A queueing analysis of max-min fairness, proportional fairness and balanced fairness

We compare the performance of three usual allocations, namely max-min fairness, proportional fairness and balanced fairness, in a communication network whose resources are shared by a random number of data flows. The model consists of a network of processor-sharing queues. The vector of service rates, which is constrained by some compact, convex capacity set representing the network resources, is a function of the number of customers in each queue. This function determines the way network resources are allocated. We show that this model is representative of a rich class of wired and wireless networks. We give in this general framework the stability condition of max-min fairness, proportional fairness and balanced fairness and compare their performance on a number of toy networks.     

Stochastic orderings, folded beta distributions and fairness in coin flips

This paper studies stochastic orderings for folded beta distributions. A necessary and sufficient condition for one folded beta random variable to be larger than another in likelihood ratio order is obtained. The work is motivated by the recent results of Porzio and Ragozini on stochastic orderings for folded binomial distributions. Beta distributions are commonly used to model proportions or probabilities, particularly as conjugate distributions for binomial random variables. Stochastic orderings may arise, for instance, in considerations of fairness in coin flips.     

Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes

For a couple of lifetimes (X₁,X₂) with an exchangeable joint survival function , attention is focused on notions of bivariate aging that can be described in terms of properties of the level curves of . We analyze the relations existing among those notions of bivariate aging, univariate aging, and dependence. A goal and, at the same time, a method to this purpose is to define axiomatically a correspondence among those objects; in fact, we characterize notions of univariate and bivariate aging in terms of properties of dependence. Dependence between two lifetimes will be described in terms of their survival copula. The language of copulæ turns out to be generally useful for our purposes; in particular, we shall introduce the more general notion of semicopula. It will be seen that this is a natural object for our analysis. Our definitions and subsequent results will be illustrated by considering a few remarkable cases; in particular, we find some necessary or sufficient conditions for Schur-concavity of , or for IFR properties of the one-dimensional marginals. The case characterized by the condition that the survival copula of (X₁,X₂) is Archimedean will be considered in some detail. For most of our arguments, the extension to the case of n>2 is straightforward.     

Insensitive,maximum stable allocations converge to proportional fairness

We describe a queuing model where service is allocated as a function of queue sizes. We consider allocations policies that are insensitive to service requirements and have a maximal stability region. We take a limit where the queuing model becomes congested. We study how service is allocated under this limit. We demonstrate that the only possible limit allocation is one that maximizes a proportionally fair optimization problem.     

Nash equilibrium based fairness

There are several approaches of sharing resources among users. There is a noncooperative approach wherein each user strives to maximize its own utility. The most common optimality notion is then the Nash equilibrium. Nash equilibria are generally Pareto inefficient. On the other hand, we consider a Nash equilibrium to be fair as it is defined in a context of fair competition without coalitions (such as cartels and syndicates). We show a general framework of systems wherein there exists a Pareto optimal allocation that is Pareto superior to an inefficient Nash equilibrium. We consider this Pareto optimum to be ??Nash equilibrium based fair.?? We further define a ??Nash proportionately fair?? Pareto optimum. We then provide conditions for the existence of a Pareto-optimal allocation that is, truly or most closely, proportional to a Nash equilibrium. As examples that fit in the above framework, we consider noncooperative flow-control problems in communication networks, for which we show the conditions on the existence of Nash-proportionately fair Pareto optimal allocations.     

Extremal lifetimes of persistent cycles

Extremes - Persistent homology captures the appearances and disappearances of topological features such as loops and cavities when growing disks centered at a Poisson point process. We study...     

Sequences with adjacent elements unequal

Aequationes mathematicae -     

Rate control for communication networks: shadow prices,proportional fairness and stability

This paper analyses the stability and fairness of two classes of rate control algorithm for communication networks. The algorithms provide natural generalisations to large-scale networks of simple additive increase/multiplicative decrease schemes, and are shown to be stable about a system optimum characterised by a proportional fairness criterion. Stability is established by showing that, with an appropriate formulation of the overall optimisation problem, the network's implicit objective function provides a Lyapunov function for the dynamical system defined by the rate control algorithm. The network's optimisation problem may be cast in primal or dual form: this leads naturally to two classes of algorithm, which may be interpreted in terms of either congestion indication feedback signals or explicit rates based on shadow prices. Both classes of algorithm may be generalised to include routing control, and provide natural implementations of proportionally fair pricing.     

Conditional residual lifetimes of coherent systems

In this paper, we derive mixture representations for the reliability function of the conditional residual lifetime of a coherent system with n$n$ independent and identically distributed (i.i.d.) components under the condition that at least j$j$ and at most k−1$k-1$ (j<k$j<k$) components have failed by time t$t$. Based on these mixture representations, we then discuss stochastic comparisons of the conditional residual lifetimes of two coherent systems with independent and identical components.     

