Estimating the probability that a simulated system will be the best

Consider a stochastic simulation experiment consisting of v independent vector replications consisting of an observation from each of k independent systems. Typical system comparisons are based on mean (long‐run) performance. However, the probability that a system will actually be the best is sometimes more relevant, and can provide a very different perspective than the systems' means. Empirically, we select one system as the best performer (i.e., it wins) on each replication. Each system has an unknown constant probability of winning on any replication and the numbers of wins for the individual systems follow a multinomial distribution. Procedures exist for selecting the system with the largest probability of being the best. This paper addresses the companion problem of estimating the probability that each system will be the best. The maximum likelihood estimators (MLEs) of the multinomial cell probabilities for a set of v vector replications across k systems are well known. We use these same v vector replications to form v^k unique vectors (termed pseudo‐replications) that contain one observation from each system and develop estimators based on AVC (All Vector Comparisons). In other words, we compare every observation from each system with every combination of observations from the remaining systems and note the best performer in each pseudo‐replication. AVC provides lower variance estimators of the probability that each system will be the best than the MLEs. We also derive confidence intervals for the AVC point estimators, present a portion of an extensive empirical evaluation and provide a realistic example. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 341–358, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10019     

Estimating critical path and arc probabilities in stochastic activity networks

This article describes a new procedure for estimating parameters of a stochastic activity network of N arcs. The parameters include the probability that path m is the longest path, the probability that path m is the shortest path, the probability that arc i is on the longest path, and the probability that arc i is on the shortest path. The proposed procedure uses quasirandom points together with information on a cutset ? of the network to produce an upper bound of O[(log K)^N?|?|+1/K] on the absolute error of approximation, where K denotes the number of replications. This is a deterministic bound and is more favorable than the convergence rate of 1/K^1/2 that one obtains from the standard error for K independent replications using random sampling. It is also shown how series reduction can improve the convergence rate by reducing the exponent on log K. The technique is illustrated using a Monte Carlo sampling experiment for a network of 16 relevant arcs with a cutset of ? = 7 arcs. The illustration shows the superior performance of using quasirandom points with a cutset (plan A) and the even better performance of using quasirandom points with the cutset together with series reduction (plan B) with regard to mean square error. However, it also shows that computation time considerations favor plan A when K is small and plan B when K is large.     

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.     

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.     

On a sequential rule for estimating the location parameter of an exponential distribution

Let us assume that observations are obtained at random and sequentially from a population with density function In this paper we consider a sequential rule for estimating μ when σ is unknown corresponding to the following class of cost functions In this paper we consider a sequential rule for estimating μ when σ is unknown corresponding to the following class of cost functions Where δ(X_I,…,X_N) is a suitable estimator of μ based on the random sample (X₁,…, X_N), N is a stopping variable, and A and p are given constants. To study the performance of the rule it is compared with corresponding “optimum fixed sample procedures” with known σ by comparing expected sample sizes and expected costs. It is shown that the rule is “asymptotically efficient” when absolute loss (p=-1) is used whereas the one based on squared error (p = 2) is not. A table is provided to show that in small samples similar conclusions are also true.     

Adaptive competitive decision in repeated play of a matrix game with uncertain entries

This study is concerned with a game model involving repeated play of a matrix game with unknown entries; it is a two-person, zero-sum, infinite game of perfect recall. The entries of the matrix ((pij)) are selected according to a joint probability distribution known by both players and this unknown matrix is played repeatedly. If the pure strategy pair (i, j) is employed on day k, k = 1, 2, …, the maximizing player receives a discounted income of β^{k - 1} X_ij, where β is a constant, 0 ≤ β ? 1, and X_ij assumes the value one with probability p_ij or the value zero with probability 1 - p_ij. After each trial, the players are informed of the triple (i, j, X_ij) and retain this knowledge. The payoff to the maximizing player is the expected total discounted income. It is shown that a solution exists, the value being characterized as the unique solution of a functional equation and optimal strategies consisting of locally optimal play in an auxiliary matrix determined by the past history. A definition of an ?-learning strategy pair is formulated and a theorem obtained exhibiting ?-optimal strategies which are ?-learning. The asymptotic behavior of the value is obtained as the discount tends to one.     

