A Randomly Weighted Minimum Arborescence with a Random Cost Constraint

We study the minimum spanning arborescence problem on the complete digraph [Formula: see text], where an edge e has a weight We and a cost Ce, each of which is an independent uniform random variable Us, where [Formula: see text] and U is uniform [Formula: see text]. There is also a constraint that the spanning arborescence T must satisfy [Formula: see text]. We establish, for a range of values for [Formula: see text], the asymptotic value of the optimum weight via the consideration of a dual problem.

Debris Flow Risk Assessment Method Based on Combination Weight of Probability Analysis

Risk assessment of debris flow is conducted by multicriteria decisions. Based on the shortcomings of the existing methods in determining the weight of assessment factors, this paper proposes a new approach to conduct a risk assessment of debris flow. This new approach regards the weight of factors as a uniform random variable, whose bounds could be determined by the equal weight method, maximal deviation method, and entropy method. The results of this new approach are obtained by Monte Carlo simulation. According to the risk of 72 debris flows collected in Beichuan, Sichuan, China, this new approach proves convergent. It is suggested that the minimum sample amount of Monte Carlo simulation should be 63095. The result also demonstrates that sorted results with different weights of factors vary a lot, so it is not convincing to sort samples with a specific weight.

REGULARIZATION OF NON-NORMAL MATRICES BY GAUSSIAN NOISE—THE BANDED TOEPLITZ AND TWISTED TOEPLITZ CASES

We consider the spectrum of additive, polynomially vanishing random perturbations of deterministic matrices, as follows. Let$M_{N}$be a deterministic$N\times N$matrix, and let$G_{N}$be a complex Ginibre matrix. We consider the matrix${\mathcal{M}}_{N}=M_{N}+N^{-\unicode[STIX]{x1D6FE}}G_{N}$, where$\unicode[STIX]{x1D6FE}>1/2$. With$L_{N}$the empirical measure of eigenvalues of${\mathcal{M}}_{N}$, we provide a general deterministic equivalence theorem that ties$L_{N}$to the singular values of$z-M_{N}$, with$z\in \mathbb{C}$. We then compute the limit of$L_{N}$when$M_{N}$is an upper-triangular Toeplitz matrix of finite symbol: if$M_{N}=\sum _{i=0}^{\mathfrak{d}}a_{i}J^{i}$where$\mathfrak{d}$is fixed,$a_{i}\in \mathbb{C}$are deterministic scalars and$J$is the nilpotent matrix$J(i,j)=\mathbf{1}_{j=i+1}$, then$L_{N}$converges, as$N\rightarrow \infty$, to the law of$\sum _{i=0}^{\mathfrak{d}}a_{i}U^{i}$where$U$is a uniform random variable on the unit circle in the complex plane. We also consider the case of slowly varying diagonals (twisted Toeplitz matrices), and, when$\mathfrak{d}=1$, also of independent and identically distributed entries on the diagonals in$M_{N}$.

TEMPORAL SHAPING OF SIMULATED TIME SERIES WITH CYCLICAL SAMPLE PATHS

Temporal shaping of time series is the activity of deriving a time series model with a prescribed marginal distribution and some sample path characteristics. Starting with an empirical sample path, one often computes from it an empirical histogram (a step-function density) and empirical autocorrelation function. The corresponding cumulative distribution function is piecewise linear, and so is the inverse distribution function. The so-called inversion method uses the latter to generate the corresponding distribution from a uniform random variable on [0,1), histograms being a special case. This paper shows how to manipulate the inverse histogram and an underlying marginally uniform process, so as to “shape” the model sample paths in an attempt to match the qualitative nature of the empirical sample paths, while maintaining a guaranteed match of the empirical marginal distribution. It proposes a new approach to temporal shaping of time series and identifies a number of operations on a piecewise-linear inverse histogram function, which leave the marginal distribution invariant. For cyclical processes with a prescribed marginal distribution and a prescribed cycle profile, one can also use these transformations to generate sample paths which “conform” to the profile. This approach also improves the ability to approximate the empirical autocorrelation function.

On Some Compound Random Variables Motivated by Bulk Queues

We consider the distribution of the number of customers that arrive in an arbitrary bulk arrival queue system. Under certain conditions on the distributions of the time of arrival of an arriving group (Y(t)) and its size (X) with respect to the considered bulk queue, we derive a general expression for the probability mass function of the random variableQ(t)which expresses the number of customers that arrive in this bulk queue during any considered periodt. Notice thatQ(t)can be considered as a well-known compound random variable. Using this expression, without the use of generating function, we establish the expressions for probability mass function for some compound distributionsQ(t)concerning certain pairs(Y(t),X)of discrete random variables which play an important role in application of batch arrival queues which have a wide range of applications in different forms of transportation. In particular, we consider the cases whenY(t)and/orXare some of the following distributions: Poisson, shifted-Poisson, geometrical, or uniform random variable.