Discrete compound poisson processes and tables of the geometric poisson distribution

This paper discusses the properties of positive, integer valued compound Poisson processes and compares two members of the family: the geometric Poisson (stuttering Poisson) and the logarithmic Poisson. It is shown that the geometric Poisson process is particularly convenient when the analyst is interested in a simple model for the time between events, as in simulation. On the other hand, the logarithmic Poisson process is more convenient in analytic models in which the state probabilities (probabilities for the number of events in a specified time period) are required. These state probabilities have a negative binomial distribution. The state probabilities of the geometric Poisson process, known as the geometric Poisson distribution, are tabled for 160 sets of parameter values. The values of mean demand range from 0.10 to 10; those for variance to mean ratio from 1.5 to 7. It is observed that the geometric Poisson density is bimodal.     

The relationship between material failures and flight hours

If material failures follow a Poisson distribution, then the expected number of failures is exactly proportional to flight hours. However, this article demonstrates that proportionality will not be revealed by simple correlation or regression analysis between monthly flight hours and the number of monthly failures. To test for proportionality, one must instead test the underlying hypothesis that the data follow a Poisson distribution. This article presents three simple tests that may be used for this purpose. The Poisson distribution requires that the mean and variance of the number of failures be equal. This article suggests several alternative models that may be used for samples in which the variance exceeds the mean. First, the mean of the Poisson distribution may itself be randomly distributed across the observational units according to a gamma distribution. If so, the number of failures will have a negative binomial distribution. Second, the mean of the Poisson distribution may depend systematically upon a set of observable explanatory variables. In this case, the Poisson regression model is appropriate. Finally, the mean of the Poisson distribution may contain both a systematic component that depends upon observable variables and a random component. This situation yields a generalized Poisson regression model.     

Simulation of nonhomogeneous poisson processes by thinning

A simple and relatively efficient method for simulating one-dimensional and two-dimensional nonhomogeneous Poisson processes is presented The method is applicable for any rate function and is based on controlled deletion of points in a Poisson process whose rate function dominates the given rate function In its simplest implementation, the method obviates the need for numerical integration of the rate function, for ordering of points, and for generation of Poisson variates.     

(R.Q) inventory problem with unknown mean demand and learning

We address a single product, continuous review model with stationary Poisson demand. Such a model has been effectively studied when mean demand is known. However, we are concerned with managing new items for which only a Bayesian prior distribution on the mean is available. As demand occurs, the prior is updated and our control parameters are revised. These include the reorder point (R) and reorder quantity (Q). Deemer, taking a clue from some earlier RAND work, suggested using a model appropriate for known mean, but using a Compound Poisson distribution for demand rather than Poisson to reflect uncertainty about the mean. Brown and Rogers also used this approach but within a periodic review context. In this paper we show how to compute optimum reorder points for a special problem closely related to the problem of real interest. In terms of the real problem, subject to a qualification to be discussed, the reorder points found are upper bounds for the optimum. At the same time, the reorder points found can never exceed those found by the Compound Poisson (Deemer) approach. And they can be smaller than those found when there is no uncertainty about the mean. As a check, the Compound Poisson and proposed approach are compared by simulation.     

Compound poisson approximation in systems reliability

The compound Poisson “local” formulation of the Stein-Chen method is applied to problems in reliability theory. Bounds for the accuracy of the approximation of the reliability by an appropriate compound Poisson distribution are derived under fairly general conditions, and are applied to consecutive-2 and connected-s systems, and the 2-dimensional consecutive-k-out-ofn system, together with a pipeline model. The approximations are usually better than the Poisson “local” approach would give. © 1996 John Wiley & Sons, Inc.     

Modeling lead-time demand for lumpy demand and variable lead time

Slow-moving items that occasionally exhibit large demand transactions are known as lumpy demand items. In modeling lumpy demand patterns, it is often assumed that the arrival of customer orders follows a Poisson process and that the order sizes are given by the geometric distribution. This gives rise to a stuttering Poisson (sP) model of lumpy demand. If lead times are constant, the result is a stuttering Poisson model of lead-time demand. Heretofore, authors such as Ward [18] and Mitchell, Rappold, and Faulkner [12] have assumed constant lead times and thus stopped at the sP model. We develop this model further by introducing the effect of lead-time variability. For illustration, we use the normal and the gamma distributions as characterizations of lead time. The resulting models of lead-time demand are referred to as the geometric Poisson normal (GPN) and the geometric Poisson gamma (GPG). For both these models, the article derives tractable expressions for calculating probabilities. Errors introduced by using the sP, constant lead-time model instead of the exact, variable lead-time model are also illustrated.     

