THE M/G/1-TYPE MARKOV CHAIN WITH RESTRICTED TRANSITIONS AND ITS APPLICATION TO QUEUES WITH BATCH ARRIVALS

2011 ◽  
Vol 25 (4) ◽  
pp. 487-517 ◽  
Author(s):  
Juan F. Pérez ◽  
Benny Van Houdt
Keyword(s):  
Markov Chain ◽  
Block Size ◽  
Computation Time ◽  
Priority Queue ◽  
Small Subset ◽  
Probability Vector ◽  
Preemptive Priority ◽  
Batch Arrivals ◽  
Speed Up ◽  
Stationary Probability Vector

We consider M/G/1-type Markov chains where a transition that decreases the value of the level triggers the phase to a small subset of the phase space. We show how this structure—referred to as restricted downward transitions—can be exploited to speed up the computation of the stationary probability vector of the chain. To this end we define a new M/G/1-type Markov chain with a smaller block size, the G matrix of which is used to find the original chain's G matrix. This approach is then used to analyze the BMAP/PH/1 queue and the BMAP[2]/PH[2]/1 preemptive priority queue, yielding significant reductions in computation time.

scholarly journals Light-tailed asymptotics of stationary probability vectors of Markov chains of GI/G/1 type

2005 ◽  
Vol 37 (04) ◽  
pp. 1075-1093 ◽  
Author(s):  
Quan-Lin Li ◽  
Yiqiang Q. Zhao
Keyword(s):  
Asymptotic Behavior ◽  
Markov Chain ◽  
Markov Chains ◽  
Asymptotic Analysis ◽  
Positive Solution ◽  
Generating Function ◽  
Spectral Properties ◽  
Stationary Probability ◽  
Probability Vector ◽  
Stationary Probability Vector

In this paper, we consider the asymptotic behavior of stationary probability vectors of Markov chains of GI/G/1 type. The generating function of the stationary probability vector is explicitly expressed by theR-measure. This expression of the generating function is more convenient for the asymptotic analysis than those in the literature. TheRG-factorization of both the repeating row and the Wiener-Hopf equations for the boundary row are used to provide necessary spectral properties. The stationary probability vector of a Markov chain of GI/G/1 type is shown to be light tailed if the blocks of the repeating row and the blocks of the boundary row are light tailed. We derive two classes of explicit expression for the asymptotic behavior, the geometric tail, and the semigeometric tail, based on the repeating row, the boundary row, or the minimal positive solution of a crucial equation involved in the generating function, and discuss the singularity classes of the stationary probability vector.

Geometric bounds on iterative approximations for nearly completely decomposable Markov chains

1990 ◽  
Vol 27 (03) ◽  
pp. 521-529 ◽  
Author(s):  
Guy Louchard ◽  
Guy Latouche
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Perturbation Analysis ◽  
Stochastic Matrix ◽  
Stationary Probability ◽  
Probability Vector ◽  
Finite Markov Chain ◽  
Stationary Probability Vector

We consider a finite Markov chain with nearly-completely decomposable stochastic matrix. We determine bounds for the error, when the stationary probability vector is approximated via a perturbation analysis.

scholarly journals Light-tailed asymptotics of stationary probability vectors of Markov chains of GI/G/1 type

2005 ◽  
Vol 37 (4) ◽  
pp. 1075-1093 ◽  
Author(s):  
Quan-Lin Li ◽  
Yiqiang Q. Zhao
Keyword(s):  
Asymptotic Behavior ◽  
Markov Chain ◽  
Markov Chains ◽  
Asymptotic Analysis ◽  
Positive Solution ◽  
Generating Function ◽  
Spectral Properties ◽  
Stationary Probability ◽  
Probability Vector ◽  
Stationary Probability Vector

In this paper, we consider the asymptotic behavior of stationary probability vectors of Markov chains of GI/G/1 type. The generating function of the stationary probability vector is explicitly expressed by the R-measure. This expression of the generating function is more convenient for the asymptotic analysis than those in the literature. The RG-factorization of both the repeating row and the Wiener-Hopf equations for the boundary row are used to provide necessary spectral properties. The stationary probability vector of a Markov chain of GI/G/1 type is shown to be light tailed if the blocks of the repeating row and the blocks of the boundary row are light tailed. We derive two classes of explicit expression for the asymptotic behavior, the geometric tail, and the semigeometric tail, based on the repeating row, the boundary row, or the minimal positive solution of a crucial equation involved in the generating function, and discuss the singularity classes of the stationary probability vector.

