This article considers the asset price movements in a financial market when risky asset prices are modeled by marked point processes. Their dynamics depend on an underlying event arrivals process—a marked point process having common jump times with the risky asset price process. The problem of utility maximization of terminal wealth is dealt with when the underlying event arrivals process is assumed to be unobserved by the market agents using, as the main tool, backward stochastic differential equations. The dual problem is studied. Explicit solutions in a particular case are given.

We consider a single-server queue with exponential service time and two types of arrivals: positive and negative. Positive customers are regular ones who form a queue and a negative arrival has the effect of removing a positive customer in the system. In many applications, it might be more appropriate to assume the dependence between positive arrival and negative arrival. In order to reflect the dependence, we assume that the positive arrivals and negative arrivals are governed by a finite-state Markov chain with two absorbing states, say 0 and 0′. The epoch of absorption to the states 0 and 0′ corresponds to an arrival of positive and negative customers, respectively. The Markov chain is then instantly restarted in a transient state, where the selection of the new state is allowed to depend on the state from which absorption occurred.

The Laplace–Stieltjes transforms (LSTs) of the sojourn time distribution of a customer, jointly with the probability that the customer completes his service without being removed, are derived under the combinations of service disciplines FCFS and LCFS and the removal strategies RCE and RCH. The service distribution of phase type is also considered.

In a job-shop manufacturing environment, consider a model in which the “job” is processed at several stages with random yields. Nondefective items move through the stages to supply a possibly random demand. At each stage there are linear costs of processing and work-in-process inventories and there are convex costs of end-item inventory. Decisions specify the lot sizes that are processed at the various stages. Easy computations specify a simply structured decision rule that minimizes the expected total cost.

This article is concerned with a loading-dependent model of cascading failure proposed recently by Dobson, Carreras, and Newman [6]. The central problem is to determine the distribution of the total number of initial components that will have finally failed. A new approach based on a closed connection with epidemic modeling is developed. This allows us to consider a more general failure model in which the additional loads caused by successive failures are arbitrarily fixed (instead of being constant as in [6]). The key mathematical tool is provided by the partial joint distributions of order statistics for a sample of independent uniform (0,1) random variables.

This paper deals with stochastic comparisons of the largest order statistics arising from two sets of independent and heterogeneous gamma samples. It is shown that the weak supermajorization order between the vectors of scale parameters together with the weak submajorization order between the vectors of shape parameters imply the reversed hazard rate ordering between the corresponding maximum order statistics. We also establish sufficient conditions of the usual stochastic ordering in terms of the p-larger order between the vectors of scale parameters and the weak submajorization order between the vectors of shape parameters. Numerical examples and applications in auction theory and reliability engineering are provided to illustrate these results.

We study an asymmetric cyclic polling system with Poisson arrivals, general service-time and switch-over time distributions, and so-called two-phase gated service at each queue, an interleaving scheme that aims to enforce some level of “fairness” among the different customer classes. For this model, we use the classical theory of multitype branching processes to derive closed-form expressions for the Laplace–Stieltjes transform of the waiting-time distributions when the load tends to 1, in a general parameter setting and under proper heavy-traffic scalings. This result is strikingly simple and provides new insights in the behavior of two-phase polling systems. In particular, the result provides insight in the waiting-time performance and the trade-off between efficiency and fairness of two-phase gated polling compared to the classical one-phase gated service policy.

The distribution of the linear combination αX + βY is derived when X and Y are independent Laplace random variables. Extensive tabulations of the associated percentage points are also given. The work is motivated by examples in automation, control, fuzzy sets, neurocomputing, and other areas of informational sciences.

This paper studies the allocation of active redundancies to coherent systems on the context that the base and redundancy components have mutual independent lifetimes. For systems with two symmetric components and systems with one component's minimal cut sets (minimal path sets) covering those of another, we derive sufficient conditions to compare the resultant system lifetimes. Some numerical examples are also presented to illustrate the theoretical results.

We consider the problem of determining when to exit an investment whose cumulative return follows a Brownian motion with drift μ and volatility σ2. After an unobserved exponential amount of time, the drift drops from μH > 0 to μL < 0. Using results from stochastic differential equations, we are able to show that it is optimal to exit the first time the posterior probability of being in the low state falls to p*, where the value of p* is given implicitly. We effect a complete comparative statics analysis; one surprising result is that a decrease in μL is beneficial when |μL| is large.

Linear combinations of independent random variables have been extensively studied in the literature. Xu & Hu [21] and Pan, Xu, & Hu [16] unified the study of linear combinations of independent random variables under the general setup. This paper is a companion one of these two papers. In this paper, we will further study this topic. The results are further generalized to the cases of permutation invariant random variables and of independent but not necessarily identically distributed random variables which are ordered in the likelihood ratio or the hazard ratio order.

Birth and death processes are useful in a wide range of disciplines from computer networks and telecommunications to chemical kinetics and epidemiology. Data from many different areas such as linguistics, music, or warfare fit Zipf's law surprisingly well. The Lerch distribution generalizes Zipf's law and is applicable in survival and dispersal processes. In this article we construct a birth and death process that converges to the Lerch distribution in the limit as time becomes large, and we investigate the speed of convergence. This is achieved by employing continued fractions. Numerical illustrations are presented through tables and graphs.

