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Accurate calculations of Stationary Distributions and Mean First Passage Times in Markov Renewal Processes and Markov Chains

100%
Hunter J. J.
Special Matrices
|
2016
|
tom 4
|
nr 1
151-175
EN
This article describes an accurate procedure for computing the mean first passage times of a finite irreducible Markov chain and a Markov renewal process. The method is a refinement to the Kohlas, Zeit fur Oper Res, 30, 197–207, (1986) procedure. The technique is numerically stable in that it doesn’t involve subtractions. Algebraic expressions for the special cases of one, two, three and four states are derived.Aconsequence of the procedure is that the stationary distribution of the embedded Markov chain does not need to be derived in advance but can be found accurately from the derived mean first passage times. MatLab is utilized to carry out the computations, using some test problems from the literature.
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Analysis of an MAP/PH/1 queue with flexible group service

88%
Brugno A., D'Apice C., Dudin A., Manzo R.
International Journal of Applied Mathematics and Computer Science
|
2017
|
tom 27
|
nr 1
119-131
EN
A novel customer batch service discipline for a single server queue is introduced and analyzed. Service to customers is offered in batches of a certain size. If the number of customers in the system at the service completion moment is less than this size, the server does not start the next service until the number of customers in the system reaches this size or a random limitation of the idle time of the server expires, whichever occurs first. Customers arrive according to a Markovian arrival process. An individual customer's service time has a phase-type distribution. The service time of a batch is defined as the maximum of the individual service times of the customers which form the batch. The dynamics of such a system are described by a multi-dimensional Markov chain. An ergodicity condition for this Markov chain is derived, a stationary probability distribution of the states is computed, and formulas for the main performance measures of the system are provided. The Laplace-Stieltjes transform of the waiting time is obtained. Results are numerically illustrated.
3
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An asymptotic expansion for the distribution of the supremum of a random walk

63%
Sgibnev M. S.
Studia Mathematica
|
2000
|
tom 140
|
nr 1
41-55
EN
Let ${S_n}$ be a random walk drifting to -∞. We obtain an asymptotic expansion for the distribution of the supremum of ${S_n}$ which takes into account the influence of the roots of the equation $1-∫_ℝe^{sx}F(dx)=0,F$ being the underlying distribution. An estimate, of considerable generality, is given for the remainder term by means of submultiplicative weight functions. A similar problem for the stationary distribution of an oscillating random walk is also considered. The proofs rely on two general theorems for Laplace transforms.
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