On dams with continuous semi-Markovian inputs

An infinite capacity dam subject to semi-Markovian inputs and a content dependent release rule will be discussed. The content process will be constructed, the distributions of the content at time t and time to first emptiness will be computed, and the limiting distribution of the content process will be obtained in a special case. Our methods rely heavily on Markov renewal theory with continuous state spaces.     

Some problems in finite dams with an application to insurance risk

Summary This paper considers a finite dam in continuous time fed by inputs, with a negative exponential distribution, whose arrival times form a Poisson process; there is a continuous release at unit rate, and overflow is allowed. Various results have been obtained by appropriate limiting methods from an analogous discrete time process, for which it is possible to find some solutions directly by determinantal methods.First the stationary dam content distribution is found. The distribution of the probability of first emptiness is obtained both when overflow is, and is not allowed. This is followed by the probability the overflow before emptiness, which is then applied to determine the exact solution for an insurance risk problem with claims having a negative exponential distribution. The time-dependent content distribution is found, and the analogy with queueing theory is discussed.     

Finite dams with double level of release

We considered a finite dam with discrete additive input and double level of release. If the current dam content is not greater than a certain boundM, the release is one unit unless the dam is empty; and if the current dam content is greater thanM, the release isr (? 1) units provided it is available, otherwise the whole content will be withdrawn. We derive all the expressions of the distributions of first emptiness with and without overflow, the distributions of emptiness with and without overflow, the time dependent distributions of dam content with and without overflow, and the distributions of overflow times and quantities. IfM is equal to the dam capacity, the results are reduced to the case of unit release; and ifM=0, the results are reduced to the case of releaser.     

Stochastic Dynamics for Passage Times and Diffusion Approximations for Finite Capacity Storage Models with Different Kinds of Barriers

In this article, we discuss a number of storage models of finite capacity with random inputs, random outputs, and linear release policy. They form a class of one-dimensional master equations with separable kernels. For this class of problems, the integral equations for first overflow or first emptiness can be transformed exactly into ordinary differential equations. Analysis is done with separable kernel. For all the stochastic models, two barriers are considered: one at X = 0 and the other at X = k, and the barriers are treated as absorbing or reflecting. The imbedding method is used to derive a third order differential equation. We consider first passage times for overflow without or with emptiness of the dam. We also study the passage times for first emptiness with and without overflows. The expected amount of overflows in a given time is also calculated. Finally, by suitable statistical features, all these models are converted into diffusion process with drift. Closed form solutions are obtained for all the problems in terms of Laplace transform functions. For the diffusion process with drift first passage time density is arrived at by treating X = 0 and X = k as absorbing barriers. One of the barriers as reflecting is also studied.     

Storage problems in continuous time with random inputs,random outputs,and deterministic release

The subject of study here is the model of a dam, with random inputs and outputs along with a deterministic release. The amounts of the Poisson jumps, either up or down, are independently and identically distributed. Closed form solutions are obtained for the Laplace transforms of first passage densities to different situations of overflow or emptiness. These results can throw insights regarding different threshold studies in storage, inventory, biological, and environmental problems. The closed form solutions are obtained by applying imbedding methods for different types of densities conceptualized in novel ways.     

First exit times for compound Poisson dams with a general release rule

An infinite dam with compound Poisson inputs and a state-dependent release rate is considered. For this dam, we solve Kolmogorov’s backward differential equation to obtain the Laplace transforms of the first exit times in terms of a certain positive kernel. This allows us to provide an explicit expression for the Laplace transform of the wet period for a finite dam.     

Stochastic Models for the Amount of Overflow in a Finite Dam with Random Inputs,Random Outputs,and Exponential Release Policy

In this article, we discuss finite dam models to study the expected amount of overflow in a given time. The inputs into the dam are taken as random and there are two types of outputs—one is random and the other is deterministic which is proportional to the content of the dam. The master equation for the expected amount of overflow is an one dimensional equation with separable kernel. For this class of master equation, the integral equation for the expected amount of overflow has been transformed exactly into ordinary differential equation with variable coefficients. The imbedding method is used to study the expected amount of overflow in a given time without emptiness in this period. We also consider the model for the expected amount of overflow in a given time with any number of emptiness of the dam in this period. The results are derived in the form of a third order differential Equation for the Laplace transformation function for the expected overflow. The closed form analytical solutions are obtained in terms of beta functions and degenerate hyper-geometric functions of two variables.     