Let τk|n denote the lifetime of a k-out-of-n system, where the n components have independent lifetimes Ti with completely arbitrary distribution Fi, i = 1,..., n. It is shown that τk+1|n ≤hr τk|n, τk|n ≤hr τk−1|n−1, and τk|n−1 ≤hr τk|n if Ti ≤hrTn, i = 1,..., n − 1; τk+1|n ≤rh τk|n, τk−1|n ≤rh τk|n, and τk|n ≤rh τk−1|n−1 if Tn ≤rhTi, i = 1,..., n − 1. These results are available in the literature for the special case of Fi's being absolutely continuous. Also, even in this case, the proofs are often tedious and use the concept of “totally positive of order infinity in differences of k.” In contrast, the proofs given here are simple and elegant and do not use the above concept.

We consider a queue fed by a large number, say n, on–off sources with generally distributed on- and off-times. The queueing resources are scaled by n: The buffer is B ≡ nb and the link rate is C ≡ nc. The model is versatile. It allows one to model both long-range-dependent traffic (by using heavy-tailed on-periods) and short-range-dependent traffic (by using light-tailed on-periods). A crucial performance metric in this model is the steady state buffer overflow probability.

This probability decays exponentially in n. Therefore, if n grows large, naive simulation is too time-consuming and fast simulation techniques have to be used. Due to the exponential decay (in n), importance sampling with an exponential change of measure goes through, irrespective of the on-times being heavy or light tailed. An asymptotically optimal change of measure is found by using large deviations arguments. Notably, the change of measure is not constant during the simulation run, which is different from many other studies (usually relying on large buffer asymptotics).

Numerical examples show that our procedure improves considerably over naive simulation. We present accelerations, we discuss the influence of the shape of the distributions on the overflow probability, and we describe the limitations of our technique.

We study the optimal admission of arriving customers to a Markovian finite-capacity queue (e.g., M/M/c/N queue) with several customer types. The system managers are paid for serving customers and penalized for rejecting them. The rewards and penalties depend on customer types. The penalties are modeled by a K-dimensional cost vector, K ≥ 1. The goal is to maximize the average rewards per unit time subject to the K constraints on the average costs per unit time. Let Km denote min{K,m − 1}, where m is the number of customer types. For a feasible problem, we show the existence of a Km-randomized trunk reservation optimal policy, where the acceptance thresholds for different customer types are ordered according to a linear combination of the service rewards and rejection costs. Additionally, we prove that any Km-randomized stationary optimal policy has this structure.

This article analyzes a generic class of queuing systems with server vacation. The special feature of the models considered is that the duration of the vacations explicitly depends on the buffer content evolution during the previous active period (i.e., the time elapsed since the previous vacation). During both active periods and vacations, the buffer content evolves as a Lévy process. For two specific classes of models, the Laplace–Stieltjes transform of the buffer content distribution at switching epochs between successive vacations and active periods, and in steady state, is derived.

In this paper we present a unified approach to the optimal dimensioning and tariffing of loss networks. In our formulation the optimum is chosen to maximize the profit for the company operating the loss network. We assume that the operating company has the flexibility to determine tariffs and grade of service — although both of these can possibly be subject to regulatory constraints. The fact that the tariffing may affect demand and, hence, the dimensioning makes it essential that the operating company include the tariff/demand trade-off in determining the optimal way to dimension the loss network. A consequence of our formulation is that the optimal tariff structure has a particularly simple form, with the optimal tariff on a particular route separating into a term related to the tariff/demand trade-off on that route and a term that reflects the cost of the circuits used by the route.

There is a growing interest in networks of queues with customers and signals. The signals in these models carry commands to the service nodes and trigger customers to move instantaneously within the network. In this note we consider networks of queues with signals and random triggering times; that is, when a signal arrives at a node, it takes a random amount of time to trigger a customer to move with distribution depending on the source of the signal. By appropriately choosing the triggering times, we can obtain network models such that a signal changes a customer's service time distribution – for example, the signal increases or decreases the service time of a customer. We show that the stationary distribution of this model has a product form solution.

Most of the real-life populations are heterogeneous and homogeneity is often just a simplifying assumption for the relevant statistical analysis. Mixtures of lifetime distributions that correspond to homogeneous subpopulations were intensively studied in the literature. Various distributional and stochastic properties of finite and continuous mixtures were discussed. In this paper, following recent publications, we develop further a mixture concept in the form of the generalized α-mixtures that include all mixture models that are widely explored in the literature. We study some main stochastic properties of the suggested mixture model, that is, aging and appropriate stochastic comparisons. Some relevant examples and counterexamples are given to illustrate our findings.

We study a model of queue storage in which items (requests for single units of storage) arrive in a Poisson stream and are accommodated by the first available location in a linear scan of storage. The processing times of items are independent, exponentially distributed random variables. The set of occupied locations (identified by their indices) at time t forms a random subset Si, of [1,2,.…]. The extent of the fragmentation in Si, i.e., the alternating holes and occupied regions of storage, is measured by Wt, = max St, – |St|.

The total hazard Monte Carlo estimator has been successfully applied to estimate the unreliability of multicomponent systems. Nevertheless, there are some cases where it is less efficient than the crude estimator. We propose in this paper a simple modification that improves the performance of the total hazard estimator.