Some probability functions in reliability and their applications

There has been much research on the general failure model recently. In the general failure model, when the unit fails at its age t, Type I failure (minor failure) occurs with probability 1 ? p(t) and Type II failure (catastrophic failure) occurs with probability p(t). In the previous research, some specific shapes (constant, non‐decreasing, or bathtub‐shape) on the probability function p(t) are assumed. In this article, general results on some probability functions are obtained and applied to study the shapes of p(t). The results are also applied to determining the optimal inspection and allocation policies in maintenance problems. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Periodic inspection policy with preventive maintenance

This article considers a modified inspection policy with periodic check intervals, where the unit after check has the same age as before with probability p and is as good as new with probability q. The mean time to failure and the expected number of checks before failure are derived, forming renewal-type equations. The total expected cost and the expected cost per unit of time until detection of failure are obtained. Optimum inspection policies which minimize the expected costs are given as a numerical example.     

Distribution of number of point targets killed and higher moments of coverage of area targets

The coverage C of area targets by salvos of weapons generally varies randomly, because of random target location and weapon impact point fluctuations. A third source of variation appears when, instead of an area target, a multiple-element target is considered, consisting of m point targets distributed randomly and independently of one another around the target center. A multiple-integral expression is derived for the probability p_k of killing exactly k target elements. It is shown that p_k is a linear function of the higher moments, of the order k to m, of the area coverage C. More explicit expressions are derived for the case of two weapons and for circular-symmetric functions. Similar to well-known results for the expectation and variance of coverage of area targets, these expressions can be evaluated by numerical quadrature. Furthermore, the coverage problem in which all underlying functions are Gaussian can be completely solved in closed form. For such a problem, with two weapons, numerical results are presented. They show that the distribution of k can be approximated by a binomial distribution only if the target center and weapon impact point fluctuations are small.     

A note on sojourn times in M/G/1 queues with instantaneous,bernoulli feedback

Queueing systems which include the possibility for a customer to return to the same server for additional service are called queueing systems with feedback. Such systems occur in computer networks for example. In these systems a chosen customer will wait in the queue, be serviced and then, with probability p, return to wait again, be serviced again and continue this process until, with probability (1 – p) = q, it departs the system never to return. The time of waiting plus service time, the nth time the customer goes through, we will call his nth sojourn time. The (random) sum of these sojourn times we will call the total sojourn time (abbreviated, sojourn time when there is no confusion which sojourn time we are talking about). In this paper we study the total sojourn time in a queueing system with feedback. We give the details for M/G/1 queues in which the decision to feedback or not is a Bernoulli process. While the details of the computations can be more difficult, the structure of the sojourn time process is unchanged for the M/G/1 queue with a more general decision process as will be shown. We assume the reader is familiar with Disney, McNickle and Simon [1].     

Confidence intervals in discrete event simulation: A comparison of replication and batch means

Suppose that we have enough computer time to make n observations of a stochastic process by means of simulation and would like to construct a confidence interval for the steady-state mean. We can make k independent runs of m observations each (n=k.m) or, alternatively, one run of n observations which we then divide into k batches of length m. These methods are known as replication and batch means, respectively. In this paper, using the probability of coverage and the half length of a confidence interval as criteria for comparison, we empirically show that batch means is superior to replication, but that neither method works well if n is too small. We also show that if m is chosen too small for replication, then the coverage may decrease dramatically as the total sample size n is increased.     