The limiting distribution for the infinitely deep dam with a Markovian input

The paper outlines a case for taking greater interest in the bottomless, or infinitely deep, dam model in Hydrology. It then shows that for such a model with unit withdrawals and an ergodic Markov chain input process the limiting distribution of depletion, when this exists, is a zero modified geometric distribution. This result generalises the well known result for independent inputs. The technical conditions required for the proof are satisfied for finite state space input processes and are shown to be satisfied by certain infinite state space input processes. These include as special cases examples which have a negative binomial limiting input distribution.     

On a Replacement Problem of a Cumulative Damage Model

Items are assumed to fail only by degradation. An appropriate stochastic model of such items is a cumulative process in which an item can fail only when the total amount of damage exceeds a prespecified failure level. This paper introduces a replacement policy in which an item is replaced at a certain level of damage before failure or at failure, whichever occurs first. The optimum replacement level of damage which will minimize the total expected cost per unit of time for an infinite time span is obtained. A numerical example is also presented. The total expected cost for a finite time span is also discussed.     

Skorohod–Loynes Characterizations of Queueing,Fluid, and Inventory Processes

We consider queueing, fluid and inventory processes whose dynamics are determined by general point processes or random measures that represent inputs and outputs. The state of such a process (the queue length or inventory level) is regulated to stay in a finite or infinite interval – inputs or outputs are disregarded when they would lead to a state outside the interval. The sample paths of the process satisfy an integral equation; the paths have finite local variation and may have discontinuities. We establish the existence and uniqueness of the process based on a Skorohod equation. This leads to an explicit expression for the process on the doubly-infinite time axis. The expression is especially tractable when the process is stationary with stationary input–output measures. This representation is an extension of the classical Loynes representation of stationary waiting times in single-server queues with stationary inputs and services. We also describe several properties of stationary processes: Palm probabilities of the processes at jump times, Little laws for waiting times in the system, finiteness of moments and extensions to tandem and treelike networks.     

On some applications of Wald's identity to dams

It is known that the main difficulty in applying the Markovian analogue of Wald's Identity is the presence, in the Identity, of the last state variable before the random walk is terminated. In this paper we show that this difficulty can be overcome if the underlying Markov chain has a finite state space. The absorption probabilities thus obtained are used, by employing a duality argument, to derive time-dependent and limiting probabilities for the depletion process of a dam with Markovian inputs.The second problem that is considered here is that of a non-homogeneous but cyclic Markov chain. An analogue of Wald's Identity is obtained for this case, and is used to derive time- dependent and limiting probabilities for the depletion process with inputs forming a non- homogeneous (cyclic) Markov chain.     

How does innovation''''s tail risk determine marginal tail risk of a stationary financial time series?

We discuss the relationship between the marginal tail risk probability and theinnovation's tail risk probability for some stationary financial time series models. We firstgive the main results on the tail behavior of a class of infinite weighted sums of randomvariables with heavy-tailed probabilities. And then, the main results are applied to threeimportant types of time series models; infinite order moving averages, the simple bilineartime series and the solutions of stochastic difference equations. The explicit formulasare given to describe how the marginal tail probabilities come from the innovation's tailprobabilities for these time series. Our results can be applied to the tail estimation of timeseries and are useful for risk analysis in finance.     

Importance sampling algorithms for first passage time probabilities in the infinite server queue

This paper applies importance sampling simulation for estimating rare event probabilities of the first passage time in the infinite server queue with renewal arrivals and general service time distributions. We consider importance sampling algorithms which are based on large deviations results of the infinite server queue, and we consider an algorithm based on the cross-entropy method, where we allow light-tailed and heavy-tailed distributions for the interarrival times and the service times. Efficiency of the algorithms is discussed by simulation experiments.     

