Conservation Laws, Extended Polymatroids and Multi-Armed Bandit Problems; A unified Approach to Indexabel Systems

We show that if performance measures in stochastic and dynamic scheduling problems satisfy generalized conservation laws, then the feasible space of achievable performance is a polyhedron called an extended polymatroid that generalizes the usual polymatroids introduced by Edmonds. Optimization of a linear objective over an extended polymatroid is solved by an adaptive greedy algorithm, which leads to an optimal solution having an indexability property (indexable systems). Under a certain condition, then the indices have a stronger decomposition property (decomposable systems). The following classical problems can be analyzed using our theory: multi-armed bandit problems, branching bandits. multiclass queues, multiclass queues with feedback, deterministic scheduling problemls. Interesting consequences of our results include: (1) a characterization of indexable systems as systems that satisfy generalized conservation laws, (2) a. sufficient condition for idexable systems to be decomposable, (3) a new linear programming proof of the decomposability property of Gittins indices in multi-armed bandit problems, (4) a unified and practical approach to sensitivity analysis of indexable systems, (5) a new characterization of the indices of indexable systems as sums of dual variables and a new interpretation of the indices in terms of retirement options in the context of branching bandits, (6) the first rigorous analysis of the indexability of undiscounted branching bandits, (7) a new algorithm to compute the indices of indexable systems (in particular Gittins indices), which is as fast as the fastest known algorithm, (8) a unification of the algorithm of Klimov for multiclass queues and the algorithm of Gittins for multi-armed bandits as special cases of the same algorithm. (9) closed form formulae for the performance of the optimal policy, and (10) an understanding of the nondependence of the indices on some of the parameters of the stochastic schediiuling problem. Most importantly, our approach provides a unified treatment of several classical problems in stochastic and dynamic scheduling and is able to address in a unified way their variations such as: discounted versus undiscounted cost criterion, rewards versus taxes. preemption versus nonpreemption, discrete versus continuous time, work conserving versus idling policies, linear versus nonlinear objective functions

Restless Bandits, Linear Programming Relaxations and a Primal-Dual Heuristic

We propose a mathematical programming approach for the classical PSPACE - hard problem of n restless bandits in stochastic optimization. We introduce a series of n increasingly stronger linear programming relaxations, the last of which is exact and corresponds to the formulation of the problem as a Markov decision process that has exponential size, while other relaxations provide bounds and are efficiently solvable. We also propose a heuristic for solving the problem that naturally arises from the first of these relaxations and uses indices that are computed through optimal dual variables from the first relaxation. In this way we propose a policy and a suboptimality guarantee. We report computational results that suggest that the value of the proposed heuristic policy is extremely close to the optimal value. Moreover, the second order relaxation provides strong bounds for the optimal solution value

Unlocking legal validity. Some remarks on the artificial ontology of law

Following Kelsen’s influential theory of law, the concept of validity has been used in the literature to refer to different properties of law (such as existence, membership, bindingness, and more) and so it is inherently ambiguous. More importantly, Kelsen’s equivalence between the existence and the validity of law prevents us from accounting satisfactorily for relevant aspects of our current legal practices, such as the phenomenon of ‘unlawful law’. This chapter addresses this ambiguity to argue that the most important function of the concept of validity is constituting the complex ontological paradigm of modern law as an institutional-normative practice. In this sense validity is an artificial ontological status that supervenes on that of existence of legal norms, thus allowing law to regulate its own creation and creating the logical space for the occurrence of ‘unlawful law’. This function, I argue in the last part, is crucial to understanding the relationship between the ontological and epistemic dimensions of the objectivity of law. For given the necessary practice-independence of legal norms, it is the epistemic accessibility of their creation that enables the law to fulfill its general action-guiding (and thus coordinating) function

Effects of hospital facilities on patient outcomes after cancer surgery: an international, prospective, observational study

Background Early death after cancer surgery is higher in low-income and middle-income countries (LMICs) compared with in high-income countries, yet the impact of facility characteristics on early postoperative outcomes is unknown. The aim of this study was to examine the association between hospital infrastructure, resource availability, and processes on early outcomes after cancer surgery worldwide.Methods A multimethods analysis was performed as part of the GlobalSurg 3 study-a multicentre, international, prospective cohort study of patients who had surgery for breast, colorectal, or gastric cancer. The primary outcomes were 30-day mortality and 30-day major complication rates. Potentially beneficial hospital facilities were identified by variable selection to select those associated with 30-day mortality. Adjusted outcomes were determined using generalised estimating equations to account for patient characteristics and country-income group, with population stratification by hospital.Findings Between April 1, 2018, and April 23, 2019, facility-level data were collected for 9685 patients across 238 hospitals in 66 countries (91 hospitals in 20 high-income countries; 57 hospitals in 19 upper-middle-income countries; and 90 hospitals in 27 low-income to lower-middle-income countries). The availability of five hospital facilities was inversely associated with mortality: ultrasound, CT scanner, critical care unit, opioid analgesia, and oncologist. After adjustment for case-mix and country income group, hospitals with three or fewer of these facilities (62 hospitals, 1294 patients) had higher mortality compared with those with four or five (adjusted odds ratio [OR] 3.85 [95% CI 2.58-5.75]; p<0.0001), with excess mortality predominantly explained by a limited capacity to rescue following the development of major complications (63.0% vs 82.7%; OR 0.35 [0.23-0.53]; p<0.0001). Across LMICs, improvements in hospital facilities would prevent one to three deaths for every 100 patients undergoing surgery for cancer.Interpretation Hospitals with higher levels of infrastructure and resources have better outcomes after cancer surgery, independent of country income. Without urgent strengthening of hospital infrastructure and resources, the reductions in cancer-associated mortality associated with improved access will not be realised

Assessing an intuitive condition for stability under a range of traffic conditions via a generalised Lu-Kumar network

We argue the importance both of developing simple sufficient conditions for the stability of general multiclass queueing networks and also of assessing such conditions under a range of assumptions on the weight of the traffic flowing between service stations. To achieve the former, we review a peak-rate stability condition and extend its range of application and for the latter, we introduce a generalisation of the Lu–Kumar network on which the stability condition may be tested for a range of traffic configurations. The peak-rate condition is close to exact when the between-station traffic is light, but degrades as this traffic increases.</jats:p

A linear programming approach to stability, optimisation and performance analysis for Markovian multiclass queueing networks

Our object of study is a multiclass queueing network (MQNET) which consists of a collection of (connected) single-server stations. Exogenous arrivals into the system form independent Poisson streams, service times are exponential and we have Markovian routing of customers between stations. Recent results concerning linear programming (LP) based approaches enable us to establish a simple and intuitive stability condition. This is of interest in its own right, but also enables us to progress with a study of optimal scheduling and performance analysis. Our methodology here is also based on LP. A primal-dual approach exploits the fact that the system satisfies (approximate) conservation laws to yield perform-ance guarantees for a natural index-based scheduling heuristic. We are also able to analyse the performance of an arbitrary priority policy