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177
Coupon Replication Systems
 in Proc. ACM SIGMETRICS
, 2005
"... Abstract—Motivated by the study of peertopeer 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 ..."
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Cited by 94 (2 self)
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Abstract—Motivated by the study of peertopeer 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 leftovers 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.
Optimization of Convex Risk Functions
, 2004
"... 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 functio ..."
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Cited by 51 (11 self)
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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.
Optimization with stochastic dominance constraints
 SIAM Journal on Optimization
"... 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 the ..."
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Cited by 37 (5 self)
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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.
Polyhedral risk measures in stochastic programming
 SIAM JOURNAL ON OPTIMIZATION
, 2005
"... 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 polyhedra ..."
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Cited by 36 (12 self)
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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 righthand 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 ConditionalValueatRisk.
Staffing of timevarying queues to achieve timestable performance
, 2005
"... Continuing research by Jennings, Mandelbaum, Massey and Whitt (1996), we investigate methods to perform timedependent staffing for manyserver queues. Our aim is to achieve timestable performance in face of general timevarying arrival rates. It turns out that it suffices to target a stable probab ..."
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Cited by 30 (21 self)
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Continuing research by Jennings, Mandelbaum, Massey and Whitt (1996), we investigate methods to perform timedependent staffing for manyserver queues. Our aim is to achieve timestable performance in face of general timevarying arrival rates. It turns out that it suffices to target a stable probability of delay. That procedure tends to produce timestable performance for several other operational measures. Motivated by telephone call centers, we focus on manyserver models with customer abandonment, especially the Markovian Mt/M/st + M model, having an exponential timetoabandon 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 simulationbased 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 squarerootstaffing rule, proposed by Jennings et al., which is based on the associated infiniteserver 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 infiniteserver 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
Scenario approximations of chance constraints
 PROBABILISTIC AND RANDOMIZED METHODS FOR DESIGN UNDER UNCERTAINTY
, 2004
"... 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 biaffine mapping and ξ is a vector of random perturbations with known distribution. When C is multidimensional and ε is small, ..."
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Cited by 27 (4 self)
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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 biaffine mapping and ξ is a vector of random perturbations with known distribution. When C is multidimensional 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).
Generalised Shot noise Cox processes
 ADVANCES IN APPLIED PROBABILITY 35
, 2003
"... We introduce a new class of Cox cluster processes called generalised shotnoise Cox processes (GSNCPs), which extends the definition of shot noise Cox processes (SNCPs) in two directions: the point process which drives the shot noise is not necessarily Poisson, and the kernel of the shot noise can b ..."
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Cited by 23 (6 self)
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We introduce a new class of Cox cluster processes called generalised shotnoise Cox processes (GSNCPs), which extends the definition of shot noise Cox processes (SNCPs) in two directions: the point process which drives the shot noise is not necessarily Poisson, and the kernel of the shot noise can be random. Thereby a very large class of models for aggregated or clustered point patterns is obtained. Due to the structure of GSNCPs, a number of useful results can be established. We focus first on deriving summary statistics for GSNCPs and next on how to make simulation for GSNCPs. Particularly, results for first and second order moment measures, reduced Palm distributions, the Jfunction, simulation with or without edge effects, and conditional simulation of the intensity function driving a GSNCP are given. Our results are exemplified for special important cases of GSNCPs, and we discuss the relation to corresponding results for SNCPs.
Staffing a Call Center with Uncertain Arrival Rate and Absenteeism
 Production and Operations Management
"... This paper proposes simple methods for staffing a singleclass 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 multiserver queue with customer aba ..."
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Cited by 23 (4 self)
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This paper proposes simple methods for staffing a singleclass 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 multiserver queue with customer abandonment, allowing nonexponential servicetime and timetoabandon distributions. The goal is to maximize the expected net return, given throughput benefit and server, customerabandonment and customerwaiting 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 modelparameter vector, and then uncondition to get the desired results. Two recentlydeveloped approximations are used for the conditional performance measures: first, a deterministic fluid approximation and, second, a numerical algorithm based on a purely Markovian birthanddeath model, having statedependent death rates. Key words: modelparameter uncertainty; contact centers; employee absenteeism; customer abandonment; fluid models
How mobility impacts the flowlevel performance of wireless data networks
 In Proceedings of IEEE Infocom
, 2004
"... 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, mobilityinduced rate variations impact performance at flow level, accounting fo ..."
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Cited by 21 (3 self)
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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, mobilityinduced 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 quasistationary, 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 quasistationary bounds are remarkably tight in certain usual cases. I.
Analysis of stochastic service guarantees in communication networks: A server model
 In Proc. of the International Workshop on Quality of Service (IWQoS 2005
, 2005
"... Abstract. The arrival curve has been used as a powerful concept for deterministic service guarantee analysis in communication networks. Since many applications and networks do not require or provide deterministic service guarantees, stochastic service guarantee analysis is becoming increasingly impo ..."
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Cited by 20 (8 self)
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Abstract. The arrival curve has been used as a powerful concept for deterministic service guarantee analysis in communication networks. Since many applications and networks do not require or provide deterministic service guarantees, stochastic service guarantee analysis is becoming increasingly important and has attracted a lot of research attention in recent years. For this, several probabilistic versions of the arrival curve have been proposed in the literature. They extend the concept of arrival curve to the stochastic case based on its traffic amount property. In this paper, we explore another property, called the virtual backlog property, of an arrival curve. Based on the virtual backlog property, we introduce the concept of virtualbacklogcentric (v.b.c) stochastic arrival curve to facilitate the analysis of stochastic service guarantees. We prove that a v.b.c stochastic arrival curve has a similar duality as a (deterministic) arrival curve. With the concept of v.b.c stochastic arrival curve, we derive results for stochastic service guarantee analysis of systems with the timevarying setting. In addition, we prove that many wellknown types of traffic can be readily represented using v.b.c stochastic arrival curves.