Continuous Time Homogeneous Markov System Research Articles

Transient Analysis of the M/M/k/N/N Queue using a Continuous Time Homogeneous Markov System with Finite State Size Capacity

In this article, the M/M/k/N/N queue is modeled as a continuous-time homogeneous Markov system with finite state size capacity (HMS/cs). In order to examine the behavior of the queue a continuous-time homogeneous Markov system (HMS) constituted of two states is used. The first state of this HMS corresponds to the source and the second one to the state with the servers. The second state has a finite capacity which corresponds to the number of servers. The members of the system which can not enter the second state, due to its finite capacity, enter the buffer state which represents the system's queue. In order to examine the variability of the state sizes formulae for their factorial and mixed factorial moments are derived in matrix form. As a consequence, the pmf of each state size can be evaluated for any t ∈ ℝ+. The theoretical results are illustrated by a numerical example.

The Homogeneous Markov System (HMS) as an Elastic Medium. The Three-Dimensional Case

Every attainable structure of the so-called continuous-time Homogeneous Markov System (HMS) with fixed size and state space S = {1, 2,…, n} is considered as a particle of R n and, consequently, the motion of the structure corresponds to the motion of the particle. Under the assumption that “the motion of every particle-structure at every time point is due to its interaction with its surroundings,” R n becomes a continuum (Tsaklidis, 1998). Then the evolution of the set of the attainable structures corresponds to the motion of the continuum. For the case of a three-state HMS it is stated that the concept of the two-dimensional isotropic elasticity can further be used to interpret the three-state HMS's evolution.

On the Distributions of the State Sizes of the Continuous Time Homogeneous Markov System with Finite State Capacities

In the present paper we study the evolution of a continuous time homogeneous Markov system whose states have finite capacities. We assume that the members of the system who overflow, due to the finite state capacities, leave the system. In order to investigate the variability of the state sizes we provide the evaluation of the intensity matrix for any time point and then we derive a formula concerning the derivate of the moments of the state sizes. As a consequence the distributions of the state sizes can be evaluated. Moreover, an alternative method of calculating the distributions of the state sizes by means of the interval transition probabilities is given. Finally we examine the distribution of the time needed for the leavers to leave the system. The theoretical results are illustrated by a numerical example.

On the Distributions of the State Sizes of Closed Continuous Time Homogeneous Markov Systems

The evolution of the state sizes of a closed continuous-time homogeneous Markov system is determined by the convolution of multinomial distributions expressing the number of transitions between the states of the system. In order to investigate the distributions of the state sizes, we provide the computation of their moments, at any time point, via a recursive formula concerning the derivative of the moments. The basic result is given by means of a new vector product which is similar to the Kronecker product. Finally, a formula for the computation of the state sizes distributions is given.

Modeling of continuous time homogeneous Markov system with fixed size as elastic solid: the two-dimensional case

Every attainable structure of a continuous time homogeneous Markov system (HMS) with n states and fixed size is considered as a point-particle of R n . Thus, the motion of an attainable structure corresponds to the motion of the respective point-particle in R n . Under the assumption that “the motion of every particle at every time point is due to its interaction with its surroundings”, R n becomes a continuum [Appl. Stochastic Models Data Anal. 13 (1998) 177]. The evolution of the set of the attainable structures corresponds to the motion of the continuum. For the particular case of a two-dimensional HMS ( n=2), it is examined to what extend the HMS could be considered as a linear elastic continuum (rod). Hence, it is concluded that the general concept of multidimensional anisotropic elasticity can further interpret the evolution of HMSs with n>3.

The size order of the state vector of a continuous-time homogeneous Markov system with fixed size

The variation of the state vectors p(t) = (pi(t)) of a continuous-time homogeneous Markov system with fixed size is examined. A specific time t0 after which the size order of the elements pi(t) becomes stable provides a criterion of the system's convergence rate. A method is developed to find t0 and a quickly evaluated lower bound for t0. This method is based on the geometric characteristics and the volumes of the attainable structures. Moreover, a condition concerning the selection of starting vectors p(0) is given so that the vector functions p(t) retain the same size order for every time greater than a given time t.

The size order of the state vector of a continuous-time homogeneous Markov system with fixed size

The variation of the state vectors p (t) = (p i (t)) of a continuous-time homogeneous Markov system with fixed size is examined. A specific time t 0 after which the size order of the elements p i (t) becomes stable provides a criterion of the system's convergence rate. A method is developed to find t 0 and a quickly evaluated lower bound for t 0. This method is based on the geometric characteristics and the volumes of the attainable structures. Moreover, a condition concerning the selection of starting vectors p (0) is given so that the vector functions p (t) retain the same size order for every time greater than a given time t.

The stress tensor and the energy of a continuous time homogeneous Markov system with fixed size

The stress tensor and the energy of a continuous time homogeneous Markov system with fixed size

The stress tensor and the energy of a continuous time homogeneous Markov system with fixed size

The set of the attainable structures of a continuous time homogeneous Markov system (HMS) with fixed size, is considered as a continuum and the evolution of the HMS in the Euclidean space corresponds to its motion. Taking account of the velocity field of the HMS, a suitable model of continuum–defined by its stress tensor–is proposed in order to explain the motion of the system. The adoption of this model (equivalently of its stress tensor) enables us to establish the concept of the energy of a structure of the HMS.

The stress tensor and the energy of a continuous time homogeneous Markov system with fixed size

The set of the attainable structures of a continuous time homogeneous Markov system (HMS) with fixed size, is considered as a continuum and the evolution of the HMS in the Euclidean space corresponds to its motion. Taking account of the velocity field of the HMS, a suitable model of continuum–defined by its stress tensor–is proposed in order to explain the motion of the system. The adoption of this model (equivalently of its stress tensor) enables us to establish the concept of the energy of a structure of the HMS.

The continuous-time homogeneous Markov system with fixed size as a Newtonian fluid

Given the velocity field of a continuous-time homogeneous Markov system (HMS) with fixed size, it is examined if the system could be considered as a continuum. The model examined is that of a Newtonian fluid. The set of attainable structures is the continuum and its evolution in Euclidean space corresponds to the motion of the continuum. It is proved that the Newtonian fluid assumption cannot generally explain the motion of the HMS. If it is possible, then we come to the conclusion that the ‘viscosity’ is a function of density and position. The dependence of viscosity on position implies that a fluid conforming to the HMS cannot be real. © 1998 John Wiley & Sons, Ltd.

The evolution of the attainable structures of a continuous time homogeneous Markov system with fixed size

In order to describe the evolution of the attainable structures of a continuous time homogeneous Markov system (HMS) with fixed size, we evaluate the volume of the sets of the attainable structures in Euclidean space in the course of time, and we find the value of the volume asymptotically. Then, using the concept of the volume of the attainable structures, we provide a method to evaluate the ‘age' of the system in continuous and discrete time. We also estimate the evolution of the distance of two (attainable) structures of the system as it changes following the transformations of the structures.

The evolution of the attainable structures of a continuous time homogeneous Markov system with fixed size

In order to describe the evolution of the attainable structures of a continuous time homogeneous Markov system (HMS) with fixed size, we evaluate the volume of the sets of the attainable structures in Euclidean space in the course of time, and we find the value of the volume asymptotically. Then, using the concept of the volume of the attainable structures, we provide a method to evaluate the ‘age' of the system in continuous and discrete time. We also estimate the evolution of the distance of two (attainable) structures of the system as it changes following the transformations of the structures.