Non-binding agreements and fairness in commons dilemma games

Usually, common pool games are analyzed without taking into account the cooperative features of the game, even when communication and non-binding agreements are involved. Whereas equilibria are inefficient, negotiations may induce some cooperation and may enhance efficiency. In the paper, we propose to use tools of cooperative game theory to advance the understanding of results in dilemma situations that allow for communication. By doing so, we present a short review of earlier experimental evidence given by Hackett, Schlager, and Walker 1994 (HSW) for the conditional stability of non-binding agreements established in face-to-face multilateral negotiations. For an experimental test, we reanalyze the HSW data set in a game-theoretical analysis of cooperative versions of social dilemma games. The results of cooperative game theory that are most important for the application are explained and interpreted with respect to their meaning for negotiation behavior. Then, theorems are discussed that cooperative social dilemma games are clear (alpha- and beta-values coincide) and that they are convex (it follows that the core is “large”): The main focus is on how arguments of power and fairness can be based on the structure of the game. A second item is how fairness and stability properties of a negotiated (non-binding) agreement can be judged. The use of cheap talk in evaluating experiments reveals that besides the relation of non-cooperative and cooperative solutions, say of equilibria and core, the relation of alpha-, beta- and gamma-values are of importance for the availability of attractive solutions and the stability of the such agreements. In the special case of the HSW scenario, the game shows properties favorable for stable and efficient solutions. Nevertheless, the realized agreements are less efficient than expected. The realized (and stable) agreements can be located between the equilibrium, the egalitarian solution and some fairness solutions. In order to represent the extent to which the subjects obey efficiency and fairness, we present and discuss patterns of the corresponding excess vectors.     

Channel coordination under fairness concerns and nonlinear demand

The supply chain literature analyzing supplier–retailer contracts and channel coordination has typically focused on profit or revenue maximization as the members’ sole objective. In such settings, it is well known that a simple wholesale price contract is not effective in coordinating the channel due to double marginalization. Recently, Cui et al. [Cui, T.H., Raju, J.S., Zhang, Z.J., 2007. Fairness and channel coordination. Management Science 53 (8) 1303–1314] introduced the members’ fairness concerns into channel coordination. Assuming a linear demand function, the authors show that a coordinating wholesale price contract can be designed when only the retailer or both parties are concerned about fairness. In this paper, we extend the authors’ results to other nonlinear demand functions that are commonly used in the literature. Our analysis reveals that, compared to the linear demand, the exponential demand function requires less stringent conditions to achieve coordination when only the retailer is fairness-concerned.     

Stable marriage assignment for unequal sets

The theory and algorithms developed by the authors in a previous paper for the stable marriage assignment for equal sets are extended to the case of unequal sets. The main theorem in this paper shows that a person unmarried in one stable marriage solution will be unmarried in any other stable marriage solution.     

Integrable derivations and formal groups in unequal characteristic

In this paper we study the link between formal group of dimension one and integrable derivations of a ring of unequal characteristic. Partially supported by M.U.R.S.T. (60%) and members of C.N.R.-G.N.S.A.G.A.     

Expectations and fairness in a simple bargaining experiment

We evaluate two competing hypotheses that try to account for robust violations of the predictions of game theory in Ultimatum bargaining experiments. One popular hypothesis is that the subjects are motivated by considerations of fairness that are not modelled by traditional theory. The alternative hypothesis is that the subjects do not have common knowledge of the rationality, beliefs and motives of other players. Each hypothesis can explain existing data. We design several experiments to discriminate between these two hypotheses. The results provide strong support for the alternative hypothesis.     

Utility-based two-stage models with fairness concern

This paper studies the impact of social comparison and fairness concerns on the efficiency of a two-stage process. The decision maker of each stage cares about not only the absolute score of his efficiency, but also the relative status when comparing with the other. By incorporating the utility theory and the concept of fairness, an efficiency-based Neumann–Morgenstern cardinal utility is defined to compose of basic utility from his self-efficiency and additional utility from the fairness concern of the other’s efficiency. Utility-based two-stage models are proposed to optimize the utilities of the stages rather than only the efficiencies instead. We characterize the concern of fairness as advantageous and disadvantageous inequity based on equitable outcome comparison. By investigating the non-cooperating relationship between two stages, we show that the stage dominating the process has the incentive to optimize his efficiency without ignoring that of the other, which is contrary to the conventional situation. In addition, the coefficients of equitable outcome and inequity significantly affect the efficiencies of both stages. We further investigate the cooperation between two stages and find that the efficiencies of the stages vary with the coefficients of unfairness perceptions. Numerical analysis verifies the validity of the proposed models and identifies the impacts of the coefficients of equitable outcome and inequity on the stages’ and overall efficiencies.     

Extremes of censored and uncensored lifetimes in survival data

Extremes - We consider a random censoring model for survival analysis, allowing the possibility that only a proportion of individuals in the population are susceptible to death or failure, and the...     

Comparisons and bounds for expected lifetimes of reliability systems

Sharp bounds on expectations of lifetimes of coherent and mixed systems composed of elements with independent and either identically or non-identically distributed lifetimes are expressed in terms of expected lifetimes of components. Similar evaluations are concluded for the respective mean residual lifetimes. In the IID case, improved inequalities dependent on a concentration parameter connected to the Gini dispersion index are obtained. The results can be used to compare systems with component lifetimes ordered in the convex ordering. In the INID case, some refined bounds are derived in terms of the expected lifetimes of series systems of smaller sizes, and the expected lifetime of single unit for the equivalent systems with IID components. The latter can be further simplified in the case of weak Schur-concavity and Schur-convexity of the system generalized domination polynomial.