Multiple dispatches in a poisson process

An optimal policy is characterized for operating the following system. Customers arrive in [O, T] according to a homogeneous Poisson process. Instantaneous services are provided at times O and T. Additional instantaneous services can be provided at N intermediate stop ping times. These times must be chosen to minimize the total expected customer-hours in [O, T] spent waiting for service.     

On the bivariate negative binomial distribution of mitchell and paulson

The bivariate negative binomial distribution of Mitchell and Paulson [17] for the case b = c = 0 is shown to be equivalent to the accident proneness model of Edwards and Gurland [4] and Subrahmaniam [19,20]. The diagonal series expansion of its joint probability function is then derived. Two other formulations of this distribution are also considered: (i) as a mixture model, which showed how it arises as the discrete analogue to the Wicksell-Kibble bivariate gamma distribution, and (ii) as a consequence of the linear birth-and-death process with immigration.     

A note on the optimal replacement time of damaged devices

Abdel Hameed and Shimi [1] in a recent paper considered a shock model with additive damage. This note generalizes the work of Abdel Hameed and Shimi by showing that the a-priori restriction to replacement at a shock time made in [1] is unnecessary.     

Evaluating cost and reliability integrated performance of stochastic logistics systems

The significance of integrating reliability into logistics performance has been established [The Logistics Performance Index and Its Indicators, World Bank International Trade and Transport Departments, (2010)]. Hence, as a response to the work by the World Bank, the present article aims to evaluate the performance index R_b,d of logistics systems as the probability that a specified demand d can be distributed successfully through multistate arc capacities from the source to the destination under the constraint that the total distribution cost should not exceed the cost limitation b. This article provides a pioneering approach for a straightforward computation of the performance index R_b,d. The proposed algorithm is a hybrid between the polynomial time capacity‐scaling algorithm, which was presented by Edmonds and Karp [JACM 19 (1972)], and the decomposition algorithm, which was presented by Jane and Laih [IEEE (2008)]. Currently, the proposed approach is the only algorithm that can directly compute R_b,d. An illustration of the proposed algorithm is presented. The results of the computational experiments indicate that the presented algorithm outperforms existing algorithms. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

Argentina and the Bomb

Atomic Obsession: Nuclear Alarmism from Hiroshima to Al Qaeda, by John Mueller. Oxford University Press, 2009. 336 pages, $27.95.

Les armes nucléaires: Mythes et réalités [Nuclear Weapons: Myths and Realities], by Georges Le Guelte. Actes Sud, 2009. 390 pages, [euro]25.     

New results on a Ruckle problem in discrete games of ambush

This article deals with a two‐person zero‐sum game in which player I chooses in integer interval [1, N] two integer intervals consisting of p and q points where p + q < N, and player II chooses an integer point in [1, N]. The payoff to player I equals 1 if the point chosen by player II is at least in one of the intervals chosen by player II and 0 otherwise. This paper complements the results obtained by Ruckle, Baston and Bostock, Lee, Garnaev, and Zoroa, Zoroa and Fernández‐Sáez. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 98–106, 2001     

Estimating value in a uniform auction

Consider an auction in which increasing bids are made in sequence on an object whose value θ is known to each bidder. Suppose n bids are received, and the distribution of each bid is conditionally uniform. More specifically, suppose the first bid X₁ is uniformly distributed on [0, θ], and the i^th bid is uniformly distributed on [X_i?1, θ] for i = 2, …?, n. A scenario in which this auction model is appropriate is described. We assume that the value θ is un known to the statistician and must be esimated from the sample X₁, X₂, …?, X_n. The best linear unbiased estimate of θ is derived. The invariance of the estimation problem under scale transformations in noted, and the best invariant estimation problem under scale transformations is noted, and the best invariant estimate of θ under loss L(θ, a) = [(a/θ) ? 1]² is derived. It is shown that this best invariant estimate has uniformly smaller mean-squared error than the best linear unbiased estimate, and the ratio of the mean-squared errors is estimated from simulation experiments. A Bayesian formulation of the estimation problem is also considered, and a class of Bayes estimates is explicitly derived.     