Simulating nonhomogeneous poisson processes with proportional intensities

The purpose of this research is to investigate simulation algorithms for nonhomogeneous Poisson processes with proportional intensities. Two algorithmic approaches are studied: inversion and thinning. Motivated by industrial practices, the covariate vector involved in the simulation is permitted to change after every event (or observation). The algorithms are extended to permit the simulation of general nonhomogeneous Poisson processes with possible discontinuities both in baseline intensity and covariate vector. This latter extension can be used to facilitate a wide range of failure situations that can arise with repairable systems. © 1994 John Wiley & Sons, Inc.     

Dynamic service rate control for a single‐server queue with Markov‐modulated arrivals

We consider the problem of service rate control of a single‐server queueing system with a finite‐state Markov‐modulated Poisson arrival process. We show that the optimal service rate is nondecreasing in the number of customers in the system; higher congestion levels warrant higher service rates. On the contrary, however, we show that the optimal service rate is not necessarily monotone in the current arrival rate. If the modulating process satisfies a stochastic monotonicity property, the monotonicity is recovered. We examine several heuristics and show where heuristics are reasonable substitutes for the optimal control. None of the heuristics perform well in all the regimes and the fluctuation rate of the modulating process plays an important role in deciding the right heuristic. Second, we discuss when the Markov‐modulated Poisson process with service rate control can act as a heuristic itself to approximate the control of a system with a periodic nonhomogeneous Poisson arrival process. Not only is the current model of interest in the control of Internet or mobile networks with bursty traffic, but it is also useful in providing a tractable alternative for the control of service centers with nonstationary arrival rates. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 60: 661–677, 2013     

Aggregating courier deliveries

We consider the problem of efficiently scheduling deliveries by an uncapacitated courier from a central location under online arrivals. We consider both adversary‐controlled and Poisson arrival processes. In the adversarial setting we provide a randomized (3βΔ/2δ ? 1) ‐competitive algorithm, where β is the approximation ratio of the traveling salesman problem, δ is the minimum distance between the central location and any customer, and Δ is the length of the optimal traveling salesman tour overall customer locations and the central location. We provide instances showing that this analysis is tight. We also prove a 1 + 0.271Δ/δ lower‐bound on the competitive ratio of any algorithm in this setting. In the Poisson setting, we relax our assumption of deterministic travel times by assuming that travel times are distributed with a mean equal to the excursion length. We prove that optimal policies in this setting follow a threshold structure and describe this structure. For the half‐line metric space we bound the performance of the randomized algorithm in the Poisson setting, and show through numerical experiments that the performance of the algorithm is often much better than this bound.     

Renewal processes of phase type

This paper discusses a class of analytically and numerically tractable renewal processes, which generalize the Poisson process. When used to describe interarrival or service times in queues, these renewal processes lead to computationally explicit solutions which involve only real arithmetic. Previous modifications of the Poisson process, based on the Erlang or the hyperexponential distributions, appear as particular cases.     

Many‐server loss models with non‐poisson time‐varying arrivals

This article proposes an approximation for the blocking probability in a many‐server loss model with a non‐Poisson time‐varying arrival process and flexible staffing (number of servers) and shows that it can be used to set staffing levels to stabilize the time‐varying blocking probability at a target level. Because the blocking probabilities necessarily change dramatically after each staffing change, we randomize the time of each staffing change about the planned time. We apply simulation to show that (i) the blocking probabilities cannot be stabilized without some form of randomization, (ii) the new staffing algorithm with randomiation can stabilize blocking probabilities at target levels and (iii) the required staffing can be quite different when the Poisson assumption is dropped. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 177–202, 2017     

