1. On the generality of binary tree-like Markov 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.

2. MAM2006On the generality of binary tree-like Markov chains 2 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

3. MAM2006On the generality of binary tree-like Markov chains 3 Tree-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, D k, U k, k = 1,...,d.

4. MAM2006On the generality of binary tree-like Markov chains 4 Tree-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 

5. MAM2006On the generality of binary tree-like Markov chains 5 Tree-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

6. MAM2006On the generality of binary tree-like Markov chains 6 Constructing the binary tree-like MC Tree-like MC (X t,N t )  X t : nodes d-ary tree  N t : auxiliary variable Nodes are denoted by strings (symbols 1,...,d) J = j 1 j 2... j n-1 j n 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

7. MAM2006On the generality of binary tree-like Markov chains 7 Constructing 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 (X t,N t ) and states (ψ(J),(0,i)) of Every possible transition in (X t,N t ) 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’))... ø  00   00 00 0  000  00 0000  0  00  0  0  ø 1 2 3 4 31 22 211 21 11 12 13121 111 1111 112

8. MAM2006On the generality of binary tree-like Markov chains 8 Constructing the binary tree-like MC d-ary tree: transition from a node to its k-th child  (J,i)  (J+k,j) with prob. (U k ) i,j binary tree:  (ψ(J),(0,i))  (ψ(J) ,(k-1,j)) with prob. (U  ) (0,i),(k-1,j) = (U k ) i,j   (ψ(J)  0,(k-2,j)) with prob. (U 0 ) (k-1,j),(k-2,j) = 1  ...  (ψ(J)  0...0,(1,j)) with prob. (U 0 ) (k-2,j),(k-3,j)... (U 0 ) (2,j),(1,j) = 1   (ψ(J)  0...00,(0,j)) = (ψ(J+k),(0,j)) with prob. (U 0 ) (1,j),(0,j) = 1... ø  00   00 00 0  000  00 0000  0  00  0  0  ø 1 2 3 4 31 22 211 21 11 12 13121 111 1111 112

9. MAM2006On the generality of binary tree-like Markov chains 9 Constructing the binary tree-like MC d-ary tree: transition from a child k to its parent  (J+k,i)  (J,j) with prob. (D k ) i,j binary tree:  (ψ(J+k),(0,i)) = (ψ(J)  0...00,(0,i))  (ψ(J)  0...0,(-1,i)) with prob. (D 0 ) (0,i),(-1,i) =  i,j,  = diag(D 1 e)  ...  (ψ(J) ,(-(k-1),i)) with prob. (D 0 ) (-1,i),(-2,i)... (D 0 ) (-(k-2),i),(-(k-1),i) = 1   (ψ(J),(0,j)) with prob. (D  ) (-(k-1),i),(0,j) = (  -1 D k ) i,j... ø  00   00 00 0  000  00 0000  0  00  0  0  ø 1 2 3 4 31 22 211 21 11 12 13121 111 1111 112

10. MAM2006On the generality of binary tree-like Markov chains 10 Constructing the binary tree-like MC d-ary tree: transition from a node to itself  root node: (ø,i)  (ø,j) with prob. F i,j  other node: (J,i)  (J,j) with prob. B i,j binary tree:  root node: (ø,(0,i))  (ø,(0,j)) with prob.  other node: (J,(0,i))  (J,(0,j)) with prob.

11. MAM2006On the generality of binary tree-like Markov chains 11 Calculating 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 All G k stochastic  G 0 and G  stochastic Algorithms for calculating the steady state probabilities:  Fixed point iteration (FPI)  Reduction to quadratic equations (RQE)  Newton’s iteration (NI)

12. MAM2006On the generality of binary tree-like Markov chains 12 Calculating the steady state probabilities The matrices corresponding to the constructed binary tree-like MC, e.g.,  U 0, U , D 0, D ,  have a structure that is related to the matrices that correspond to the original d-ary tree-like MC Example (d=4)

13. MAM2006On the generality of binary tree-like Markov chains 13 Calculating 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

14. MAM2006On the generality of binary tree-like Markov chains 14 Calculating the steady state probabilities Reduction to quadratic equations (RQE)  iterative algorithm: G i [0] = 0, i = 1,...,d d quadratic matrix equations  solve for G i [N+1] G i [N] converges to G i, i = 1,...,d Applied to binary tree: G 0 [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

15. MAM2006On the generality of binary tree-like Markov chains 15 Calculating the steady state probabilities Newton’s iteration (NI)  iterative algorithm  computes the matrices G i, 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 ???

16. MAM2006On the generality of binary tree-like Markov chains 16Conclusions 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