Reliability evaluation according to a routing scheme for multi-state computer networks under assured accuracy rate

In many real-time networks such as computer networks, each arc has stochastic capacity, lead time, and accuracy rate. Such a network is named a multi-state computer network (MSCN). Under the strict assumption that the capacity of each arc is deterministic, the quickest path (QP) problem is to find a path that sends a specific amount of data with minimum transmission time. From the viewpoint of internet quality, the transmission accuracy rate is one of critical performance indicators to assess internet network for system administrators and customers. Under both assured accuracy rate and time constraint, this paper extends the QP problem to discuss the flow assignment for a MSCN. An efficient algorithm is proposed to find the minimal capacity vector meeting such requirements. The system reliability, the probability to send \(d\) units of data through multiple minimal paths under both assured accuracy rate and time constraint, can subsequently be computed. Furthermore, two routing schemes with spare minimal paths are adopted to reinforce the system reliability. The enhanced system reliability according to the routing scheme is calculated as well. The computational complexity in both the worst case and average case are analyzed.     

Time version of the shortest path problem in a stochastic-flow network

Many studies on hardware framework and routing policy are devoted to reducing the transmission time for a flow network. A time version of the shortest path problem thus arises to find a quickest path, which sends a given amount of data from the unique source to the unique sink with minimum transmission time. More specifically, the capacity of each arc in the flow network is assumed to be deterministic. However, in many real-life networks, such as computer systems, telecommunication systems, etc., the capacity of each arc should be stochastic due to failure, maintenance, etc. Such a network is named a stochastic-flow network. Hence, the minimum transmission time is not a fixed number. We extend the quickest path problem to evaluating the probability that d$d$ units of data can be sent under the time constraint T$T$. Such a probability is named the system reliability. In particular, the data are transmitted through two minimal paths simultaneously in order to reduce the transmission time. A simple algorithm is proposed to generate all (d,T)$(d,T)$-MPs and the system reliability can then be computed in terms of (d,T)$(d,T)$-MPs. Moreover, the optimal pair of minimal paths with highest system reliability could be obtained.     

On transmission time through k minimal paths of a capacitated-flow network

The quickest path problem is to minimize the transmission time for sending a specified amount of data through a single minimal path. Two deterministic attributes are involved herein; the capacity and the lead time. However, in many real-life networks such as computer systems, urban traffic systems, etc., the arc capacity should be multistate due to failure, maintenance, etc. Such a network is named a capacitated-flow network. The minimum transmission time is thus not a fixed number. This paper is mainly to evaluate system reliability that d units of data can be transmitted through k minimal paths under time constraint T. A simple algorithm is proposed to generate all minimal capacity vectors meeting the demand and time constraints. The system reliability is subsequently computed in terms of such vectors. The optimal k minimal paths with highest system reliability can further be derived.     

Calculation of minimal capacity vectors through k minimal paths under budget and time constraints

Reducing the transmission time is an important issue for a flow network to transmit a given amount of data from the source to the sink. The quickest path problem thus arises to find a single path with minimum transmission time. More specifically, the capacity of each arc is assumed to be deterministic. However, in many real-life networks such as computer networks and telecommunication networks, the capacity of each arc is stochastic due to failure, maintenance, etc. Hence, the minimum transmission time is not a fixed number. Such a network is named a stochastic-flow network. In order to reduce the transmission time, the network allows the data to be transmitted through k minimal paths simultaneously. Including the cost attribute, this paper evaluates the probability that d units of data can be transmitted under both time threshold T and budget B. Such a probability is called the system reliability. An efficient algorithm is proposed to generate all of lower boundary points for (d, T, B), the minimal capacity vectors satisfying the demand, time, and budget requirements. The system reliability can then be computed in terms of such points. Moreover, the optimal combination of k minimal paths with highest system reliability can be obtained.     

A single server queueing system with two phases of service subject to server breakdown and Bernoulli vacation

This paper deals with a single server M/G/1 queue with two phases of heterogeneous service and unreliable server. We assume that customers arrive to the system according to a Poisson process with rate λ. After completion of two successive phases of service the server either goes for a vacation with probability p(0 ? p ? 1) or may continue to serve the next unit, if any, with probability q(=1 ? p). Otherwise it remains in the system until a customer arrives. While the server is working with any phase of service, it may breakdown at any instant and the service channel will fail for a short interval of time. For this model, we first derive the joint distribution of state of the server and queue size, which is one of the chief objectives of the paper. Secondly, we derive the probability generating function of the stationary queue size distribution at a departure epoch. Next, we derive Laplace Stieltjes transform of busy period distribution and waiting time distribution. Finally we obtain some important performance measures and reliability indices of this model.     

