Marginal productivity index policies for problems of admission control and routing to parallel queues with delay

In this paper we consider the problem of admission control of Bernoulli arrivals to a buffer with geometric server, in which the controller’s actions take effect one period after the actual change in the queue length. An optimal policy in terms of marginal productivity indices (MPI) is derived for this problem under the following three performance objectives: (i) minimization of the expected total discounted sum of holding costs and rejection costs, (ii) minimization of the expected time-average sum of holding costs and rejection costs, and (iii) maximization of the expected time-average number of job completions. Our employment of existing theoretical and algorithmic results on restless bandit indexation together with some new results yields a fast algorithm that computes the MPI for a queue with a buffer size of I performing only O(I) arithmetic operations. Such MPI values can be used both to immediately obtain the optimal thresholds for the admission control problem, and to design an index policy for the routing problem (with possible admission control) in the multi-queue system. Thus, this paper further addresses the problem of designing and computing a tractable heuristic policy for dynamic job admission control and/or routing in a discrete time Markovian model of parallel loss queues with one-period delayed state observation and/or action implementation, which comes close to optimizing an infinite-horizon problem under the above three objectives. Our approach seems to be tractable also for the analogous problems with larger delays and, more generally, for arbitrary restless bandits with delays

Marginal productivity index policies for problems of admission control and routing to parallel queues with delay

In this paper we consider the problem of admission control of Bernoulli arrivals to a buffer with geometric server, in which the controller’s actions take effect one period after the actual change in the queue length. An optimal policy in terms of marginal productivity indices (MPI) is derived for this problem under the following three performance objectives: (i) minimization of the expected total discounted sum of holding costs and rejection costs, (ii) minimization of the expected time-average sum of holding costs and rejection costs, and (iii) maximization of the expected time-average number of job completions. Our employment of existing theoretical and algorithmic results on restless bandit indexation together with some new results yields a fast algorithm that computes the MPI for a queue with a buffer size of I performing only O(I) arithmetic operations. Such MPI values can be used both to immediately obtain the optimal thresholds for the admission control problem, and to design an index policy for the routing problem (with possible admission control) in the multi-queue system. Thus, this paper further addresses the problem of designing and computing a tractable heuristic policy for dynamic job admission control and/or routing in a discrete time Markovian model of parallel loss queues with one-period delayed state observation and/or action implementation, which comes close to optimizing an infinite-horizon problem under the above three objectives. Our approach seems to be tractable also for the analogous problems with larger delays and, more generally, for arbitrary restless bandits with delays.Admission control, Routing, Parallel queues, Delayed information, Delayed action implementation, Index policy, Restless bandits, Marginal productivity index

Admission control and routing to parallel queues with delayed information via marginal productivity indices

This paper addresses the problem of designing and computing a tractable index policy for dynamic job admission control and/or routing in a discrete time Markovian model of parallel loss queues with one-period delayed state observation, which comes close to optimizing an infinite-horizon discounted or average performance objective involving linear holding costs and rejection costs. Instead of devising some ad hoc indices, we deploy a unifying fundamental design principle for design of priority index policies in dynamic resource allocation problems of multiarmed restless bandit type, based on decoupling the problem into subproblems and defining an appropriate marginal productivity index (MPI) for each subproblem. In the model of concern, such subproblems represent admission control problems to a single queue with one-period feedback delay, for which the structure of optimal policies has been characterized in previous work as being of bi-threshold type, yet without giving an algorithm to compute the optimal thresholds. We deploy in such subproblems theoretical and algorithmic results on restless bandit indexation, which yields a fast algorithm that computes the MPI for a subproblem with a buffer size of n performing only O(n) arithmetic operations. Such MPI values can be used both to immediately obtain the optimal thresholds for the subproblem, and to design an index policy for the admission control and/or routing problem in the multi-queue system. The results readily extend to models with infinite buffer space

Dynamic priority allocation via restless bandit marginal productivity indices

This paper surveys recent work by the author on the theoretical and algorithmic aspects of restless bandit indexation as well as on its application to a variety of problems involving the dynamic allocation of priority to multiple stochastic projects. The main aim is to present ideas and methods in an accessible form that can be of use to researchers addressing problems of such a kind. Besides building on the rich literature on bandit problems, our approach draws on ideas from linear programming, economics, and multi-objective optimization. In particular, it was motivated to address issues raised in the seminal work of Whittle (Restless bandits: activity allocation in a changing world. In: Gani J. (ed.) A Celebration of Applied Probability, J. Appl. Probab., vol. 25A, Applied Probability Trust, Sheffield, pp. 287-298, 1988) where he introduced the index for restless bandits that is the starting point of this work. Such an index, along with previously proposed indices and more recent extensions, is shown to be unified through the intuitive concept of ``marginal productivity index'' (MPI), which measures the marginal productivity of work on a project at each of its states. In a multi-project setting, MPI policies are economically sound, as they dynamically allocate higher priority to those projects where work appears to be currently more productive. Besides being tractable and widely applicable, a growing body of computational evidence indicates that such index policies typically achieve a near-optimal performance and substantially outperform benchmark policies derived from conventional approaches.Comment: 7 figure

