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The consecutive hit probability of antiaircraft artillery corresponds to the multivariate normal probability distribution. The computational complexity depends on the length of the firing error sequence (i.e., the integral dimension may exceed 100). The traditional numerical integration and the Monte Carlo method are too slow for this calculation. This paper established the state equation of the firing error sequence, which was the bridge between the multivariate normal probability distribution and stochastic process theory. The recursive calculation model was given after the rigorous derivation process. The accuracy and computational complexity of the model were quantified by theoretical analysis and expressed intuitively by examples. The model shows the upper bound for the absolute error, and the computational efficiency is significantly improved. [ABSTRACT FROM AUTHOR]

In this paper, we propose an algorithm to fit heavy-tailed (HT) distribution functions by generalized hyperexponential (GH) distribution functions. A discussion of the steps, usage, and accuracy of the GH algorithm is given. Several examples in this paper show that the proposed method can be applied to fit HT distributions with a completely monotone probability density function (pdf) very well, like the Pareto distribution and the Weibull distribution with the shape parameter less than one, as well as HT distributions whose pdf is not completely monotone, like the lognormal distribution. In addition, we provide an example that shows that the proposed method can be applied to density estimation of real data presenting a heavy tail. [ABSTRACT FROM AUTHOR]

Existing models of stochastic network equilibrium route choice in transport networks are able to represent exogenously specified variations in drivers' actual or perceived travel costs, but assume throughout that flows are deterministic. In this paper, a new notion of equilibrium is presented based on stochastic flow variables, in which the impact of variable flows on the variability in travel costs is endogenously handled. Firstly, a very general notion of equilibrium is deduced as a fixed point condition on the joint probability distribution of network flows. Then, an approximation to this condition is derived, which operates by equilibrating moments of order n and below of the joint flow probability distribution, and is termed a Generalized Stochastic User Equilibrium of order n, being denoted GSUE(n). The GSUE(1) model is seen to be a conventional Stochastic User Equilibrium. The paper goes on to focus on the second order model, GSUE(2). Conditions are presented to guarantee the existence of GSUE(2) solutions. Conditions are deduced to guarantee (a) uniqueness of solutions in networks with a single interzonal movement, and (b) proximity of solutions in networks with multiple interzonal movements. Finally, a simple example is presented. [ABSTRACT FROM AUTHOR]

This paper extends the results of a previous study concerning distribution with time windows. It is recognized that customers who do not need to be allocated to a time window should receive different service than the rest. Three strategies were studied to accomplish that: stratified service, discriminating service, and staggered and discriminating service. Of these, the last strategy yields the lowest local distribution distance per customer, a distance which can be less than hall that for the strategy explained in the previous paper (joint service). Stratified service, however, can yield a lower line-haul distance per customer than staggered and discriminating service. In a specific case, a numerical comparison between stratified service and staggered and discriminating service should determine the overall best choice. This choice notwithstanding, the best strategy involves overlapping vehicle tours. The conventional wisdom of dividing customers into non-overlapping clusters for vehicle muting does not seem attractive for distribution problems with time windows. [ABSTRACT FROM AUTHOR]

This paper shows how distribution problems with delivery time constraints can be modeled approximately with just a few variables. Its objective is neither to develop a scheduling algorithm nor an exact predictive method; rather, it is to illustrate some trade-offs and principles that can be used for planning and algorithm development. A workday is divided into time periods. Time windows are modeled by specifying the period in which a vehicle should visit each customer. (The companion paper explores scenarios where many customers do not specify a time window, and thus, it is advantageous not to allocate all the customers to periods.) Travel distance expressions are provided for a "cluster-first, route-second" strategy, similar to some routing methods currently in use. Travel distance expressions are also provided for refinements of the strategy, including one in which tours are systematically staggered, overlapping. The consequent reductions in travel distance can be quite significant. We suggest here that more attention should be paid to the clustering part of algorithm construction, and point to ways in which the customers served by one vehicle should be selected. This paper shows how distribution problems with delivery time constraints can be modeled approximately with just a few variables. Its objective is neither to develop a scheduling algorithm nor an exact predictive method; rather, it is to illustrate some trade-offs and principles that can be used for planning and algorithm development. A workday is divided into time periods. Time windows are modeled by specifying the period in which a vehicle should visit each customer. (The companion paper explores scenarios where many customers do not specify a time window, and thus, it is advantageous not to allocate all the customers to periods.) Travel distance expressions are provided for a "cluster-first, route-second" strategy, similar to some routing methods currently in use. Travel distance expressions are... [ABSTRACT FROM AUTHOR]

