Search

• The algorithms can generate a Matrix-Geometric (MG) representation of the system lifetime distribution from its components lifetime distributions for any structures. • The resulting MG representation of the system lifetime distribution can be used to calculate several functional reliability measures at once. • The algorithms can be applied for any structures with discrete-time components. • The algorithms can reduce computation time and memory significantly. • The model can be generated using simple algorithms in MATLAB codes. • It applies to dynamic reliability analysis for many engineering systems. • The computation method is simple and fundamentals that they can be used to enrich course material on applied computations and system reliability. System reliability analysis in discrete phase-type (DPH) distributions requires longer computation time as the matrix order increases with the number of components and the complexity of the system structure. This paper presents a method to reduce the computation time by performing a similarity transformation on the DPH distribution model of the component lifetimes. Similarity transformation produces a matrix-geometric (MG) distribution whose generator matrix is in Jordan canonical form with fewer non-zero elements, so the computation time is faster. We modified the algorithms for system reliability in DPH distributions to make them applicable to MG distributions. Our experiments using several Jordan canonical forms show significant reductions in computation times. [ABSTRACT FROM AUTHOR]

The Liouvillian exceptional points for a quantum Markovian master equation of an oscillator in a generic environment are obtained. They occur at the points when the modified frequency of the oscillator vanishes, whereby the eigenvalues of the Liouvillian become real. In a generic system there are two parameters that modify the oscillator's natural frequency. One of the parameters can be the damping rate. The exceptional point then corresponds to critical damping of the oscillator. This situation is illustrated by the Caldeira–Leggett (CL) equation and the Markovian limit of the Hu–Paz–Zhang (HPZ) equation. The other parameter changes the oscillator's effective mass whereby the exceptional point is reached in the limit of extremely heavy oscillator. This situation is illustrated by a modified form of the Kossakowski–Lindblad (KL) equation. The eigenfunctions coalesce at the exceptional points and break into subspaces labelled by a natural number N. In each of the N -subspace, there is a (N + 1) -fold degeneracy and the Liouvillian has a Jordan block structure of order- (N + 1). We obtain the explicit form of the generalized eigenvectors for a few Liouvillians. Because of the degeneracies, there is a freedom of choice in the generalized eigenfunctions. This freedom manifests itself as an invariance in the Jordan block structure under a similarity transformation whose form is obtained. We compare the relaxation of the first excited state of an oscillator in the underdamped region, critically damped region which corresponds to the exceptional point, and overdamped region using the generalized eigenvectors of the CL equation. • For a natural frequency, exceptional points lie on a circle in a two-parameter space. • Exceptional points of arbitrary orders occur simultaneously. • Degeneracy permits a freedom of choice in the basis of generalized eigenvectors. • Freedom manifests as invariance of Jordan block under similarity transformation. [ABSTRACT FROM AUTHOR]