/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Hamiltonian dynamics can be used to produce distant proposals for the Metropolis algorithm, thereby avoiding the slow exploration of the state space that results from the diffusive behaviour of simple random-walk proposals. Though originating in physics, Hamiltonian dynamics can be applied to most problems with continuous state spaces by simply introducing fictitious "momentum" variables. A key to its usefulness is that Hamiltonian dynamics preserves volume, and its trajectories can thus be used to define complex mappings without the need to account for a hard-to-compute Jacobian factor - a property that can be exactly maintained even when the dynamics is approximated by discretizing time. In this review, I discuss theoretical and practical aspects of Hamiltonian Monte Carlo, and present some of its variations, including using windows of states for deciding on acceptance or rejection, computing trajectories using fast approximations, tempering during the course of a trajectory to handle isolated modes, and short-cut methods that prevent useless trajectories from taking much computation time.

MCMC using Hamiltonian dynamics

This paper is a review of research in product development, which we define as the transformation of a market opportunity into a product available for sale. Our review is broad, encompassing work in the academic fields of marketing, operations management, and engineering design. The value of this breadth is in conveying the shape of the entire research landscape. We focus on product development projects within a single firm. We also devote our attention to the development of physical goods, although much of the work we describe applies to products of all kinds. We look inside the "black box" of product development at the fundamentaldecisions that are made by intention or default. In doing so, we adopt the perspective of product development as a deliberate business process involving hundreds of decisions, many of which can be usefully supported by knowledge and tools. We contrast this approach to prior reviews of the literature, which tend to examine the importance of environmental and contextual variables, such as market growth rate, the competitive environment, or the level of top-management support.

/pdf/product-development-decisions-a-review-of-the-literature-1jhatri7bn.pdf

Product Development Decisions: A Review of the Literature

The subject of inverse problems in differential equations is of enormous practical importance, and has also generated substantial mathematical and computational innovation. Typically some form of regularization is required to ameliorate ill-posed behaviour. In this article we review the Bayesian approach to regularization, developing a function space viewpoint on the subject. This approach allows for a full characterization of all possible solutions, and their relative probabilities, whilst simultaneously forcing significant modelling issues to be addressed in a clear and precise fashion. Although expensive to implement, this approach is starting to lie within the range of the available computational resources in many application areas. It also allows for the quantification of uncertainty and risk, something which is increasingly demanded by these applications. Furthermore, the approach is conceptually important for the understanding of simpler, computationally expedient approaches to inverse problems.

/pdf/inverse-problems-a-bayesian-perspective-1dknxttxjb.pdf

Inverse problems: A Bayesian perspective

While product development efforts are often viewed a unique configurations of idiosyncratic tasks, in reality different projects within an organization often exhibit substantial similarity in the flow of their constituent activities. Moreover, while most of the planning tools available to managers assume that projects are independent clusters of activities, in reality many organizations must manage concurrent projects that place competing demands on shared human and technical resources. This study develops an empirically-based framework for analyzing development time in such contexts. We model the product development organization as a stochastic processing network in which engineering resources are “workstations” and projects are “jobs” that flow between the workstations. At any given time, a job is either receiving service or queueing for access to a resource. Our model’s spreadsheets quantify this division of time, and our simulation experiments investigate the determinants of development cycle time. Th...

https://www.jstor.org/stable/pdfplus/2632976.pdf

From project to process management: an empirically-based framework for analyzing product development time

