The well-known Marshall–Olkin model is known for its extension of exponential distribution preserving lack of memory property. Based on shock models, a new generalization of the bivariate Marshall–Olkin exponential distribution is given. The proposed model allows wider range tail dependence which is appealing in modeling risky events. Moreover, a stochastic comparison according to this shock model and also some properties, such as association measures, tail dependence and Kendall distribution, are presented. The new shock model is analytically quite tractable, and it can be used quite effectively, to analyze discrete–continuous data. This has been shown on real data. Finally, we propose the multivariate extension of the Marshall–Olkin model that has some intersection with the well-known multivariate Archimax copulas.

A probabilistic generative network model with $n$ nodes and $m$ overlapping layers is obtained as a superposition of $m$ mutually independent Bernoulli random graphs of varying size and strength. When $n$ and $m$ are large and of the same order of magnitude, the model admits a sparse limiting regime with a tunable power-law degree distribution and nonvanishing clustering coefficient. In this article, we prove an asymptotic formula for the joint degree distribution of adjacent nodes. This yields a simple analytical formula for the model assortativity and opens up ways to analyze rank correlation coefficients suitable for random graphs with heavy-tailed degree distributions. We also study the effects of power laws on the asymptotic joint degree distributions.

Artificial Neural Networks (ANNs)-based techniques have dominated state-of-the-art results in most problems related to computer vision, audio recognition, and natural language processing in the past few years, resulting in strong industrial adoption from all leading technology companies worldwide. One of the major obstacles that have historically delayed large-scale adoption of ANNs is the huge computational and power costs associated with training and testing (deploying) them. In the mean-time, Neuromorphic Computing platforms have recently achieved remarkable performance running the bio-realistic Spiking Neural Networks at high throughput and very low power consumption making them a natural alternative to ANNs. Here, we propose using the Random Neural Network, a spiking neural network with both theoretical and practical appealing properties, as a general purpose classifier that can match the classification power of ANNs on a number of tasks while enjoying all the features of being a spiking neural network. This is demonstrated on a number of real-world classification datasets.

This paper analyzes a scheduling system where a fixed number of nonpreemptive jobs is to be processed on multiple parallel processors with different processing speeds. Each processor has an exponential processing time distribution and the processors are ordered in ascending order of their mean processing times. Each job has its own deadline that is exponentially distributed with rate ß1, independent of the deadlines of other jobs and also independent of job processing times. A job departs the system as soon as either its processing completes or its deadline occurs. We show that there exists a simple threshold strategy that slochastically minimizes the total delay of all jobs. The policy depends on distributions of processing times and deadlines, but is independent of the rate of deadlines. When the rate of the deadline distribution is 0 (no deadlines), the total delay reduces to the flowtime (the sum of completion times of all jobs). If each job has its own probability of being correctly processed, then an extension of this policy stochastically maximizes the total number of correctly processed, nontardy jobs. We discuss possible generalizations and limitations of this result.

We consider M/G/1-type Markov chains where a transition that decreases the value of the level triggers the phase to a small subset of the phase space. We show how this structure—referred to as restricted downward transitions—can be exploited to speed up the computation of the stationary probability vector of the chain. To this end we define a new M/G/1-type Markov chain with a smaller block size, the G matrix of which is used to find the original chain's G matrix. This approach is then used to analyze the BMAP/PH/1 queue and the BMAP[2]/PH[2]/1 preemptive priority queue, yielding significant reductions in computation time.

We consider the filtering problem for a doubly stochastic Poisson or Cox process, where the intensity follows the Cox–Ingersoll–Ross model. In this article we assume that the Brownian motion, which drives the intensity, is not observed. Using filtering theory for point process observations, we first derive the dynamics for the intensity and its moment-generating function, given the observations of the Cox process. A transformation of the dynamics of the conditional moment-generating function allows us to solve in closed form the filtering problem, between the jumps of the Cox process as well as at the jumps, which constitutes the main contribution of the article. Assuming that the initial distribution of the intensity is of the Gamma type, we obtain an explicit solution to the filtering problem for all t>0. We conclude the article with the observation that the resulting conditional moment-generating function at time t, after Nt jumps, corresponds to a mixture of Nt+1 Gamma distributions. Currently, the model that we analyze has become popular in credit risk modeling, where one uses the intensity-based approach for the modeling of default times of one or more companies. In this approach, the default times are defined as the jump times of a Cox process. In such a model, one only has access to observations of the Cox process, and thus filtering comes in as a natural technique in credit risk modeling.

In this paper, we investigate executive stock options with endogenous departure and time-varying variances. We use a “Generalized Autoregressive Conditional Heteroskedasticity” process to capture the variance process of the log stock price. In addition, we take into consideration the departure risk of the executive and assume that the probability of remaining employed has a power form of stock price ratios. After deriving the closed-form pricing formulae of executive stock options, we illustrate the effects of the departure risk on the values of executive stock options.

A large lot is to be sampled by a number of inspectors for the purpose of ascertaining its number of defective units. However, instead of a defective unit always being detected if it is sampled, there are unknown probabilities, depending on the inspectors, that the unit will not be identified as a defective. We offer an approach to estimating the number of defective units in the lot.

This paper deals with the mean residual life function (MRLF) and its monotonicity in the case of additive and multiplicative hazard rate models. It is shown that additive (multiplicative) hazard rate does not imply reduced (proportional) MRLF and vice versa. Necessary and sufficient conditions are obtained for the two models to hold simultaneously. In the case of non-monotonic failure rates, the location of the turning points of the MRLF is investigated in both the cases. The case of random additive and multiplicative hazard rate is also studied. The monotonicity of the mean residual life is studied along with the location of the turning points. Examples are provided to illustrate the results.

