Phase Enlargement of Semi-Markov Systems without Determining Stationary Distribution of Embedded Markov Chain

A phase enlargement of semi-Markov systems that does not require determining stationary distribution of the embedded Markov chain is considered. Phase enlargement is an equivalent replacement of a semi-Markov system with a common phase state space by a system with a discrete state space. Finding the stationary distribution of an embedded Markov chain for a system with a continuous phase state space is one of the most time-consuming and not always solvable stage, since in some cases it leads to a solution of integral equations with kernels containing sum and difference of variables. For such equations there is only a particular solution and there are no general solutions to date. For this purpose a lemma on a type of a distribution function of the difference of two random variables, provided that the first variable is greater than the subtracted variable, is used. It is shown that the type of the distribution function of difference of two random variables under the indicated condition depends on one constant, which is determined by a numerical method of solving the equation presented in the lemma. Based on the lemma, a theorem on the difference of a random variable and a complicated recovery flow is built up. The use of this method is demonstrated by the example of modeling a technical system consisting of two series-connected process cells, provided that both cells cannot fail simultaneously. The distribution functions of the system residence times in enlarged states, as well as in a subset of working and non-working states, are determined. The simulation results are compared by the considered and classical method proposed by V. Korolyuk, showed the complete coincidence of the sought quantities.

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On a generalized finite-capacity storage model

Journal of Applied Probability ◽

10.2307/3213901 ◽

1983 ◽

Vol 20 (3) ◽

pp. 663-674 ◽

Cited By ~ 1

Author(s):

Samuel W. Woolford

Keyword(s):

Distribution Function ◽

Markov Chain ◽

State Space ◽

Random Variables ◽

Time Dependent ◽

Finite Capacity ◽

Random Input ◽

Storage Model ◽

Limit Distributions

This paper considers a finite-capacity storage model defined on a Markov chain {Xn; n = 0, 1, ·· ·}, having state space J ⊆ {1, 2, ·· ·}. If Xn = j, then there is a random ‘input' Vn(j) (a negative input implying a demand) of ‘type' j, having a distribution function Fj(·). We assume that {Vn(j)} is an i.i.d. sequence of random variables, taken to be independent of {Xn} and of {Vn (k)}, for k ≠ j. Here, the random variables Vn(j) represent instantaneous ‘inputs' of type j for our storage model. Within this framework, we establish certain limit distributions for the joint processes (Zn, Xn) and (Zn, Qn, Ln), where Zn (defined in (1.2)) is the level of storage at time n, Qn (defined in (1.3)) is the cumulative overflow at time n, and Ln (defined in (1.4)) is the cumulative demand lost due to shortage of supply up to time n. In addition, an expression for the time-dependent distribution of (Zn, Xn) is obtained.

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On a generalized finite-capacity storage model

Advances in Applied Probability ◽

10.2307/1427221 ◽

1984 ◽

Vol 16 (1) ◽

pp. 23-23

Author(s):

Samuel W. Woolford

Keyword(s):

Distribution Function ◽

Markov Chain ◽

State Space ◽

Random Variables ◽

Time Dependent ◽

Finite Capacity ◽

Random Input ◽

Storage Model ◽

Limit Distributions

This paper considers a finite-capacity storage model defined on a Markov chain {Xn; n = 0, 1, …}, having state space J ⊆ {1, 2, …}. If Xn, = j, then there is a random. ‘input’ Vn(j) (a negative input implying a demand) of ‘type’ j, having a distribution function Fj (·). We assume that {Vn (j)} is an i.i.d. sequence of random variables, taken to be independent of {Xn} and of {Vn(k)}, for k ≠ = j. Here, the random variables Vn(j) represent instantaneous ‘inputs’ of type j for our storage model. Within this framework, we establish certain limit distributions for the joint processes (zn, Xn) and (Zn, OnLn), where Zn, is the level of storage at time n, Qn is the cumulative overflow at time n, and Ln is the cumulative demand lost due to shortage of supply up to time n. In addition, an expression for the time-dependent distribution of (Zn, Xn) is obtained.