Marginal productivity index policies for dynamic priority allocation in restless bandit models

Esta tesis estudia tres complejos problemas dinámicos y estocásticos de asignación de recursos: (i) Enrutamiento y control de admisión con información retrasada, (ii) Promoción dinámica de productos y el Problema de la mochila para artículos perecederos, y (iii) Control de congestión en “routers” con información del recorrido futuro. Debido a que la solución óptima de estos problemas no es asequible computacionalmente a gran y mediana escala, nos concentramos en cambio en diseñar políticas heurísticas de prioridad que sean computacionalmente tratables y cuyo rendimiento sea cuasi-óptimo. Modelizamos los problemas arriba mencionados como problemas de “multi-armed restless bandit” en el marco de procesos de decisión Markovianos con estructura especial. Empleamos y enriquecemos resultados existentes en la literatura, que constituyen un principio unificador para el diseño de políticas de índices de prioridad basadas en la relajación Lagrangiana y la descomposición de dichos problemas. Esta descomposición permite considerar subproblemas de optimización paramétrica, y en ciertos casos “indexables”, resolverlos de manera óptima mediante el índice de productividad marginal (MP). El índice MP es usado como medida de prioridad dinámica para definir reglas heurísticas de prioridad para los problemas originales intratables. Para cada uno de los problemas bajo consideración realizamos tal descomposición, identificamos las condiciones de indexabilidad, y obtenemos fórmulas para los índices MP o algoritmos computacionalmente tratables para su cálculo. Los índices MP correspondientes a cada uno de estos tres problemas pueden ser interpretados en términos de prioridades como el nivel de: (i) la penalización de dirigir un trabajo a una cola particular, (ii) la necesidad de promocionar un cierto artículo perecedero, y (iii) la utilidad de una transmisión de flujo particular. Además de la contribución práctica de la obtención de reglas heurísticas de prioridad para los tres problemas analizados, las principales contribuciones teóricas son las siguientes: (i) un algoritmo lineal en el tiempo para el cómputo de los índices MP en el problema de control de admisión con información retrasada, igualando, por lo tanto, la complejidad del mejor algoritmo existente para el caso sin retrasos, (ii) un nuevo tipo de política de índice de prioridad basada en la resolución de un problema (determinista) de la mochila, y (iii) una nueva extensión del modelo existente de “multi-armed restless bandit” a través de la incorporación de las llegadas aleatorias de los “restless bandits”.This dissertation addresses three complex stochastic and dynamic resource allocation problems: (i) Admission Control and Routing with Delayed Information, (ii) Dynamic Product Promotion and Knapsack Problem for Perishable Items, and (iii) Congestion Control in Routers with Future-Path Information. Since these problems are intractable for finding an optimal solution at middle and large scale, we instead focus on designing tractable and well-performing heuristic priority rules. We model the above problems as the multi-armed restless bandit problems in the framework of Markov decision processes with special structure. We employ and enrich existing results in the literature, which identified a unifying principle to design dynamic priority index policies based on the Lagrangian relaxation and decomposition of such problems. This decomposition allows one to consider parametric-optimization subproblems and, in certain “indexable” cases, to solve them optimally via the marginal productivity (MP) index. The MP index is then used as a dynamic priority measure to define heuristic priority rules for the original intractable problems. For each of the problems considered we perform such a decomposition, identify indexability conditions, and obtain formulae for the MP indices or tractable algorithms for their computation. The MP indices admit the following priority interpretations in the three respective problems: (i) undesirability for routing a job to a particular queue, (ii) promotion necessity of a particular perishable product, and (iii) usefulness of a particular flow transmission. Apart from the practical contribution of deriving the heuristic priority rules for the three intractable problems considered, our main theoretical contributions are the following: (i) a linear-time algorithm for computing MP indices in the admission control problem with delayed information, matching thus the complexity of the best existing algorithm under no delays, (ii) a new type of priority index policy based on solving a (deterministic) knapsack problem, and (iii) a new extension of the existing multi-armed restless bandit model by incorporating random arrivals of restless bandits

