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Efficient Algorithms for Computation of Equilibrium/Transient Probability Distribution of Finite Markov Chains: Potential Lower Bound on Computational Complexity

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Efficient Algorithms for Computation of Equilibrium/Transient Probability Distribution of Finite Markov Chains: Potential Lower Bound on Computational Complexity EasyChair Preprint № 490 Efficient Algorithms for Computation of Equilibrium/Transient Probability Distribution of Finite Markov Chains: Potential Lower Bound on Computational Complexity Rama Murthy Garimella EasyChair preprints are intended for rapid dissemination of research results and are integrated with the rest of EasyChair. March 8, 2020 EFFICIENT ALGORITHMS FOR COMPUTATION OF EQUILIBRIUM/TRANSIENT PROBABILITY DISTRIBUTION OF ARBITRARY FINITE CONTINUOUS TIME MARKOV CHAINS: Garimella Rama Murthy, Professor, Department of Computer Science & Engineering, Mahindra Ecole Centrale, Hyderabad, India ABSTRACT In this research paper, efficient algorithms for computation of equilibrium as well as transient probability distribution of arbitrary finite state space Continuous / Discrete Time Markov Chains are proposed. The effective idea is to solve a structured systems of linear equations efficiently. It is realized that the approach proposed in this research paper generalizes to infinite state space Continuous Time Markov Chains. The ideas of this research paper could be utilized in solving structured system of linear equations that arise in other applications. 1. Introduction: Markov chains provide interesting stochastic models of natural/artificial phenomena arising in science and engineering. The existence of equilibrium behavior enables computation of equilibrium performance measures. Thus, researchers invested considerable effort in EFFICIENTLY computing the equilibrium probability distribution of Markov chains and thus, the equilibrium performance measures. Traditionally, computation of transient probability distribution of Continuous Time Markov Chains ( CTMCs ) was considered to be a difficult open problem. It requires computation of MATRIX EXPONENTIAL associated with the generator matrix of CTMC. Even in the case of finite state space CTMCs, efficient computation of transient probability distribution was considered to be a difficult open problem. In [ Rama1 ], an interesting approach for recursive computation of transient probability distribution of arbitrary finite state space CTMCs was proposed. In the case of infinite state space, Quasi-birth-and Death (QBD) processes, matrix geometric recursion for transient probability distribution was found in [ZhC]. It is well known that computation of equilibrium distribution of CTMCs reduces to solution of linear system of equations. The approach proposed in [ Rama 2] reduces the computation of transient probability distribution of finite state space CTMCs to solving linear system of equations in the transform ( Laplace Transform ) domain. Thus, an interesting question that remained deals with efficient solution of such STRUCTURED linear system of equations in the Laplace transform domain. In fact, a more interesting problem is to design an algorithm which meets a LOWER BOUND on the computation of solution of structured system of linear equations arising in the transient/equilibrium analysis of CTMCs. This research paper is organized as follows. In Section 2, algorithm for efficient computation of equilibrium probability mass function of Continuous Time Markov Chains (CTMCs ) is discussed. In Section 3, algorithm for efficient computation of transient probability mass Function of Continuous Time Markov Chains is discussed. In section 4, we provide numerical results. The research paper concludes in Section 5. 