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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)

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)

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)

- Nikolai K. Krivulin
- ArXiv
- 2012

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)

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.

- Nikolai K. Krivulin
- ArXiv
- 2011

Multidimensional minimax single facility location problems with Chebyshev distance are examined within the framework of idempotent algebra. A new algebraic solution based on an extremal property of the eigenvalues of irreducible matrices is given. The solution reduces both unconstrained and constrained location problems to evaluation of the eigenvalue and… (More)

- Nikolai K. Krivulin
- ArXiv
- 1996

A class of queueing networks which consist of single-server fork-join nodes with infinite 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 the network nodes in an explicit vector form… (More)

- Nikolai K. Krivulin
- ArXiv
- 2002

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)

- Nikolai K. Krivulin
- ArXiv
- 1995

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)

- Nikolai K. Krivulin
- ArXiv
- 2002

Simple lower and upper bounds on mean cycle time in stochastic 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. Key-Words:… (More)