For the birth–death process on a finite state space with bilateral boundaries, we give a simpler derivation of the hitting time distributions by h-transform and φ-transform. These transforms can then be used to construct a quick derivation of the hitting time distributions of the minimal birth–death process on a denumerable state space with exit/regular boundaries.

In this paper, we study the instant availability A(t) of a repairable system by integral equation. We have proved initial monotonicity of availability, and derived lower bounds of A(t) and average availability. The availabilities of two systems are compared.

We discuss a rich family of directed series–parallel (SP) graphs grown by the simultaneous random series or parallel development of multiple edges. The family portrays a spectrum that spans a wide range of SP graphs: from simple models, where only as few as one edge is chosen for evolution at each discrete point in time, to complex hierarchical lattice networks grown by a take-all strategy, where all the edges in the existing network are developed.

The family of SP graphs we discuss is grown from an initial seed graph with τ0 edges under an arbitrary building sequence, $\{k_{n}\}_{n=1}^{\infty}$, of nonnegative integers (with $k_n \le \tau _0 + \sum\nolimits_{i = 1}^n {k_i} $, for arbitrary τ0 ≥ 1), that specifies the number of edges subjected to evolution at time n. We study the average north polar degree and show that we can go beyond averages to strong laws. We also find the exact average number of critical edges. The asymptotics of the critical edges are facilitated under the regularity condition that $k_n/\sum\nolimits_{i = 1}^n {k_i} $ converges to a constant (as n → ∞), a natural condition easily met by practical strategies, such as single-edge evolution and take-all choice, and much in between.

One of the most fundamental results in inventory theory is the optimality of (s, S) policy for inventory systems with setup cost. This result is established under a key assumption of infinite ordering/production capacity. Several studies have shown that, when the ordering/production capacity is finite, the optimal policy for the inventory system with setup cost is very complicated and indeed, only partial characterization for the optimal policy is possible. In this paper, we consider a continuous review production/inventory system with finite capacity and setup cost. The demand follows a Poisson process and a demand that cannot be satisfied upon arrival is backlogged. We show that the optimal control policy has a very simple structure when the holding/shortage cost rate is quasi-convex. We also develop efficient algorithms to compute the optimal control parameters.

De Bruijn's identity shows a link between two fundamental concepts in information theory: entropy and Fisher information. In the literature, De Bruijn's identity has been stated under the assumption of independence between input signal and an additive noise. However, in the real world, the noise could be highly dependent on the main signal. The main aim of this paper is, firstly, to extend De bruijn's identity for signal-dependent noise channels and, secondly, to study how Stein and heat identities are related to De bruijn's identity. Thus, new versions of De Bruijn's identity are introduced in the case when input signal and additive noise are dependent and are jointly distributed according to Archimedean and Gaussian copulas. It is shown that in this generalized model, the derivatives of the differential entropy can be expressed in terms of a function of Fisher information. Our finding enfolds the conventional De Bruijn's identity as some remarks. Then, the equivalence among the new De Bruijn-type identity, Stein's identity and heat equation identity is established. The paper concludes with an application of the developed results in information theory.

Let X be a random vector and A a matrix. Let M be the maximal coordinate of the vector AX. For given marginal distributions of the coordinates of X, we present sharp bounds on the expectations of convex increasing functions of M. We derive joint distributions of X that achieve some of these bounds, and under these “worst case” distributions we study the joint distribution of M and the index of the largest coordinate of AX. Some possible applications are PERT network analysis and design of experiments.

We give a new bound on the spectral gap of an ergodic time reversible Markov chain in term of the conductance of the chain.

In the present article, we consider a class of reliability structures that can be efficiently described through a finite Markov chain (Markov chain imbeddable systems) and investigate its closeness with respect to the increasing failure rate (IFR) property. More specifically we derive a sufficient condition for the system's lifetime to have increasing failure rate when the identical and independent components comprising it own this property. As an application of the general theory, we establish an alternative proof of the IFR property preservation for the k-out-of-n system and derive some related results for the family of weighted k-out-of-n systems.

An efficient simulation estimator of the expected time until all states of a finite state semi-Markov process have been visited is determined.

The classical regenerative method of simulation output analysis exploits the regenerative structure of a stochastic process to break up a path into independent and identically distributed cycles based on a single sequence of regeneration times. If a process is regenerative with respect to more than one sequence of regeneration times, the classical regenerative method does not exploit the additional structure, and the variance of the resulting estimator for certain performance measures (e.g., the time-average variance constant) can vary greatly, depending on the particular regeneration sequence chosen. In a previous article, we introduced an efficiency-improvement technique for regenerative simulation of processes having two sequences of regeneration times based on permuting regenerative cycles associated with the second sequence of regeneration points. In this article, we show how to exploit more than two regeneration sequences. In particular, for birth–death Markov chains, the regenerations associated with hitting times to each state can all be exploited. We present empirical results that show significant variance reductions in some cases, and the results seem to indicate that the permuted estimator for the time-average variance constant can have a variance that is independent of the primary regeneration sequence used to run the simulation.

Mixture distributions are a frequently used tool in modelling random phenomena. We consider mixtures of densities from a one-parameter exponenvial family of distributions. Using the tools of totally positive functions and the variation-diminishing property of such, we study the effect of sign-crossing properties of two mixing densities μ1 and μ2 on the resulting mixture distributions f1 and f2. The results enable us to make stochastic and variability cornparisons for binomial-beta, mixed Weibull, and mixed gamma distributions.