A duel complicated by false targets and uncertainty as to opponent type

An attacker, being one of two types, initiates an attack at some time in the interval [-T, 0]. The a priori probabilities of each type are known. As time elapses the defender encounters false targets which occur according to a known Poisson process and which can be properly classified with known probability. The detection and classification probabilities for each type attacker are given. If the defender responds with a weapon at the time of attack, he survives with a probability which depends on the number of weapons in his possession and on attacker type. If he does not respond, his survival probability is smaller. These probabilities are known, as well as the current number of weapons in the defender's possession. They decrease as the number of weapons decreases. The payoff is the defender's survival probability. An iterative system of first-order differential equations is derived whose unique solution V₁(t),V₂(t),…,V_k(t) is shown to be the value of the game at time t, when the defender has 1, 2,…, k,… weapons, respectively. The optimal strategies are determined. Limiting results are obtained as t→-∞, while the ratio of the number of weapons to the expected number of false targets remaining is held constant.     

Preservation of unimodality under shock models

A number of results pertaining to preservation of aging properties (IFR, IFRA etc.) under various shock models are available in the literature. Our aim in this paper is to examine in the same spirit, the preservation of unimodality under various shock models. For example, it is proved that in a non-homogeneous Poisson shock model if {p_k}_K≥0, the sequence of probabilities with which the device fails on the kth shock, is unimodal then under some suitable conditions on the mean value function Λ (t), the corresponding survival function is also unimodal. The other shock models under which the preservation of unimodality is considered in this paper are pure birth shock model and a more general shock model in which shocks occur according to a general counting process. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 952–957, 1999     

On the behavior of the batch arrival queue Mx/M/1 during a busy period

The queue size process (^t)⁰≤^t≤^t0 of the batch arrival queue M^X/M/1 is studied under the condition that the duration of its busy period is larger than t⁰. Explicit formulas for the transition probabilities are given and the limiting Markov process for t⁰ → ∞ is investigated. Several properties of this process are considered. Its transition probabilities and moments and the distribution of its minimum are derived and a functional limit theorem for the rescaled process is proved. © 1994 John Wiley & Sons, Inc.     

An M/M/1 queue with delayed feedback

We present some results for M/M/1 queues with finite capacities with delayed feedback. The delay in the feedback to an M/M/1 queue is modelled as another M-server queue with a finite capacity. The steady state probabilities for the two dimensional Markov process {N(t), M(t)} are solved when N(t) = queue length at server 1 at t and M(t) = queue length at server 2 at t. It is shown that a matrix operation can be performed to obtain the steady state probabilities. The eigenvalues of the operator and its eigenvectors are found. The problem is solved by fitting boundary conditions to the general solution and by normalizing. A sample problem is run to show that the solution methods can be programmed and meaningful results obtained numerically.     

A simulation study of the efficiency of empirical Bayes' estimators of multiple correlated probability vectors

Empirical Bayes' methods had been used by Brier, Zacks, and Marlow [1] for estimating performance characteristic vectors of success probabilities. The problem is that of estimating k-dimensional success probabilities of dependent binomial random variables, which are highly correlated. The present study reinforces the results of the previous one by showing, via simulations, that the relative efficiency of the empirical Bayes estimators, compared to the Stein-type and to the maximum-likelihood ones, is very high. This holds even if the success proportions are based on a small number of trials. We study the case of equicorrelation structure with positive correlations.     

The single server queue in discrete time-numerical analysis I

This is the first of a sequence of papers dealing with the computational aspects of the transient behavior of queues in discrete time It is shown that for a substantial class of queues of practical interest, a wealth of numerical information may be obtained by relatively unsophisticated methods This approach should prove useful in the analysis of unstable queues which operate over a limited time interval, but is by no means limited to such queues Mathematically the service unit is modeled in terms of a multivariate Markov chain, whose particular structure is used in iterative computation. Many important queue features may then be derived from the n-step transition probabilities of this chain.     

Aircraft sortie effectiveness model

A mathematical model describing the sortie of a single aircraft under enemy threats, attacking a single passive target, is developed. Emphasis is placed on the determination of the probabilities associated with the various events in the sortie. These probabilities are then used to derive appropriate measures of effectiveness. The optimum sortie time is analyzed.     

