We show that in preferential attachment models with power-law exponent τ ∈ (2, 3) the distance between randomly chosen vertices in the giant component is asymptotically equal to (4 + o(1))log log N / (-log(τ − 2)), where N denotes the number of nodes. This is twice the value obtained for the configuration model with the same power-law exponent. The extra factor reveals the different structure of typical shortest paths in preferential attachment graphs.

The notion of the signature is a basic concept and a powerful tool in the analysis of networks and reliability systems of binary type. An appropriate definition of this concept has recently been introduced for systems that have ν possible states (with ν ≥ 3). In this paper we analyze in detail several properties and the most relevant aspects of such a general definition. For simplicity's sake, we focus our attention on the case ν = 3. Our analysis will however provide a number of hints for understanding the basic aspects of the general case.

We study the dynamic assignment of cross-trained workers in serial production lines characterized by stochastic process times and inventory buffers between stations. Throughput maximization is the objective. Each worker is trained for a subset of tasks, where emphasis is placed on systems with each worker trained for a zone of stations with stations near the zone boundaries being served (shared) by one or more other workers as well. Using sample path comparisons, we identify structural properties of optimal worker allocation policies. We identify when (i) a worker can prioritize the job in the most downstream station (last-buffer-first-served), and (ii) only the downstream (as opposed to upstream) server should serve a single task.

Two classical stochastic processes are considered, the Ehrenfest process, introduced in 1907 in the kinetic theory of gases to describe the heat exchange between two bodies, and the Engset process, one of the early (1918) stochastic models of communication networks. In this paper we investigate the asymptotic behavior of the distributions of hitting times of these two processes when the number of particles/sources goes to infinity. Results concerning the hitting times of boundaries in particular are obtained. We rely on martingale methods; a key ingredient is an important family of simple nonnegative martingales, an analogue, for the Ehrenfest process, of the exponential martingales used in the study of random walks or of Brownian motion.

We propose an efficient semi-numerical approach to compute the steady-state probability distribution for the number of requests at arbitrary and at arrival time instants in PH/M/c-like systems with homogeneous servers in which the inter-arrival time distribution is represented by an acyclic set of memoryless phases. Our method is based on conditional probabilities and results in a simple computationally stable recurrence. It avoids the explicit manipulation of potentially large matrices and involves no iteration. Owing to the use of conditional probabilities, it delays the onset of numerical issues related to floating-point underflow as the number of servers and/or phases increases. For generalized Coxian distributions, the computational complexity of the proposed approach grows linearly with the number of phases in the distribution.

We consider the problem of twenty questions with noisy answers, in which we seek to find a target by repeatedly choosing a set, asking an oracle whether the target lies in this set, and obtaining an answer corrupted by noise. Starting with a prior distribution on the target's location, we seek to minimize the expected entropy of the posterior distribution. We formulate this problem as a dynamic program and show that any policy optimizing the one-step expected reduction in entropy is also optimal over the full horizon. Two such Bayes optimal policies are presented: one generalizes the probabilistic bisection policy due to Horstein and the other asks a deterministic set of questions. We study the structural properties of the latter, and illustrate its use in a computer vision application.

In a random graph, counts for the number of vertices with given degrees will typically be dependent. We show via a multivariate normal and a Poisson process approximation that, for graphs which have independent edges, with a possibly inhomogeneous distribution, only when the degrees are large can we reasonably approximate the joint counts as independent. The proofs are based on Stein's method and the Stein-Chen method with a new size-biased coupling for such inhomogeneous random graphs, and, hence, bounds on the distributional distance are obtained. Finally, we illustrate that apparent (pseudo-)power-law-type behaviour can arise in such inhomogeneous networks despite not actually following a power-law degree distribution.

Multistate monotone systems are used to describe technological or biological systems when the system itself and its components can perform at different operationally meaningful levels. This generalizes the binary monotone systems used in standard reliability theory. In this paper we consider the availabilities and unavailabilities of the system in an interval, i.e. the probabilities that the system performs above or below the different levels throughout the whole interval. In complex systems it is often impossible to calculate these availabilities and unavailabilities exactly, but it is possible to construct lower and upper bounds based on the minimal path and cut vectors to the different levels. In this paper we consider systems which allow a modular decomposition. We analyse in depth the relationship between the minimal path and cut vectors for the system, the modules, and the organizing structure. We analyse the extent to which the availability bounds are improved by taking advantage of the modular decomposition. This problem was also treated in Butler (1982) and Funnemark and Natvig (1985), but the treatment was based on an inadequate analysis of the relationship between the different minimal path and cut vectors involved, and as a result was somewhat inaccurate. We also extend to interval bounds that have previously only been given for availabilities at a fixed point of time.

In many service, production, and traffic systems there are multiple types of customers requiring different types of ‘servers’, i.e. different services, products, or routes. Often, however, a proportion of the customers are flexible, i.e. they are willing to change their type in order to achieve faster service, and even if this proportion is small, it has the potential of achieving large performance gains. We generalize earlier results on the optimality of ‘join the shortest queue’ (JSQ) for flexible arrivals to the following: arbitrary arrivals where only a subset are flexible, multiple-server stations, and abandonments. Surprisingly, with abandonments, the optimality of JSQ for minimizing the number of customers in the system depends on the relative abandonment and service rates. We extend our model to finite buffers and resequencing. We assume exponential service. Our optimality results are very strong; we minimize the queue length process in the weak majorization sense.