A Bijective Approach to the Area of Generalized Motzkin Paths

For fixed positive integer k, let E_n denote the set of lattice paths using the steps (1, 1), (1, − 1), and (k, 0) and running from (0, 0) to (n, 0) while remaining strictly above the x-axis elsewhere. We first prove bijectively that the total area of the regions bounded by the paths of E_n and the x-axis satisfies a four-term recurrence depending only on k. We then give both a bijective and a generating function argument proving that the total area under the paths of E_n equals the total number of lattice points on the x-axis hit by the unrestricted paths running from (0, 0) to (n − 2, 0) and using the same step set as above.     

Cost optimization of a repairable M/G/1 queue with a randomized policy and single vacation

This paper deals with the (p, N)-policy M/G/1 queue with an unreliable server and single vacation. Immediately after all of the customers in the system are served, the server takes single vacation. As soon as N customers are accumulated in the queue, the server is activated for services with probability p or deactivated with probability (1 − p). When the server returns from vacation and the system size exceeds N, the server begins serving the waiting customers. If the number of customers waiting in the queue is less than N when the server returns from vacation, he waits in the system until the system size reaches or exceeds N. It is assumed that the server is subject to break down according to a Poisson process and the repair time obeys a general distribution. This paper derived the system size distribution for the system described above at a stationary point of time. Various system characteristics were also developed. We then constructed a total expected cost function per unit time and applied the Tabu search method to find the minimum cost. Some numerical results are also given for illustrative purposes.     

Reliability of multi-component stress–strength models

In this paper, we estimate the reliability of some parallel and series multi-component stress–strength models. We determine the reliability of a system composed of k dependent components subjected to n dependent stresses. We study the cases, when the components are either arranged in series or in parallel. The components strengths are assumed to have (k + 1)-parameter multivariate Marshall–Olkin exponential distribution, while the stresses are (n + 1)-parameter multivariate Marshall–Olkin exponentially distributed.     

Extended optimal age-replacement policy with minimal repair of a system subject to shocks

An operating system is subject to shocks that arrive according to a non-homogeneous Poisson process. As shocks occur the system has two types of failure: type I failure (minor) or type II failure (catastrophic). A generalization of the age replacement policy for such a system is proposed and analyzed in this study. Under such a policy, if an operating system suffers a shock and fails at age y (⩽t), it is either replaced by a new system (type II failure) or it undergoes minimal repair (type I failure). Otherwise, the system is replaced when the first shock after t arrives, or the total operating time reaches age T (0 ⩽ t ⩽ T), whichever occurs first. The occurrence of those two possible actions occurring during the period [0, t] is based on some random mechanism which depends on the number of shocks suffered since the last replacement. The aim of this paper is to find the optimal pair (t¹, T¹) that minimizes the long-run expected cost per unit time of this policy. Various special cases are included, and a numerical example is given.     

An algebraic equation for linear operators

Let T be a linear operator on a vector space V, possibly of infinite dimension, over a general field K. We solve the functional equation p(T) = F where p ∈ K[x] and F, an algebraic operator on V, are given. For nilpotent F we give an explicit linear system which determines the solutions by their similarity classes. The method is based on a canonical decomposition theorem.     

Computation of the solutions of the Fokker–Planck equation for one and two DOF systems

Uncertainty in structures may come from unknowns in the modelisation and in the properties of the materials, from variability with time, external noise, etc. This leads to uncertainty in the dynamic response. Moreover, the consequences are issues in safety, reliability, efficiency, etc. of the structure. So an issue is the gain of information on the response of the system taking into account the uncertainties [Mace BR, Worden K, Manson G. Uncertainty in structural dynamics. J Sound Vib 2005;288(3):423–9].If the forcing or the uncertainty can be modelled through a white noise, the Fokker–Planck (or Kolmogorov forward) equation exists. It is a partial differential linear equation with unknown p(X, t), where p(X, t) is the probability density function of the state X at time t.In this article, we solve this equation using the finite differences method, for one and two DOF systems. The numerical solutions obtained are proved to be nearly correct.     