The paper studies the problem of predicting traffic equilibrium (TE) in a transportation network within the framework of decision making among discrete choices in a probabilistic and uncertain environment. Conventional approaches to predict TE typically assume that travel times are deterministic and perceived accurately by the travelers; and some new TE models have considered probabilistic travel times or inaccurate perceptions but not both. In our generalized model, which we refer to as GTESP, the travel time on each route is random and each traveler perceives, possibly inaccurately, a travel time probability distribution for each route which may vary from traveler to traveler Each traveler uses a disutility function of travel time to evaluate each route and chooses that route which minimizes his expected disutility. GTESP is difficult to solve for general nonlinear disutility functions. However, special cases—in particular when arc travel times are statistically independent and the disutility functions to evaluate route travel times are linear, exponential, or quadratic—are solvable, at least approximately. GTESP is general in the sense that most existing TE models can be shown to be special or limiting cases of GTESP. Furthermore, this paper demonstrates, with illustrative examples, that GTESP appears to capture travelers' risk-taking behavior more realistically than existing TE models. [ABSTRACT FROM AUTHOR]

This paper derives a novel, asymptotic statistical test of the Karush-Kuhn-Tucker first-order necessary optimality conditions in random simulation models with multiple responses. This test combines a simple form of the delta method and a generalized version of Wald's statistic. The test is applied to both a toy problem and an (s, S) inventory-optimization problem with a service-level constraint; its numerical results are encouraging. [ABSTRACT FROM AUTHOR]

This paper presents a new randomized search method called evolutionary random policy search (ERPS) for solving infinite-horizon discounted-cost Markov-decision-process (MDP) problems. The algorithm is particularly targeted at problems with large or uncountable action spaces. ERPS approaches a given MDP by iteratively dividing it into a sequence of smaller, random, sub-MDP problems based on information obtained from random sampling of the entire action space and local search. Each sub-MDP is then solved approximately by using a variant of the standard policy-improvement technique, where an elite policy is obtained. We show that the sequence of elite policies converges to an optimal policy with probability one. Some numerical studies are carried out to illustrate the algorithm and compare it with existing procedures. [ABSTRACT FROM AUTHOR]

We demonstrate a numerically exact method for evaluating the time-dependent mean, variance, and higher- order moments of the number of entities in the multiclass [Pht/Pht/∞]K queueing network system, as well as at the individual network nodes. We allow for multiple, independent, time-dependent entity classes and develop time-dependent performance measures by entity class at the nodal and network levels. We also demonstrate a numerically exact method for evaluating the distribution function and moments of virtual sojourn time through the network for virtual entities, by entity class, arriving to the system at time t. We include an example using software that we have developed and have put in downloadable form in the Online Supplement to this paper on the journal's website. [ABSTRACT FROM AUTHOR]

We develop a numerically exact method for evaluating the time-dependent mean, variance, and higher- order moments of the number of entities in a Pht/Pht/∞ queueing system. We also develop a numerically exact method for evaluating the distribution function and moments of the virtual sojourn time for any time t; in our setting, the virtual sojourn time is equivalent to the service time for virtual entities arriving to the system at that time t. We include several examples using software that we have developed and have put in downloadable form in the Online Supplement to this paper on the journal's website. [ABSTRACT FROM AUTHOR]

This paper considers a service station consisting of s identical, independent exponential servers and m buffers. Customers arrive according to a Coxian process with a stages. Blocked customers may either leave the system forever or return, after an exponentially distributed amount of time, to the service station to retry for services. An efficient algorithm is developed to evaluate system steady-state probabilities, and formulas for certain system performance measures are derived in terms of the steady-state probabilities. The computational complexity of the algorithm is shown to be O(4N), where N is the total number of state probabilities evaluated. The influence of the form of interarrival time distributions on system performance measures is examined through numerical results. Issues concerning numerical stability of the algorithm are also discussed. [ABSTRACT FROM AUTHOR]

Many transportation problems can be formulated as a linearly-constrained convex programming problem whose objective function consists of entropy functions and other cost-related terms. In this paper, we propose an unconstrained convex programming dual approach to solving these problems. In particular, we focus on a class of linearly-constrained entropy maximization problem with quadratic cost, study its Lagrangian dual, and provide a globally convergent algorithm with a quadratic rate of convergence. The theory and algorithm can be readily applied to the trip distribution problem with quadratic cost and many other entropy-based formulations, including the conventional trip distribution problem with linear cost, the entropy-based modal split model, and the decomposed problems of the combined problem of trip distribution and assignment. The efficiency and the robustness of this approach are confirmed by our computational experience. [ABSTRACT FROM AUTHOR]

The multinomial logit formulation of travel demand models is analytically transparent and computationally tractable, but it is based on relatively restrictive behavioral assumptions. The validity of these assumptions for a given choice situation can be tested empirically by comparing the logit model with a general multinomial pro bit model This paper describes a Lagrangian multiplier test that enables the comparison to be carried out without estimating the values of the parameters of a general probit model. [ABSTRACT FROM AUTHOR]

This paper presents an explicit procedure for finding improving cycles or paths in the distribution model. The procedure developed may be incorporated in the row-column sum method. When this is done, both the row and column numbers and the predecessor index numbers may be determined simultaneously, a fact which contributes to the rapidity of the method. The chief difference between our method and other labeling procedures derives from the fact that the ordinary procedures are implemented in a 'dual' framework and are concerned with identifying a flow augmenting path that maintains dual feasibility. [ABSTRACT FROM AUTHOR]