek: Fluid models 77 3.3 Transient description of fluid models Using i ( t, x ) + u 2 p i ( t, x − u ) = p i ( t, x ) − up ′ 2 p ′′ i ( t, x ) + O ( u ) 3 we have: ∗∗ = � ∞ � ∞ − p ′ p i ( t, x ) i ) ( u ) du i ( t, x ) i ) ( u ) du + f N (∆ r i , ∆ σ 2 uf N (∆ r i , ∆ σ 2 −∞ −∞ � �� � � �� � 1 ∆ r i � ∞ � ∞ u 2 O ( u ) 3 f N (∆ r i , ∆ σ 2 p ′′ i ( t, x ) i ) ( u ) du + i ) ( u ) du 2 f N (∆ r i , ∆ σ 2 . −∞ −∞ � �� � � �� � ∆ 2 r 2 i +∆ σ 2 i / 2=∆ σ 2 O (∆) 2 = σ (∆) i / 2+ σ (∆)
Gribaudo, Telek: Fluid models 78 3.3 Transient description of fluid models From which: p i ( t + ∆ , x ) = � � p i ( t, x ) − p ′ i ( t, x )∆ r i + p ′′ i ( t, x )∆ σ 2 (1 + q ii ∆) i / 2 + � q ki ∆ p k ( t, x − O (∆)) + σ (∆) , k ∈S ,k � = i p i ( t + ∆ , x ) − p i ( t, x ) = q ii ∆ p i ( t, x ) − p ′ i ( t, x )∆ r i + p ′′ i ( t, x )∆ σ 2 i / 2+ � q ki ∆ p k ( t, x − O (∆)) + σ (∆) , k ∈S ,k � = i ∂xp i ( t, x ) r i − ∂ 2 ∂x 2 p i ( t, x ) σ 2 ∂tp i ( t, x ) + ∂ ∂ � i = q ki p k ( t, x ) . 2 k ∈S
Gribaudo, Telek: Fluid models 79 3.3 Transient description of fluid models Second order , infinite buffer, reflecting barrier , homogeneous behaviour. Boundary condition: Reflecting barrier − → ℓ i ( t ) = 0. Fluid density at 0: � � ∞ ∂ p i ( t, x ) dx = π i ( t ) ∂t 0
Gribaudo, Telek: Fluid models 80 3.3 Transient description of fluid models � ∞ ∂ ∂ ∂tp i ( t, x ) dx = ∂tπ i ( t ) x =0 � �� � � �� � � �� � � �� � � r i + ∂ 2 p i ( t, x ) σ 2 − ∂p i ( t, x ) � q ki π i ( t ) i 2 + q ki p k ( t, x ) ∂x 2 ∂x k ∈S k ∈S ∞ ∞ � ∞ + σ 2 � � i p ′ − r i p i ( t, x ) i ( t, x ) + p k ( t, x ) dx = q ki π i ( t ) q ki 2 x =0 k ∈S k ∈S x =0 x =0 � �� � � �� � � �� � πi ( t ) − p ′ − pi ( t, 0) i ( t, 0) r i p i ( t, 0) − σ 2 2 p ′ i i ( t, 0) = 0
Gribaudo, Telek: Fluid models 81 3.3 Transient description of fluid models First order, infinite buffer, inhomogeneous behaviour . Fluid density: ∂tp i ( t, x ) + r i ( x ) ∂ ∂ � ∂xp i ( t, x ) = q ki ( x ) p k ( t, x ) k ∈S Empty buffer probability: if r i (0) < 0 (and r i ( x ) is continuous): d � dtℓ i ( t ) = − r i (0) p i ( t, 0) + q ki (0) ℓ k ( t ) , k ∈S if r i (0) > 0 (and r i ( x ) is continuous): ℓ i ( t ) = 0 .
