At first, I would like to thank the authors for giving the detailed survey on the current state of stochastic orderings of order statistics from heterogeneous populations. Their review illustrates the fast development of research in this area during the recent years, in particular including many important contributions by the authors themselves. Besides the concrete open problems given in the different sections of the review, the following general topics may be also of interest for further studies.

The P–P plot is a powerful graphical tool to compare stochastically the magnitudes of two random variables. In this note, we introduce a new partial order, called P–P order based on P–P plots. For a pair of random variables (X1, Y1) and (X2, Y2) one can see the relative precedence of Y2 over X2 versus that of Y1 over X1 using P–P order. We show that several seemingly very technical and difficult concepts like convex transform order and super-additive ordering can be easily explained with the help of this new partial order. Several concepts of positive dependence can also be expressed in terms of P–P orders of the conditional distributions.

This paper studies a multiple-recipe predictive maintenance problem with M/G/1 queueing effects. The server degrades according to a discrete-time Markov chain and we assume that the controller knows both the machine status and the current number of jobs in the system. The controller's objective is to minimize total discounted costs or long-run average costs which include preventative and corrective maintenance costs, holdings costs, and possibly production costs. An optimal policy determines both when to perform maintenance and which type of job to process. Since the policy takes into account the machine's degradation status, such control decisions are known as predictive maintenance policies. In the single-recipe case, we prove that the optimal policy is monotone in the machine status, but not in the number of jobs in the system. A similar monotonicity result holds in the two-recipe case. Finally, we provide computational results indicating that significant savings can be realized when implementing a predictive maintenance policies instead of a traditional job-based threshold policy for preventive maintenances.

Queues operated by a processor-sharing mode have important applications in many modern systems. However, because of the simultaneous sharing of service capacity by all customers, the distribution function and moments of the sojourn time are difficult to derive, even with a given initial condition. In addition, when a limit on the number of customers in the system is enforced to ensure the quality of service, the sojourn time becomes more complicated. In recent literature, the distribution function is obtained via the Laplace–Stieltjes transform. In this paper, we take a pure algebraic approach to derive the moments of the sojourn time. We obtain an iterative formula and use it to investigate properties of the conditional sojourn time. The approach is simple and intuitive, and applies to queues with multiple class customers as well.

We study the asymptotic behavior of finite horizon ruin probabilities for random walks with heavy tailed increment via corrected diffusion approximation. We follow the main idea in [4] of inverting Fourier transformation, and the Fourier transformation is calculated through optimal stopping and a central limit theorem for renewal process.

First of all, we express our sincere thanks to all the discussants for their valuable comments and suggestions as well as for their own significant contributions to the area of order statistics in general, and to the topic of stochastic comparison in particular. We shall now provide our response to the comments and suggestions of all the discussants.

We establish several such new inequalities that improve the accuracy of bounding the probability content of bivariate normal distribution function by the Slepian's inequalities. Apart from the improvement of the accuracy, their easy–to–compute form is another attractive feature of these new inequalities which fact is of considerable interest for both theoretical and practical applications. A novel application possibility is mentioned, too.

This note introduces bivariate dual generalized Marshall–Olkin distribution and builds a comparison result on the copula of this distribution. Several applications in survival analysis and actuarial science are presented as well.

Motivated by the fact that many investors have limited ability to update the expectation regarding future stock returns with the arrival of new information instantly, this paper provides a continuous-time model to study the performance of passive trading strategies. We derive the true Sharp ratio of the passive strategies in terms of the mean and variance of an explicit stochastic process. Based on this expression, we quantify the impact of partial information by performing a thorough comparative static analysis. Such an analysis provides a rationale for why investors with inaccurate information about stock return behave better in the mean-reverting environment than in the i.i.d. environment and why pessimistic investors can achieve better performance than optimistic ones. As a by-product, we propose an analytical approach to compute the “implied” parameters in stock return predictor for both i.i.d. and mean-reverting dynamics, which seems interesting for future research.

Several limit laws for the Zagreb indices of the classical Erdös–Rényi random graphs are investigated in this paper. We have obtained the necessary and sufficient condition for the asymptotic normality of the two Zagreb indices (suitably normalized), as well as the explicit values for the means and variances of both the indices. Besides, the limiting joint distribution of the numbers of paths of various lengths is also studied under several conditions.

Let F be a bathtub-shaped (BT) hazard rate distribution function. It has been shown that the hazard rate function of the order statistics may be BT, increasing, etc. Then, we have carried out a graphical study for some useful lifetime models.

Moreover, we are interested to compare the time that maximizes the mean residual life (MRL) function of F with the one related to a general weighted model in terms of their locations. Also, the times maximizing the conditional reliability proposed by Mi [13] of F have been compared with the corresponding times of a general weighted model. As special cases, we consider order statistics and the proportional hazard rate model.

When waiting times cannot be observed directly, Little's law can be applied to estimate the average waiting time by the average number in system divided by the average arrival rate, but that simple indirect estimator tends to be biased significantly when the arrival rates are time-varying and the service times are relatively long. Here it is shown that the bias in that indirect estimator can be estimated and reduced by applying the time-varying Little's law (TVLL). If there is appropriate time-varying staffing, then the waiting time distribution may not be time-varying even though the arrival rate is time varying. Given a fixed waiting time distribution with unknown mean, there is a unique mean consistent with the TVLL for each time t. Thus, under that condition, the TVLL provides an estimator for the unknown mean wait, given estimates of the average number in system over a subinterval and the arrival rate function. Useful variants of the TVLL estimator are obtained by fitting a linear or quadratic function to arrival data. When the arrival rate function is approximately linear (quadratic), the mean waiting time satisfies a quadratic (cubic) equation. The new estimator based on the TVLL is a positive real root of that equation. The new methods are shown to be effective in estimating the bias in the indirect estimator and reducing it, using simulations of multi-server queues and data from a call center.

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 portfolio of n identically distributed risks X1, …, Xn with dependence structure modelled by an Archimedean survival copula. It is known that the probability of a large aggregate loss of $\sum\nolimits_{i=1}^{n} X_{i}$ is in proportion to the probability of a large individual loss of X1. The proportionality factor depends on the dependence strength and the tail behavior of the individual risk. In this paper, we establish analogous results for an aggregate loss of the form g(X1, …, Xn) under the more general model in which the Xi's have different but tail-equivalent distributions and the copula remains unchanged, where g is a homogeneous function of order 1. Properties of these factors are studied, and asymptotic Value-at-Risk behaviors of functions of dependent risks are also given. The main results generalize those in Wüthrich [16], Alink, Löwe, and Wüthrich [2], Barbe, Fougères, and Genest [4], and Embrechts, Nešlehová, and Wüthrich [9].

This work focuses on Exchangeable Occupancy Models (EOMs) and their relations with the Uniform Order Statistics Property (UOSP) for point processes in discrete time. As our main purpose, we show how definitions and results presented in Shaked, Spizzichino, and Suter [8] can be unified and generalized in the frame of occupancy models. We first show some general facts about EOMs. Then we introduce a class of EOMs, called ℳ(a)-models, and a concept of generalized Uniform Order Statistics Property in discrete time. For processes with this property, we prove a general characterization result in terms of ℳ(a)-models. Our interest is also focused on properties of closure w.r.t. some natural transformations of EOMs.