Sensitivity analysis represents an important aspect of network flow design problems. For example, what is the incremental increase in system flow of increasing the diameters of specified pipes in a water distribution network? Although methods exist for solving this problem in the deterministic case, no comparable methodology has been available when the network's arc capacities are subject to random variation. This paper provides this methodology by describing a Monte Carlo sampling plan that allows one to conduct a sensitivity analysis for a variable upper bound on the flow capacity of a specified arc. The proposed plan has two notable features. It permits estimation of the probabilities of a feasible flow for many values of the upper bound on the arc capacity from a single data set generated by the Monte Carlo method at a single value of this upper bound. Also, the resulting estimators have considerably smaller variancesthan crude Monte Carlo sampling would produce in the same setting. The success of the technique follows from the use of lower and upper bounds on eachprobability of interest where the bounds are generated from an established method of decomposing the capacity state space.

New integral inequality characterizations of the NBUC and NBU(2) classes of lifetime distributions are given. These results are then used to prove preservation under mixture results for the NBUC and NBU(2) classes.

Recently, proportional mean residual life model has received a lot of attention after the importance of the model was discussed by Zahedi [17]. In this paper, we define dynamic proportional mean residual life model and study its properties for different aging classes. The closure of this model under different stochastic orders is also discussed. Many examples are presented to illustrate different properties of the model.

We introduce a natural growth model for directed series-parallel (SP) graphs and look at some of the graph properties under this stochastic model. Specifically, we look at the degrees of certain types of nodes in the random SP graph. We examine the degree of a pole and will find its exact distribution, given by a probability formula with alternating signs. We also prove that, for a fixed value s, the number of nodes of outdegree 1, …, s asymptotically has a joint multivariate normal distribution. Pólya urns will systematically provide a working tool.

This paper studies optimal switching on and off of the entire service capacity of an M/M/∞ queue with holding, running and switching costs. The running costs depend only on whether the system is on or off, and the holding costs are linear. The goal is to minimize average costs per unit time. The main result is that an average-cost optimal policy either always runs the system or is an (M, N)-policy defined by two thresholds M and N, such that the system is switched on upon an arrival epoch when the system size accumulates to N and is switched off upon a departure epoch when the system size decreases to M. It is shown that this optimization problem can be reduced to a problem with a finite number of states and actions, and an average-cost optimal policy can be computed via linear programming. An example, in which the optimal (M, N)-policy outperforms the best (0, N)-policy, is provided. Thus, unlike the case of single-server queues studied in the literature, (0, N)-policies may not be average-cost optimal.

A new stopping problem and the critical exercise price of American fractional lookback option are developed in the case where the stock price follows a special mixed jump diffusion fractional Brownian motion. By using Itô formula and Wick-Itô-Skorohod integral a new market pricing model is built, and the fundamental solutions of stochastic parabolic partial differential equations are deduced under the condition of Merton assumptions. With an optimal stopping problem and the exercise boundary, the explicit integral representation of early exercise premium and the critical exercise price are also derived. Numerical simulation illustrates the asymptotic behavior of this critical boundary.

In this paper, we extend the strong laws of large numbers and entropy ergodic theorem for partial sums for tree-indexed nonhomogeneous Markov chains fields to delayed versions of nonhomogeneous Markov chains fields indexed by a homogeneous tree. At first we study a generalized strong limit theorem for nonhomogeneous Markov chains indexed by a homogeneous tree. Then we prove the generalized strong laws of large numbers and the generalized asymptotic equipartition property for delayed sums of finite nonhomogeneous Markov chains indexed by a homogeneous tree. As corollaries, we can get the similar results of some current literatures. In this paper, the problem settings may not allow to use Doob's martingale convergence theorem, and we overcome this difficulty by using Borel–Cantelli Lemma so that our proof technique also has some new elements compared with the reference Yang and Ye (2007).

