A dynamical system which is described in terms of an idempotent algebra by means of a vector equation with random irreducible matrix is considered. An approach based on approximation of the matrix of the system by means of matrices of simple structure is applied to evaluate bounds on the mean rate of growth of the state vector of the system. The process of… (More)
We consider a multidimensional extremal problem formulated in terms of tropical mathematics. The problem is to minimize a nonlinear objective function, which is defined on a finite-dimensional semimodule over an idempotent semifield, subject to linear inequality constraints. An efficient solution approach is developed which reduces the problem to that of… (More)
The problem on the solutions of homogeneous and nonhomogeneous generalized linear vector equations in idempotent algebra is considered. For the study of equations, an idempotent analog of matrix determinant is introduced and its properties are investigated. In the case of irreducible matrix, existence conditions are found and the general solutions of… (More)
New recursive equations designed for the G/G/m queue are presented. These equations describe the queue in terms of recursions for the arrival and departure times of customers, and involve only the operations of maximum, minimum and addition.
The eigenvalue problem for the mattix of a generalized linear operator is considered. In the case of irreducible mattices, the problem is reduced to the analysis of an idempotent analogue of the charactetistic polynomial of the mattix. The eigenvectors are obtained as solutions to a homogeneous equation. The results are then extended to cover the case of an… (More)
The application of the max-algebra to describe queueing systems by both linear scalar and vector equations is discussed. It is shown that these equations may be handled using ordinary algebraic manipulations. Examples of solving the equations representing the G/G/1 queue and queues in tandem are also presented.
A class of queueing networks which may have an arbitrary topology, and consist of single-server fork-join nodes with both infinite and finite buffers is examined to derive a representation of the network dynamics in terms of max-plus algebra. For the networks , we present a common dynamic state equation which relates the departure epochs of customers from… (More)
Max-algebra models of tandem single-server queueing systems with both finite and infinite buffers are developed. The dynamics of each system is described by a linear vector state equation similar to those in the conventional linear systems theory, and it is determined by a transition matrix inherent in the system. The departure epochs of a customer from the… (More)
Simple lower and upper bounds on mean cycle time in stochas-tic acyclic fork-join queueing networks are derived using a (max, +)-algebra based representation of network dynamics. The behaviour of the bounds under various assumptions concerning the service times in the networks is discussed, and related numerical examples are presented .
A linear vector equation is considered defined in terms of idempotent mathematics. To solve the equation, we apply an approach that is based on the analysis of distances between vectors in idempotent vector spaces and reduces the solution of the equation to that of a tropical optimization problem. Based on the approach, existence and uniqueness conditions… (More)