Gribaudo, Telek: Fluid models 82 3.3 Transient description of fluid models General case: Second order , finite buffer , inhomogeneous behaviour . Differential equations: R ( x ) − ∂ 2 p ( t, x ) ∂p ( t, x ) + ∂p ( t, x ) S ( x ) = p ( t, x ) Q ( x ) , ∂x 2 ∂t ∂x p ( t, 0) R (0) − p ′ ( t, 0) S (0) = ℓ ( t ) Q (0) , − p ( t, B ) R ( B ) + p ′ ( t, B ) S ( B ) = u ( t ) Q ( B ) , σ 2 i ( x ) where R ( x ) = Diag � r i ( x ) � and S ( x ) = Diag � � . 2
Gribaudo, Telek: Fluid models 83 3.3 Transient description of fluid models General case: Second order , finite buffer , inhomogeneous behaviour . Bounding behaviour: σ i = 0 and positive/negative drift: ℓ i ( t )=0/ u i ( t )=0. σ i > 0 , reflecting lower/upper barrier: ℓ i ( t ) = 0/ u i ( t ) = 0. σ i > 0 , absor. lower/upper barrier: p i ( t, 0)=0/ p i ( t, B )=0. Normalizing condition: � B p ( t, x ) dx 1 I + ℓ ( t )1 I + u ( t )1 I = 1 . 0
Gribaudo, Telek: Fluid models 84 3.4 Stationary description of fluid models Condition of ergodicity: For ∀ x, y ∈ R + , ∀ i, j ∈ S the transition time t> 0 ( X ( t ) = y, S ( t ) = j | X (0) = x, S (0) = i ) T = min has a finite mean (i.e., E ( T ) < ∞ ).
Gribaudo, Telek: Fluid models 85 3.4 Stationary description of fluid models Notations: π i = lim t →∞ Pr ( S ( t ) = i ) – state probability, u i = lim t →∞ Pr ( X ( t ) = B, S ( t ) = i ) – buffer full probability, ℓ i = lim t →∞ Pr ( X ( t ) = 0 , S ( t ) = i ) – buffer empty probability, 1 p i ( x ) = lim t →∞ lim ∆ Pr ( x < X ( t ) < x + ∆ , S ( t ) = i ) ∆ → 0 – fluid density, F i ( x ) = lim t →∞ Pr ( X ( t ) < x, S ( t ) = i ) – fluid distribution.
Gribaudo, Telek: Fluid models 86 3.4 Stationary description of fluid models First order, infinite buffer, homogeneous behaviour. Fluid density: r i ∂ � ∂xp i ( x ) = q ki p k ( x ) . k ∈S Empty buffer probability: if r i < = 0: � 0 = − r i p i (0) + q ki ℓ k , k ∈S if r i > 0: ℓ i = 0 .
Gribaudo, Telek: Fluid models 87 3.4 Stationary description of fluid models First order, finite buffer , homogeneous behaviour. Fluid density: r i ∂ � ∂xp i ( x ) = q ki p k ( x ) . k ∈S Boundary equations: � if r i ≤ 0 , r i p i (0) = q ki ℓ k , k ∈S ℓ i = 0 , if r i > 0 . � − r i p i ( B ) = if r i ≥ 0 , q ki u k , k ∈S u i = 0 , if r i < 0 .
Gribaudo, Telek: Fluid models 88 3.4 Stationary description of fluid models Second order , infinite buffer, reflecting boundary , homogeneous behaviour. Fluid density: ∂xp i ( x ) − ∂ 2 ∂x 2 p i ( x ) σ 2 r i ∂ � i 2 = q ki p k ( x ) . k ∈S Empty buffer probability: ℓ i = 0 . Boundary equation: r i p i (0) − σ 2 � 2 p ′ i i (0) = q ki ℓ k = 0 . k ∈S
Gribaudo, Telek: Fluid models 89 3.4 Stationary description of fluid models Second order , infinite buffer, absorbing boundary , homogeneous behaviour. Fluid density: ∂xp i ( x ) − ∂ 2 ∂x 2 p i ( x ) σ 2 r i ∂ � i 2 = q ki p k ( x ) . k ∈S Empty buffer probability: p i (0) = 0 . Boundary equation: − σ 2 � 2 p ′ i i (0) = q ki ℓ k . k ∈S
Gribaudo, Telek: Fluid models 90 3.4 Stationary description of fluid models General case: Second order , finite buffer , inhomogeneous behaviour . p ′ ( x ) R ( x ) − p ′′ ( x ) S ( x ) = p ( x ) Q ( x ) , p (0) R (0) − p ′ (0) S (0) = ℓ Q (0) , − p ( B ) R ( B ) + p ′ ( B ) S ( B ) = u Q ( B ) , σ i =0 and positive/negative drift: ℓ i = 0/ u i = 0. σ i > 0, reflecting lower/upper barrier: ℓ i = 0/ u i = 0. σ i > 0, absorbing lower/upper barrier: p i (0) = 0/ p i ( B ) = 0.
