On the moments of gamma order statistics

A recurrence relation between the moments of order statistics from the gamma distribution having an integer parameter r is obtained. It is shown that if the negative moments of orders -(r?1), …, ?1 of the smallest order statistic in random samples of size n are known, then one can obtain all the moments. Tables of negative moments for r = 2 (1) 5 are also given.     

The asymptotic distribution of order statistics

For each n., X₁(n), X₂(n), …, X_n(n) are IID, with common pdf f_n(x). y₁(n) < … < Y_n (n) are the ordered values of X₁ (n), …, X_n(n). K_n is a positive integer, with lim K_n = ∞. Under certain conditions on K_n and f_n (x), it was shown in an earlier paper that the joint distribution of a special set of K_n + 1 of the variables Y₁ (n), …, Y_n (n) can be assumed to be normal for all asymptotic probability calculations. In another paper, it was shown that if f_n (x) approaches the pdf which is uniform over (0, 1) at a certain rate as n increases, then the conditional distribution of the order statistics not in the special set can be assumed to be uniform for all asymptotic probability calculations. The present paper shows that even if f_n (x) does not approach the uniform distribution as n increases, the distribution of the order statistics contained between order statistics in the special set can be assumed to be the distribution of a quadratic function of uniform random variables, for all asymptotic probability calculations. Applications to statistical inference are given.     

Inventory policies for a make‐to‐order system with a perishable component and fixed ordering cost

We consider a make‐to‐order production system where two major components, one nonperishable (referred to as part 1) and one perishable (part 2), are needed to fulfill a customer order. In each period, replenishment decisions for both parts need to be made jointly before demand is realized and a fixed ordering cost is incurred for the nonperishable part. We show that a simple (s_n,S,S) policy is optimal. Under this policy, S along with the number of backorders at the beginning of a period if any and the availability of the nonperishable part (part 1) determines the optimal order quantity of the perishable part (part 2), while (s_n,S) guide when and how much of part 1 to order at each state. Numerical study demonstrates that the benefits of using the joint replenishment policy can be substantial, especially when the unit costs are high and/or the profit margin is low. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

Covariance of weibull order statistics

Let X₁ < X₂ <… < X_n denote an ordered sample of size n from a Weibull population with cdf F(x) = 1 - exp (?x^p), x > 0. Formulae for computing Cov (X_i, X_j) are well known, but they are difficult to use in practice. A simple approximation to Cov(X_i, X_j) is presented here, and its accuracy is discussed.     

Estimating moments of the minimum order statistic from normal populations—a bayesian approach

Let Yⁱ ∽ N(θⁱ, σ), i = 1, …, p, be independently distributed, where θⁱ and σ are unknown. A Bayesian approach is used to estimate the first two moments of the minimum order statistic, W = min (Y¹, …, Y^p). In order to compute the Bayes estimates, one has to evaluate the predictive densities of the Yⁱ's conditional on past data. Although the required predictive densities are complicated in form, an efficient algorithm to calculate them has been developed and given in the article. An application of the Bayesian method in a continuous-review control model with multiple suppliers is discussed. © 1994 John Wiley & Sons, Inc.     

Coherent systems based on sequential order statistics

The sequential order statistics (SOS) are a good way to model the lifetimes of the components in a system when the failure of a component at time t affects the performance of the working components at this age t. In this article, we study properties of the lifetimes of the coherent systems obtained using SOS. Specifically, we obtain a mixture representation based on the signature of the system. This representation is used to obtain stochastic comparisons. To get these comparisons, we obtain some ordering properties for the SOS, which in this context represent the lifetimes of k‐out‐of‐n systems. In particular, we show that they are not necessarily hazard rate ordered. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Combination setwise-Bonferroni-type bounds

We consider three classes of lower bounds to P(c) = P (X₁ ≤ c₁,…, X_n ≤ c); Bonferroni-type bounds, product-type bounds and setwise bounds. Setwise probability inequalities are shown to be a compromise between product-type and Bonferroni-type probability inequalities. Bonferroni-type inequalities always hold. Product-type inequalities require positive dependence conditions, but are superior to the Bonferroni-type and setwise bounds when these conditions are satisfied. Setwise inequalities require less stringent positive dependence bound conditions than the product-type bounds. Neither setwise nor Bonferroni-type bounds dominate the other. Optimized setwise bounds are developed. Results pertaining to the nesting of setwise bounds are obtained. Combination setwise-Bonferroni-type bounds are developed in which high dimensional setwise bounds are applied and second and third order Bonferroni-type bounds are applied within each subvector of the setwise bounds. These new combination bounds, which are applicable for associated random variables, are shown to be superior to Bonferroni-type and setwise bounds for moving averages and runs probabilities. Recently proposed upper bounds to P(c) are reviewed. The lower and upper bounds are tabulated for various classes of multivariate normal distributions with banded covariance matrices. The bounds are shown to be surprisingly accurate and are much easier to compute than the inclusion-exclusion bounds. A strategy for employing the bounds is developed. © 1996 John Wiley & Sons, Inc.     