评估航空重力系统误差及向下延拓的逆泊松半参数方法

常用航空重力系统误差事后处理方法需外部重力数据,但很多地区无外部重力数据。研究发现,半参数模型可在无外部数据时估计系统误差。先用自然样条函数为系统误差建模,后用补偿最小二乘法和光滑参数求解,最后用广义交叉核实法(不需要先验信息)选取光滑参数。将半参数模型用于向下延拓逆泊松积分,建立逆泊松半参数混合模型,既可无外部重力时估计系统误差,又可向下延拓。实验结果表明:无外部重力时逆泊松积分和最小二乘配置法受系统误差影响最大,向下延拓精度最差;正则化算法可减弱系统误差影响,向下延拓精度较好;逆泊松半参数混合模型可估计系统误差,向下延拓精度最好。     

Optimal control for entry of many classes of customers to an M/M/1 queue

A method is developed for determining the optimal policy for entry of customers from many independent classes of Poisson arrivals to a first-come, first-serve (for customers admitted to the queue) single-server queue with exponential service times. The solution technique utilizes a semi-Markov formulation or the decision problem.     

相依风险下的超额赔款再保险

本文通过求二次效用函数的货币期望效用的最大值来确定自留额,认为索赔次数服从二元泊松分布,超额赔款再保险的保费按期望值原理进行计算。     

成数超额混合再保险中最优自留额的确定

Centeno在Sparre Anderson模型中研究了调节系数关于再保险自留额的函数的性质,得到了保险人的调节系数是关于其自留额的单峰函数的结论。本文给出带扩散扰动项的复合Poisson过程的索赔时调节系数与再保险自留额的函数,得出保险公司的最优再保险自留额。     

分段试验的可行性

运用随机点过程方法,从理论上对可修系统能否进行分段试验进行了研究.论证了对于泊松过程模型分段试验的可行性,并研究了分段试验对参数估计的影响.     

Estimating negative binomial demand for retail inventory management with unobservable lost sales

The importance of effective inventory management has greatly increased for many major retailers because of more intense competition. Retail inventory management methods often use assumptions and demand distributions that were developed for application areas other than retailing. For example, it is often assumed that unmet demand is backordered and that demand is Poisson or normally distributed. In retailing, unmet demand is often lost and unobserved. Using sales data from a major retailing chain, our analysis found that the negative binomial fit significantly better than the Poisson or the normal distribution. A parameter estimation methodology that compensates for unobserved lost sales is developed for the negative binomial distribution. The method's effectiveness is demonstrated by comparing parameter estimates from the complete data set to estimates obtained by artificially truncating the data to simulate lost sales. © 1996 John Wiley & Sons, Inc.     

An MX/G/1 queueing system with disasters and repairs under a multiple adapted vacation policy

We consider a queueing system with batch Poisson arrivals subject to disasters which occur independently according to a Poisson process but affect the system only when the server is busy, in which case the system is cleared of all customers. Following a disaster that affects the system, the server initiates a repair period during which arriving customers accumulate without receiving service. The server operates under a Multiple Adapted Vacation policy. The stationary regime of this process is analyzed using the supplementary variables method. We obtain the probability generating function of the number of customers in the system, the fraction of customers who complete service, and the Laplace transform of the system time of a typical customer in stationarity. The stability condition for the system and the Laplace transform of the time between two consecutive disasters affecting the system is obtained by analyzing an embedded Markov renewal process. The statistical characteristics of the batches that complete service without being affected by disasters and those of the partially served batches are also derived. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 171–189, 2015     

Peak rate of occurrence of a poisson process

Suppose that a nonhomogeneous Poisson process is observed for a length of time T, say Let λ (t) denote the mean value function of the process. It is assumed that λ (t) is first increasing then decreasing inside the interval (0, T) with peak at t = t₀, say. Three methods are given for estimating to. One of these methods is nonparametric, and the other two methods are based on the standard regression technique and the maximum likelihood principle The given resull has application in a problem of determining the azimuth of a target from the radar-impulse data. The time series of incoming signals may be approximated by the occurrence of a nonhomogeneous Poisson process with mean value function λ (t). The azimuth of the target is reasonably determined from the direction of the axis of the radar beam at the instant to, corresponding to the peak value of λ (t).     

Application of the stein-chen method for bounds and limit theorems in the reliability of coherent structures

The Stein-Chen method for establishing Poisson convergence is used to approximate the reliability of coherent systems with exponential-type distribution functions. These bounds lead to quite general limit theorems for the lifetime distribution of large coherent systems. © 1993 John Wiley & Sons, Inc.