Gribaudo, Telek: Fluid models 91 4 Solution methods Numerical techniques: reward fluid differential equations (+) + spectral decomposition (+) + randomization + + transform domain + + matrix exponent + + moments + -
Gribaudo, Telek: Fluid models 92 4 Solution methods Transient analysis: • initial condition , • set of differential equations, • bounding behaviour. Stationary analysis: • set of differential equations, • bounding behaviour, • normalizing condition .
Gribaudo, Telek: Fluid models 93 4.1 Transient solution methods • Numerical solution of differential equations, • Randomization, • Markov regenerative approach, • Transform domain.
Gribaudo, Telek: Fluid models 94 4.1 Transient solution methods Numerical solution of differential equations (Chen et al.) All cases. The approach • starts from the initial condition, and • follows the evolution of the fluid distribution in the ( t, t + ∆) interval at some fluid levels based on the differential equations and the boundary condition. This is the only approach for inhomogeneous models.
Gribaudo, Telek: Fluid models 95 4.1 Transient solution methods Randomization (Sericola) First order, infinite buffer, homogeneous behaviour. � � ∞ n e − λt ( λt ) n n � � j (1 − x j ) n − k b ( j ) F c x k i ( t, x ) = i ( n, k ) , n ! k n =0 k =0 where F c i ( t, x ) = Pr ( X ( t ) > x, S ( t ) = i ), x − r + j − 1 t j − 1 t if x ∈ [ r + x j = j − 1 t, r j t ), and r j t − r + b ( j ) i ( n, k ) is defined by initial value and a simple recursion.
Gribaudo, Telek: Fluid models 96 4.1 Transient solution methods Properties of the randomization based solution method: • the expression with the given recursive formulas is a solution of the differential equation, the initial value of b ( j ) i ( n, k ) is set to fulfill the boundary condition, • 0 ≤ x j ≤ 1 − → convex combination of non-negative numbers − → numerical stability, • the initial fluid level is X (0) = 0. (extension to X (0) > 0 and to finite buffer is not available.)
Gribaudo, Telek: Fluid models 97 4.1 Transient solution methods First order, infinite buffer, homogeneous case. Markov regenerative approach (Ahn-Ramaswami) Busy/idle period: interval when the buffer is non-empty/empty. T i : the beginning of the i th busy period. ⇒ ( S ( t i ) , T i ) is a Markov renewal sequence. = The idle period is PH distributed. Analysis of a single busy period: similar analysis as in Matrix geometric models.
Gribaudo, Telek: Fluid models 98 4.1 Transient solution methods First order, infinite/finite buffer, homogeneous case. Transform domain description (Ren-Kobayashi) The Laplace transform of R − ∂ 2 p ( t, x ) ∂p ( t, x ) + ∂p ( t, x ) = p ( t, x ) Q , S ∂x 2 ∂t ∂x is p ∗ (0 , v ) + p ∗ ( s, 0) p ∗∗ ( s, v ) = ( R )( s I + v R − Q ) − 1 . � �� � � �� � initial condition unknown p ∗ ( s, 0) eliminates the roots of det( s I + v R − Q ).
Gribaudo, Telek: Fluid models 99 4.2 Stationary solution methods Condition of stability of infinite buffer first/second order homogeneous fluid models. Suppose S ( t ) is a finite state irreducible CTMC with stationary distribution π . The fluid model is stable if the overall drift is negative: � d = π i r i < 0 . i ∈S − → the variance does not play role.
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