On the generality of binary tree-like Markov chains K. Spaey - B. Van Houdt - C. Blondia Performance...
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Transcript of On the generality of binary tree-like Markov chains K. Spaey - B. Van Houdt - C. Blondia Performance...
On the generality of binary tree-like On the generality of binary tree-like Markov chainsMarkov chains
K. Spaey - B. Van Houdt - C. Blondia
Performance Analysis of Telecommunication Systems (PATS) Research Group
University of Antwerp - IBBT
MAM2006 - June 12-14, 2006 - Charleston, S.C.
MAM2006MAM2006 On the generality of binary tree-like On the generality of binary tree-like Markov chainsMarkov chains
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On the generality of binary tree-like On the generality of binary tree-like Markov chainsMarkov chains
Aim of the paper:
Show that an arbitrary tree-like Markov chain can be embedded in a binary tree-like Markov chain with a special structure
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Tree-like QBD Markov chainsTree-like QBD Markov chains
• States are grouped into sets of m states: “nodes”
• The nodes form a d-ary tree
• Transitions: from a node to itself, to its parent node, to its child nodes
• Characterized by the matrices B and F, Dk, Uk, k = 1,...,d.
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Tree-like QBD Markov chainsTree-like QBD Markov chains
• Key equations:
for k = 1,...,d
• Stability condition:
needs to be stochastic for all k
• Steady state probabilities:
for all J, for all k
d
1kk
1k DV)(IUBV
1kk V)(IUR
k1
k DV)(IG
kR (J)k)(J ππ
)DR(F )()(d
1kkk
ππ
1 )eR)(I(d
1kk
π
MAM2006MAM2006 On the generality of binary tree-like On the generality of binary tree-like Markov chainsMarkov chains
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Tree-like QBD Markov chainsTree-like QBD Markov chains
• Tree-like Markov chains were introduced as a special case of the tree-structured Markov chains (Bini, Latouche & Meini, Solving nonlinear matrix equations arising in tree-like stochastic processes, Linear Algebra Appl. 366, 2003)
• Any tree-structured Markov chain can be reduced to a tree-like Markov chain(Van Houdt & Blondia, Tree structured QBD Markov chains and tree-like QBD processes, Stochastic Models 19(4), 2003)
• Any tree-like Markov chain can be embedded in a binary (d=2) tree-like Markov chain
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Constructing the binary tree-like MCConstructing the binary tree-like MC
• Tree-like MC (Xt,Nt) Xt: nodes d-ary tree
Nt: auxiliary variable
• Nodes are denoted by strings (symbols 1,...,d) J = j1 j2 ... jn-1 jn
Root node: ø
• Auxiliary variable i = 1,...,m
• Binary tree-like MC : nodes binary tree : 2D auxiliary variable
• Nodes are denoted by binary strings (symbols 0, ) starting with a “”
Root node: ø
• Auxiliary variable (0,i) corresponding to node ø (a,i) for other nodes
i = 1,...,m a = -(d-1),...,-1,0,1,...,d-1
))M,Q(N,X( ttttˆˆˆˆ
tX̂)M,Q(N ttt
ˆˆˆ
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Constructing the binary tree-like MCConstructing the binary tree-like MC
• Binary notation ψ of the nodes of the d-ary tree: and ψ(ø) = ø
• 1-1 correspondence between states (J,i) of (Xt,Nt) and states (ψ(J),(0,i)) of
• Every possible transition in (Xt,Nt) between (J,i) and (J’,i’) will be mimicked by a path of transitions in between (ψ(J),(0,i)) and (ψ(J’),(0,i’))
1j1j1j1j
n1-n21
n1-n1
00000000 )jjjψ(j
2
)N,X( ttˆˆ
...
ø
0
00 0 0
000 00 00 0 00 0 0
ø
1
2
3
4 31 22 211
21
11
12
13 121
111
1111112
)N,X( ttˆˆ
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Constructing the binary tree-like MCConstructing the binary tree-like MC
• d-ary tree: transition from a node to its k-th child (J,i) (J+k,j) with prob. (Uk)i,j
• binary tree: (ψ(J),(0,i)) (ψ(J),(k-1,j)) with prob. (U)(0,i),(k-1,j) = (Uk)i,j
(ψ(J)0,(k-2,j)) with prob. (U0)(k-1,j),(k-2,j) = 1
... (ψ(J)0...0,(1,j)) with prob. (U0)(k-2,j),(k-3,j) ... (U0)(2,j),(1,j) = 1
(ψ(J)0...00,(0,j)) = (ψ(J+k),(0,j)) with prob. (U0)(1,j),(0,j) = 1
...