Geometric bounds on iterative approximations for nearly completely decomposable Markov chains

1990 ◽  
Vol 27 (3) ◽  
pp. 521-529 ◽  
Author(s):  
Guy Louchard ◽  
Guy Latouche
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Perturbation Analysis ◽  
Stochastic Matrix ◽  
Stationary Probability ◽  
Probability Vector ◽  
Finite Markov Chain ◽  
Stationary Probability Vector

We consider a finite Markov chain with nearly-completely decomposable stochastic matrix. We determine bounds for the error, when the stationary probability vector is approximated via a perturbation analysis.

scholarly journals A New GMRES(m) Method for Markov Chains

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Bing-Yuan Pu ◽  
Ting-Zhu Huang ◽  
Chun Wen
Keyword(s):  
Markov Chain ◽  
Convergence Rate ◽  
Numerical Experiments ◽  
Gmres Method ◽  
Stationary Probability ◽  
Probability Vector ◽  
Vector Extrapolation ◽  
Restarted Gmres ◽  
Stationary Probability Vector ◽  
Irreducible Markov Chain

This paper presents a class of new accelerated restarted GMRES method for calculating the stationary probability vector of an irreducible Markov chain. We focus on the mechanism of this new hybrid method by showing how to periodically combine the GMRES and vector extrapolation method into a much efficient one for improving the convergence rate in Markov chain problems. Numerical experiments are carried out to demonstrate the efficiency of our new algorithm on several typical Markov chain problems.

On a Markovian queue with weakly correlated interarrival times

1981 ◽  
Vol 18 (01) ◽  
pp. 190-203 ◽  
Author(s):  
Guy Latouche
Keyword(s):  
Time Distribution ◽  
Queueing System ◽  
Interarrival Time ◽  
Probability Vector ◽  
Common Value ◽  
Markovian Queue ◽  
The Stability ◽  
Number Of Customers ◽  
Stationary Probabilities ◽  
Stationary Probability Vector

A queueing system with exponential service and correlated arrivals is analysed. Each interarrival time is exponentially distributed. The parameter of the interarrival time distribution depends on the parameter for the preceding arrival, according to a Markov chain. The parameters of the interarrival time distributions are chosen to be equal to a common value plus a factor ofε, where ε is a small number. Successive arrivals are then weakly correlated. The stability condition is found and it is shown that the system has a stationary probability vector of matrix-geometric form. Furthermore, it is shown that the stationary probabilities for the number of customers in the system, are analytic functions ofε, for sufficiently smallε, and depend more on the variability in the interarrival time distribution, than on the correlations.

International Real Estate Review

2016 ◽  
Vol 19 (3) ◽  
pp. 265-296
Author(s):  
Richard D. Evans ◽  
◽  
Glenn R. Mueller ◽  
Keyword(s):  
Markov Chain ◽  
Real Estate ◽  
Cash Flow ◽  
Discounted Cash Flow ◽  
First Order ◽  
Order Markov Chain ◽  
Office Markets ◽  
Speed Up ◽  
Staying Time ◽  
Over Time

Metro market real estate cycles for office, industrial, retail, apartment, and hotel properties may be specified as first order Markov chains, which allow analysts to use a well-developed application, ¡§staying time¡¨. Anticipations for time spent at each cycle point are consistent with the perception of analysts that these cycle changes speed up, slow down, and pause over time. We find that these five different property types in U.S. markets appear to have different first order Markov chain specifications, with different staying time characteristics. Each of the five property types have their longest mean staying time at the troughs of recessions. Moreover, industrial and office markets have much longer mean staying times in very poor trough conditions. Most of the shortest mean staying times are in hyper supply and recession phases, with the range across property types being narrow in these cycle points. Analysts and investors should be able to use this research to better estimate future occupancy and rent estimates in their discounted cash flow (DCF) models.

scholarly journals Parallel Algorithms for Spatial Rainfall Distribution

Jurnal INKOM ◽  
2014 ◽  
Vol 8 (1) ◽  
pp. 29 ◽  
Author(s):  
Arnida Lailatul Latifah ◽  
Adi Nurhadiyatna
Keyword(s):  
Parallel Algorithms ◽  
Computation Time ◽  
Computational Time ◽  
Rainfall Distribution ◽  
Flood Modelling ◽  
Computation Efficiency ◽  
Distance Weighting ◽  
Speed Up ◽  
Important Input ◽  
Serial Algorithms

This paper proposes parallel algorithms for precipitation of flood modelling, especially applied in spatial rainfall distribution. As an important input in flood modelling, spatial distribution of rainfall is always needed as a pre-conditioned model. In this paper two interpolation methods, Inverse distance weighting (IDW) and Ordinary kriging (OK) are discussed. Both are developed in parallel algorithms in order to reduce the computational time. To measure the computation efficiency, the performance of the parallel algorithms are compared to the serial algorithms for both methods. Findings indicate that: (1) the computation time of OK algorithm is up to 23% longer than IDW; (2) the computation time of OK and IDW algorithms is linearly increasing with the number of cells/ points; (3) the computation time of the parallel algorithms for both methods is exponentially decaying with the number of processors. The parallel algorithm of IDW gives a decay factor of 0.52, while OK gives 0.53; (4) The parallel algorithms perform near ideal speed-up.

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