# Analyzing Markov Chains using Kronecker Products: Theory and by Tugrul Dayar

By Tugrul Dayar

Kronecker items are used to outline the underlying Markov chain (MC) in a variety of modeling formalisms, together with compositional Markovian versions, hierarchical Markovian versions, and stochastic approach algebras. the incentive in the back of utilizing a Kronecker based illustration instead of a flat one is to relieve the garage standards linked to the MC. With this procedure, structures which are an order of value better will be analyzed at the related platform. The advancements within the resolution of such MCs are reviewed from an algebraic viewpoint and attainable components for additional examine are indicated with an emphasis on preprocessing utilizing reordering, grouping, and lumping and numerical research utilizing block iterative, preconditioned projection, multilevel, decompositional, and matrix analytic tools. Case experiences from closed queueing networks and stochastic chemical kinetics are supplied to inspire decompositional and matrix analytic equipment, respectively.

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2 involving rectangular factors. 44 4 Decompositional Methods We now turn to a class of problems that can be treated with approximative decompositional iterative methods using the understanding developed in this section. 3 Case Study from Closed Queueing Networks We consider closed networks of first-come first-served (FCFS) queues with phasetype (PH) service distributions and arbitrary buffer sizes [56]. Queueing networks (QNs) have been used in the literature to model and analyze a variety of systems involving customers, packets, or jobs waiting to get service [90, 140].

2; 2/g: pD1 As observed, excluding the isomorphic cases there are eight different ways in which one can obtain jSj D 5 from a two-dimensional product state space using 2 Ä N Ä 5 partitions. However, there are certain state-space sizes that never lead to jSj D 5 no matter how the state spaces are partitioned. 2/ j D 4, for instance. h/ , of each subsystem for h D 1; : : : ; H , the number of possibilities available in choosing the number of partitions, N , is large enough to accommodate a representation of the state space without unreachable states.

I1 ; : : : ; il // in BSOR must be solved in lexicographical order. If there is space, one can generate and p factorize in sparse storage these blocks as in BJOR at the outset and solve the bl systems directly at each iteration. Otherwise, one can use an iterative method such as BSOR since the off-diagonal parts of diagonal blocks are also sums of Kronecker products. i1 ; : : : ; il //, b is updated by multiplying the computed subvector by the corresponding row of blocks above the diagonal. 1 in [146]).