Recall Jackson s theorem. Performance Analysis. PEPA and Product Form. PEPA and RCAT

Transcription

1 Recall Jackson s theorem Performance Analysis Peter Harrison, Maria Vigliotti and Jeremy Bradley Room jb@doc.ic.ac.uk Department of Computing, Imperial College ondon Produced with prosper and AT E X For a steady-state probability π(r 1,...,r N ) of there being r 1 jobs in node 1, r 2 nodes at node 2, etc.: π(r 1,r 2,...,r N ) = = N (1 ρ i )ρ r i i i=1 N π i (r i ) i=1 where π i (r i ) is the steady-state probability there being n i jobs at node i independently 336 JTB [02/2007] p. 1/ JTB [02/2007] p. 2/26 PEPA and Product Form A product form result links the overall steady-state of a system to the product of the steady state for the components of that system e.g. Jackson s theorem In PEPA, a simple product form can be got from: P 1 P 2 P n n r1 i=1 π(p1 ) π(p r n n ) π(p r1 r2 1,P2,...,P r n n ) = 1 G where π(p ri i ) is steady state prob. that component P i is in state r i PEPA and RCAT RCAT: Reversed Compound Agent Theorem RCAT can take the more general cooperation: P Q...and find a product form, given structural conditions, in terms of the individual components P and Q 336 JTB [02/2007] p. 3/ JTB [02/2007] p. 4/26

2 What does RCAT do? RCAT expresses the reversed component P Q in terms of P and Q (almost) This is powerful since it avoids the need to expand the state space of P Q This is useful since from the forward and reversed processes, P Q and P Q, we can find the steady state distribution π(p i,q i ) π(p i,q i ) is the steady state distribution of both the forward and reversed processes (by inition) Recall: Reversed processes The reversed process of a stationary Markov process {X t : t 0} with state space S, generator matrix Q and stationary probabilities π is a stationary Markov process with generator matrix Q ined by: q ij = π jq ji π i : i,j S and with the same stationary probabilities π. 336 JTB [02/2007] p. 5/ JTB [02/2007] p. 6/26 Kolmogorov s Generalised Criteria A stationary Markov process with state space S and generator matrix Q has reversed process with generator matrix Q if and only if: 1. q i = q i for every state i S 2. For every finite sequence of states i 1,i 2,...,i n S, q i1 i 2 q i2 i 3...q in 1 i n q in i 1 = q i 1 i n q i n i n 1...q i 3 i 2 q i 2 i 1 where q i = q ii = j : j i q ij Finding π from the reversed process Once reversed process rates Q have been found, can be used to extract π In an irreducible Markov process, choose a reference state 0 arbitrarily Find a sequence of connected states, in either the forward or reversed process, 0,...,j (i.e. with either q i,i+1 > 0 or q i,i+1 > 0 for 0 i j 1) for any state j and calculate: j 1 q i,i+1 j 1 π j = π 0 q i+1,i = π 0 i=0 q i,i+1 q i=0 i+1,i 336 JTB [02/2007] p. 7/ JTB [02/2007] p. 8/26

3 Reversing a sequential component Reversing a sequential component, S, is straightforward: S = (a i,λ i ).R i S 2 S 1 S 4 (a,λ) S 3 (c,λ) i : R i (a i,λ i ) S S 2 S 1 S 4 (a,λ) S 3 (c,λ) Activity substitution We need to be able to substitute a PEPA activity α = (a,r) for another α = (a,r ): { α (β.p){α α.(p {α α }) : if α = β } = β.(p {α α }) : otherwise (P + Q){α α } = P {α α } + Q{α α } (P Q){α α } = P {α α } Q{α α } {α α } where {(a,λ) (a,λ )} = ( \ {a}) {a } if a and otherwise A set of substitutions can be applied with: P {α α,β β } 336 JTB [02/2007] p. 9/ JTB [02/2007] p. 10/26 RCAT Conditions (Informal) For a cooperation P Q, the reversed process P Q can be created if: 1. Every passive action in P or Q that is involved in the cooperation must always be enabled in P or Q respectively. 2. Every reversed action a in P or Q, where a is active in the original cooperation, must: (a) always be enabled in P or Q respectively (b) have the same rate throughout P or Q respectively 336 JTB [02/2007] p. 11/26 RCAT Notation In the cooperation, P Q: A P () is the set of actions in that are also active in the component P A Q () is the set of actions in that are also active in the component Q P P () is the set of actions in that are also passive in the component P P Q () is the set of actions in that are also passive in the component Q is the reversed set of actions in, that is = {a a } 336 JTB [02/2007] p. 12/26

