In this work we provide sufficient conditions for the arrival times of a renewal process so that the number of its events occurring before a randomly distributed time, T, independent of the process preserves the aging properties of T.

In the present article we provide a formula that facilitates the evaluation of the signature of a reliability structure by a generating function approach. A simple sufficient condition is also derived for proving the nonpreservation of the IFR property for the system's lifetime (when the components are IFR) by exploiting the signature of the system. As an application of the general results, we deduce recurrence relations for the signature of a linear consecutive k-out-of-n: F system. We establish a simple relation between the signature of a linear and a circular system and investigate the IFR preservation property under the formulation of such systems.

We model the weight (e.g., delay, distance, or cost) from an arbitrary node to the nearest (in weight) peer in a peer-to-peer (P2P) network. The exact probability generating function and an asymptotic analysis is presented for a random graph with independent and identically distributed exponential link weights. The asymptotic distribution function is a Fermi–Dirac distribution that frequently appears in statistical physics. The good agreement with simulation results for relatively small P2P networks makes the asymptotic formula for the probability density function useful for estimating the minimal number of peers to offer an acceptable quality (delay or latency).

To determine optimal investment and maintenance decisions, the total costs should be minimized over the whole life of a system or structure. In minimizing life-cycle costs, it is important to account for the time value of money by discounting and to consider the uncertainties involved. This article presents new results in renewal theory with costs that can be discounted according to any discount function that is nonincreasing and monotonic over time (such as exponential, hyperbolic, generalized hyperbolic, and no discounting). The main results include expressions for the first and second moment of the discounted costs over a bounded and unbounded time horizon as well as asymptotic expansions for nondiscounted costs.

We use induction to derive the distribution of the waiting time of the ith waiting customer in a busy period for a G/M/1 queue with a first come–first serve service. A trivial implication gives the law for the ith waiting time in a busy period for a G/M/c queue. Finally, we use the Lindley recursion to relate our results to the distribution of random walks.

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.

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.

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.

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.