W e study a single-stage, continuous-time inventory model where unit-sized demands arrive according to a renewal process and show that an (s, S) policy is optimal under minimal assumptions on the ordering/procurement and holding/backorder cost functions. To our knowledge, the derivation of almost all existing (s, S)-optimality results for stochastic inventory models assume that the ordering cost is composed of a fixed setup cost and a proportional variable cost; in contrast, our formulation allows virtually any reasonable ordering-cost structure. Thus, our paper demonstrates that (s, S)-optimality actually holds in an important, primitive stochastic setting for all other practically interesting ordering cost structures such as well-known quantity discount schemes (e.g., all-units, incremental and truckload), multiple setup costs, supplier-imposed size constraints (e.g., batch-ordering and minimum-order-quantity), arbitrary increasing and concave cost, as well as any variants of these. It is noteworthy that our proof only relies on elementary arguments.
We consider a system of N queues served by a single server in cyclic order. Each queue has its own distinct Poisson arrival stream and its own distinct general service-time distribution (asymmetric queues); and each queue has its own distinct distribution of switchover time (the time required for the server to travel from that queue to the next). We consider two versions of this classical polling model: In the first, which we refer to as the zeroswitchover-times model, it is assumed that all switchover times are zero and the server stops traveling whenever the system becomes empty. In the second, which we refer to as the nonzero-switchover-times model, it is assumed that the sum of all switchover times in a cycle is nonzero and the server does not stop traveling when the system is empty. After providing a new analysis for the zero-switchover-times model, we obtain, for a host of service disciplines, transform results that completely characterize the relationship between the waiting times in these two, operationally-different, polling models. These results can be used to derive simple relations that express (all) waiting-time moments in the nonzeroswitchover-times model in terms of those in the zero-switchover-times model. Our results, therefore, generalize corresponding results for the expected waiting times obtained recently by Fuhrmann  and Cooper, Niu, and Srinivasan .
A set of n jobs is to be processed by two machines in series that are separated by an infinite waiting room; each job requires a (known) fixed amount of processing from each machine. In a classic paper, Johnson gave a simple rule for ordering of the set of jobs to minimize the time until the system becomes empty, i.e., the makespan. This paper studies a stochastic generalization of this problem in which job processing times are independent random variables. Our main result is a sufficient condition on the processing time distributions that implies that the makespan becomes stochastically smaller when two adjacent jobs in a given job sequence are interchanged. We also give an extension of the main result to job shops.
We develop, in this article, a sales model for movie and game products at Blockbuster. The model assumes that there are three sales components: the first is from consumers who have already committed to purchasing (or renting) a product (e.g., based on promotion of, or exposure to, the product prior to its launch); the second comes from consumers who are potential buyers of the product; and the third comes from either a networking effect on closely tied (as in a social group) potential buyers from previous buyers (in the case of movie rental and all retail products) or re‐rents (in the case of game rental). In addition, we explicitly formulate into our model dynamic interactions between these sales components, both within and across sales periods. This important feature is motivated by realism, and it significantly contributes to the accuracy of our model. The model is thoroughly tested against sales data for rental and retail products from Blockbuster. Our empirical results show that the model offers excellent fit to actual sales activity. We also demonstrate that the model is capable of delivering reasonable sales forecasts based solely on environmental data (e.g., theatrical sales, studio, genre, MPAA ratings, etc.) and actual first‐period sales. Accurate sales forecasts can lead to significant cost savings. In particular, it can improve the retail operations at Blockbuster by determining appropriate order quantities of products, which is critical in effective inventory management (i.e., it can reduce the extent of over‐stocking and under‐stocking). While our model is developed specifically for product sales at Blockbuster, we believe that with context‐dependent modifications, our modeling approach could also provide a reasonable basis for the study of sales for other short‐Life‐Cycle products.
We consider the classical polling model: queues served in cyclic order with either exhaustive or gated service, each with its own distinct Poisson arrival stream, service-time distribution, and switchover-time (the server's travel time from that queue to the next) distribution. Traditionally, models with zero switchover times (the server travels at infinite speed) and nonzero switchover times have been considered separately because of technical difficulties reflecting the fact that in the latter case the mean cycle time approaches zero as the travel speed approaches infinity. We argue that the zero-switchover-times model is the more fundamental model: the mean waiting times in the nonzero-switchover-times model decompose (reminiscent of vacation models) into a sum of two terms, one being a simple function of the sum of the mean switchover times, and the other the mean waiting time in a “corresponding” model obtained from the original by setting the switchover times to zero and modifying the service-time variances. This generalizes a recent result of S. W. Fuhrmann for the case of constant switchover times, where no variance modification is necessary. The effect of these studies is to reduce computation and to improve theoretical understanding of polling models.
The Bass Model (BM) is a widely-used framework in marketing for the study of new-product sales growth. Its usefulness as a demand model has also been recognized in production, inventory, and capacity-planning settings. The BM postulates that the cumulative number of adopters of a new product in a large population approximately follows a deterministic trajectory whose growth rate is governed by two parameters that capture (i) an individual consumer's intrinsic interest in the product, and (ii) a positive force of influence on other consumers from existing adopters. A finite-population purebirth-process (re)formulation of the BM, called the Stochastic Bass Model (SBM), was proposed recently by the author in a previous paper, and it was shown that if the size of the population in the SBM is taken to infinity, then the SBM and the BM agree (in probability) in the limit. Thus, the SBM "expands" the BM in the sense that for any given population size, it is a well-defined model. In this paper, we exploit this expansion and introduce a further extension of the SBM in which demands of a product in successive time periods are governed by a history-dependent family of SBMs (one for each period) with different population sizes. A sampling theory for this extension, which we call the Piecewise-Diffusion Model (PDM), is also developed. We then apply the theory to a typical product example, demonstrating that the PDM is a remarkably accurate and versatile framework that allows us to better understand the underlying dynamics of new-product demands over time. Joint movement of price and advertising levels, in particular, is shown to have a significant influence on whether or not consumers are "ready" to participate in product purchase.
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