The Optimum Repair Limit Replacement Policies

This paper considers a single unit system which is first repaired if it fails. If the repair is not completed up to the fixed repair limit time then the unit under repair is replaced by a new one. The cost functions are introduced for the repair and the replacement of the failed unit. The optimum repair limit replacement time minimizing the expected cost per unit of time for an infinite time span is obtained analytically under suitable conditions. Two special cases where the repair cost functions are proportional to time and are exponential are discussed in detail with numerical examples.     

Computational algorithm and parameter optimization for a multi-server system with unreliable servers and impatient customers

We consider an infinite capacity M/M/c queueing system with c unreliable servers, in which the customers may balk (do not enter) and renege (leave the queue after entering). The system is analyzed as a quasi-birth-and-death (QBD) process and the necessary and sufficient condition of system equilibrium is obtained. System performance measures are explicitly derived in terms of computable forms. The useful formulae for computing the rate matrix and stationary probabilities are derived by means of a matrix analytical approach. A cost model is derived to determine the optimal values of the number of servers, service rate and repair rate simultaneously at the minimal total expected cost per unit time. The parameter optimization is illustrated numerically by the Quasi-Newton method.     

Optimal control of a finite dam with a sample path constraint

In this paper, we discuss the 2-stage output procedure of a finite dam under the condition that water must be released by a fixed time. From this standpoint, the reservoir model we consider is subject to a sample path constraint and has a more general cost function than the earlier contributions. We analytically derive explicit formulas for the long-run average and the expected total discounted costs for an infinite time span and numerically calculate the optimal control policy. Finally, the optimal policy is compared with one by Zuckerman [1] and the effect of the fixed release time is discussed further.     

Exponential Bounds for Ruin Probability in Two Moving Average Risk Models with Constant Interest Rate

The authors consider two discrete-time insurance risk models. Two moving average risk models are introduced to model the surplus process, and the probabilities of ruin are examined in models with a constant interest force. Exponential bounds for ruin probabilities of an infinite time horizon are derived by the martingale method.     

基于故障树分析的堰塞湖溃坝概率估计方法

堰塞湖排险的一个关键问题是如何针对实施不同应对措施情况下的堰塞湖溃坝概率进行估计,这是一个值得关注的重要研究课题。本文提出了一种基于故障树分析(FTA)的堰塞湖溃坝概率估计方法。首先,通过堰塞湖排险问题的实际背景分析,基于FTA构建了堰塞湖溃坝故障树的基本架构;然后,通过相关领域知识、历史案例分析、专家主观判断和多位专家主观判断信息的融合,可以确定实施某一应对措施情形下故障树中各基本事件在不同时段内发生的概率;进一步地,依据构建的故障树和基本事件发生的概率,给出了在不同时段内堰塞湖溃坝事件发生的概率的估计方法。最后,通过一个实例分析说明了本文所提出方法的可行性与有效性。     

Tail probabilities for M/G/∞ input processes (I): Preliminary asymptotics

The infinite server model of Cox with arbitrary service time distribution appears to provide a large class of traffic models - Pareto and log-normal distributions have already been reported in the literature for several applications. Here we begin the analysis of the large buffer asymptotics for a multiplexer driven by this class of inputs. To do so we rely on recent results by Duffield and O’Connell on overflow probabilities for the general single server queue. In this paper we focus on the key step in this approach: The appropriate large deviations scaling is shown to be related to the forward recurrence time of the service time distribution, and a closed form expression is derived for the corresponding generalized limiting log-moment generating function associated with the input process. Three different regimes are identified. In a companion paper we apply these results to obtain the large buffer asymptotics under a variety of service time distributions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Ruin probabilities for an insurance company based on some stochastic risk models

The paper is concerned with a stochastic risk model with independent random claims and premiums. Recurrence formulas for the ruin probabilities of an insurance company at times of claim payments are obtained. Both the random premiums and the insurance damages are assumed to be independent and identically distributed. The number of claims and premiums are independent Poisson processes, both of which are independent of the size of premiums and claims. We consider the case when the random premiums and insurance damages are exponentially distributed and the more general case when they are gamma distributed with integer parameters. Based on the probabilities obtained in this paper, it is possible to calculate the ruin probabilities on infinite and finite time intervals. Examples are given.