Let $\{Y_{1},\ldots ,Y_{n}\}$ be a collection of interdependent nonnegative random variables, with $Y_{i}$ having an exponentiated location-scale model with location parameter $\mu _i$, scale parameter $\delta _i$ and shape (skewness) parameter $\beta _i$, for $i\in \mathbb {I}_{n}=\{1,\ldots ,n\}$. Furthermore, let $\{L_1^{*},\ldots ,L_n^{*}\}$ be a set of independent Bernoulli random variables, independently of $Y_{i}$'s, with $E(L_{i}^{*})=p_{i}^{*}$, for $i\in \mathbb {I}_{n}.$ Under this setup, the portfolio of risks is the collection $\{T_{1}^{*}=L_{1}^{*}Y_{1},\ldots ,T_{n}^{*}=L_{n}^{*}Y_{n}\}$, wherein $T_{i}^{*}=L_{i}^{*}Y_{i}$ represents the $i$th claim amount. This article then presents several sufficient conditions, under which the smallest claim amounts are compared in terms of the usual stochastic and hazard rate orders. The comparison results are obtained when the dependence structure among the claim severities are modeled by (i) an Archimedean survival copula and (ii) a general survival copula. Several examples are also presented to illustrate the established results.

In several queueing systems the service rate of a server is affected by the work load present in the system. For example, a teller at a bank or a checker at a check-out counter in a supermarket may change the service rate depending on the number of customers present in the system. But the service rate as a function of the number in the system can rarely be measured. Consequently, in a typical model of such a system it is assumed that the service rate is constant. Hence, such systems with a single stage are often modeled by GI/GI/c queueing systems with mutually independent arrival and service processes. Then the observed service times are used to find a sample distribution that will represent the distribution of the assumed i.i.d. service times. The purpose of this paper is to explore the effect of this black box queue inference (BBQI) in its ability to predict the performance of the actual system. In this regard, we have shown that when the arrival process is Poisson, if the servers react favorably [unfavorably] to higher work loads (i.e., if the server increases [decreases] the service rate as the number of customers in the system increases) then the BBQI predictions will be pessimistic [optimistic]. This result can be used to identify the server's attitude toward higher work load.

Fixed point approximations for blocking probabilities arising from a link independence assumption and a light retrial rate limit are derived for circuitswitched network models that incorporate caller retrials. This approximation is a generalization of the well-known reduced load approximation for loss networks, which retains much of the versatility of this approximation.

This paper studies the variability of both series and parallel systems comprised of heterogeneous (and dependent) components. Sufficient conditions are established for the star and dispersive orderings between the lifetimes of parallel [series] systems consisting of dependent components having multiple-outlier proportional hazard rates and Archimedean [Archimedean survival] copulas. We also prove that, without any restriction on the scale parameters, the lifetime of a parallel or series system with independent heterogeneous scaled components is larger than that with independent homogeneous scaled components in the sense of the convex transform order. These results generalize some corresponding ones in the literature to the case of dependent scenarios or general settings of components lifetime distributions.

We introduce an incremental growth model for directed binary series-parallel (SP) graphs. The vertices of a directed binary SP graphs can only have outdgrees 1 or 2. We show that the number of vertices of outdegree 1 have a normal distribution (so, necessarily, the vertices of outdegree 2 have a normal distribution, too). Furthermore, we study the average length of a random walk between the poles of the graph. The asymptotic equivalent of the latter property includes the golden ratio. Pólya urns will systematically provide a coding method to initiate the studies.

In this article we consider a noncooperative n-person optimal stopping game of Showcase Showdown, in which each player observes the sum of independent and identically distributed random variables uniformly distributed in [0, 1]. Players can decide to stop the draw in each moment. The objective of a player is to get the maximal number of scores that does is not exceeded level 1. If the scores of all players exceed 1, then the winner is the player whose score is closest to 1. We derive the equilibrium in this game on the basis of the dynamic programming approach.