On the reliability,availability and bayes confidence intervals for multicomponent systems

The problem of computing reliability and availability and their associated confidence limits for multi-component systems has appeared often in the literature. This problem arises where some or all of the component reliabilities and availabilities are statistical estimates (random variables) from test and other data. The problem of computing confidence limits has generally been considered difficult and treated only on a case-by-case basis. This paper deals with Bayes confidence limits on reliability and availability for a more general class of systems than previously considered including, as special cases, series-parallel and standby systems applications. The posterior distributions obtained are exact in theory and their numerical evaluation is limited only by computing resources, data representation and round-off in calculations. This paper collects and generalizes previous results of the authors and others. The methods presented in this paper apply both to reliability and availability analysis. The conceptual development requires only that system reliability or availability be probabilities defined in terms acceptable for a particular application. The emphasis is on Bayes Analysis and the determination of the posterior distribution functions. Having these, the calculation of point estimates and confidence limits is routine. This paper includes several examples of estimating system reliability and confidence limits based on observed component test data. Also included is an example of the numerical procedure for computing Bayes confidence limits for the reliability of a system consisting of N failure independent components connected in series. Both an exact and a new approximate numerical procedure for computing point and interval estimates of reliability are presented. A comparison is made of the results obtained from the two procedures. It is shown that the approximation is entirely sufficient for most reliability engineering analysis.     

Optimal location of a facility relative to area demands

This paper deals with the Weber single-facility location problem where the demands are not only points but may be areas as well. It provides an iterative procedure for solving the problem with l^p distances when p > 1 (a method of obtaining the exact solution when p = 1 and distances are thus rectangular already exists). The special case where the weight densities in the areas are uniform and the areas are rectangles or circles results in a modified iterative process that is computationally much faster. This method can be extended to the simultaneous location of several facilities.     

One-facility location with rectilinear tour distances

This article concerns the location of a facility among n points where the points are serviced by “tours” taken from the facility. Tours include m points at a time and each group of m points may become active (may need a tour) with some known probability. Distances are assumed to be rectilinear. For m ≤ 3, it is proved that the objective function is separable in each dimension and an exact solution method is given that involves finding the median of numbers appropriately generated from the problem data. It is shown that the objective function becomes multimodal when some tours pass through four or more points. A bounded heuristic procedure is suggested for this latter case. This heuristic involves solving an auxiliary three-point tour location problem.     

Estimating the accuracy of the coverages of outer β-content tolerance intervals,which control both tails of the normal distribution

Tolerance limits which control both tails of the normal distribution so that there is no more than a proportion β₁ in one tail and no more than β₂ in the other tail with probability γ may be computed for any size sample. They are computed from X? - k₁S and X? - k₂S, where X? and S are the usual sample mean and standard deviation and k₁ and k₂ are constants previously tabulated in Odeh and Owen [3]. The question addressed is, “Just how accurate are the coverages of these intervals (– Infin;, X? – k₁S) and (X? + k₂S, ∞) for various size samples?” The question is answered in terms of how widely the coverage of each tail interval differs from the corresponding required content with a given confidence γ′.     

Confidence intervals for ranked means

Suppose that observations from populations π₁, …, π_k (k ≥ 1) are normally distributed with unknown means μ₁., μ_k, respectively, and a common known variance σ². Let μ_[1] μ … ≤ μ_[k] denote the ranked means. We take n independent observations from each population, denote the sample mean of the n observation from π₁ by X _i (i = 1, …, k), and define the ranked sample means X _[1] ≤ … ≤ X _[k]. The problem of confidence interval estimation of μ₍₁₎, …,μ_[k] is stated and related to previous work (Section 1). The following results are obtained (Section 2). For i = 1, …, k and any γ(0 < γ < 1) an upper confidence interval for μ_[i] with minimal probability of coverage γ is (? ∞, X _[i]+ h) with h = (σ/n^1/2) Φ^?1(γ^1/k-i+1), where Φ(·) is the standard normal cdf. A lower confidence interval for μ_[i] with minimal probability of coverage γ is (X _i[i] – g, + ∞) with g = (σ/n^1/2) Φ^?1(γ^1/i). For the upper confidence interval on μ_[i] the maximal probability of coverage is 1– [1 – γ^1/k-i+1]ⁱ, while for the lower confidence interval on μ_[i] the maximal probability of coverage is 1–[1– γ^1/i] ^k-i+¹. Thus the maximal overprotection can always be calculated. The overprotection is tabled for k = 2, 3. These results extend to certain translation parameter families. It is proven that, under a bounded completeness condition, a monotone upper confidence interval h(X ₁, …, X _k) for μ_[i] with probability of coverage γ(0 < γ < 1) for all μ = (μ_[1], …,μ_[k]), does not exist.     