2. Review of Research Literature: It is well known that computation of equilibrium probability vector (PMF) 휋̅ of a Homogeneous Continuous Time Markov Chain ( CTMC ) with generator 푄̅ reduces to the solution of following linear system of equations ( i.e. computation of vector 휋̅ in the left null space of the generator matrix 푄̅ ) i.e. 휋̅ 푄̅ ≡ 0̅ . It should be noted that in the case of homogeneous Discrete Time Markov Chain ( DTMC) with state transition matrix 푃̅, the computation of equilibrium probability vector (PMF) 휋̅ reduces to the solution of following linear system of equations i.e. 휋̅ 푃̅ ≡ 휋̅ 표푟 푒푞푢푖푣푎푙푒푛푡푙푦 휋̅ ( 푃̅ − 퐼 ) ≡ 0̅ . 퐼푡 푟푒푎푑푖푙푦 푓표푙푙표푤푠 푡ℎ푎푡 ( 푃̅ − 퐼 ) ℎ푎푠 푡ℎ푒 푝푟표푝푒푟푡푖푒푠 표푓 푎 푔푒푛푒푟푎푡표푟 푚푎푡푟푖푥. 푇ℎ푢푠, 푖푛 푡ℎ푒 푓표푙푙표푤푖푛푔 푑푖푠푐푢푠푠푖표푛, 푤푒 푐표푛푠푖푑푒푟 표푛푙푦 푡ℎ푒 푐표푚푝푢푡푎푡푖표푛 표푓 푒푞푢푖푙푖푏푟푖푢푚 푝푟표푏푎푏푖푙푖푡푦 푣푒푐푡표푟 표푓 푎 퐶푇푀퐶. In computational linear algebra, there are efficient algorithms to solve linear system of equations ,[Gall], [Sto], [Wil] . Volker Strassen made fundamental contributions to this problem [Str1], [Str2]. He showed that the complexity of matrix inversion is equivalent to that of matrix multiplication. It readily follows that the matrix ‘-Q’ is a Laplacian matrix. There are efficient algorithms to solve Laplacian system of equations [Coh], [DGA]. 3. Efficient Computation of Equilibrium Probability Mass Function of Continuous Time Markov Chains: But, from the point of view of computing 휋̅ , earlier researchers donot fully take into account the structure of generator matrix 푄̅̅̅ ̅. Thus, the goal of this research paper is to design efficient algorithm for computing 휋̅, taking into account the structure of generator matrix 푄̅ . Some related effort dealing with Laplacian system of equations is discussed in [DGA]. In fact, we would like to design a COMPUTATIONALLY OPTIMAL ALGORITHM for computation of 휋̅ ( 푖푛 푡푒푟푚푠 표푓 Computational Complexity ). We illustrate the essential idea with a Continuous Time Markov chain ( CTMC ) with 4 states. First consider the generator matrix of a CTMC with 4 states ( i.e. 4 x 4 matrix ) partitioned using blocks of size 2 x 2 ( i.e. there are 4 such 2 x 2 matrices in the generator ) [RaR1], [RaR2]. Specifically, we have 퐴11 퐴12 푄 = [ ] , 푤ℎ푒푟푒 { 퐴푖푗; 푖, 푗 ∈ {1,2} } 푎푟푒 퐴21 퐴22 2 푥 2 푠푞푢푎푟푒 푚푎푡푟푖푐푒푠 푖푛 푡ℎ푒 4 푥 4 푏푙표푐푘 푚푎푡푟푖푥 푄. Claim: For a positive recurrent ( recurrent non-null ) CTMC with generator matrix Q, it is well known that the matrices { 퐴11, 퐴22 } are non-singular. Hence, we have that −1 퐴21 = 퐴22 푋 푤푖푡ℎ 푋 = 퐴22 퐴21. Thus, by means of elementary column operations, the generator matrix Q can be converted into the following form i.e. 퐴̃11 퐴̃12 푄̃ = [ ] 푖. 푒. 푏푙표푐푘 푢푝푝푒푟 푡푟푖푎푛푔푢푙푎푟 푚푎푡푟푖푥. 0 퐴22 In fact, let 푋 = [ 푓1 푓2 ]. Hence, column operations to arrive at 푄̃ ( from Q ) are determined by the column vectors { 푓1 푓2 } . Also, it follows from linear algebra, that the equilibrium probability vector 휋̅ is unaffected by such column operations. Hence, with 휋̅ = [ 휋̅̅1̅ 휋̅̅2̅ ], we have the following system of linear equations i.e. 휋̅̅1̅ 퐴̃11 = 0̅ 푖. 푒. 