On the numerical solution of Kronecker-based infinite level-dependent QBD processes

Baumann, Hendrik; Dayar, Tuǧrul; Orhan, M. Can; Sandmann, Werner

doi:10.1016/j.peva.2013.05.001

Cited by 10 publications

(19 citation statements)

References 38 publications

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“…One of the MCs is a queueing network model of a call center having five subsystems; it is a parallel service system known as the N-model under the threshold routing control policy proposed in [5]. Further information regarding these seven models may be obtained from [3,19]. The remaining eight MCs are models of the Courier protocol introduced in [34], the multiserver multiqueue (MSMQ) models discussed in [1], models of manufacturing systems having Kanban control [27], and a model of token based scheduling in a queueing network called qh realcontrol [11].…”

Section: Numerical Resultsmentioning

confidence: 99%

On Vector-Kronecker Product Multiplication with Rectangular Factors

Dayar¹,

Orhan²

2015

SIAM J. Sci. Comput.

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Abstract. The infinitesimal generator matrix underlying a multidimensional Markov chain can be represented compactly by using sums of Kronecker products of small rectangular matrices. For such compact representations, analysis methods based on vector-Kronecker product multiplication need to be employed. When the factors in the Kronecker product terms are relatively dense, vectorKronecker product multiplication can be performed efficiently by the shuffle algorithm. When the factors are relatively sparse, it may be more efficient to obtain nonzero elements of the generator matrix in Kronecker form on the fly and multiply them with corresponding elements of the vector. This work proposes a modification to the shuffle algorithm that multiplies relevant elements of the vector with submatrices of factors in which zero rows and columns are omitted. This approach avoids unnecessary floating-point operations that evaluate to zero during the course of the multiplication and possibly reduces the amount of memory used. Numerical experiments on a large number of models indicate that in many cases the modified shuffle algorithm performs a smaller number of floating-point operations than the shuffle algorithm and the algorithm that generates nonzeros on the fly, sometimes with a minimum number of floating-point operations and as little of memory possible.

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Section: Numerical Resultsmentioning

confidence: 99%

On Vector-Kronecker Product Multiplication with Rectangular Factors

Dayar¹,

Orhan²

2015

SIAM J. Sci. Comput.

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“…The separators in this graph are Z (2,1) = Z (1,2) and Z (2,2) = Z (1,5) . The priorities of both separators are 1, and either of them can be chosen for refinement.…”

Section: Example 32: (Cont'd) Algorithm 2 First Constructs the Unit mentioning

confidence: 99%

Cartesian Product Partitioning of Multi-Dimensional Reachable State Spaces

Dayar¹,

Orhan²

2016

Prob. Eng. Inf. Sci.

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Markov chains (MCs) are widely used to model systems which evolve by visiting the states in their state spaces following the available transitions. When such systems are composed of interacting subsystems, they can be mapped to a multi-dimensional MC in which each subsystem normally corresponds to a different dimension. Usually the reachable state space of the multi-dimensional MC is a proper subset of its product state space, that is, Cartesian product of its subsystem state spaces. Compact storage of the matrix underlying such a MC and efficient implementation of analysis methods using Kronecker operations require the set of reachable states to be represented as a union of Cartesian products of subsets of subsystem state spaces. The problem of partitioning the reachable state space of a three or higher dimensional system with a minimum number of partitions into Cartesian products of subsets of subsystem state spaces is shown to be NP-complete. Two algorithms, one merge based the other refinement based, that yield possibly non-optimal partitionings are presented. Results of experiments on a set of problems from the literature and those that are randomly generated indicate that, although it may be more time and memory consuming, the refinement based algorithm almost always computes partitionings with a smaller number of partitions than the merge-based algorithm. The refinement based algorithm is insensitive to the order in which the states in the reachable state space are processed, and in many cases it computes partitionings that are optimal.

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“…So, from (16) we conclude that the probabilities of merged states in the second stage of hierarchy π 2 (<n 1 >), n 1 ∈ 0, 1, ..., R 1 are calculated as state probabilities of the model…”

Section: Q( )mentioning

confidence: 99%

“…We developed an effective method for the calculation of state probabilities of the large dimension 3-D MC. There is much research on solving similar problems for the large dimension 2-D MC [9][10][11][12][13][14][15][16][17][18][19][20]. However, almost all of them have similar problems related to ill-conditioned matrices which arise during computational procedures.…”

Section: Introductionmentioning

confidence: 99%

Hierarchical Space Merging Algorithm for Analysis of Two Stage Queueing Network with Feedback

Melikov¹,

Пономаренко

Rustamov

et al. 2016

Communications in Computer and Information Science

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Abstract. A Markov model of two stage queuing network with feedback is proposed. Poisson flows arriving to both stages from outside and part of already serviced calls in the first node instantaneously enter to the second node (if there is free space here) while the remaining part leaves the network. At the completion of call processing in the second node there are three possibilities: (1) it leaves the network; (2) it instantaneously feeds back to the first node (if there is free space here); (3) it feeds back to the first node after some delay in orbit. All feedbacks are determined by known probabilities. Both nodes have finite capacities. The mathematical model of the investigated network is a three dimensional Markov chain (3-D MC) and hierarchical space merging algorithm to calculate its steady-state probabilities is developed. This algorithm allows asymptotic analysis of the quality of service (QoS) metrics of the investigated network as well.

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On the numerical solution of Kronecker-based infinite level-dependent QBD processes

Cited by 10 publications

References 38 publications

On Vector-Kronecker Product Multiplication with Rectangular Factors

On Vector-Kronecker Product Multiplication with Rectangular Factors

Cartesian Product Partitioning of Multi-Dimensional Reachable State Spaces

Hierarchical Space Merging Algorithm for Analysis of Two Stage Queueing Network with Feedback

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