On the first time a separately maintained parallel system has been down for a fixed time

Consider a system consisting of n separately maintained independent components where the components alternate between intervals in which they are “up” and in which they are “down”. When the i^th component goes up [down] then, independent of the past, it remains up [down] for a random length of time, having distribution F_i[G_i], and then goes down [up]. We say that component i is failed at time t if it has been “down” at all time points s ?[t-A.t]: otherwise it is said to be working. Thus, a component is failed if it is down and has been down for the previous A time units. Assuming that all components initially start “up,” let T denote the first time they are all failed, at which point we say the system is failed. We obtain the moment-generating function of T when n = l, for general F and G, thus generalizing previous results which assumed that at least one of these distributions be exponential. In addition, we present a condition under which T is an NBU (new better than used) random variable. Finally we assume that all the up and down distributions F_i and G_i i = l,….n, are exponential, and we obtain an exact expression for E(T) for general n; in addition we obtain bounds for all higher moments of T by showing that T is NBU.     

Strategic defense and attack for series and parallel reliability systems: reply 1 to comment 1

Kovenock and Roberson's [2011] comment provides initial work which has the potential, when suitably extended, to advance the research frontier. Kovenock and Roberson's paper consists of three sections. The first section is an interesting introduction. The second section, titled ‘Model and Main Result,’ provides no contribution beyond Hausken [2008a]. It consists of Equations (1)–(10) which are equivalent to equations developed by Hausken, and Equation (11) which is equivalent to the utility requirements u???0 and U???0 provided after Equation (17) in Hausken. The third section provides interesting ideas about mixed-strategy equilibria that can be extended in future research.     

A good simple percentile estimator of the weibull shape parameter for use when all three parameters are unknown

In this paper we consider a simple three-order-statistic asymptotically unbiased estimator of the Weibull shape parameter c for the case in which all three parameters are unknown. Optimal quantiles that minimize the asymptotic variance of this estimator, c? are determined and shown to depend only on the true (unknown) shape parameter value c and in a rather insensitive way. Monte Carlo studies further verified that, in practice where the true shape parameter c is unknown, using always c? with the optimal quantities that correspond to c = 2.0 produces estimates, c?, remarkably close to the theoretical optimal. A second stage estimation procedure, namely recalculating c? based on the optimal quantiles corresponding to c?, was not worth the additional effort. Benchmark simulation comparisons were also made with the best percentile estimator of Zanakis [20] and with a new estimator of Wyckoff, Bain and Engelhardt [18], one that appears to be the best of proposed closed-form estimators but uses all sample observations. The proposed estimator, c?, should be of interest to practitioners having limited resources and to researchers as a starting point for more accurate iterative estimation procedures. Its form is independent of all three Weibull parameters and, for not too large sample sizes, it requires the first, last and only one other (early) ordered observation. Practical guidelines are provided for choosing the best anticipated estimator of shape for a three-parameter Weibull distribution under different circumstances.     

Nonlinear one-parametric linear programming and t-norm transportation problems

Previous methods for solving the nonlinear one-parametric linear programming problem min {c(t)^Tx |Ax = b, x ≥ 0} for t ? [α,β] were based on the simplex method using a considerably extended tableau. The proposed method avoids such an extension. A finite sequence of feasible bases (B^k | k = 1, 2, …, r) optimal in [t^k, t^k+1] for k = 1, 2, …,^r with α = t¹ < t² < … < t^r+1 = β is determined using the zeroes of a set of nonlinear functions. Computational experience is discussed in the special case of t-norm transportation problems.     

Test for the extreme value and weibull distributions based on normalized spacings

Discussed in this article are tests for the extreme-value distribution, or, equivalently, for the two-parameter Weibull distribution when parameters are unknown and the sample may be censored. The three tests investigated are based on the median, the mean, and the Anderson-Darling A² statistic calculated from a set zⁱ of values derived from the spacings of the sample. The median and the mean have previously been discussed by Mann, Scheuer, and Fertig [10] and by Tiku and Singh [14]. Asymptotic distributions and points are given for the test statistics, based on recently developed theory, and power studies are conducted to compare them with each other and with two other statistics suitable for the test. Of the normalized spacings tests, A² is recommended overall; the mean also gives good power in many situations, but can be nonconsistent.     

