Browsing by Subject "Stochastic processes"

Now showing 1 - 3 of 3
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    Essays on Stochastic Inventory Systems
    (2015-07) Chen, Rui
    This thesis consists of three essays in stochastic inventory systems. The first essay is on the impact of input price variability and correlation on stochastic inventory systems. For a general class of such systems, we show that the expected cost function is concave in the input price. From this, it follows that higher input price variability in the sense of the convex order always leads to lower expected cost. We show that this is true under a wide range of assumptions for price evolution, including cases with i.i.d. prices and cases where prices are correlated and evolve according to an AR(1) process, a geometric Brownian motion, or a Markovian martingale. In addition, the result holds in cases where there is just a single period. We also examine the impact of price correlation over time and across inputs, and we find that expected cost is increasing in price correlation over time and decreasing in price correlation across components. We present results of a numerical study that provide insights on how various parameters influence the effects of price variability and correlation. The second essay is on the optimal control of inventory systems with stochastic and independent leadtimes. We show that a fixed base-stock policy is sub-optimal and can perform poorly. For the case of exponentially distributed leadtimes, we show that the optimal policy is state-dependent and specified in terms of an inventory-dependent threshold function. Moreover, we show that this threshold function is non-increasing in the inventory level and characterized by at most m parameters. That is, once the threshold function starts to decrease it continues to decrease with a rate that is at least one. Taking advantage of this structure, we develop an efficient algorithm for computing these parameters. In characterizing the structure of the optimal policy, we rely on an application of the Banach fixed point theorem. We compare the performance of the optimal policy to that of simpler heuristics. We also extend our analysis to systems with lost sales and systems with order cancellations. The third essay is on the optimal policies for inventory systems with concave ordering costs. By extending the Scarf (1959} model to systems with piecewise linear concave ordering costs, we characterize the structure of optimal policies for periodic review inventory systems with concave ordering costs and general demand distributions. We show that, except for a bounded region, the generalized (s,S) policy is optimal. We do so by (a) introducing a conditional monotonicity property for the optimal order-up-to levels and (b) applying the notion of c-convexity. We also provide conditions under which the generalized (s, S) policy is optimal for all regions of the state space.
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    Real-Time Traffic Prediction for Advanced Traffic Management Systems: Phase I
    (Intelligent Transportation Systems Institute, University of Minnesota, 1995-10) Davis, Gary A.; Stephanedes, Yorgos J.; Kang, Jeong-Gyu
    It has been recommended that Advanced Traffic Management Systems (ATMS) must work in real-time, must respond to and predict changes in traffic conditions, and must included areawide detection surveillance. To support such ATMS, this project developed a tractable, stochastic model of freeway traffic flow and travel demand which satisfies three primary objectives. First, the model should generate real-time estimates of traffic state variables from loop detector data, which can in turn be used as time-varying initial conditions for more comprehensive simulation models, such as KRONOS or FREESIM. Second, the model should generate its own predictions of mainline and off-ramp traffic volumes, as well as calculate the expected error associated with these predictions, thus supporting the use of both deterministic and stochastic optimization for determining traffic management actions. Third, the model should be capable of full on-line implementation, in that it should be capable of estimating required parameters from traffic detector data. The basic model was developed by combining ideas from the theory of Markov population processes with a new for the relationship between traffic flow and density, producing a stochastic version of a simple-continuum model. Kalman filtering was then applied to the basic model to develop algorithms for (1) estimating from loop detector counts the traffic density in freeway sections broken down by destination off-ramp, (2) predicting main-line and off-ramp traffic volumes from given on-ramp volumes and, (3) computing adaptive estimates of the freeway's origin destination matrix from loop detector counts. Monte Carlo simulation tests were used to evaluate three different methods for off-line estimation of model parameters, as well as to assess the accuracy of the density estimates and volume predictions. The results indicated that the estimation and prediction model tends to be robust with respect to the parameter estimation scheme, and that the model generates a reasonable characterization of estimation and prediction uncertainty. Limited tests with field data tended to confirm the simulation results, and to emphasize the importance of real-time estimation of freeway origin-destination matrices in generating accurate predictions.
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    Stochastic Models of Epithelial Cancer Initiation and Glioblastoma Recurrence
    (2018-06) Storey, Kathleen
    Cancer development involves the inherently stochastic accumulation of genetic mutations, conferring growth advantages to the cells affected by these mutations. Thus, stochastic modeling provides useful insight when studying the evolutionary processes of cancer initiation and tumor progression. This thesis consists of three projects within the field of stochastic modeling of cancer evolution. First we explore the temporal dynamics of spatial heterogeneity during the process of carcinogenesis from healthy tissue. We utilize a spatial stochastic model of mutation accumulation and clonal expansion to describe this process. Under a two-step carcinogenesis model, we analyze two new measures of spatial population heterogeneity. In particular, we study the typical length-scale of genetic heterogeneity during carcinogenesis and estimate the size of the clone surrounding a sampled premalignant cell. Next we study the propagation speed of a premalignant clone during carcinogenesis. We approximate a premalignant clone in epithelial tissue containing w layers of proliferating cells (referred to as a ``basal zone'') with a biased voter model on a set of w stacked integer lattices. Using the dual process of the biased voter model, we determine the asymptotic propagation speed of the premalignant clone in this setting and compare it to the previously determined speed in epithelial tissue with a single layer of proliferating cells. We then use this speed to investigate clinical implications for primary tumors detected in various types of epithelial tissue. Finally we develop a multi-type branching process model of the tumor progression and treatment response in glioblastoma multiforme (GBM). GBM recurrence is often attributed to acquired resistance to the standard chemotherapeutic agent temozolomide (TMZ). Promoter methylation of the DNA repair gene MGMT is frequently linked to TMZ sensitivity. We develop and parameterize a model using clinical and experimental data, to investigate the interplay between TMZ and MGMT methylation during GBM treatment. Our model suggests that TMZ may have an inhibitory effect on maintenance methylation of MGMT after cell division. Incorporating this effect, we study the optimal TMZ dosing regimen for GBM patients with high and low levels of MGMT methylation at diagnosis.