Asymptotic joint normality of an increasing number of multivariate order statistics and associated cell frequencies

Take n independent identically distributed (IID) observations from a continuous r-variate population, and choose some order statistics from each of the r variates. These order statistics are used to construct a grid in r-dimensional space. Under certain conditions, it is shown that as n increases we can choose an increasing number of order statistics in such a way that the asymptotic joint distribution of the chosen order statistics and of the frequencies of sample points falling in the cells of the grid can be assumed to be a normal distribution. An application to testing independence of random variables is given.     

A note on circular error probabilities

An approximation for P(X² + Y² ≤ K²σ²₁) based on an unpublished result of Kleinecke is derived, where X and Y are independent normal variables having zero means and variances σ²₁ and σ²₂ and σ₁ ≥ σ₂. Also, we provide asymptotic expressions for the probabilities for large values of β = K²(1 - c²)/4c² where c = σ₂/σ₁. These are illustrated by comparing with values tabulated by Harter [6]. Solution of K for specified P and c is also considered. The main point of this note is that simple and easily calculable approximations for P and K can be developed and there is no need for numerical evaluation of integrals.     

A remark on a helicopter and submarine game

The article considers a two-person zero-sum game in which the movement of the players is constrained to integer points …, −1, 0, 1, … of a line L. Initially the searcher (hider) is at point x = 0 (x = d, d > 0). The searcher and the hider perform simple motion on L with maximum speeds w and u, respectively, where w > u > 0. Each of the players knows the other's initial position but not the other's subsequent positions. The searcher has a bomb which he can drop at any time during his search. Between the dropping of the bomb and the bomb exploding there is a T time lag. If the bomb explodes at point i and the hider is at point i − 1, or i, or i + 1, then the destruction probability is equal to P, or 1, or P, respectively, where 0 < P < 1. d, w, u, and T are integer constants. The searcher can drop the bomb at integer moments of time t = 0, 1, … . The aim of the searcher is to maximize the probability of the destruction of the hider. © 1993 John Wiley & Sons, Inc.     

Large deviation probabilities for order statistics

Asymptotic representations are found for the large deviation probabilities that the nα-th order statistic exceeds δ, where δ>α. The probabilities are first expressed in terms of the empirical distribution function, and then the 1960 theorem of Bahadur and Ranga Rao is applied. The result is then shown to be more precise than a logarithmic statement in a 1969 paper of Sievers dealing with the asymptotic relative efficiency of the sample median test.     

Pricing bridges to cross a river

We consider a pricing problem in directed, uncapacitated networks. Tariffs must be defined by an operator, the leader, for a subset of m arcs, the tariff arcs. Costs of all other arcs in the network are assumed to be given. There are n clients, the followers, and after the tariffs have been determined, the clients route their demands independent of each other on paths with minimal total cost. The problem is to find tariffs that maximize the operator's revenue. Motivated by applications in telecommunication networks, we consider a restricted version of this problem, assuming that each client utilizes at most one of the operator's tariff arcs. The problem is equivalent to pricing bridges that clients can use in order to cross a river. We prove that this problem is APX‐hard. Moreover, we analyze the effect of uniform pricing, proving that it yields both an m approximation and a (1 + lnD)‐approximation. Here, D is upper bounded by the total demand of all clients. In addition, we consider the problem under the additional restriction that the operator must not reject any of the clients. We prove that this problem does not admit approximation algorithms with any reasonable performance guarantee, unless P = NP, and we prove the existence of an n‐approximation algorithm. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Tail hazard rate ordering properties of order statistics and coherent systems

We study tail hazard rate ordering properties of coherent systems using the representation of the distribution of a coherent system as a mixture of the distributions of the series systems obtained from its path sets. Also some ordering properties are obtained for order statistics which, in this context, represent the lifetimes of k‐out‐of‐n systems. We pay special attention to systems with components satisfying the proportional hazard rate model or with exponential, Weibull and Pareto type II distributions. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Technical note: Finite-time regret analysis of Kiefer-Wolfowitz stochastic approximation algorithm and nonparametric multi-product dynamic pricing with unknown demand

We consider the problem of nonparametric multi-product dynamic pricing with unknown demand and show that the problem may be formulated as an online model-free stochastic program, which can be solved by the classical Kiefer-Wolfowitz stochastic approximation (KWSA) algorithm. We prove that the expected cumulative regret of the KWSA algorithm is bounded above by where κ₁, κ₂ are positive constants and T is the number of periods for any T = 1, 2, … . Therefore, the regret of the KWSA algorithm grows in the order of , which achieves the lower bounds known for parametric dynamic pricing problems and shows that the nonparametric problems are not necessarily more difficult to solve than the parametric ones. Numerical experiments further demonstrate the effectiveness and efficiency of our proposed KW pricing policy by comparing with some pricing policies in the literature.     