We develop a framework for asymptotic optimization of a queueing system. The motivation is the staffing problem of call centers with 100's of agents (or more). Such a call center is modeled as an M/M/N queue, where the number of agents~$N$ is large. Within our framework, we determine the asymptotically optimal staffing level~$N^*$ that trades off agents' costs with service quality: the higher the latter, the more expensive is the former. As an alternative to this optimization, we also develop a constraint satisfaction approach where one chooses the least~$N^*$ that adheres to a given constraint on waiting cost. Either way, the analysis gives rise to three regimes of operation: quality-driven, where the focus is on service quality; efficiency-driven, which emphasizes agents' costs; and a rationalized regime that balances, and in fact unifies, the other two. Numerical experiments reveal remarkable accuracy of our asymptotic approximations: over a wide range of parameters, from the very small to the extremely large, $N^*$ is {em exactly/ optimal, or it is accurate to within a single agent. We demonstrate the utility of our approach by revisiting the square-root safety staffing principle, which is a long-existing rule-of-thumb for staffing the M/M/N queue. In its simplest form, our rule is as follows: if $c$ is the hourly cost of an agent, and $a$ is the hourly cost of customers' delay, then $N^* = R + y^*({a over c) sqrt R$, where $R$ is the offered load, and $y^*(cdot)$ is a function that is easily computable.

Dimensioning large call centers

We develop a framework for asymptotic optimization of a queueing system. The motivation is the staffing problem of call centers with 100''s of agents (or more). Such a call center is modeled as an M/M/N queue, where the number of agents~$N$ is large. Within our framework, we determine the asymptotically optimal staffing level~$N^*$ that trades off agents'' costs with service quality: the higher the latter, the more expensive is the former. As an alternative to this optimization, we also develop a constraint satisfaction approach where one chooses the least~$N^*$ that adheres to a given constraint on waiting cost. Either way, the analysis gives rise to three regimes of operation: quality-driven, where the focus is on service quality; efficiency-driven, which emphasizes agents'' costs; and a rationalized regime that balances, and in fact unifies, the other two. Numerical experiments reveal remarkable accuracy of our asymptotic approximations: over a wide range of parameters, from the very small to the extremely large, $N^*$ is {\em exactly\/} optimal, or it is accurate to within a single agent. We demonstrate the utility of our approach by revisiting the square-root safety staffing principle, which is a long-existing rule-of-thumb for staffing the M/M/N queue. In its simplest form, our rule is as follows: if $c$ is the hourly cost of an agent, and $a$ is the hourly cost of customers'' delay, then $N^* = R + y^*({a \over c}) \sqrt R$, where $R$ is the offered load, and $y^*(\cdot)$ is a function that is easily computable.

/pdf/dimensioning-large-call-centers-2sfewyijzg.pdf

Inspired by service systems such as telephone call centers, we develop limit theorems for a large class of stochastic service network models. They are a special family of nonstationary Markov processes where parameters like arrival and service rates, routing topologies for the network, and the number of servers at a given node are all functions of time as well as the current state of the system. Included in our modeling framework are networks of M_t/M_t/n_t queues with abandonment and retrials. The asymptotic limiting regime that we explore for these networks has a natural interpretation of scaling up the number of servers in response to a similar scaling up of the arrival rate for the customers. The individual service rates, however, are not scaled. We employ the theory of strong approximations to obtain functional strong laws of large numbers and functional central limit theorems for these networks. This gives us a tractable set of network fluid and diffusion approximations. A common theme for service network models with features like many servers, priorities, or abandonment is “non-smooth” state dependence that has not been covered systematically by previous work. We prove our central limit theorems in the presence of this non-smoothness by using a new notion of derivative.

Strong approximations for Markovian service networks

We conduct bottleneck analysis of a deterministic dynamic discrete-flow network. The analysis presupposes only the existence of long-run averages, and is based on a continuous fluid approximation to the network in terms of these averages. The results provide functional strong laws-of-large-numbers for stochastic Jackson queueing networks since they apply to their sample paths with probability one.

https://www.jstor.org/stable/pdfplus/3689886.pdf

Avi Mandelbaum

Papers

From project to process management: an empirically-based framework for analyzing product development time

Dimensioning large call centers

Dimensioning large call centers

Strong approximations for Markovian service networks

Discrete flow networks: bottleneck analysis and fluid approximations