4 RCAT Conditions (Formal) For a cooperation P Q, the reversed process P Q can be created if: 1. Every passive action type in P P () or P Q () is always enabled in P or Q respectively (i.e. enabled in all states of the transition graph) 2. Every reversed action of an active action type in A P () or A Q () is always enabled in P or Q respectively 3. Every occurrence of a reversed action of an active action type in A P () or A Q () has the same rate in P or Q respectively RCAT (I) For P Q, the reversed process is: where: R S P Q = R S = R{(a,p a ) (a, ) a A P ()} = S{(a,q a ) (a, ) a A Q ()} R = P {(a, ) (a,x a ) a P P ()} S = Q{(a, ) (a,x a ) a P Q ()} where the reversed rates, p a and q a, of reversed actions are solutions of Kolmogorov equations. 336 JTB [02/2007] p. 13/ JTB [02/2007] p. 14/26 RCAT (II) x a are solutions to the linear equations: { q x a = a : if a P P () : if a P Q () p a and p a, q a are the symbolic rates of action types a in P and Q respectively. To obtain P RCAT in words Q = R S : 1. substitute all the cooperating passive rates in P, Q with symbolic rates, x action, to get R, S 2. reverse R and S, to get R and S 3. solve non-linear equations to get reversed rates, {r} in terms of forward rates {r} 4. solve non-linear equations to get symbolic rates {x action } in terms of forward rates 5. substitute all the cooperating active rates in R, S with to get R, S 336 JTB [02/2007] p. 15/ JTB [02/2007] p. 16/26

5 Example: Tandem queues (I) Example: Tandem queues (II) γ P Q γ P Q Jobs arrive to node P with activity (e,γ) Jobs are serviced at node P with rate µ 1 Jobs move between node P and Q with action a Jobs are serviced at node Q with rate µ 2 Jobs depart Q with action d PEPA description, P 0 Q 0, where: {a} P 0 P n Q 0 Q n = (e,γ).p 1 = (e,γ).p n+1 + (a,µ 1 ).P n 1 : n > 0 = (a, ).Q 1 = (a, ).Q n+1 + (d,µ 2 ).Q n 1 : n > JTB [02/2007] p. 17/ JTB [02/2007] p. 18/26 Example: Tandem queues (III) Replace passive rates in cooperation with variables: R = P {(a, ) (a,x a ) a P P ()} S = Q{(a, ) (a,x a ) a P Q ()} Transformed PEPA model: R 0 R n S 0 S n = (e,γ).r 1 = (e,γ).r n+1 + (a,µ 1 ).R n 1 : n > 0 = (a,x a ).S 1 = (a,x a ).S n+1 + (d,µ 2 ).S n 1 : n > 0 Example: Tandem queues (IV) Reverse components R and S to get: R 0 R n S 0 S n = (a,µ 1 ).R 1 = (a,µ 1 ).R n+1 + (e,γ).r n 1 : n > 0 = (d,µ 2 ).S 1 = (d,µ 2 ).S n+1 + (a,x a ).S n 1 : n > 0 Now need to find in this order: 1. reverse rates in terms of forward rates 2. variable x a in terms of forward rates 336 JTB [02/2007] p. 19/ JTB [02/2007] p. 20/26

6 Example: Tandem queues (V) Finding reverse rates using Kolmogorov Compare forward/reverse leaving rate from states R 0, S 0 : exit_rate(r 0 ) = exit_rate(r 0 ) : exit_rate(s 0 ) = exit_rate(s 0 ) : µ 1 = γ Compare rate cycles in R, R and S, S: R 0 R 1 R 0 : S 0 S 1 S 0 : γµ 1 = µ 1 γ x a µ 2 = µ 2 x a µ 2 = x a Example: Tandem queues (VI) Finding symbolic rates recall: { q x a = a : if a P P () : if a P Q () p a In this case, a P Q (), so x a = p a = reversed rate of a-action in R Thus x a = µ 1 = γ This agrees with rate of customers leaving forward network why? Giving: γ = µ 1 and x a = µ JTB [02/2007] p. 21/ JTB [02/2007] p. 22/26 Example: Tandem queues (VII) Constructing P Q P 0 {a} R 0 R n S 0 S n Q 0 = R0 S0 where: {a} = (a, ).R1 = (a, ).Rn+1 + (e,µ 1 ).Rn 1 : n > 0 = (d,γ).s1 = (d,γ).sn+1 + (a,µ 2).Sn 1 : n > 0 Example: Tandem queues (VIII) Finding the steady state distribution: Need to use the following formula: j 1 q i,i+1 π j = π 0 q i+1,i i=0...to find the steady state distribution First need to construct a sequence of events to a generic state (n,m) in network where (n,m) represents n jobs in node P and m in node Q 336 JTB [02/2007] p. 23/ JTB [02/2007] p. 24/26

7 Example: Tandem queues (IX) Generic state can be reached by: 1. n + m arrivals or e-actions to node P (forward rate = γ, reverse rate = µ 1 ) 2. followed by m departures or a-actions from node P and arrivals to node Q (forward rate = µ 1, reverse rate = µ 2 ) n+m 1 Thus: π(n,m) = π 0 i=0 m 1 γ µ 1 µ 1 µ i=0 2 = π 0 ( γ µ 1 ) n ( γ µ 2 ) m References RCAT Turning back time in Markovian Process Algebra. Peter Harrison. TCS 290(3), pp January Generalised RCAT: less strict structural conditions Reversed processes, product forms and a non-product form. Peter Harrison. AA 386, pp July MARCAT: N-way cooperation extension: Separable equilibrium state probabilities via time-reversal in Markovian process algebra. Peter Harrison and Ting-Ting ee. TCS, pp November JTB [02/2007] p. 25/ JTB [02/2007] p. 26/26