The queue G/M/m/N: Busy period and cycle,waiting time under reverse and random order service

The busy period, busy cycle, and the numbers of customers served and lost therein, of the G/M/m queue with balking is studied via the embedded Markov chain approach. It is shown that the expectations of the two discrete variables give the loss probability. For the special case G/M/1/N a closed expression in terms of contour integrals is obtained for the Laplace transform of these four variables. This yields as a byproduct the LIFO waiting time distribution for the G/M/m/N queue. The waiting time under random order service for this queue is also studied.     

Optimal dispatching strategies for vehicles having exponentially distributed trip times

A transportation system has N vehicles with no capacity constraint which take passengers from a depot to various destinations and return to the depot. The trip times are considered to be independent and identically distributed random variables. The dispatch strategy at the depot is to dispatch immediately, or to hold any returning vehicles with the objective of minimizing the average wait per passenger at the depot, if passengers arrive at a uniform rate. Optimal control strategies and resulting waits are determined in the special case of exponentially distributed trip time for various N up to N = 15. For N ? 1, the nature of the solution is always to keep a reservoir of vehicles in the depot, and to decrease (increase) the time headway between dispatches as the size of the reservoir gets larger (smaller). For sufficiently large N, one can approximate the number of vehicles in the reservoir by a continuum and obtain analytic experession for the optimal dispatch rate as a function of the number of vehicles in the reservoir. For the optimal strategy, it is shown that the average number of vehicles in the depot is of order N^1/3. These limit properties are expected to be quite insensitive to the actual trip time distribution, but the convergence of the exact properties to the continuum approximation as N → ∞ is very slow.     

Discrete‐time analysis of MAP/PH/1 multiclass general preemptive priority queue

We use the matrix‐geometric method to study the MAP/PH/1 general preemptive priority queue with a multiple class of jobs. A procedure for obtaining the block matrices representing the transition matrix P is presented. We show that the special upper triangular structure of the matrix R obtained by Miller [Computation of steady‐state probabilities for M/M/1 priority queues, Oper Res 29(5) (1981), 945–958] can be extended to an upper triangular block structure. Moreover, the subblock matrices of matrix R also have such a structure. With this special structure, we develop a procedure to compute the matrix R. After obtaining the stationary distribution of the system, we study two primary performance indices, namely, the distributions of the number of jobs of each type in the system and their waiting times. Although most of our analysis is carried out for the case of K = 3, the developed approach is general enough to study the other cases (K ≥ 4). © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 662–682, 2003.     

Openshop scheduling under linear resources constraints

Consider n jobs (J₁, …, J_n), m working stations (M₁, …, M_m) and λ linear resources (R₁, …, R_λ). Job J_i consists of m operations (O_i1, …, O_im). Operation O_ij requires P_k(i, j) units of resource R_k to be realized in an M_j. The availability of resource R_k and the ability of the working station M_h to consume resource R_k, vary over time. An operation involving more than one resource consumes them in constant proportions equal to those in which they are required. The order in which operations are realized is immaterial. We seek an allocation of the resources such that the schedule length is minimized. In this paper, polynomial algorithms are developed for several problems, while NP-hardness is demonstrated for several others. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 51–66, 1998     

Asymptotically optimal component assembly plans in repairable systems and server allocation in parallel multiserver queues

T identical exponential lifetime components out of which G are initially functioning (and B are not) are to be allocated to N subsystems, which are connected either in parallel or in series. Subsystem i, i = 1,…, N, functions when at least K_i of its components function and the whole system is maintained by a single repairman. Component repair times are identical independent exponentials and repaired components are as good as new. The problem of the determination of the assembly plan that will maximize the system reliability at any (arbitrary) time instant t is solved when the component failure rate is sufficiently small. For the parallel configuration, the optimal assembly plan allocates as many components as possible to the subsystem with the smallest K_i and allocates functioning components to subsystems in increasing order of the K_i's. For the series configuration, the optimal assembly plan allocates both the surplus and the functioning components equally to all subsystems whenever possible, and when not possible it favors subsystems in decreasing order of the K_i's. The solution is interpreted in the context of the optimal allocation of processors and an initial number of jobs in a problem of routing time consuming jobs to parallel multiprocessor queues. © John Wiley & Sons, Inc. Naval Research Logistics 48: 732–746, 2001