Characterization and computation of restless bandit marginal productivity indices

The Whittle index [P. Whittle (1988). Restless bandits: Activity allocation in a changing world. J. Appl. Probab. 25A, 287-298] yields a practical scheduling rule for the versatile yet intractable multi-armed restless bandit problem, involving the optimal dynamic priority allocation to multiple stochastic projects, modeled as restless bandits, i.e., binary-action (active/passive) (semi-) Markov decision processes. A growing body of evidence shows that such a rule is nearly optimal in a wide variety of applications, which raises the need to efficiently compute the Whittle index and more general marginal productivity index (MPI) extensions in large-scale models. For such a purpose, this paper extends to restless bandits the parametric linear programming (LP) approach deployed in [J. Niño-Mora. A (2/3)$n^{3}$ fast-pivoting algorithm for the Gittins index and optimal stopping of a Markov chain, INFORMS J. Comp., in press], which yielded a fast Gittins-index algorithm. Yet the extension is not straightforward, as the MPI is only defined for the limited range of socalled indexable bandits, which motivates the quest for methods to establish indexability. This paper furnishes algorithmic and analytical tools to realize the potential of MPI policies in largescale applications, presenting the following contributions: (i) a complete algorithmic characterization of indexability, for which two block implementations are given; and (ii) more importantly, new analytical conditions for indexability — termed LP-indexability — that leverage knowledge on the structure of optimal policies in particular models, under which the MPI is computed faster by the adaptive-greedy algorithm previously introduced by the author under the more stringent PCL-indexability conditions, for which a new fast-pivoting block implementation is given. The paper further reports on a computational study, measuring the runtime performance of the algorithms, and assessing by a simulation study the high prevalence of indexability and PCL-indexability.

Empirical Studies in Hospital Emergency Departments

This dissertation focuses on the operational impacts of crowding in hospital emergency departments. The body of this work is comprised of three essays. In the first essay, Waiting Patiently: An Empirical Study of Queue Abandonment in an Emergency Department, we study queue abandonment, or left without being seen. We show that abandonment is not only influenced by wait time, but also by the queue length and the observable queue flows during the waiting exposure. We show that patients are sensitive to being jumped in the line and that patients respond differently to people more sick and less sick moving through the system. This study shows that managers have an opportunity to impact abandonment behavior by altering what information is available to waiting customers. In the second essay, Doctors Under Load: An Empirical Study of State-Dependent Service Times in Emergency Care, we show that when crowded, multiple mechanisms in the emergency department act to retard patient treatment, but care providers adjust their clinical behavior to accelerate the service. We identify two mechanisms that providers use to accelerate the system: early task initiation and task reduction. In contrast to other recent works, we find the net effect of these countervailing forces to be an increase in service time when the system is crowded. Further, we use simulation to show that ignoring state-dependent service times leads to modeling errors that could cause hospitals to overinvest in human and physical resources. In the final essay, The Financial Consequences of Lost Demand and Reducing Boarding in Hospital Emergency Departments, we use discrete event simulation to estimate the number of patients lost to Left Without Being Seen and ambulance diversion as a result of patients waiting in the emergency department for an inpatient bed (known as boarding). These lost patients represent both a failure of the emergency department to meet the needs of those seeking care and lost revenue for the hospital. We show that dynamic bed management policies that proactively cancel some non-emergency patients when the hospital is near capacity can lead to reduced boarding, increased number of patients served, and increased hospital revenue

Asymptotically optimal priority policies for indexable and non-indexable restless bandits

We study the asymptotic optimal control of multi-class restless bandits. A restless bandit is a controllable stochastic process whose state evolution depends on whether or not the bandit is made active. Since finding the optimal control is typically intractable, we propose a class of priority policies that are proved to be asymptotically optimal under a global attractor property and a technical condition. We consider both a fixed population of bandits as well as a dynamic population where bandits can depart and arrive. As an example of a dynamic population of bandits, we analyze a multi-class M/M/S+M queue for which we show asymptotic optimality of an index policy.We combine fluid-scaling techniques with linear programming results to prove that when bandits are indexable, Whittle's index policy is included in our class of priority policies. We thereby generalize a result of Weber and Weiss (1990) about asymptotic optimality of Whittle's index policy to settings with (i) several classes of bandits, (ii) arrivals of new bandits, and (iii) multiple actions. Indexability of the bandits is not required for our results to hold. For non-indexable bandits we describe how to select priority policies from the class of asymptotically optimal policies and present numerical evidence that, outside the asymptotic regime, the performance of our proposed priority policies is nearly optimal