Matrix-Analytic Methods in Stochastic Models

This paper will describe an analytical approach to the solutions of queueing models with finites capacity. The methods is chosen because its abilitiy to model exactly complex non-Markovian model which have correlated, non-renewal input process which is nearly impossible to get exact expressions by using classical methods of generating functions. We applied this methodology to compute the loss performance a of a queueing item specifically in the scenario of Asynchronous Transfer Mode (ATM) networks.

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Stochastic and substochastic solutions for infinite-state Markov chains with applications to matrix-analytic methods

Advances in Applied Probability ◽

10.1017/s0001867800003001 ◽

2008 ◽

Vol 40 (04) ◽

pp. 1157-1173

Author(s):

Winfried K. Grassmann ◽

Javad Tavakoli

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Transition Matrices ◽

Matrix Analytic Methods ◽

Analytic Methods ◽

Finite Queues ◽

Qbd Processes ◽

Stochastic Solution ◽

Number Of Visits ◽

Infinite State

This paper deals with censoring of infinite-state banded Markov chains. Censoring involves reducing the time spent in states outside a certain set of states to 0 without affecting the number of visits within this set. We show that, if all states are transient, there is, besides the standard censored Markov chain, a nonstandard censored Markov chain which is stochastic. Both the stochastic and the substochastic solutions are found by censoring a sequence of finite transition matrices. If all matrices in the sequence are stochastic, the stochastic solution arises in the limit, whereas the substochastic solution arises if the matrices in the sequence are substochastic. We also show that, if the Markov chain is recurrent, the only solution is the stochastic solution. Censoring is particularly fruitful when applied to quasi-birth-and-death (QBD) processes. It turns out that key matrices in such processes are not unique, a fact that has been observed by several authors. We note that the stochastic solution is important for the analysis of finite queues.

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Matrix-analytic methods for fluid queues with finite buffers

Performance Evaluation ◽

10.1016/j.peva.2005.02.002 ◽

2006 ◽

Vol 63 (4-5) ◽

pp. 295-314 ◽

Cited By ~ 65

Author(s):

Ana da Silva Soares ◽

Guy Latouche

Keyword(s):

Matrix Analytic Methods ◽

Finite Buffers ◽

Fluid Queues ◽

Analytic Methods

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