A One-step Method of Order 10 for y' = f(x, y)

In some situations, especially if one demands the solution ofthe differential equation with a great precision, it is preferableto use high-order methods. The methods considered here are similarto Runge—Kutta methods, but for the second-order equationy'= f(x, y). As for Runge—Kutta methods, the complexityof the order conditions grows rapidly with the order, so thatwe have to solve a non—linear system of 440 algebraicequations to obtain a tenth—order method. We demonstratehow this system can be solved. Finally we give the coefficients(20 decimals) of two methods with small local truncation errors.     

Shape-preserving interpolation by G1 and G2 PH quintic splines

The interpolation of a planar sequence of points p₀, ..., p_Nby shape-preserving G¹ or G² PH quintic splines with specifiedend conditions is considered. The shape-preservation propertyis secured by adjusting ‘tension’ parameters thatarise upon relaxing parametric continuity to geometric continuity.In the G² case, the PH spline construction is based on applyingNewton–Raphson iterations to a global system of equations,commencing with a suitable initialization strategy—thisgeneralizes the construction described previously in NumericalAlgorithms 27, 35–60 (2001). As a simpler and cheaperalternative, a shape-preserving G¹ PH quintic spline schemeis also introduced. Although the order of continuity is lower,this has the advantage of allowing construction through purelylocal equations.     

Percolation and limit theory for the poisson lilypond model

The lilypond model on a point process in d ‐space is a growth‐maximal system of non‐overlapping balls centred at the points. We establish central limit theorems for the total volume and the number of components of the lilypond model on a sequence of Poisson or binomial point processes on expanding windows. For the lilypond model over a homogeneous Poisson process, we give subexponentially decaying tail bounds for the size of the cluster at the origin. Finally, we consider the enhanced Poisson lilypond model where all the balls are enlarged by a fixed amount (the enhancement parameter), and show that for d > 1 the critical value of this parameter, above which the enhanced model percolates, is strictly positive. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Optimal smoothing of accounting earnings

Accountants seeking to estimate the profitability of a firmvia the calculation of earnings utilize information about thenumber and current lifespan of unfinished activities by incorporatinga smoothing device—depreciation—into their calculations.This paper considers a firm in steady state, carrying out alarge number of similar activities with random starts and completiondates. By developing a stochastic model for the firm's activities,a difference equation for the minimum-variance smoothing functionis obtained. This can be solved explicitly in the case of aperiodic Poisson process of start times and independent exponentialdurations and is a variant of declining-balance depreciation.     

On the Evaluation of Cost Functionals, with Particular Emphasis on Time-delay Systems

It is useful to have closed-form expressions for the cost functional where e(t) is a function representing errorin a control system. The paper shows how the well-known resultsfor delay-free systems may be extended to the time-delay caseand a survey of various methods is given. Finally, an alternativederivation of the Routh—Hurwitz stability criteria forthe delay-free case is given. This makes use of the close relationshipbetween the above integrals and stability theory.     

The Limiting Behavior for Observations That Change with Time

Consider a system where units have random magnitude entering according to a homogeneous or nonhomogeneous Poisson process, while in the system, a unit's magnitude may change with time. In this paper, the authors obtain some results for the limiting behavior of the sum process of all unit magnitudes present in the system at time t.     

带有故障的休假排队模型

研究了一个复合的休假排队模型,工作或休假时服务台都有可能故障,服务台一旦修好可立即进行服务,而且每个忙期结束就开始一次休假,顾客到达服从Poisson过程,到达率依赖于系统状态,修理时间、服务时间和休假长度都服从指数分布.给出了系统状态的平衡方程,利用概率母函数求出队长,并做了数值分析.     

Analysis of the M [X]/G/1 queues with second multi-optional service and unreliable server

A bulk-arrival single server queueing system with second multi-optional service and unreliable server is studied in this paper. Customers arrive in batches according to a homogeneous Poisson process, all customers demand the first ＂essential＂ service, whereas only some of them demand the second ＂multi-optional＂ service. The first service time and the second service all have general distribution and they are independent. We assume that the server has a service-phase dependent, exponentially distributed life time as well as a servicephase dependent, generally distributed repair time. Using a supplementary variable method, we obtain the transient and the steady-state solutions for both queueing and reliability measures of interest.     

