A serial production line with an arbitrary number of stations is analyzed using the holding-time model. In contrast to Markovian state models, the holding-time model does not require the service times to be exponentially distributed. The distribution of the interdemand time at the first station is obtained as a function of the distribution of station service times which are of a general nature. The throughput rate of the line is the reciprocal of the mean interdemand time. For service times of phase type, two numerical solution methods, one exact and the other approximate, are presented. Numerical results are given for up to ten stations.

This paper proposes a method that allows the construction of discrete-state Markov chains approximating an Ito-diffusion process. The transition probabilities of the Markov chains are chosen in such a way that functionals converge with a desired weak order with respect to vanishing step size under sufficient smoothness assumptions.

Stochastic ordering of random variables may be defined by the relative convexity of the tail functions. This has been extended to higher order stochastic orderings, by iteratively reassigning tail-weights. The actual verification of stochastic orderings is not simple, as this depends on inverting distribution functions for which there may be no explicit expression. The iterative definition of distributions, of course, contributes to make that verification even harder. We have a look at the stochastic ordering, introducing a method that allows for explicit usage, applying it to the Gamma and Weibull distributions, giving a complete description of the order of relations within each of these families.

When launching a new product, a firm has to set an initial price with incomplete information about demand. However, after observing the demand over a period of time, the firm might decide to mark down the price, especially when the Bayesian updated belief about demand is lower than originally anticipated. We consider the case in which the manufacturer makes three decisions: initial price, when to mark down the price, and the markdown price. Modeling the cumulative demand as a Brownian motion with an unknown drift, we compute the posterior probability distribution of the unknown drift. We then show that it is optimal to mark down the price when the posterior probability is below a computable threshold. This threshold policy enables us to determine the optimal (a) regular price, (b) markdown price, and (c) markdown time. Additionally, we examine the impact of demand volatility and evaluate the value of learning.

We consider here a general class of bivariate distributions from reliability point of view, and refer to it as generalized Marshall–Olkin bivariate distributions. This class includes as special cases the Marshall–Olkin bivariate exponential distribution and the class of bivariate distributions studied recently by Sarhan and Balakrishnan [25]. For this class, the reliability, survival, hazard, and mean residual life functions are all derived, and their monotonicity is discussed for the marginal as well as the conditional distributions. These functions are also studied for the series and parallel systems based on this bivariate distribution. Finally, the Clayton association measure for this bivariate model is derived in terms of the hazard gradient.

Consider a tandem queuing network with an infinite supply of jobs in front of the first station, infinite room for completed jobs after the last station, finite buffers between stations, and a number of flexible servers who are subject to failures. We study the dynamic assignment of servers to stations with the goal of maximizing the long-run average throughput. Our main conclusion is that the presence of server failures does not have a major impact on the optimal assignment of servers to stations for the systems we consider. More specifically, we show that when the servers are generalists, any nonidling policy is optimal, irrespective of the reliability of the servers. We also provide theoretical and numerical results for Markovian systems with two stations and two or three servers that suggest that the structure of the optimal server assignment policy does not depend on the reliability of the servers and that ignoring server failures when assigning servers to stations yields near-optimal throughput. Finally, we present numerical results that illustrate that simple server assignment heuristics designed for larger systems with reliable servers also yield good throughput performance in Markovian systems with three stations and three failure-prone servers.

GIaz and Johnson [14] introduce ith-order product-type approximations, βi, i =1,…n − 1, for Pn = P(X1 ≤c1, X2 ≤ c2,…Xn≤ cn) and show that Pn≥ βn−1 ≥ βn−2 ≥… ≥ β2 ≥ β1 when X is MTP2. In this article, it is shown that

under weaker positive dependence conditions. For multivariate normal distributions, these conditions reduce to cov(Xi,Xj) ≥ 0 for 1 ≤ i < j ≤ n and cov(Xi,Xj| Xj−1) ≥ 0 for 1 ≤ i < j − 1, j = 3,…,n. This is applied to group sequential analysis with bivariate normal responses. Conditions for Pn ≥ β3 ≥ β2 ≥β1 are also derived. Bound conditions are also obtained that ensure that product-type approximations are nested lower bounds to upper orthant probabilities P(X1 > C1,…,Xn > cn). It is shown that these conditions are satisfied for the multivariate exponential distribution of Marshall and Olkin [20].

We characterize the probabilistic nature of the maximum queue length and the maximum waiting time in a multiserver G/G/c queue. We assume a general i.i.d. interarrival process and a general i.i.d. service time process for each server with the possibility of having different service time distributions for different servers. Under a weak additional condition, we will prove that the maximum queue length and waiting time grow asymptotically in probability as logωn−1 and log n1/θ, respectively, where ω < 1 and θ > 0 are parameters of the queuing system. Furthermore, it is shown that the maximum waiting time – when appropriately normalized – converges in distribution to the extreme distribution Λ(x) = exp(−e−x). The maximum queue length exhibits similar behavior, except that some oscillation caused by the discrete nature of the queue length must be taken into account. The first results of this type were obtained for the G/M/l queue by Heyde and for the G/G/l queue by Iglehart. Our analysis is similar to that of Heyde and Iglehart. The generalization to c > 1 servers is made possible due to the recent characterization of the tail of the stationary queue length and waiting time in a G/G/c queue (cf. Sadowsky and Szpankowski [17]).

