Chaithanya Bandi

Tractable stochastic analysis in high dimensions via robust optimization

Modern probability theory, whose foundation is based on the axioms set forth by Kolmogorov, is currently the major tool for performance analysis in stochastic systems. While it offers insights in understanding such systems, probability theory, in contrast to optimization, has not been developed with computational tractability as an objective when the dimension increases. Correspondingly, some of its major areas of application remain unsolved when the underlying systems become multidimensional: Queueing networks, auction design in multi-item, multi-bidder auctions, network information theory, pricing multi-dimensional options, among others. We propose a new approach to analyze stochastic systems based on robust optimization. The key idea is to replace the Kolmogorov axioms and the concept of random variables as primitives of probability theory, with uncertainty sets that are derived from some of the asymptotic implications of probability theory like the central limit theorem. In addition, we observe that several desired system properties such as incentive compatibility and individual rationality in auction design are naturally expressed in the language of robust optimization. In this way, the performance analysis questions become highly structured optimization problems (linear, semidefinite, mixed integer) for which there exist efficient, practical algorithms that are capable of solving problems in high dimensions. We demonstrate that the proposed approach achieves computationally tractable methods for (a) analyzing queueing networks, (b) designing multi-item, multi-bidder auctions with budget constraints, and (c) pricing multi-dimensional options.

Efficiency of linear supply function bidding in electricity markets

We study the efficiency loss caused by strategic bidding behavior from power generators in electricity markets. In the considered market, the demand of electricity is inelastic, the generators submit their supply functions (i.e., the amount of electricity willing to supply given a unit price) to bid for the supply of electricity, and a uniform price is set to clear the market. We aim to understand how the total generation cost increases under strategic bidding, compared to the minimum total cost. Existing literature has answers to this question without regard to the network structure of the market. However, in electricity markets, the underlying physical network (i.e., the electricity transmission network) determines how electricity flows through the network and thus influences the equilibrium outcome of the market. Taking into account the underlying network, we prove that there exists a unique equilibrium supply profile, and derive an upper bound on the efficiency loss of the equilibrium supply profile compared to the socially optimal one that minimizes the total cost. Our upper bound provides insights on how the network topology affects the efficiency loss.

Fairness considerations in network flow problems

In most of the physical networks, such as power, water and transportation systems, there is a system-wide objective function, typically social welfare, and an underlying physics constraint governing the flow in the networks. The standard economics and optimization theories suggest that at optimal operating point, the price in the system should correspond to the optimal dual variables associated with those physical constraint. While this set of prices can achieve the best social welfare, they may feature significant differences even for neighboring agents in the system. This work addresses fairness considerations in network flow problems, where we not only care about the standard social welfare maximization, but also distribution of prices. We first interpret the network flow problem as an economic market problem. We then show that by tuning a design parameter, we can achieve a spectrum of price-fairness, where the gap between prices satisfy certain design objective. We derive the required physical means to implement the fairness adjustment and show that the adjusted optimal solution depends on the original network topology.

Robust transient analysis of multi-server queueing systems and feed-forward networks

We propose an analytically tractable approach for studying the transient behavior of multi-server queueing systems and feed-forward networks. We model the queueing primitives via polyhedral uncertainty sets inspired by the limit laws of probability. These uncertainty sets are characterized by variability parameters that control the degree of conservatism of the model. Assuming the inter-arrival and service times belong to such uncertainty sets, we obtain closed-form expressions for the worst case transient system time in multi-server queues and feed-forward networks with deterministic routing. These analytic formulas offer rich qualitative insights on the dependence of the system times as a function of the variability parameters and the fundamental quantities in the queueing system. To approximate the average behavior, we treat the variability parameters as random variables and infer their density by using ideas from queues in heavy traffic under reflected Brownian motion. We then average the worst case values obtained with respect to the variability parameters. Our averaging approach yields approximations that match the diffusion approximations for a single queue with light-tailed primitives and allows us to extend the framework to heavy-tailed feed-forward networks. Our methodology achieves significant computational tractability and provides accurate approximations for the expected system time relative to simulated values.

Robust Multiclass Queuing Theory for Wait Time Estimation in Resource Allocation Systems

In this paper, we study systems that allocate different types of scarce resources to heterogeneous allocatees based on predetermined priority rules—the U.S. deceased-donor kidney allocation system or the public housing program. We tackle the problem of estimating the wait time of an allocatee who possesses incomplete system information with regard, for example, to his relative priority, other allocatees’ preferences, and resource availability. We model such systems as multiclass, multiserver queuing systems that are potentially unstable or in transient regime. We propose a novel robust optimization solution methodology that builds on the assignment problem. For first-come, first-served systems, our approach yields a mixed-integer programming formulation. For the important case where there is a hierarchy in the resource types, we strengthen our formulation through a drastic variable reduction and also propose a highly scalable heuristic, involving only the solution of a convex optimization problem (usually...

Supply function equilibrium in power markets: Mesh networks

We study decentralized power markets with strategic power generators. In decentralized markets, each generator submits its supply function (i.e., the amount of electricity it is willing to produce at various unit prices) to the independent system operator (ISO), who takes the submitted supply functions as the true marginal cost functions, and dispatches the generators to clear the market. If all generators reported their true marginal cost functions, the market outcome would be efficient (i.e., the total generation cost is minimized). However, when generators are strategic and aim to maximize their own profits, the reported supply functions are not necessarily the marginal cost functions, and the resulting market outcome may be inefficient. The efficiency loss depends on the topology of the underlying transmission network, because the topology sets constraints on the feasible power supply from generators. This paper provides an analytical upper bound of the efficiency loss due to strategic generators. Our upper bound sheds light on how the efficiency loss depends on the (mesh) transmission network topology (e.g., the degrees of buses, the admittances and flow limits of transmission lines).

Efficiency of supply function equilibrium in networked markets

Motivated by the deregulation of electricity market, we study the efficiency loss of the supply function equilibrium (SFE). Specifically, we consider a market where the demand is inelastic, the suppliers submit their supply functions, and a uniform price is set to clear the market. Existing literature has answers to this question without regard to the network structure of the market. However, in many markets (such as electricity markets), there is an underlying physical network that limits the market operations (e.g., in electricity markets, the transmission network determines how electricity flows through the network and thus puts constraints on the supply profile). Motivated by electricity markets, we study how network topology affects the efficiency of SFE. For general mesh networks, we show that the efficiency loss is upper bounded as in the case without networks. Interestingly, we find a class of radial networks, for which the efficiency loss is independent of the local network topology.