푏표푢푛푑푎푟푦 푠푦푠푡푒푚 표푓 푡푤표 푒푞푢푎푡푖표푛푠 ̃ ̅ ̃ −1 휋̅̅1̅ 퐴12 + 휋̅̅2̅ 퐴22 = 0 . 푇ℎ푢푠, 푤푒 ℎ푎푣푒 휋̅̅2̅ = −휋̅̅1̅ 퐴12 퐴22 . The boundary system of linear equations leads to a single linear equation in 2 variables 1 2 i.e. denoting 휋̅̅1̅ = [휋1 휋1 ], we have a linear equation of the form 훼 휋1훼 + 휋2 훽 = 0. 푇ℎ푢푠, 휋2 = −휋1 . 1 1 1 1 훽 Further, since 휋̅̅2̅ can be expressed in terms of 휋̅̅1̅ , we utilize the normalizing equation ( 휋̅̅1̅ + 휋̅̅2̅) 푒̅ = 1, 푤ℎ푒푟푒 푒̅ is a column vector of ones, 1 to determine 휋1 and hence all the other equilibrium probabilities. Remark 1: We realize that the inverse of a 2 x 2 matrix can be computed by inspection and essentially only requires computation of the determinant. For instance, let us consider 푏 푏 1 푏 −푏 a 2 x 2 matrix B i.e. 퐵 = [ 11 12] . 퐼푡 푖푠 푤푒푙푙 푘푛표푤푛 푡ℎ푎푡 퐵−1 = [ 22 12] , 푏21 푏22 ∆ −푏21 푏11 푤ℎ푒푟푒 ∆ = 푑푒푡푒푟푚푖푛푎푛푡(퐵) = ( 푏11푏22 − 푏12푏21 ). COMPUTATIONAL COMPLEXITY: Now, we determine the computational complexity of our algorithm: The number of arithmetic operations required to convert the generator matrix Q into block upper triangular matrix 푄̃ are as follows. We need one 2 x 2 matrix inversion, one 2 x 2 matrix multiplication and one addition of 2 x 2 matrices. 2 휋1 푐표푚푝푢푡푎푡푖표푛 푟푒푞푢푖푟푒푠 푂푁퐸 푑푖푣푖푠푖표푛 . 휋̅̅2̅ 푐표푚푝푢푡푎푡푖표푛 푟푒푞푢푖푟푒푠 푖푛푣푒푟푠푖표푛 표푓 2 푥 2 푚푎푡푟푖푥 퐴22 푎푛푑 표푛푒 푚푎푡푟푖푥 푚푢푙푡푖푝푙푖푐푎푡푖표푛 ( 표푓 2 푥 2 푚푎푡푟푖푐푒푠 퐴̃12 푎푛푑 퐴22 ). 1 퐴푙푠표, 푡ℎ푒 푛표푟푚푎푙푖푧푖푛푔 푒푞푢푎푡푖표푛 푡표 푑푒푡푒푟푚푖푛푒 휋1 푟푒푞푢푖푟푒푠 3 푎푑푑푖푡푖표푛푠 푎푛푑 푂푁퐸 푑푖푣푖푠푖표푛. 1 퐹푖푛푎푙푙푦, 푡표 푑푒푡푒푟푚푖푛푒 휋̅̅1̅ , 휋̅̅2̅ 푖푛 푡푒푟푚푠 표푓 휋1, 푤푒 푟푒푞푢푖푟푒 3 푚푢푙푡푖푝푙푖푐푎푡푖표푛푠. Now, we generalize the above algorithm for the case, where the number of states, N = 2 m where m > 1. In such case, the generator matrix is of the following form: 퐴11 퐴12 퐴13 … 퐴1푚 퐴 퐴 퐴 … 퐴 푄 = [ 21 22 23 2푚 ] 푤ℎ푒푟푒 퐴′ 푠 푎푟푒 2 푥 2 푚푎푡푟푖푐푒푠. ⋮ ⋮ ⋮ … ⋮ 푖푗 퐴푚1 퐴푚2 퐴푚3 … 퐴푚푚 Lemma 1 : For a positive recurrent CTMC, the following sub-matrices of generator i.e. { 퐴푖푖: 1 ≤ 푖 ≤ 푚 } are all non-singular matrices. Proof: Refer [Rama1]. The proof utilizes the fact that strictly diagonally dominant matrix is non-singular. Hence, as in the m=1 case, by means of elementary column operations on the generator matrix Q, we arrive at the following block upper triangular matrix 푄̃ . 퐴̃ 퐴̃ 퐴̃ ⋯ ̃ ̃ 11 12 13 퐴1 푚−1 퐴1푚 ̃ ̃ 0 퐴22 퐴23 ⋯ 퐴̃2 푚−1 퐴̃2푚 푄̃ = ⋮ ⋮ . ⋮ ⋮ 0 ⋯ 퐴 ̃ 퐴̃ 0 0 푚−1 푚−1 푚−1 푚 [ 0 0 0 … . 0 퐴̃푚푚 ] Thus, the equilibrium probability vector 휋̅ satisfies the following linear system of equations: 휋̅̅1̅ 퐴̃11 ≡ 0̅ 휋̅̅1̅ 퐴̃12 + 휋̅̅2̅ 퐴̃22 ≡ 0̅ ⋮ 휋̅̅1̅ 퐴̃1푚−1 + 휋̅̅2̅ 퐴2 푚−1 ̃+ ⋯ . 휋̅̅푚̅̅̅−̅1̅ 퐴푚̃−1 푚−1 ≡ 0̅ 휋̅̅1̅ 퐴̃1푚 + 휋̅̅2̅ 퐴̃2푚 + ⋯ + 휋̅̅푚̅̅ 퐴푚푚 ≡ 0̅ . The above system of linear equations is recursively solved to compute the equilibrium probability vector i.e. 휋̅̅1̅ 퐴̃11 ≡ 0̅ −1 휋̅̅2̅ = −휋̅̅1̅ 퐴̃12 퐴̃22 . ⋮ −1 −1 −1 휋̅푗 = −휋̅̅1̅ 퐴̃1푗 퐴̃푗푗 − 휋̅̅2̅ 퐴̃2푗 퐴̃푗푗 − ⋯ ⋯ − 휋̅̅푗̅̅−̅1̅ 퐴̃푗−1 푗 퐴̃푗푗 푓표푟 2 ≤ 푗 ≤ (푚 − 1). −1 −1 −1 휋̅̅푚̅̅ = −휋̅̅1̅ 퐴̃1푚 퐴̃푚푚 − 휋̅̅2̅ 퐴̃2푚 퐴̃푚푚 − ⋯ ⋯ − 휋̅̅푚̅̅̅−̅1̅ 퐴푚̃−1 푚 퐴푚 푚 . Note: As in the 2 x2 case, using non-singular matrices on the diagonal of generator matix, Q, it can be converted to a block upper-triangular matrix. Since inverse of 2 x 2 matrices on the diagonal of Q can be computed efficiently, an efficient algorithm exists for such a computational problem. Definition: Finite Memory Recursion of order “L” for the equilibrium probability vector is of the following form ′ 흅̅ (푳 + ퟏ) = 휋̅ (1) 푊1 + 휋̅ (2) 푊2 + ⋯ + 휋̅ (퐿) 푊퐿 ,, 푤ℎ푒푟푒 푊푖 푠 푎푟푒 푟푒푐푢푟푠푖표푛 푚푎푡푟푖푐푒푠.
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