The maintenance and availability of unidirectionally patrolled stations

This paper models the problem of the maintenance of N identicalstations looked after by one operative who patrols the stationsunidirectionally in the order 1, 2, ..., N, 1, 2,... The traveltime between adjacent stations is a random variable, identicallyand independently distributed for each adjacent pair. The timetaken to repair station i, should it be found in a failed state,is a random variable, independently and identically distributedfor each i. In addition it is assumed that is the probabilitythat a repair attempt is successful. It is further assumed thateach station fails at random at an average in running time. It is shown that the important characteristics of the system—responsetime, waiting time, and availability for example—all dependon the quantity , the probability that a station is found inthe failed state. The model for the system shows how can befound from the values of N,i,, and the Laplace transforms ofthe travel-time and repair-time variables.     

A model for maintenance service contract design,negotiation and optimization

A model is presented in this paper for maintenance service contract design, negotiation and optimization. The model was developed under the assumption that there are one customer and one unique service provider who is the Original Equipment Manufacturer (OEM) and is called the agent in this paper. This is typically applied to the situation where the OEM is the only possible service supplier such as in the case of major military equipment in the defense sector. Three contract options were considered, depending on the extent of outsourced maintenance activities. From an agent point of view, they are, (1), the agent carries out all repairs and inspections; (2), the agent carries out failure based repairs, and (3), the agent does inspections and repairs to the defects identified at inspections. For options two and three, the customer does the rest of maintenance. The relationship between inspections and failures was modeled using the delay time concept and a numerical example was illustrated. The cases of perfect information to both parties and information asymmetry were also discussed in the example. The model developed can be used for contract design, negotiation and optimization.     

On the Time-Dependent Delta-Shock Model Governed by the Generalized PóLya Process

One of the widely discussed in the literature and relevant in practice shock models is the delta-shock model that is described by the constant time of a system’s recovery after a shock. However, in practice, as time progresses and due to deterioration of a system, this recovery time is gradually increasing. This important phenomenon was not discussed in the literature so far. Therefore, in this paper, we are considering a time-dependent delta-shock model, i.e., the recovery time becomes an increasing function of time. Moreover, we assume that shocks occur according to the generalized Pólya process that contains the homogeneous Poisson process, the non-homogeneous Poisson process and the Pólya process as particular cases. For the defined survival model, we derive the corresponding survival function and the mean lifetime and study the related optimal replacement policy along with some relevant stochastic properties.

     

The Computation of all the Derivatives of a B-spline Basis

Splines are currently much used in the field of interpolationto functions and their derivatives. In this context for a givenargument two relationships between derivatives of B-spline basesof consecutive orders are derived. Using these relationshipsit is shown there are (K—1)!((K—m1)!m!) schemesfor the evaluation of the mth derivative of a B-spline basisof order k. Analyses of error growth in terms of a matrix notationare carried out in order to see which of the schemes is themost numerically stable, for uniform or highly non-uniform knotsets. The computation of the B-spline basis of order ^K and its(K—1)th derivative are shown to have small a priori relativeerror bounds.     

Developments in delay time analysis for modelling plant maintenance

Delay time (DT) analysis is a pragmatic mathematical concept readily embraced by engineers which has been developed as a means to model maintenance decision problems. Attention is focused upon the maintenance engineering decisions of what to do, as opposed to the logistical decisions of how to do it. This paper reviews the cumulative knowledge and experience of delay time modelling. The decision environment within which delay time (DT) models are intended as decision aids is briefly reviewed, and the initial development of simple DT models for a repairable component and a complex plant presented. Variations on the basic model are outlined and discussed including perfect and non-perfect inspection, steady state and non-steady state conditions, and homogeneous and non-homogeneous Poisson arrival rate of defects. Attention is given to the parameter estimation process, and both subjective and objective estimation techniques are outlined. Case sketches present practical experience in using the DT concept to model actual plant, to assess the benefits obtained, and to validate modelling and parameter assessment.     