Consider a large number of interacting queues with symmetric interactions. In the asymptotic limit, the interactions between any fixed finite subcollection become negligible, and the overall effect of interactions can be replaced by an empirical rate. The evolution of each queue is given by a time inhomogeneous Markov process. This may be considered a technique for writing dynamic Erlang fixed-point equations. We explore this as a tool to approximate sojourn time distributions.

This article derives a closed-form pricing formula for the European exchange option in a stochastic volatility framework. Firstly, with the Feynman–Kac theorem's application, we obtain a relation between the price of the European exchange option and a European vanilla call option with unit strike price under a doubly stochastic volatility model. Then, we obtain the closed-form solution for the vanilla option using the characteristic function. A key distinguishing feature of the proposed simplified approach is that it does not require a change of numeraire in contrast with the usual methods to price exchange options. Finally, through numerical experiments, the accuracy of the newly derived formula is verified by comparing with the results obtained using Monte Carlo simulations.

In the standard (S – 1, S) stock model, demand follows a Poisson process. It has appeared to many stock analysts that this model causes an abundance of stock in reality. In case demand is caused by failure or is derived from another process, demand typically does not follow a Poisson process. In this paper, we discuss the (S – 1, S) stock model where demand follows a renewal process and the lead time is deterministic. Moreover, we will extend this to compound renewal demand and multi-echelon inventory systems. Our goal is to show the severe influence of taking the Poisson process for granted.

To determine optimal investment and maintenance decisions, the total costs should be minimized over the whole life of a system or structure. In minimizing life-cycle costs, it is important to account for the time value of money by discounting and to consider the uncertainties involved. This article presents new results in renewal theory with costs that can be discounted according to any discount function that is nonincreasing and monotonic over time (such as exponential, hyperbolic, generalized hyperbolic, and no discounting). The main results include expressions for the first and second moment of the discounted costs over a bounded and unbounded time horizon as well as asymptotic expansions for nondiscounted costs.

Let X1, X2,… be i.i.d. random points in R2 with distribution v, and let Ln be the length of the shortest path through X1,…, Xn. The exact almost sure rate of growth of Ln, is obtained under the assumption that v is self-similar in an appropriate sense. This extends a well-known theorem of Beardwood, Halton, and Hammersley.

Consider the random Dirichlet partition of the interval into n fragments at temperature θ > 0. Explicit results on the law of its size-biased permutation are first supplied. Using these, new results on the comparative search cost distributions from Dirichlet partition and from its size-biased permutation are obtained.

In this article we study the joint optimization of finished goods inventory and pricing in a make-to-stock production system with long-run average profit criterion. The production time is random with controllable rate and the demand is Markovian with rate depending on the sale price. The objective is to dynamically adjust the production rate and the sale price to maximize the long-run average profit. We obtain the optimal dynamic pricing and production control policy and present an efficient bisection algorithm for computing the policy parameters.

The investigation in this article was motivated by an extended generalized inverse Gaussian (EGIG) distribution, which has more than one turning point of the failure rate for certain values of the parameters. In order to study the turning points of a failure rate, we appeal to Glaser's eta function, which is much simpler to handle. We present some general results for studying the reationship among the change points of Glaser's eta function, the failure rate, and the mean residual life function (MRLF). Additionally we establish an ordering among the number of change points of Glaser's eta function, the failure rate, and the MRLF. These results are used to investigate, in detail, the monotonicity of the three functions in the case of the EGIG. The EGIG model has one additional parameter, δ, than the generalized inverse Gaussian (GIG) model's three parameters; see Jorgensen [7]. It has been observed that the EGIG model fits certain datasets better than the GIG of Jorgensen [7]. Thus, the purpose of this article is to present some general results dealing with the relationship among the change points of the three functions described earlier. The EGIG model is used as an illustration.

Recently, Belzunce, Ortega, Pellerey, and Ruiz [3] have obtained stochastic comparisons in increasing componentwise convex order sense for vectors of random sums when the summands and number of summands depend on a common random environment, which prove how the dependence among the random environmental parameters influences the variability of vectors of random sums. The main results presented here generalize the results in Belzunce et al. [3] by considering vectors of parameters instead of a couple of parameters and the increasing directionally convex order. Results on stochastic directional convexity of families of random sums under appropriate conditions on the families of summands and number of summands are obtained, which lead to the convex comparisons between random sums mentioned earlier. Different applications in actuarial science, reliability, and population growth are also provided to illustrate the main results.

This paper deals with two ideas appeared during the last developing phase in Artificial Intelligence: Reservoir Computing (RC) and Random Neural Networks. Both have been very successful in many applications. We propose a new model belonging to the first class, taking the structure of the second for its dynamics. The new model is called Echo State Queuing Network. The paper positions the model in the global Machine Learning area, and provides examples of its use and performances. We show on largely used benchmarks that it is a very accurate tool, and we illustrate how it compares with standard RC models.

In this paper the objective is to find a job sequence that minimizes, stochastically or in expectation, the sum of the total weighted job earliness and the total weighted job tardiness on one machine, when the job processing times are independent and identically distributed random variables. We first derive results for the case in which the jobs share a common, random due date. We then obtain results when the job due dates are distinct, independent random variables.

This paper deals with a two-unit reliability model having one operating unit and one cold standby unit and with ample repair facilities. An approximate method is given for the calculation of the probability distribution of the proportion of time that the system is available in a given time period. The approximate method first computes the mean-time-to-failure and the meantime-to-repair and next approximates the up-and-down process of the two-unit reliability system by an alternating renewal process having exponentially distributed on and off periods. Numerical investigations show a satisfactory performance of the approximation. Also, a sensitivity analysis is given.