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On a generalized finite-capacity storage model

Journal of Applied Probability ◽

10.1017/s0021900200023901 ◽

1983 ◽

Vol 20 (03) ◽

pp. 663-674

Author(s):

Samuel W. Woolford

Keyword(s):

Distribution Function ◽

Markov Chain ◽

State Space ◽

Random Variables ◽

Time Dependent ◽

Finite Capacity ◽

Random Input ◽

Storage Model ◽

Limit Distributions

This paper considers a finite-capacity storage model defined on a Markov chain {Xn ; n = 0, 1, ·· ·}, having state space J ⊆ {1, 2, ·· ·}. If Xn = j, then there is a random ‘input' Vn (j) (a negative input implying a demand) of ‘type' j, having a distribution function Fj (·). We assume that {Vn (j)} is an i.i.d. sequence of random variables, taken to be independent of {Xn } and of {Vn (k)}, for k ≠ j. Here, the random variables Vn (j) represent instantaneous ‘inputs' of type j for our storage model. Within this framework, we establish certain limit distributions for the joint processes (Zn, Xn ) and (Zn, Qn, Ln ), where Zn (defined in (1.2)) is the level of storage at time n, Qn (defined in (1.3)) is the cumulative overflow at time n, and Ln (defined in (1.4)) is the cumulative demand lost due to shortage of supply up to time n. In addition, an expression for the time-dependent distribution of (Zn, Xn ) is obtained.

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On a generalized finite-capacity storage model

Advances in Applied Probability ◽

10.1017/s0001867800022187 ◽

1984 ◽

Vol 16 (01) ◽

pp. 23

Author(s):

Samuel W. Woolford

Keyword(s):

Distribution Function ◽

Markov Chain ◽

State Space ◽

Random Variables ◽

Time Dependent ◽

Finite Capacity ◽

Random Input ◽

Storage Model ◽

Limit Distributions

This paper considers a finite-capacity storage model defined on a Markov chain {Xn ; n = 0, 1, …}, having state space J ⊆ {1, 2, …}. If Xn , = j, then there is a random. ‘input’ Vn (j) (a negative input implying a demand) of ‘type’ j, having a distribution function Fj (·). We assume that {Vn (j)} is an i.i.d. sequence of random variables, taken to be independent of {Xn } and of {Vn(k)}, for k ≠ = j. Here, the random variables Vn (j) represent instantaneous ‘inputs’ of type j for our storage model. Within this framework, we establish certain limit distributions for the joint processes (zn , Xn ) and (Zn , On Ln ), where Zn , is the level of storage at time n, Qn is the cumulative overflow at time n, and Ln is the cumulative demand lost due to shortage of supply up to time n. In addition, an expression for the time-dependent distribution of (Zn , Xn ) is obtained.

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Criteria for the non-ergodicity of stochastic processes: application to the exponential back-off protocol

Journal of Applied Probability ◽

10.1017/s0021900200030990 ◽

1987 ◽

Vol 24 (02) ◽

pp. 347-354 ◽

Cited By ~ 1

Author(s):

Guy Fayolle ◽

Rudolph Iasnogorodski

Keyword(s):

Stochastic Process ◽

Markov Chain ◽

Stochastic Processes ◽

State Space ◽

Discrete Time ◽

Random Variables ◽

Countable State Space ◽

Countable State ◽

New Criteria

In this paper, we present some simple new criteria for the non-ergodicity of a stochastic process (Yn ), n ≧ 0 in discrete time, when either the upward or downward jumps are majorized by i.i.d. random variables. This situation is encountered in many practical situations, where the (Yn ) are functionals of some Markov chain with countable state space. An application to the exponential back-off protocol is described.

