semi-Markov Matrices Research Articles

The second moments of the absorption times in the Markov renewal branching process

We derive explicit formulas for the second moments of the absorption time matrices in the Markov renewal branching process. These formulas may easily be computationally implemented and are useful in the iterative computation of the semi-Markov matrices, which give the distributions of the duration and of the number of customers served in a busy period of a great variety of complex queueing models.

The second moments of the absorption times in the Markov renewal branching process

We derive explicit formulas for the second moments of the absorption time matrices in the Markov renewal branching process. These formulas may easily be computationally implemented and are useful in the iterative computation of the semi-Markov matrices, which give the distributions of the duration and of the number of customers served in a busy period of a great variety of complex queueing models.

Limit theorems for sums of a sequence of random variables defined on a Markov chain

Let {(Jn, Xn), n ≧ 0} be the standard J–X process of Markov renewal theory. Suppose {Jn, n ≧ 0} is irreducible, aperiodic and positive recurrent. It is shown using the strong mixing condition, that if converges in distribution, where an, bn > 0 (bn → ∞) are real constants, then the limit law F must be stable. Suppose Q(x) = {PijHi(x)} is the semi-Markov matrix of {(JnXn), n ≧ 0}. Then the n-fold convolution, Q∗n(bnx + anbn), converges in distribution to F(x)Π if and only if converges in distribution to F. Π is the matrix of stationary transition probabilities of {Jn, n ≧ 0}. Sufficient conditions on the Hi's are given for the convergence of the sequence of semi-Markov matrices to F(x)Π, where F is stable.

Limit theorems for sums of a sequence of random variables defined on a Markov chain

Let {(Jn, Xn),n≧ 0} be the standardJ–Xprocess of Markov renewal theory. Suppose {Jn,n≧ 0} is irreducible, aperiodic and positive recurrent. It is shown using the strong mixing condition, that ifconverges in distribution, wherean, bn>0 (bn→∞) are real constants, then the limit lawFmust be stable. SupposeQ(x) = {PijHi(x)} is the semi-Markov matrix of {(JnXn),n≧ 0}. Then then-fold convolution,Q∗n(bnx + anbn), converges in distribution toF(x)Π if and only ifconverges in distribution toF. Π is the matrix of stationary transition probabilities of {Jn,n≧ 0}. Sufficient conditions on theHi's are given for the convergence of the sequence of semi-Markov matrices toF(x)Π, whereFis stable.

A Lindeberg-type theorem

Let ( kQ ij ) k be a sequence of semi-Markov matrices arising from the nonhomogeneous J− X process of Markov renewal theory. A generalization of the sufficiency half of the Lindeberg central limit theorem is proved for sums S n =∑ n i=1 X i suitably normed, where the X i are the holding times of the J− X process. The approach used involves an adaption of the wellknown Trotter proof of the central limit theorem. Some familiar results are also obtained, by using this “convolution operator” approach.

Infinite divisibility and inbeddability of semi-Markov matrices on certain topological groups

Infinite divisibility and inbeddability of semi-Markov matrices on certain topological groups

Infinite divisibility and inbeddability of semi-Markov matrices on certain topological groups

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Infinite divisibility and stability of finite semi-Markov matrices

In this paper the elements of a semi-Markov matrix A may have support anywhere on the real line, and A(+ co) may be a sub-Markov matrix. The sub-characteristic matrix a of A is the matrix of Fourier-Stieltjes transforms corresponding to A. If A(+ co) = a(O) is a Markov matrix, A is called a matrix distribution funotion and at a characteristic matrix. Infinite divisibility and stability of semi-Markov and sub-characteristic matrices is defined as for distribution functions and characteristic functions.

Infinite divisibility and stability of finite semi-Markov matrices

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Almost sure stability of maxima

For random variables {Xn, n ≧ 1} unbounded above set Mn = max {X1, X2, …, Xn}. When do normalizing constants bn exist such that Mn/bn→ 1 a.s.; i.e., when is {Mn} a.s. stable? If {Xn} is i.i.d. then {Mn} is a.s. stable iff for all and in this case bn ∼ F–1 (1 – 1/n) Necessary and sufficient conditions for lim supn→∞, Mn/bn = l > 1 a.s. are given and this is shown to be insufficient in general for lim infn→∞Mn/bn = 1 a.s. except when l = 1. When the Xn are r.v.'s defined on a finite Markov chain, one shows by means of an analogue of the Borel Zero-One Law and properties of semi-Markov matrices that the stability problem for this case can be reduced to the i.i.d. case.

Almost sure stability of maxima

For random variables {Xn, n≧ 1} unbounded above setMn= max {X1,X2, …,Xn}. When do normalizing constantsbnexist such thatMn/bn→1 a.s.; i.e., when is {Mn} a.s. stable? If {Xn} is i.i.d. then {Mn} is a.s. stable iff for alland in this casebn∼F–1(1 – 1/n) Necessary and sufficient conditions for lim supn→∞,Mn/bn= l >1 a.s. are given and this is shown to be insufficient in general for lim infn→∞Mn/bn= 1 a.s. except whenl= 1. When theXnare r.v.'s defined on a finite Markov chain, one shows by means of an analogue of the Borel Zero-One Law and properties of semi-Markov matrices that the stability problem for this case can be reduced to the i.i.d. case.