The “signature” of a coherent system and its application to comparisons among systems

Various methods and criteria for comparing coherent systems are discussed. Theoretical results are derived for comparing systems of a given order when components are assumed to have independent and identically distributed lifetimes. All comparisons rely on the representation of a system's lifetime distribution as a function of the system's “signature,” that is, as a function of the vector p= (p₁, … , p_n), where p_i is the probability that the system fails upon the occurrence of the ith component failure. Sufficient conditions are provided for the lifetime of one system to be larger than that of another system in three different senses: stochastic ordering, hazard rate ordering, and likelihood ratio ordering. Further, a new preservation theorem for hazard rate ordering is established. In the final section, the notion of system signature is used to examine a recently published conjecture regarding componentwise and systemwise redundancy. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 507–523, 1999     

A search game with a protector

A classic problem in Search Theory is one in which a searcher allocates resources to the points of the integer interval [1, n] in an attempt to find an object which has been hidden in them using a known probability function. In this paper we consider a modification of this problem in which there is a protector who can also allocate resources to the points; allocating these resources makes it more difficult for the searcher to find an object. We model the situation as a two‐person non‐zero‐sum game so that we can take into account the fact that using resources can be costly. It is shown that this game has a unique Nash equilibrium when the searcher's probability of finding an object located at point i is of the form (1 − exp (−λ_ix_i)) exp (−μ_iy_i) when the searcher and protector allocate resources x_i and y_i respectively to point i. An algorithm to find this Nash equilibrium is given. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47:85–96, 2000     

Technical note: Find a hidden “treasure”

This paper deals with a two searchers game and it investigates the problem of how the possibility of finding a hidden object simultaneously by players influences their behavior. Namely, we consider the following two‐sided allocation non‐zero‐sum game on an integer interval [1,n]. Two teams (Player 1 and 2) want to find an immobile object (say, a treasure) hidden at one of n points. Each point i ∈ [1,n] is characterized by a detection parameter λ_i (μ_i) for Player 1 (Player 2) such that p_i(1 ? exp(?λ_ix_i)) (p_i(1 ? exp(?μ_iy_i))) is the probability that Player 1 (Player 2) discovers the hidden object with amount of search effort x_i (y_i) applied at point i where p_i ∈ (0,1) is the probability that the object is hidden at point i. Player 1 (Player 2) undertakes the search by allocating the total amount of effort X(Y). The payoff for Player 1 (Player 2) is 1 if he detects the object but his opponent does not. If both players detect the object they can share it proportionally and even can pay some share to an umpire who takes care that the players do not cheat each other, namely Player 1 gets q₁ and Player 2 gets q₂ where q₁ + q₂ ≤ 1. The Nash equilibrium of this game is found and numerical examples are given. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

A basically poisson queue with nonpoisson output

The output of the queueing system M/M/1 is well known to be Poisson. This has also been shown to be true for other more general models inclusive of M/M_n/1; the system in which arrivals and epochs of service completion are elements of a birth and death process with parameters Λ and nμ, respectively, when the system contains n ≥ 1 customers. We shall here show that this result is not true in M_nM/1; a system where arrival parameter is state dependent quantity Λ/n+1. Expressions will be given for the steady state joint density of two consecutive output intervals as well as the coefficient of correlation between them.