The signature is an important structural characteristic of a coherent system. Its computation, however, is often rather involved and complex. We analyze several cases where this complexity can be considerably reduced. These are the cases when a ‘large’ coherent system is obtained as a series, parallel, or recurrent structure built from ‘small’ modules with known signature. Corresponding formulae can be obtained in terms of cumulative notions of signatures. An algebraic closure property of families of homogeneous polynomials plays a substantial role in our derivations.

We establish heavy-traffic limits for nearly deterministic queues, such as the G/D/n many-server queue. Since waiting times before starting service in the G/D/n queue are equivalent to waiting times in an associated Gn /D/1 model, where the Gn interarrival times are the sum of n consecutive interarrival times in the original model, we focus on the Gn /D/1 model and the generalization to Gn /Gn /1, where ‘cyclic thinning’ is applied to both the arrival and service processes. We establish different limits in two cases: (i) when (1 − ρn )√n → β as n → ∞ and (ii) when (1 − ρn )n → β as n → ∞, where ρn is the traffic intensity in model n. The nearly deterministic feature leads to interesting nonstandard scaling.

We study a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = ℝ+ d , with drift r 0 ∈ ℝd and Hurst parameter H ∈ (½, 1). Under a natural stability condition on the drift vector r 0 and reflection directions, we establish a geometric drift towards a compact set for the 1-skeleton chain Ž̆ of the RFBM process Z; that is, there exist β, b ∈ (0, ∞) and a compact set C ⊂ S such that ΔV(x):= Ex [V(Ž̆(1))] − V(x) ≤ −βV(x) + b 1 C (x), x ∈ S, for an exponentially growing Lyapunov function V : S → [1, ∞). For a wide class of Markov processes, such a drift inequality is known as a necessary and sufficient condition for exponential ergodicity. Indeed, similar drift inequalities have been established for reflected processes driven by standard Brownian motions, and our result can be viewed as their fractional Brownian motion counterpart. We also establish that the return times to the set C itself are geometrically bounded. Motivation for this study is that RFBM appears as a limiting workload process for fluid queueing network models fed by a large number of heavy-tailed ON/OFF sources in heavy traffic.

We construct an integrated probabilistic model to capture interactions between atoms of a nanocomponent. We then use this model to assess reliabilities of nanocomponents with different structures. Several properties of our proposed model are also described under a sparseness condition. The model is an extension of our previous model based on Markovian random field theory. The proposed integrated model is flexible in that pairwise relationship information among atoms as well as features of individual atoms can be easily incorporated. An important feature that distinguishes the integrated probabilistic model from our previous model is that the integrated approach uses all available sources of information with different weights for different types of interaction. In this paper we consider the nanocomponent at a fixed moment of time, say the present moment, and we assume that the present state of the nanocomponent depends only on the present states of its atoms.

A forwards induction policy is a type of greedy algorithm for Markov decision processes. It is straightforward to implement and is optimal for a large class of models, especially in stochastic resource allocation. In this paper we consider a model for the optimal allocation of resources in pre-clinical pharmaceutical research. We show that although they are not always strictly optimal, forwards induction policies perform well.

We consider a stochastic control model for a queueing system driven by a two-dimensional fractional Brownian motion with Hurst parameter 0 < H < 1. In particular, when H > ½, this model serves to approximate a controlled two-station tandem queueing model with heavy-tailed ON/OFF sources in heavy traffic. We establish the weak convergence results for the distribution of the state process and construct an explicit stationary state process associated with given controls. Based on suitable coupling arguments, we show that each state process couples with its stationary counterpart and we use it to represent the long-run average cost functional in terms of the stationary process. Finally, we establish the existence result of an optimal control, which turns out to be independent of the initial data.

In this note we revisit the discussion on minimal repair in heterogeneous populations in Finkelstein (2004). We consider the corresponding stochastic intensities (intensity processes) for items in heterogeneous populations given available information on their operational history, i.e. the failure (repair) times and the time since the last failure (repair). Based on the improved definitions, the setup of Finkelstein (2004) is modified and the main results are corrected in accordance with the updating procedure for the conditional frailty distribution.

Suppose that N is a ℤ+-valued random variable and that X,X 1,X 2,… is a sequence of independent and identically distributed ℤ+ random variables independent of N. In this paper we are interested in properties of the conditional variable In particular, we want to know the asymptotic behavior of the conditional mean EN k or the conditional variance varN k as k→∞. We consider the cases when X is Poisson and when X is mixed Poisson. The problem is motivated by modeling loss reserves in nonlife insurance.

We establish some interesting duality results for Markov-modulated fluid flow models. Though fluid flow models are continuous-state analogues of quasi-birth-and-death processes, some duality results do differ by the inclusion of a scaling factor.

In this paper we extend our previous work. We consider the local-time process L of a strong Markov process X, add negative drift to L, and reflect it à la Skorokhod to obtain a process Q. The reflection of X, together with Q, is, in some sense, a macroscopic model for a service system with two priorities. We derive an expression for the joint law of the duration of an excursion, the maximum value of the process on it, and the time between successive excursions. We work with a properly constructed stationary version of the process. Examples are also given in the paper.

To enhance the performance of a system, a common practice employed by reliability engineers is to use redundant components in the system. In this paper we compare lifetimes of series (parallel) systems arising out of different allocations of one or two standby redundancies. These comparisons are made with respect to the increasing concave (convex) order, the hazard rate order, and the stochastic precedence order. The main results extend some related conclusions in the literature.