Assembly of systems having maximum reliability

The first problem considered in this paper is concerned with the assembly of independent components into parallel systems so as to maximize the expected number of systems that perform satisfactorily. Associated with each component is a probability of it performing successfully. It is shown that an optimal assembly is obtained if the reliability of each assembled system can be made equal. If such equality is not attainable, then bounds are given so that the maximum expected number of systems that perform satisfactorily will lie within these stated bounds; the bounds being a function of an arbitrarily chosen assembly. An improvement algorithm is also presented. A second problem treated is concerned with the optimal design of a system. Instead of assembling given units, there is an opportunity to “control” their quality, i.e., the manufacturer is able to fix the probability, p, of a unit performing successfully. However, his resources, are limited so that a constraint is imposed on these probabilities. For (1) series systems, (2) parallel systems, and (3) k out of n systems, results are obtained for finding the optimal p's which maximize the reliability of a single system, and which maximize the expected number of systems that perform satisfactorily out of a total assembly of J systems.     

An (s,S) inventory system with rest periods to the server

We study the (s,S) inventory system in which the server takes a rest when the level of the inventory is zero. The demands are assumed to occur for one unit at a time. The interoccurrence times between successive demands, the lead times, and the rest times are assumed to follow general distributions which are mutually independent. Using renewal and convolution techniques we obtain the state transition probabilities.     

Operations analysis during the underwater search for Scorpions

This paper discusses the operations analysis in the underwater search for the remains of the submarine Scorpion The a priori target location probability distribution for the search was obtained by monte-carlo procedures based upon nine different scenarios concerning the Scorpion loss and associated credibility weights. These scenarios and weights were postulated by others. Scorpion was found within 260 yards of the search grid cell having the largest a priori probability Frequent computations of local effectiveness probabilities (LEPs) were carried out on scene during the search and were used to determine an updated (a posteriori) target location distribution. This distribution formed the basis for recommendation of the current high probability areas for search The sum of LEPs weighted by the a priori target location probabilities is called search effectiveness probability (SEP) and was used as the overall measure of effectiveness for the operation. SEP and LEPs were used previously in the Mediterranean H-bomb search On-scene and stateside operations analysis are discussed and the progress of the search is indicated by values of SEP for various periods during the operation.     

Arrival and departure state distributions in the general bulk‐service queue

In this paper, we give an explicit relation between steady‐state probability distributions of the buffer occupancy at customer entrance and departure epochs, for the classical single‐server system G/G^[N]/1 with batch services and for the finite capacity case. The method relies on level‐crossing arguments. For the particular case of Poisson input, we also express the loss probability in terms of state probabilities at departure epochs, yielding probabilities observed by arriving customers. This work provides the “bulk queue” version of a result established by Burke, who stated the equality between probabilities at arrival and departure epochs for systems with “unit jumps.” © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 107–118, 1999     

An M/M/1 queue with a general bulk service rule

A model of an M/M/1, bulk queue with service rates dependent on the batch size is developed. The operational policy is to commence service when at least L customers are available with a maximum batch size of K. Arriving customers are not allowed to join in-process service. The solution procedure utilizes the matrix geometric methodology and reduces to obtaining the inverse of a square matrix of dimension K + 1 - L. For the case where the service rates are not batch size dependent, the limiting probabilities can be written in closed form. A numerical example illustrates the variability of the system cost as a function of the minimum batch service size L.     

机动目标IMM三维并行滤波跟踪算法

由于复杂的空中目标机动,其三维方向的机动强度是不一致的,传统IMM算法存在模型匹配不准确的问题,提出一种机动目标IMM三维并行滤波的跟踪算法。算法以CV和修正的CS模型为子集,在3个坐标轴上分别根据目标机动的分量实际更新其模型概率,并行IMM滤波方法,尽量确保模型的适配性,提高滤波精度。仿真结果表明,该算法比传统IMM方法跟踪精度更高,对空中机动目标跟踪适应性更强。     

A (Q,R) inventory model with lost sales and erlang-distributed lead times

The exact expression is derived for the average stationary cost of a (Q,R) inventory system with lost sales, unit Poisson demands, Erlang-distributed lead times, fixed order cost, fixed cost per unit lost sale, linear holding cost per unit time, and a maximum of one order outstanding. Explicit expressions for the state probabilities and a fast method of calculating them are obtained for the case of Q greater than R. Exponential lead times are analyzed as a special case. A simple cyclic coordinate search procedure is used to locate the minimum cost policy. Examples of the effect of lead time variability on costs are given.     