N. Balakrishnan and Peng Zhao have prepared an outstanding survey of recent results that stochastically compare various order statistics and some ranges based on two collections of independent heterogeneous random variables. Their survey focuses on results for heterogeneous exponential random variables and their extensions to random variables with proportional hazard rates. In addition, some results that stochastically compare order statistics based on heterogeneous gamma, Weibull, geometric, and negative binomial random variables are also given. In particular, the authors of have listed some stochastic comparisons that are based on one heterogeneous collection of random variables, and one homogeneous collection of random variables. Personally, I find these types of comparisons to be quite fascinating. Balakrishnan and Zhao have done a thorough job of listing all the known results of this kind.

In this article we introduce isotone differences stochastic ordering of Markov processes on lattice ordered state spaces as a device to compare the internal dependencies of two such processes. We derive a characterization in terms of intensity matrices. This enables us to compare the internal dependency structure of different degradable Jackson networks in which the nodes are subject to random breakdowns and repairs. We show that the performance behavior and the availability of such networks can be compared.

Recently an error-bound theorem was reported to conclude analytic error bounds for approximate Markov chains. The theorem required a uniform bound for marginal expectations of the approximate model. This note will relax this bound to steady state rather than marginal expectations as of practical interest

(i) to simplify verification and/or

(ii) to obtain a more accurate error bound.

A communication model is studied in detail to support the results.

We propose a method for approximating equivalent local volatility functions of stochastic volatility models. Enlightened by the theory of generalized Wiener functionals proposed by Watanabe and Yoshida (1987, 1992), our key technique is to propose a closed-form expansion of conditional expectations involving marginal distributions generated by stochastic differential equations. A numerical test and an illustration of application are provided to demonstrate the efficiency of our approach.

Let [script S] be a system that can be in one of two states, up or down. We can interpret the up state as the working state and the down state as the nonworking state for system [script S]. Assume the system [script S] is up at time t = 0. Denote the elapsed time from zero until [script S] enters down state by U1, and the elapsed time from then until [script S] is up again by D1. The interval [0,U1 + D1) is called cycle 1, and, at that point, cycle 2 starts and the sequence is repeated so on and so forth. In general, the up time and down time in the jth cycle are denoted by Uj and Dj, respectively, for j ≥ 1. We can introduce a binary process {X(t), t ≥ 0} to describe the state of [script S] at time t using X(t) = 1(0) to indicate [script S] is up (down). Many problems require finding the probability that [script S] is up at time t, that is, P(X(t) = 1). However, an even more interesting question to answer is what the probability is that [script S] is in the up state in an interval of length w starting at time t, that is, P(X(s) = 1, t ≤ s ≤ t + w). Due to the complexity of this problem, no explicit expressions are available for most systems even in the case where (Uj, Dj), j ≥ 1 are i.i.d. Fortunately, in practice engineers are more interested in the long-run properties of the above two probabilities.

The elementary problem of exhaustively sampling a finite population without replacement is used as a nonreversible test case for comparing two recently proposed MCMC algorithms for perfect sampling, one based on backward coupling and the other on strong stationary duality. The backward coupling algorithm runs faster in this case, but the duality-based algorithm is unbiased for user impatience. An interesting by-product of the analysis is a new and simple stochastic interpretation of a mixing-time result for the move-to-front rule.

In this paper, the multi-state survival signature is first redefined for multi-state coherent or mixed systems with independent and identically distributed (i.i.d.) multi-state components. With the assumption of independence of component lifetimes at different state levels, transformation formulas of multi-state survival signatures of different sizes are established through the use of equivalent systems and a generalized triangle rule for order statistics from several independent and non-identical distributions. The results obtained facilitate stochastic comparisons of multi-state coherent or mixed systems with different numbers of i.i.d. multi-state components. Specific examples are finally presented to illustrate the transformation formulas established here, and also their use in comparing systems of different sizes.