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A stable algorithm for stationary distribution calculation for a BMAP/SM/1 queueing system with Markovian arrival input of disasters

Journal of Applied Probability ◽

10.1017/s0021900200014492 ◽

2004 ◽

Vol 41 (02) ◽

pp. 547-556 ◽

Cited By ~ 4

Author(s):

Alexander Dudin ◽

Olga Semenova

Keyword(s):

Markov Chain ◽

Stationary State ◽

Stationary Distribution ◽

Queueing System ◽

Embedded Markov Chain ◽

State Distribution ◽

Stable Algorithm ◽

Distribution Calculation

Disaster arrival into a queueing system causes all customers to leave the system instantaneously. We present a numerically stable algorithm for calculating the stationary state distribution of an embedded Markov chain for the BMAP/SM/1 queue with a MAP input of disasters.

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Sums and weighted sums of a gamma Markov sequence

Journal of Applied Probability ◽

10.1017/s0021900200040766 ◽

1988 ◽

Vol 25 (01) ◽

pp. 204-209 ◽

Cited By ~ 2

Author(s):

Ravindra M. Phatarfod

Keyword(s):

Markov Chain ◽

Stationary Distribution ◽

Random Variables ◽

Laplace Transforms ◽

Weighted Sums ◽

Sums Of Random Variables ◽

Markov Sequence

We derive the Laplace transforms of sums and weighted sums of random variables forming a Markov chain whose stationary distribution is gamma. Both seasonal and non-seasonal cases are considered. The results are applied to two problems in stochastic reservoir theory.

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Insensitive Bounds for the Stationary Distribution of a Single Server Retrial Queue with Server Subject to Active Breakdowns

Advances in Operations Research ◽

10.1155/2014/985453 ◽

2014 ◽

Vol 2014 ◽

pp. 1-12 ◽

Cited By ~ 6

Author(s):

Mohamed Boualem

Keyword(s):

Markov Chain ◽

Stationary Distribution ◽

Retrial Queue ◽

Embedded Markov Chain ◽

Single Server ◽

Waiting Room ◽

Monotonicity Properties

The paper addresses monotonicity properties of the single server retrial queue with no waiting room and server subject to active breakdowns. The obtained results allow us to place in a prominent position the insensitive bounds for the stationary distribution of the embedded Markov chain related to the model in the study. Numerical illustrations are provided to support the results.

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On the existence of quasi-stationary distributions in denumerable R-transient Markov chains

Journal of Applied Probability ◽

10.1017/s002190020010659x ◽

1992 ◽

Vol 29 (01) ◽

pp. 21-36 ◽

Cited By ~ 3

Author(s):

Masaaki Kijima

Keyword(s):

Markov Chain ◽

Random Walk ◽

Markov Chains ◽

State Space ◽

Stationary Distribution ◽

Practical Importance ◽

Prime Concern ◽

Continuous Random Walk ◽

Transient Markov Chain ◽

Quasi Stationary Distribution

Let {Xn, n= 0, 1, 2, ···} be a transient Markov chain which, when restricted to the state space 𝒩+= {1, 2, ···}, is governed by an irreducible, aperiodic and strictly substochastic matrix𝐏= (pij), and letpij(n) =P∈Xn=j, Xk∈ 𝒩+fork= 0, 1, ···,n|X0=i],i, j𝒩+. The prime concern of this paper is conditions for the existence of the limits,qijsay, ofasn →∞. Ifthe distribution (qij) is called the quasi-stationary distribution of {Xn} and has considerable practical importance. It will be shown that, under some conditions, if a non-negative non-trivial vectorx= (xi) satisfyingrxT=xT𝐏andexists, whereris the convergence norm of𝐏, i.e.r=R–1andand T denotes transpose, then it is unique, positive elementwise, andqij(n) necessarily converge toxjasn →∞.Unlike existing results in the literature, our results can be applied even to theR-null andR-transient cases. Finally, an application to a left-continuous random walk whose governing substochastic matrix isR-transient is discussed to demonstrate the usefulness of our results.

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