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.

In this paper, some results on the complete moment convergence for arrays of rowwise negatively associated (NA, for short) random variables are established. The results obtained in this paper correct the corresponding one obtained in Ko [13] and also improve and generalize the corresponding ones of Kuczmaszewska [14] and Ko [13]. As an application of the main results, we present a result on complete consistency for the estimator in a non-parametric regression model based on NA errors. Finally, we provide a numerical simulation to verify the validity of our result.

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.

In this paper, we consider finite mixture models with components having distributions from the location-scale family. We then discuss the usual stochastic order and the reversed hazard rate order of such finite mixture models under some majorization conditions on location, scale and mixing probabilities as model parameters.

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.

In this article, we consider a compound Poisson insurance risk model with a random discount factor. This model is also known as the compound filtered Poisson model. By using some stochastic analysis techniques, a convergence result for the discounted surplus process, an expression for the ruin probability, and the upper bounds for the ruin probability are obtained.

Consider a particular bidimensional risk model, in which two insurance companies divide between them in different proportions both the premium income and the aggregate claims. In practice, it can be interpreted as an insurer–reinsurer scenario, where the reinsurer takes over a proportion of the insurer's losses. Under the assumption that the claim sizes and inter-arrival times form a sequence of independent and identically distributed random pairs, with each pair obeying a dependence structure, an asymptotic expression for the ruin probability of this bidimensional risk model with constant interest rates is established.

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|.

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.

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.

This article derives the transient behavior of the stochastic process X(t) for which in earlier articles it was proven that for α<1, the process (X(t))1/(1−α) (approximately, under low drop probability or ECN marking probability) describes the behavior of the congestion windows in certain transport protocols in the so-called TCP paradigm. The transient distribution is found explicitly and is particularly transparent for moments like E[(X(t))k |X(0)]. The purpose of this article is to be part of the mathematical foundation when comparing the many protocols in the TCP paradigm.

We consider a dam model with a compound Poisson input having positive jumps in which the release rule can be dynamically controlled between two release functions, r1 and r2. The dam process has two switchover levels, a and b, 0 < a < b < ∞, which are prescribed to switch the release rule from r2 to r1 only when the dam downcrosses levels and switch from r1 to r2 only when the dam upcrosses level b. We derive integral equations whose solutions determine the stationary distribution of the dam content and demonstrate such a determination explicitly for the case of exponential jumps. We also compute an expression for the average number of switches from r1 to r2, and vice versa; consider an approximation to the general case (with b − a being sufficiently large with respect to r1 and r2 and the jump sizes), which allows the use of asymptotic results from renewal theory; and study the wet period analysis for the case of exponential jumps.

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.

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.

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.

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.

In this paper, we study the almost sure convergence for sequences of asymptotically negative associated (ANA) random variables. As a result, we extend the classical Khintchine–Kolmogorov convergence theorem, Marcinkiewicz strong law of large numbers, and the three series theorem for sequences of independent random variables to sequences of ANA random variables without necessarily adding any extra conditions.

We consider sufficient conditions for stochastic equivalence of convex ordered random variables. Our main results apply to all convex ordered distributions on the real line and improve on a recent result of Huang and Lin [8] for equality in distribution of convex ordered survival times. Illustrative applications include testing for equality in distribution with convex ordered alternatives and demonstrating several earlier results on stochastic equivalence as special cases.

We study an online capacity planning problem in which arriving patients require a series of appointments at several departments, within a certain access time target.

This research is motivated by a study of rehabilitation planning practices at the Sint Maartenskliniek hospital (the Netherlands). In practice, the prescribed treatments and activities are typically booked starting in the first available week, leaving no space for urgent patients who require a series of appointments at a short notice. This leads to the rescheduling of appointments or long access times for urgent patients, which has a negative effect on the quality of care and on patient satisfaction.

We propose an approach for allocating capacity to patients at the moment of their arrival, in such a way that the total number of requests booked within their corresponding access time targets is maximized. The model considers online decision making regarding multi-priority, multi-appointment, and multi-resource capacity allocation. We formulate this problem as a Markov decision process (MDP) that takes into account the current patient schedule, and future arrivals. We develop an approximate dynamic programming (ADP) algorithm to obtain approximate optimal capacity allocation policies. We provide insights into the characteristics of the optimal policies and evaluate the performance of the resulting policies using simulation.

The introduction of the class of queueing networks called G-networks by Gelenbe has been a breakthrough in the field of stochastic modeling since it has largely expanded the class of models which are analytically or numerically tractable. From a theoretical point of view, the introduction of the G-networks has lead to very important considerations: first, a product-form queueing network may have non-linear traffic equations; secondly, we can have a product-form equilibrium distribution even if the customer routing is defined in such a way that more than two queues can change their states at the same time epoch. In this work, we review some of the classes of product-forms introduced for the analysis of the G-networks with special attention to these two aspects. We propose a methodology that, coherently with the product-form result, allows for a modular analysis of the G-queues to derive the equilibrium distribution of the network.