The Normal and Gamma distributions are the most popular models for analyzing symmetric and skewed data, respectively. In this article, a new multimodal distribution is introduced that contains the Normal and Gamma distributions as particular cases and thus could be a better model for both symmetric and skewed data. Various structural properties of this distribution are derived, including its moment-generating function, characteristic function, moments, entropy, asymptotic distribution of the extreme order statistics, method of moment estimates, maximum likelihood estimates, Fisher information matrix, and simulation issues. The superiority of the new distribution is illustrated by means of two real datasets.

We consider a polling system where a group of an infinite number of servers visits sequentially a set of queues. When visited, each queue is attended for a random time. Arrivals at each queue follow a Poisson process, and the service time of each individual customer is drawn from a general probability distribution function. Thus, each of the queues comprising the system is, in isolation, an M/G/∞-type queue. A job that is not completed during a visit will have a new service-time requirement sampled from the service-time distribution of the corresponding queue. To the best of our knowledge, this article is the first in which an M/G/∞-type polling system is analyzed. For this polling model, we derive the probability generating function and expected value of the queue lengths and the Laplace–Stieltjes transform and expected value of the sojourn time of a customer. Moreover, we identify the policy that maximizes the throughput of the system per cycle and conclude that under the Hamiltonian-tour approach, the optimal visiting order is independent of the number of customers present at the various queues at the start of the cycle.

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.

The discrete-time G/GI/∞ queue model is explored. Jobs arrive to an infinite-server queuing system following an arbitrary input process X; job sizes are general independent and identically distributed random variables. The system's output process Y (of job departures) and queue process N (tracking the number of jobs present in the system) are analyzed. Various statistics of the stochastic maps X↦ Y and X↦ N are explicitly obtained, including means, variances, autocovariances, cross-covariances, and multidimensional probability generating functions. In the case of stationary inputs, we further compute the spectral densities of the stochastic maps, characterize the fixed points (in the L2 sense) of the input–output map, precisely determine when the output and queue processes display either short-ranged or long-ranged temporal dependencies, and prove a decomposition result regarding the intrinsic L2 structure of general stationary G/GI/∞ systems.

We consider two-stage tandem queuing systems with dedicated servers in each station and flexible servers that can serve in both stations. We assume exponential service times, linear holding costs, and operating costs incurred by the servers at rates proportional to their speeds. Under conditions that ensure the optimality of nonidling policies, we show that the optimal allocation of flexible servers is determined by a transition-monotone policy. Moreover, we present conditions under which the optimal policy can be explicitly determined.

The question of how long to run a discrete event simulation before data collection starts is an important issue when estimating steady-state performance measures such as average queue lengths. By using experiments based on numerical (nonsimulation) methods published elsewhere, we shed light on this question. Our experiments indicate that no initialization phase should be used when starting in state with a reasonable high equilibrium probability. Delaying data collection is only justified if the starting state is highly unlikely, and data collection should start as soon as a system enters a state with reasonably high probability.

In many companies, legacy systems have been used to serve customers arriving at service counters. The demand of a customer arriving at a counter is divided into R subdemands (SDs). Each SD is processed sequentially by the legacy system. When the final SD service is completed, the customer leaves the counter. On the other hand, Internet users desire that each SD be processed through the Internet without going to a service counter. We note that the applications for operating the web system differ from those of the legacy systems. For that reason, companies might use legacy systems for customers who come to the company through the Internet, without changing those legacy systems' applications. This web system, which is integrated with the legacy system, is called a front-end web system. Suppose that, for the web system, demands are generated according to a Poisson process. Then we show that the loss probability and macrostate distribution for the front-end web system are insensitive with respect to the distribution of the discrete random variable R, aside from the continuous distributions of other uncertain factors, under a restriction.

This article presents a novel approach for the dynamic control of a signalized intersection. At the intersection, there is a number of arrival flows of cars, each having a single queue (lane). The set of all flows is partitioned into disjoint combinations of nonconflicting flows that will receive green together. The dynamic control of the traffic lights is based on the numbers of cars waiting in the queues. The problem concerning when to switch (and which combination to serve next) is modeled as a Markovian decision process in discrete time. For large intersections (i.e., intersections with a large number of flows), the number of states becomes tremendously large, prohibiting straightforward optimization using value iteration or policy iteration. Starting from an optimal (or nearly optimal) fixed-cycle strategy, a one-step policy improvement is proposed that is easy to compute and is shown to give a close to optimal strategy for the dynamic problem.

This article investigates TP2 dependence of sample spacings. It is proved that TP2 (RR2) dependence between a general spacing and a nonadjacent order statistic might be characterized by the DLR (ILR) property of the parent distribution, and TP2 dependence between any pair of consecutive spacings might be characterized by the DLR aging property of the population. Furthermore, TP2 dependence between any two consecutive spacings in multiple outliers exponential models is also derived. In addition, some applications in reliability and business auction are presented as well.

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.

The probabilistic distance clustering method of works well if the cluster sizes are approximately equal. We modify that method to deal with clusters of arbitrary size and for problems where the cluster sizes are themselves unknowns that need to be estimated. In the latter case, our method is a viable alternative to the expectation-maximization (EM) method.

To sample a typical key in a “trie,” an appropriate climbing might consider generating random edges in the same manner as the data are generated. In the absence of the probability generating the keys, an uninformed random choice among the children still provides an alternative. We are also interested in extremal sampling, achieved by following a leftmost (or a rightmost) path. Each of these climbing strategies always generates a key, but one that might not necessarily be in the database. We investigate the altitude of the position at which climbing is terminated. Analytical techniques, including poissonization and the Mellin transform, are used for the accurate calculation of moments. In all strategies, the mean is always logarithmic. For typical and uninformed climbing, the variance is bounded in unbiased tries but grows logarithmically in biased tries. Consequently, in the biased case, one can find appropriate centering and scaling to produce a limit distribution for these two climbing strategies; the limit is normal. For extremal climbing, the variance is always bounded for both biased and unbiased cases, and no nontrivial limit exists under any scaling.

In our article [3] we have found a gap in the middle of the proof of Theorem 3.2. Therefore, we do not know whether Theorem 3.2 is true for the reverse hazard rate order. However, we could prove the following weaker result for the stochastic order.

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.

This article considers portfolio credit risk models of factor type. The dependence between the individual defaults is driven by a small number of systematic factors. The present work aims to investigate the effect of increasing the strength of the dependence between systematic factors on the default indicators in standard credit risk models. The intensity of the dependence is measured by means of appropriate multivariate stochastic orderings, based on the comparison of supermodular and ultramodular functions.

This article shows that very accurate accurate approximations to performance measures in the multiserver M/D/c/c+N queue with finite buffer and deterministic service times can be obtained by replacing the deterministic service time by a two-phase process with exponential sojourn times and branching probabilities outside the interval [0, 1].