Abstract—Motivated by the study of peer-to-peer file swarming systems à la BitTorrent, we introduce a probabilistic model of coupon replication systems. These systems consist of users, aiming to complete a collection of distinct coupons. Users are characterised by their current collection of coupons, and leave the system once they complete their coupon collection. The system evolution is then specified by describing how users of distinct types meet, and which coupons get replicated upon such encounters. For open systems, with exogenous user arrivals, we derive necessary and sufficient stability conditions in a layered scenario, where encounters are between users holding the same number of coupons. We also consider a system where encounters are between users chosen uniformly at random from the whole population. We show that performance, captured by sojourn time, is asymptotically optimal in both systems as the number of coupon types becomes large. We also consider closed systems with no exogenous user arrivals. In a special scenario where users have only one missing coupon, we evaluate the size of the population ultimately remaining in the system, as the initial number of users, N, goes to infinity. We show that this decreases geometrically with the number of coupons, K. In particular, when the ratio K / log(N) is above a critical threshold, we prove that this number of left-overs is of order log(log(N)). These results suggest that performance of file swarming systems does not depend critically on either altruistic user behavior, or on load balancing strategies such as rarest first. 1.

We consider optimization problems involving convex risk functions. By employing techniques of convex analysis and optimization theory in vector spaces of measurable functions we develop new representation theorems for risk models, and optimality and duality theory for problems involving risk functions.

We consider stochastic programs with risk measures in the objective and study stability properties as well as decomposition structures. Thereby we place emphasis on dynamic models, i.e., multistage stochastic programs with multiperiod risk measures. In this context, we define the class of polyhedral risk measures such that stochastic programs with risk measures taken from this class have favorable properties. Polyhedral risk measures are defined as optimal values of certain linear stochastic programs where the arguments of the risk measure appear on the right-hand side of the dynamic constraints. Dual representations for polyhedral risk measures are derived and used to deduce criteria for convexity and coherence. As examples of polyhedral risk measures we propose multiperiod extensions of the Conditional-Value-at-Risk.

Continuing research by Jennings, Mandelbaum, Massey and Whitt (1996), we investigate methods to perform time-dependent staffing for many-server queues. Our aim is to achieve time-stable performance in face of general time-varying arrival rates. It turns out that it suffices to target a stable probability of delay. That procedure tends to produce time-stable performance for several other operational measures. Motivated by telephone call centers, we focus on many-server models with customer abandonment, especially the Markovian Mt/M/st + M model, having an exponential time-to-abandon distribution (the +M), an exponential servicetime distribution and a nonhomogeneous Poisson arrival process. We develop three different methods for staffing, with decreasing generality and decreasing complexity: First, we develop a simulation-based iterativestaffing algorithm (ISA), and conduct experiments showing that it is effective. The ISA is appealing because it applies to very general models and is automatically validating: we directly see how well it works. Second, we extend the square-root-staffing rule, proposed by Jennings et al., which is based on the associated infinite-server model. The rule dictates that the staff level at time t be st = mt + β √ mt, where mt is the offered load (mean number of busy servers in the infinite-server model) and the constant β reflects the service grade. We show that the service grade β in the staffing formula can be represented as a function of the target delay probability α by

We consider the problem of constructing a portfolio of finitely many assets whose returns are described by a discrete joint distribution. We propose a new portfolio optimization model involving stochastic dominance constraints on the portfolio return. We develop optimality and duality theory for these models. We construct equivalent optimization models with utility functions. Numerical illustration is provided.

This paper proposes simple methods for staffing a single-class call center with uncertain arrival rate and uncertain staffing due to employee absenteeism. The arrival rate and the proportion of servers present are treated as random variables. The basic model is a multi-server queue with customer abandonment, allowing non-exponential service-time and time-to-abandon distributions. The goal is to maximize the expected net return, given throughput benefit and server, customer-abandonment and customer-waiting costs, but attention is also given to the standard deviation of the return. The approach is to approximate the performance and the net return, conditional on the random model-parameter vector, and then uncondition to get the desired results. Two recently-developed approximations are used for the conditional performance measures: first, a deterministic fluid approximation and, second, a numerical algorithm based on a purely Markovian birth-and-death model, having state-dependent death rates. Key words: model-parameter uncertainty; contact centers; employee absenteeism; customer abandonment; fluid models

Abstract — The potential for exploiting rate variations to increase the capacity of wireless systems by opportunistic scheduling has been extensively studied at packet level. In the present paper, we examine how slower, mobility-induced rate variations impact performance at flow level, accounting for the random number of flows sharing the transmission resource. We identify two limit regimes, termed fluid and quasi-stationary, where the rate variations occur on an infinitely fast and an infinitely slow time scale, respectively. Using stochastic comparison techniques, we show that these limit regimes provide simple performance bounds that only depend on easily calculated load factors. Additionally, we prove that for a broad class of fading processes, performance varies monotically with the speed of the rate variations. These results are illustrated through numerical experiments, showing that the fluid and quasi-stationary bounds are remarkably tight in certain usual cases. I.

We consider an optimization problem of minimization of a linear function subject to the chance constraint Prob{G(x, ξ) ∈ C} ≥ 1 − ε, where C is a convex set, G(x, ξ) is bi-affine mapping and ξ is a vector of random perturbations with known distribution. When C is multi-dimensional and ε is small, like 10 −6 or 10 −10, this problem is, generically, a problem of minimizing under a nonconvex and difficult to compute constraint and as such is computationally intractable. We investigate the potential of conceptually simple scenario approximation of the chance constraint. That is, approximation of the form G(x, η j) ∈ C, j = 1,..., N, where {η j} N j=1 is a sample drawn from a properly chosen trial distribution. The emphasis is on the situation where the solution to the approximation should, with probability at least 1 − δ, be feasible for the problem of interest, while the sample size N should be polynomial in the size of this problem and in ln(1/ε), ln(1/δ).

Copula modeling has taken the world of finance and insurance, and well beyond, by storm. Why is this? In this paper I review the early start of this development, discuss some important current research, mainly from an applications point of view, and comment on potential future developments. An alternative title of the paper would be “Demystifying the copula craze”. The paper also contains what I would like to call the copula must-reads. Keywords: copula, extreme value theory, Fréchet–Hoeffding bounds, quantitative risk management, Value–at–Risk 1

In this paper we review and extend some key results on the stochastic ordering of risks and on bounding the influence of stochastic dependence on risk functionals. The first part of the paper is concerned with a.s. constructions of random vectors and with diffusion kernel type comparisons which are of importance for various comparison results. In the second part we consider generalizations of the classical Fréchet-bounds, in particular for the distribution of sums and maxima and for more general monotonic functionals of the risk vector. In the final part we discuss three important orderings of risks which arise from ∆-monotone, supermodular, and directionally convex functions. We give some new criteria for these orderings. For the basic results we also take care to give references to “original sources ” of these results. 1