Special Functions on the Sphere with Applications to Minimal Surfaces

A function which is homogeneous in x, y, z of degree n and satisfies V_xx + V_yy + V_zz = 0 is called a spherical harmonic. In polar coordinates, the spherical harmonics take the form rⁿf_n, where f_n is a spherical surface harmonic of degree n. On a sphere, f_n satisfies ▵ f_n + n(n + 1)f_n = 0, where ▵ is the spherical Laplacian. Bounded spherical surface harmonics are well studied, but in certain instances, unbounded spherical surface harmonics may be of interest. For example, if X is a parameterization of a minimal surface and n is the corresponding unit normal, it is known that the support function, w = X · n, satisfies ▵w + 2w = 0 on a branched covering of a sphere with some points removed. While simple in form, the boundary value problem for the support function has a very rich solution set. We illustrate this by using spherical harmonics of degree one to construct a number of classical genus-zero minimal surfaces such as the catenoid, the helicoid, Enneper's surface, and Hennenberg's surface, and Riemann's family of singly periodic genus-one minimal surfaces.     

Existence and asymptotics of traveling waves for nonlocal diffusion systems

In this paper, we deal with the existence and asymptotic behavior of traveling waves for nonlocal diffusion systems with delayed monostable reaction terms. We obtain the existence of traveling wave front by using upper-lower solutions method and Schauder’s fixed point theorem for c > c₁(τ) and using a limiting argument for c = c₁(τ). Moreover, we find a priori asymptotic behavior of traveling waves with the help of Ikehara’s Theorem by constructing a Laplace transform representation of a solution. Especially, the delay can slow the minimal wave speed for ?₂f(0, 0) > 0 and the delay is independent of the minimal wave speed for ?₂f(0, 0) = 0.     

Effects of limited interactions between individuals on cooperation in spatial evolutionary prisoner’s dilemma game

We study the spatial evolutionary prisoner’s dilemma game with limited interactions by introducing two kinds of individuals, say type-A and type-B with a fraction of p and (1 − p), respectively, distributed randomly on a square lattice. Each kind of individuals can adopt two pure strategies: either to cooperate or to defect. During the evolution, the individuals can only interact with others belonging to the same kind, but they can learn from either kinds of individuals in the nearest neighborhood. Using Monte Carlo simulations, the average frequency of cooperators ρ_C is calculated as a function of p in the equilibrium state. It is shown that, compared with the case of p = 0 (only one kind of individuals existing in the system), cooperation can be evidently promoted. In particular, the cooperator density can reach a maximum level at some moderate values of p in a wide range of payoff parameters. The results imply that certain limited interactions between individuals plays an important and nontrivial role in the evolution of cooperation.     

Equivalent Blocks of Finite General Linear Groups in Non-describing Characteristic

J. Chuang, R. Kessar, and J. Rickard have proved Broué's Abelian defect group conjecture for many symmetric groups. We adapt the ideas of Kessar and Chuang towards finite general linear groups (represented over non-describing characteristic). We then describe Morita equivalences between certain p-blocks of GL_n(q) with defect group C_p^α × C_p^α, as q varies (see Theorem 2). Here p and q are coprime. This generalizes work of S. Koshitani and M. Hyoue, who proved the same result for principal blocks of GL_n(q) when p = 3, α = 1, in a different way.     

Analysis of R out of N systems with several repairmen,exponential life times and phase type repair times: An algorithmic approach

An R out of N repairable system consisting of N independent components is operating if at least R components are functioning. The system fails whenever the number of good components decreases from R to R − 1. A failed component is sent to a repair facility having several repairmen. Life times of working components are i.i.d random variables having an exponential distribution. Repair times are i.i.d random variables having a phase type distribution. Both cold and warm stand-by systems are considered. We present an algorithm deriving recursively in the number of repairmen the generator of the Markov process that governs the process. Then we derive formulas for the point availability, the limiting availability, the distribution of the down time and the up time. Numerical examples are given for various repair time distributions. The numerical examples show that the availability is not very sensitive to the repair time distribution while the mean up time and the mean down time might be very sensitive to the repair time distributions.     