Optimal design of accelerated life tests under periodic inspection

For the exponentially distributed lifetimes, optimal accelerated life test plans are determined under the assumptions of periodic inspection and Type I censoring. Computational results indicate that for the range of parameter values considered the asymptotic variance of the estimated mean or pth quantile at the use stress is not sensitive to the number of inspections at overstress levels. Senstivity analyses are also conducted to see how senstive the asymptotic variance of the estimated mean is with respect to the uncertainties involved in the guessed failure probabilities at the use and high stress levels. Computational results show that moderate deviations (several tens of percents) of the guessed failure probabilities from their true values are fairly tolerable in terms of the relative amount of increase in the asymptotic variance of the estimated mean. Procedures for selecting a sample size and for determining whether or not to conduct an accelerated life test are also discussed.     

The N-seasons S-servers loss system

We consider a class of loss systems with exponential service times and a Poisson arrival process with a rate that varies periodically among N levels called seasons. For two special cases, we derive transient and steady-state solutions and provide simple proofs that losses are minimized when the arrival rates for all seasons are equal. In the general case, we describe a straightforward procedure to derive the steady-state probabilities. We also prove that when S=1, the server is generally busier during the high arrival rate seasons.     

Transient solution of a finite-capacity M/Ga,b/1 queueing system

In this article we consider a single-server, bulk-service queueing system in which the waiting room is of finite capacity. Arrival process is Poisson and all the arrivals taking place when the waiting room is full are lost. The service times are generally distributed independent random variables and the distribution is depending on the batch size being served. Using renewal theory, we derive the time-dependent solution for the system-size probabilities at arbitrary time points. Also we give expressions for the distribution of virtual waiting time in the queue at any time t.     

Survival distributions in crossing fields containing clusters of mines with possible detection and uncertain activation or kill

The present paper presents an algorithm for the exact determination of survival distributions in crossing mine fields. The model under consideration considers clusters of mines, scattered at random in the field around specified aim points. The scatter distributions of the various clusters are assumed to be known The encounter process allows for a possible detection and destruction of the mines, for inactivation of the mines and for the possibility that an activated mine will not destroy the object. Recursive formulae for the determination of the survival probabilities of each object (tank) in a column of n crossing at the same path are given. The distribution of the number of survivors out of n objects in a column is also determined. Numerical examples are given.     

An application of empirical bayes techniques to the simultaneous estimation of many probabilities

Consider the following situation: Each of N different combat units is presented with a number of requirements to satisfy, each requirement being classified into one of K mutually exclusive categories. For each unit and each category, an estimate of the probability of that unit satisfying any requirement in that category is desired. The problem can be generally stated as that of estimating N different K-dimensional vectors of probabilities based upon a corresponding set of K-dimensional vectors of sample proportions. An empirical Bayes model is formulated and applied to an example from the Marine Corps Combat Readiness Evaluation System (MCCRES). The EM algorithm provides a convenient method of estimating the prior parameters. The Bayes estimates are compared to the ordinary estimates, i.e., the sample proportions, by means of cross validation, and the Bayes estimates are shown to provide considerable improvement.     

Some two-on-two homogeneous stochastic combats

In this article we consider two versions of two-on-two homogeneous stochastic combat and develop expressions, in each case, for the state probabilities. The models are natural generalizations of the exponential Lanchester square law model. In the first version, a marksman whose target is killed resumes afresh the killing process on a surviving target; in the second version, the marksman whose target is killed merely uses up his remaining time to a kill on a surviving target. Using the state probabilities we then compute such important combat measures as (1) the mean and variance of the number of survivors as they vary with time for each of the sides, (2) the win probabilities for each of the sides, and (3) the mean and variance of the battle duration time. As an application, computations were made for the specific case of a gamma (2) interfiring time random variable for each side and the above combat measures were compared with the appropriate exponential and deterministic Lanchester square law approximations. The latter two are shown to be very poor approximations in this case.