ø
0
00 0 0
000 00 00 0 00 0 0
ø
1
2
3
4 31 22 211
21
11
12
13 121
111
1111112
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Constructing the binary tree-like MCConstructing the binary tree-like MC
• d-ary tree: transition from a child k to its parent (J+k,i) (J,j) with prob. (Dk)i,j
• binary tree: (ψ(J+k),(0,i)) = (ψ(J)0...00,(0,i)) (ψ(J)0...0,(-1,i))
with prob. (D0)(0,i),(-1,i) = i,j, = diag(D1e)
... (ψ(J),(-(k-1),i)) with prob. (D0)(-1,i),(-2,i) ... (D0)(-(k-2),i),(-(k-1),i) = 1
(ψ(J),(0,j)) with prob. (D)(-(k-1),i),(0,j) = (-1Dk)i,j
...
ø
0
00 0 0
000 00 00 0 00 0 0
ø
1
2
3
4 31 22 211
21
11
12
13 121
111
1111112
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Constructing the binary tree-like MCConstructing the binary tree-like MC
• d-ary tree: transition from a node to itself root node: (ø,i) (ø,j) with prob. Fi,j
other node: (J,i) (J,j) with prob. Bi,j
• binary tree: root node: (ø,(0,i)) (ø,(0,j)) with prob. other node: (J,(0,i)) (J,(0,j)) with prob.
ji,j)(0,i),(0, FF ˆ
ji,j)(0,i),(0, BB ˆ
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Calculating the steady state probabilitiesCalculating the steady state probabilities
d-ary tree like MC
•
• for k = 1,...,d
• stability condition: needs to be stochastic for all k
binary tree-like MC
•
•
• stability condition:and need to be stochastic
d
1kk
1k DV)(IUBV
1kk V)(IUR
D)V(IUD)V(IUBV 1
01
0ˆˆˆˆ
1100 )V(IUR and )V(IUR
ˆˆ
k1
k DV)(IG D)V(IG 01
0 ˆ
D)V(IG 1
ˆ
All Gk stochastic G0 and G stochastic
Algorithms for calculating the steady state probabilities: Fixed point iteration (FPI) Reduction to quadratic equations (RQE) Newton’s iteration (NI)
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Calculating the steady state probabilitiesCalculating the steady state probabilities
• The matrices corresponding to the constructed binary tree-like MC, e.g., U0, U, D0, D,
have a structure that is related to the matrices that correspond to
the original d-ary tree-like MC
• Example (d=4)
F ,B ˆˆ
U)V(IG ,U)V(IG ,)V(IUR ,)V(IUR ,V 1
01
011
00ˆˆˆˆˆ
0000ΔV)-(I00
00000ΔV)-(I0
000000ΔV)-(I
000V000
0000000
0000000
0000000
V
1-
1-
1-
ˆ
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Calculating the steady state probabilitiesCalculating the steady state probabilities
• Fixed point iteration (FPI) iterative algorithm:
V[N] monotonically converges to V
• Applied to binary tree:
more iterations needed taking the structure of into account
identical to applying FPI to d-ary tree
d
1kk
1k DV[N])(IUB1]V[N
BV[0]
D[N])V(IUD[N])V(IUB1][NV
B[0]V1
01
0ˆˆˆˆ
ˆˆ
V̂
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Calculating the steady state probabilitiesCalculating the steady state probabilities
• Reduction to quadratic equations (RQE) iterative algorithm:
• Gi[0] = 0, i = 1,...,d
• d quadratic matrix equations solve for Gi[N+1]
• Gi[N] converges to Gi, i = 1,...,d
• Applied to binary tree:• G0[0] = G[0] = 0
• slower convergence• taking the structure of the matrices into account d quadratic matrix
equations as when applying RQE to d-ary tree
01][NGU1][NG [N]GU1][NGUIBD 2iii
1i
1k
d
1ikkkkki
1][NG for solve 01][NGU1][NG 1][NGUIBD
1][NG for solve 01][NGU1][NG [N]GUIBD2
00
02
0000
ˆ
ˆ
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Calculating the steady state probabilitiesCalculating the steady state probabilities
• Newton’s iteration (NI) iterative algorithm computes the matrices Gi, i=1,...,d
converges quadratically each step requires solving an equation of the form
large linear system of equations Ax=b (inefficient)
• Applied to binary tree: each step requires solving an equation of the form
≈ Sylvester equation linear system of equations
reduction to binary tree can result in computational gain ???
d
1kkk L XK XH
L X K XH K XH 2211ˆˆˆˆˆˆˆˆ
d
1kk
Tk )H(KIA
2T21
T1 HKHK I- A ,bxA ˆˆˆˆˆˆˆˆ
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ConclusionsConclusions
• Any tree-like Markov chain can be embedded in a binary tree-like Markov chain with a special structure
any tree-structured Markov chain can be embedded in a binary tree-like Markov chain with a special structure
• Mainly of theoretical interest: FPI and RQE algorithms applied to binary tree do not speed up
calculations of the steady state probabilities NI algorithm: currently unclear