There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We consider the following natural model: customers arrive as a Poisson stream of rate /spl lambda/n, /spl lambda/<1, at a collection of n servers. Each customer chooses some constant d servers independently and uniformly at random from the n servers and waits for service at the one with the fewest customers. Customers are served according to the first-in first-out (FIFO) protocol and the service time for a customer is exponentially distributed with mean 1. We call this problem the supermarket model. We wish to know how the system behaves and in particular we are interested in the effect that the parameter d has on the expected time a customer spends in the system in equilibrium. Our approach uses a limiting, deterministic model representing the behavior as n/spl rarr//spl infin/ to approximate the behavior of finite systems. The analysis of the deterministic model is interesting in its own right. Along with a theoretical justification of this approach, we provide simulations that demonstrate that the method accurately predicts system behavior, even for relatively small systems. Our analysis provides surprising implications. Having d=2 choices leads to exponential improvements in the expected time a customer spends in the system over d=1, whereas having d=3 choices is only a constant factor better than d=2. We discuss the possible implications for system design.

/pdf/the-power-of-two-choices-in-randomized-load-balancing-4pp4939a6q.pdf

The power of two choices in randomized load balancing

The benefits of dynamic pricing methods have long been known in industries, such as airlines, hotels, and electric utilities, where the capacity is fixed in the short-term and perishable. In recent years, there has been an increasing adoption of dynamic pricing policies in retail and other industries, where the sellers have the ability to store inventory. Three factors contributed to this phenomenon: (1) the increased availability of demand data, (2) the ease of changing prices due to new technologies, and (3) the availability of decision-support tools for analyzing demand data and for dynamic pricing. This paper constitutes a review of the literature and current practices in dynamic pricing. Given its applicability in most markets and its increasing adoption in practice, our focus is on dynamic (intertemporal) pricing in the presence of inventory considerations.

Dynamic Pricing in the Presence of Inventory Considerations: Research Overview, Current Practices, and Future Directions

In this paper, we examine the research and results of dynamic pricing policies and their relation to revenue management. The survey is based on a generic revenue management problem in which a perishable and nonrenewable set of resources satisfy stochastic price sensitive demand processes over a finite period of time. In this class of problems, the owner (or the seller) of these resources uses them to produce and offer a menu of final products to the end customers. Within this context, we formulate the stochastic control problem of capacity that the seller faces: How to dynamically set the menu and the quantity of products and their corresponding prices to maximize the total revenue over the selling horizon.

https://pages.stern.nyu.edu/~rcaldent/papers/Abstract-Survey-Paper-MSOM.pdf

Commissioned Paper: An Overview of Pricing Models for Revenue Management

We study the optimal pricing of a finite quantity of a fashion-like seasonal good in the presence of forward-looking (strategic) customers. We distinguish between two classes of pricing strategies: contingent and announced fixed-discount. In both cases, the seller acts as a Stackelberg leader announcing his pricing strategy, while consumers act as followers taking the seller's strategy as given and determining their purchasing behavior. In each case, we identify a subgame-perfect Nash equilibrium and show that given the seller's strategy, the equilibrium in the consumer subgame is unique and consists of symmetric threshold purchasing policies. For both cases, we develop a benchmark model in which customers are nonstrategic (myopic). We conduct a comprehensive numerical study to explore the impact of strategic consumer behavior on pricing policies and expected revenue performance. We show that strategic customer behavior suppresses the benefits of price segmentation, particularly under medium-to-high values of heterogeneity and modest rates of decline in valuations. However, when the level of consumer heterogeneity is small, the rate of decline is medium-to-high, and the seller can optimally choose the time of discount in advance, segmentation can be used quite effectively even with strategic consumers. We find that the seller cannot avoid the adverse impact of strategic consumer behavior even under low levels of initial inventory. We argue that while the seller expects customers to be more concerned about product availability at discount time, he cannot use high-price “betting” strategies as he would in the case of low inventory and myopic customers. Under certain qualifications, announced fixed-discount strategies perform essentially the same as contingent pricing policies in the case of myopic consumers. However, under strategic consumer behavior, announced pricing policies can be advantageous to the seller, compared to contingent pricing schemes. Interestingly, those cases that announced discount strategies offer a significant advantage compared to contingent pricing policies. They appear to offer only a minimal advantage in comparison to fixed-pricing policies. Finally, when the seller incorrectly assumes that strategic customers are myopic in their purchasing decisions, it can be quite costly, reaching potential revenue losses of about 20%.

Optimal Pricing of Seasonal Products in the Presence of Forward-Looking Consumers

This paper considers the intertemporal pricing problem for a monopolist marketing a new product. The key feature differentiating this paper from the extant management science literature on intertemporal pricing is the assumption that consumers are intertemporal utility maximizers. A subgame perfect Nash equilibrium pricing policy is characterized and shown to involve intertemporal price discrimination. We compare this policy to the optimal policy for a monopolist facing myopic consumers and find that for any given state, prices are always lower with rational consumers than with myopic consumers. For plausible parameter values the assumption of consumer rationality can be shown to lead to large differences in optimal prices. Moreover, if a monopolist facing rational consumers implements the optimal myopic consumer pricing policy, profits can be significantly less than if the monopolist follows the equilibrium pricing policy for rational consumers.

Optimal price skimming by a monopolist facing rational consumers

A highly successful spreadsheet-based management science course for MBA students at Indiana University deals mainly with simulation and mathematical optimization with spreadsheet solvers and includes many applications of management science that are extremely relevant to these students. They seem to prefer the spreadsheet approach over the traditional management science approach.

The Teachers' Forum: Management Science with Spreadsheets for MBAs at Indiana University

We consider an exponential queuing system consisting of several removable servers in which the arrival rate depends on the current system state. Costs are incurred at a rate that depends on both the number of customers present and the number of servers in operation. We present conditions that ensure that the number of servers in operation is a non-decreasing function of the number of customers in the system.

Technical Note—Optimality of Monotonic Policies for Multiple-Server Exponential Queuing Systems with State-Dependent Arrival Rates

A maintenance system consists of o finite number of machines and a single-server repair facility that may be operated at several rates. Machines are subject to failure and machines that fail are sent to the repair facility. Under the assumption that costs depend on the repair rate and lost production, we derive conditions that ensure that, for a discrete time maintenance system, the optimal repair rate is a non-increasing function of the number of machines in good condition. By considering a continuous time maintenance system as a limit of a sequence of discrete time maintenance systems, we derive analogous conditions that ensure that the optimal repair rate for the continuous time maintenance system is a non-increasing function of the number of machines in good condition.

Optimal Control of Discrete and Continuous Time Maintenance Systems with Variable Service Rates

We consider a discrete time queuing system composed of a set of servers in parallel in which both customers and servers may be of several types. Customers in service may be switched from one server to another at the beginning of each period. We consider dynamic assignment rules and derive conditions that ensure that it is optimal (in a very strong sense) always to assign customers with longer service times to faster servers.

Assignment of Customers to Servers in a Heterogeneous Queuing System with Switching

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