Analysis of a Discrete-Time Queueing Model with Disasters

Queueing models with disasters can be used to evaluate the impact of a breakdown or a system reset in a service facility. In this paper, we consider a discrete-time single-server queueing system with general independent arrivals and general independent service times and we study the effect of the occurrence of disasters on the queueing behavior. Disasters occur independently from time slot to time slot according to a Bernoulli process and result in the simultaneous removal of all customers from the queueing system. General probability distributions are allowed for both the number of customer arrivals during a slot and the length of the service time of a customer (expressed in slots). Using a two-dimensional Markovian state description of the system, we obtain expressions for the probability, generating functions, the mean values, variances and tail probabilities of both the system content and the sojourn time of an arbitrary customer under a first-come-first-served policy. The customer loss probability due to a disaster occurrence is derived as well. Some numerical illustrations are given.

A Black Hole-Aided Deep-Helix Channel Model for DNA

In this article, we present a black-hole-aided deep-helix (bh-dh) channel model to enhance information bound and mitigate a multiple-helix directional issue in Deoxyribonucleic acid (DNA) communications. The recent observations of DNA do not match with Shannon bound due to their multiple-helix directional issue. Hence, we propose a bh-dh channel model in this paper. The proposed bh-dh channel model follows a similar fashion of DNA and enriches the earlier DNA observations as well as achieving a composite like information bound. To do successfully the proposed bh-dh channel model, we first define a black-hole-aided Bernoulli-process and then consider a symmetric bh-dh channel model. After that, the geometric and graphical insight shows the resemblance of the proposed bh-dh channel model in DNA and Galaxy layout. In our exploration, the proposed bh-dh symmetric channel geometrically sketches a deep-pair-ellipse when a deep-pair information bit or digit is distributed in the proposed channel. Furthermore, the proposed channel graphically shapes as a beautiful circulant ring. The ring contains a central-hole, which looks like a central-black-hole of a Galaxy. The coordinates of the inner-ellipses denote a deep-double helix, and the coordinates of the outer-ellipses sketch a deep-parallel strand. Finally, the proposed bh-dh symmetric channel significantly outperforms the traditional binary-symmetric channel and is verified by computer simulations in terms of Shannon entropy and capacity bound.

A Black Hole-Aided Deep-Helix Channel Model for DNA

In this article, we present a black-hole-aided deep-helix (bh-dh) channel model to enhance information bound and mitigate a multiple-helix directional issue in Deoxyribonucleic acid (DNA) communications. The recent observations of DNA do not match with Shannon bound due to their multiple-helix directional issue. Hence, we propose a bh-dh channel model in this paper. The proposed bh-dh channel model follows a similar fashion of DNA and enriches the earlier DNA observations as well as achieving a composite like information bound. To do successfully the proposed bh-dh channel model, we first define a black-hole-aided Bernoulli-process and then consider a symmetric bh-dh channel model. After that, the geometric and graphical insight shows the resemblance of the proposed bh-dh channel model in DNA and Galaxy layout. In our exploration, the proposed bh-dh symmetric channel geometrically sketches a deep-pair-ellipse when a deep-pair information bit or digit is distributed in the proposed channel. Furthermore, the proposed channel graphically shapes as a beautiful circulant ring. The ring contains a central-hole, which looks like a central-black-hole of a Galaxy. The coordinates of the inner-ellipses denote a deep-double helix, and the coordinates of the outer-ellipses sketch a deep-parallel strand. Finally, the proposed bh-dh symmetric channel significantly outperforms the traditional binary-symmetric channel and is verified by computer simulations in terms of Shannon entropy and capacity bound.

Generalized Bernoulli process: simulation, estimation, and application

A generalized Bernoulli process (GBP) is a stationary process consisting of binary variables that can capture long-memory property. In this paper, we propose a simulation method for a sample path of GBP and an estimation method for the parameters in GBP. Method of moments estimation and maximum likelihood estimation are compared through empirical results from simulation. Application of GBP in earthquake data during the years of 1800-2020 in the region of conterminous U.S. is provided.

Generalized Bernoulli process with long-range dependence and fractional binomial distribution

Bernoulli process is a finite or infinite sequence of independent binary variables, X i , i = 1, 2, · · ·, whose outcome is either 1 or 0 with probability P(X i = 1) = p, P(X i = 0) = 1 – p, for a fixed constant p ∈ (0, 1). We will relax the independence condition of Bernoulli variables, and develop a generalized Bernoulli process that is stationary and has auto-covariance function that obeys power law with exponent 2H – 2, H ∈ (0, 1). Generalized Bernoulli process encompasses various forms of binary sequence from an independent binary sequence to a binary sequence that has long-range dependence. Fractional binomial random variable is defined as the sum of n consecutive variables in a generalized Bernoulli process, of particular interest is when its variance is proportional to n 2 H , if H ∈ (1/2, 1).

Logarithmic Regret in the Dynamic and Stochastic Knapsack Problem with Equal Rewards

We study a dynamic and stochastic knapsack problem in which a decision maker is sequentially presented with items arriving according to a Bernoulli process over n discrete time periods. Items have equal rewards and independent weights that are drawn from a known nonnegative continuous distribution F. The decision maker seeks to maximize the expected total reward of the items that the decision maker includes in the knapsack while satisfying a capacity constraint and while making terminal decisions as soon as each item weight is revealed. Under mild regularity conditions on the weight distribution F, we prove that the regret—the expected difference between the performance of the best sequential algorithm and that of a prophet who sees all of the weights before making any decision—is, at most, logarithmic in n. Our proof is constructive. We devise a reoptimized heuristic that achieves this regret bound.

The Entropy Function for Non Polynomial Problems and Its Applications for Turing Machines

We present a general process for the halting problem, valid regardless of the time and space computational complexity of the decision problem. It can be interpreted as the maximization of entropy for the utility function of a given Shannon-Kolmogorov-Bernoulli process. Applications to non-polynomials problems are given. The new interpretation of information rate proposed in this work is a method that models the solution space boundaries of any decision problem (and non polynomial problems in general) as a communication channel by means of Information Theory. We described a sort method that order objects using the intrinsic information content distribution for the elements of a constrained solution space - modeled as messages transmitted through any communication systems. The limits of the search space are defined by the Kolmogorov-Chaitin complexity of the sequences encoded as Shannon-Bernoulli strings. We conclude with a discussion about the implications for general decision problems in Turing machines.

The Entropy Function for Non Polynomial Problems and Its Applications for Turing Machines

We present a general process for the halting problem, valid regardless of the time and space computational complexity of the decision problem. It can be interpreted as the maximization of entropy for the utility function of a given Shannon-Kolmogorov-Bernoulli process. Applications to non-polynomials problems are given. The new interpretation of information rate proposed in this work is a method that models the solution space boundaries of any decision problem (and non polynomial problems in general) as a communication channel by means of Information Theory. We described a sort method that order objects using the intrinsic information content distribution for the elements of a constrained solution space - modeled as messages transmitted through any communication systems. The limits of the search space are defined by the Kolmogorov-Chaitin complexity of the sequences encoded as Shannon-Bernoulli strings. We conclude with a discussion about the implications for general decision problems in Turing machines.