An application of the MTP2 property on bounds on system reliability

This paper is concerned with the joint prior distribution of the dependent reliabilities of the components of a binary system. When this distribution is MTP₂ (Multivariate Totally Positive of Order 2), it is shown in general that this actually makes the machinery of Natvig and Eide [7] available to arrive at the posterior distribution of the system's reliability, based on data both at the component and system level. As an illustration in a common environmental stress case, the joint prior distribution of the reliabilities is shown to have the MTP₂ property. We also show, similarly to Gåsemyr and Natvig [3], for the case of independent components given component reliabilities how this joint prior distribution may be based on the combination of expert opinions. A specific system is finally treated numerically. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 741–755, 1997     

On queues with dependent interarrival and service times

AnM/G/1 queueing system is studied in which the service time required by a customer is dependent on the interarrival time between his arrival and that of his predecessor Assuming the two variables are “associated,” we prove that the expected delay in this system is less than or equal to than of a conventional M/G/1 queue This conclusion has been verified via simulation by Mitchell and Paulson [9] for a special class of dependent M/M/1 queue. Their model is a special case of the one we consider here. We also study another modified GI/G/1 queue. where the arrival process and/or the service process are individually “associated”.     

A note on the relationship between correlation and circular error probability (CEP)

This note investigates the effects of ignoring correlation (p) as is sometimes done when estimating or placing confidence intervals on circular error probability (CEP). It is shown that better estimates of CEP can be made if the axis is rotated [5] so that the estimate of p is zero. It is also shown that ignoring variability in estimating correlation can affect the confidence intervals on CEP or the placing of a lower bound on the probability content of a circle.     

A note on allocating jobs to two machines

A recent article in this journal by Mehta, Chandrasekaran, and Emmons [1] described a dynamic programming algorithm for assigning jobs to two identical parallel processors in a way that minimizes the average delay of these jobs. Their problem has a constraint on the sequence of the jobs such that any group of jobs assigned to a processor must be processed in the order of the sequence. This note has two purposes. First, we wish to point out a relationship between this work and some prior work [2]. Second, we wish to point out that Mehta, Chandrasekaran, and Emmons formulation, slightly generalized, can be used to find the optimum assignment of jobs to two machines in a more general class of problems than they considered including a subclass in which the jobs are not constrained to be processed in a given sequence.     

A note on optimal switching between two activities

Let f₁ and f₂ map [0, T] into the real numbers. A system is following either f₁ or f₂ and earning the associated reward ∫ f₁ or ∫ f₂, respectively. It is possible at any time to switch from f_i to f_j by paying a switching cost b > 0. We determine a switching policy which maximizes the total reward. Conditions which guarantee a planning horizon are established.     

The analysis of some algorithms for generating random variates with a given hazard rate

We analyze the expected time performance of two versions of the thinning algorithm of Lewis and Shedler for generating random variates with a given hazard rate on [0,∞]. For thinning with fixed dominating hazard rate g(x) = c for example, it is shown that the expected number of iterations is cE(X) where X is the random variate that is produced. For DHR distributions, we can use dynamic thinning by adjusting the dominating hazard rate as we proceed. With the aid of some inequalities, we show that this improves the performance dramatically. For example, the expected number of iterations is bounded by a constant plus E(log₊(h(0)X)) (the logarithmic moment of X).     

On a ruckle problem in discrete games of ambush

The following zero-sum game is considered. Red chooses in integer interval [1, n] two integer intervals consisting of k and m points where k + m < n, and Blue chooses an integer point in [1, n]. The payoff to Red equals 1 if the point chosen by Blue is at least in one of the intervals chosen by Red, and 0 otherwise. This work complements the results obtained by Ruckle, Baston and Bostock, and Lee. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 353–364, 1997     

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