Simulación numérica del flujo turbulento de aire con gotas dispersas de agua a través de separadores de torres de refrigeración

This work presents a numerical study on the turbulent flow of air with dispersed water droplets in separators of mechanical cooling towers. The averaged Navier-Stokes equations are discretised through a finite volume method, using the Fluent and Phoenics codes, and alternatively employing the turbulence models k ? ?, k ? ω and the Reynolds stress model, with low-Re version and wall enhanced treatment refinements. The results obtained are compared with numerical and experimental results taken from the literature. The degree of accuracy obtained with each of the considered models of turbulence is stated. The influence of considering whether or not the simulation of the turbulent dispersion of droplets is analyzed, as well as the effects of other relevant parameters on the collection efficiency and the coefficient of pressure drop. Focusing on four specific eliminators (‘Belgian wave’, ‘H1-V’, ‘L-shaped’ and ‘Zig-zag’), the following ranges of parameters are outlined: 1 ≤ U_e ≤ 5 m/s for the entrance velocity, 2 ≤ D_p ≤ 50 μm for the droplet diameter, 650 ≤ Re ≤ 8.500 for Reynolds number, and 0.05 ≤ P_i ≤ 5 for the inertial parameter. Results reached alternately with Fluent and Phoenics codes are compared. The best results correspond to the simulations performed with Fluent, using the SST k ? ω turbulence model, with values of the dimensionless scaled distance to wall y⁺ in the range 0.2 to 0.5. Finally, correlations are presented to predict the conditions for maximum collection efficiency (100 %), depending on the geometric parameter of removal efficiency of each of the separators, which is introduced in this work.     

An interval nonlinear program for the planning of waste management systems with economies-of-scale effects—A case study for the region of Hamilton,Ontario, Canada

In practical environmental systems with the effects of economies-of-scale (EOS), most relationships among different system components are nonlinear in nature, which can be described precisely only if a nonlinear model is employed. In this study, an interval nonlinear programming (INLP) model is developed and applied to the planning of a municipal solid waste (MSW) management system with EOS effects on system costs. The INLP has a nonlinear objective function and linear constraints. It handles nonlinearity presented as exponential functions. When exponential term p = 1 (in the INLP’s objective function), the model becomes an interval linear program; when p = 2, it becomes an interval quadratic program. Therefore, the INLP is flexible in reflecting a variety of system complexities. A solution algorithm with satisfactory performance is proposed. Application of the proposed method to the planning of waste management activities in the Hamilton-Wentworth Region, Ontario, Canada, indicated that reasonable solutions have been generated. In general, the INLP model could reflect uncertain and nonlinear characteristics of MSW management systems with EOS effects. The modeling results provided useful decision support for the Region’s waste management activities.     

The Exponent of Discrepancy Is at Least 1.0669

LetP⊂[0, 1]^dbe ann-point set and letw: P→[0, ∞) be a weight function withw(P)=∑_z∈P w(z)=1. TheL₂-discrepancy of the weighted set (P, w) is defined as theL₂-average ofD(x)=vol(B_x)−w(P∩B_x) overx∈[0, 1]^d, where vol(B_x) is the volume of thed-dimensional intervalB_x=∏^d_k=1 [0, x_k). The exponent of discrepancyp* is defined as the infimum of numberspsuch that for all dimensionsd⩾1 and allε>0 there exists a weighted set of at mostKε^−ppoints in [0, 1]^dwithL₂-discrepancy at mostε, whereK=K(p) is a suitable number independent ofεandd. Wasilkowski and Woźniakowski proved thatp*⩽1.4779, by combining known bounds for the error of numerical integration and using their relation toL₂-discrepancy. In this note we observe that a careful treatment of a classical lower- bound proof of Roth yieldsp*⩾1.04882, and by a slight modification of the proof we getp*⩾1.0669. Determiningp* exactly seems to be quite a difficult problem.     

A semi-Markov process with an inverse Gaussian distribution as sojourn time

Let X_n denote the state of a device after n repairs. We assume that the time between two repairs is the time τ taken by a Wiener process {W(t), t ? 0}, starting from w₀ and with drift μ < 0, to reach c ∈ [0, w₀). After the nth repair, the process takes on either the value X_n?1 + 1 or X_n?1 + 2. The probability that X_n = X_n?1 + j, for j = 1, 2, depends on whether τ ? t₀ (a fixed constant) or τ > t₀. The device is considered to be worn out when X_n ? k, where k ∈ {1, 2, …}. This model is based on the ones proposed by Rishel (1991) [1] and Tseng and Peng (2007) [2]. We obtain an explicit expression for the mean lifetime of the device. Numerical methods are used to illustrate the analytical findings.