The statistical analysis of standard-cost deviations

Very little is known about the theoretical statistical propertiesof accounting numbers. In this papera probability modellingtechnique is used to analyse the volume, efficiency, and pricedeviation—more usually known in the accounting literatureas ‘variances’—of standard costing. Standard-costdeviations are defined asdifferences between certain types ofconditional and unconditional expected costs. The basic statisticsrequired to construct theoretical confidence intervals for thesedifferences are identified, and it is shown how the formulaenormally used in their calculation may sometimes be correlated.The modelling technique adopted is based on a theory of accountingmeasurement which interprets accounting numbers as specializedtypes of statistics. It illustrates the potential of the theoryfor the development of a better understanding of the statisticalproperties of accounting numbers.     

The reliability of local error estimators for convection-diffusion equations

We assess the reliability of a simple a posteriori error estimatorfor steady-state convection–diffusion equations in caseswhere convection dominates. Our estimator is computed by solvinga local Poisson problem with Neumann boundary conditions. Itgives global upper and local lower bounds on the error measuredin the H¹ semi-norm. However, the error may be overestimatedlocally within boundary layers if these are not resolved bythe mesh, that is, when the local mesh Péclet numberis significantly greater than unity. We discuss the implicationsof this overestimation in a practical context where the estimatoris used as a local error indicator within a self-adaptive meshrefinement process. Received 18 June 1999. Accepted 7 March 2000.     

A new risk model based on policy entrance process and its weak convergence properties

In this paper, we construct a new risk model based on the policy entrance process. The model is concerned with n kinds of independent policies, and each policy is allowed to claim more than once before it expires. As each kind of policy is issued according to a non‐homogeneous Poisson process, the long run behaviour of the new risk process is investigated. When the tail of the claim size distribution is regularly varying, the standardized risk process is proved to converge to a stable law. When each kind of policy is issued according to a homogeneous Poisson process, we also give a diffusion approximation of the new risk process. Copyright © 2007 John Wiley & Sons, Ltd.     

A Generalization of the Kustaanheimo-Stiefel Transform for Two-Centre Systems

We generalize the Kustaanheimo—Stiefel transform to beapplicable to two-centre problems. As a consequence, we remarkthat—as in the one-centre situation—the number ofbound states below a given negative energy E can be determined,under the usual assumptions on the absence of periodic orbits,up to second order.     

Life behavior of censored <Emphasis Type="Italic">δ</Emphasis>-shock model

The shock models play an important role in reliability theory. In a previous paper by Li et al., the δ-shock model, which is different from the classical shock model, was introduced and studied. In this paper, a censored δ-shock model is presented, which originates from the study of customer relationship management (CRM). Based on the underlying homogeneous Poisson process, we obtain the explicit distribution of system lifetime and other related properties, and demonstrate that this new system is superior to NBU. In addition, an exponential limit theorem is obtained. Finally, the application of this new system to evaluate customer lifetime values (CLV) is provided. These results are important to theory and applications.     

Transient analysis of a queue with system disasters and customer impatience

A single server queue with Poisson arrivals and exponential service times is studied. The system suffers disastrous breakdowns at an exponential rate, resulting in the loss of all running and waiting customers. When the system is down, it undergoes a repair mechanism where the repair time follows an exponential distribution. During the repair time any new arrival is allowed to join the system, but the customers become impatient when the server is not available for a long time. In essence, each customer, upon arrival, activates an individual timer, which again follows an exponential distribution with parameter ξ. If the system is not repaired before the customer’s timer expires, the customer abandons the queue and never returns. The time-dependent system size probabilities are presented using generating functions and continued fractions.     

Optimal global approximation of stochastic differential equations with additive Poisson noise

We consider strong global approximation of SDEs driven by a homogeneous Poisson process with intensity λ > 0. We establish the exact convergence rate of minimal errors that can be achieved by arbitrary algorithms based on a finite number of observations of the Poisson process. We consider two classes of methods using equidistant or nonequidistant sampling of the Poisson process, respectively. We provide a construction of optimal schemes, based on the classical Euler scheme, which asymptotically attain the established minimal errors. It turns out that methods based on nonequidistant mesh are more efficient than those based on the equidistant mesh.