以排隊理論及隨機顆粒軌跡模型模擬隨機泥砂傳輸下之泥沙傳輸率及濃度變化

Application of Queueing Theory and Stochastic Particle Tracking Model to Simulating Stochastic Sediment Transport: Concentrations and Transport Rates

洪毓茹(Yu-Ju Hung)

指導教授 ： 蔡宛珊

國立臺灣大學/工學院/土木工程學研究所/碩士(2015年)

https://doi.org/10.6342/NTU.2015.01725

摘要

泥砂傳輸在傳統上通常使用定律模型進模擬。然而其間歇性(intermittent)與其隨機(stochastic)的特性使泥砂傳輸更為適合以離散隨機的方法進行描述。本研究運用排隊理論(queueing theory) 以離散隨機的方式設計出一隨機模型的框架，此框架可以用來模擬泥砂顆粒在水中隨機的進入控制體積的行為。泥砂顆粒、控制體積、泥砂傳輸的機制(包含懸浮、沉澱以及從底床被帶起的機制)分別對應於排隊理論中的顧客、商店以及服務。在本研究中主要以隨機擴散粒子追蹤模型(SD-PTM)以及底床顆粒被帶起為主要的泥砂傳輸機制。排隊理論最大的特點在於可以模擬顧客數量及到顧客達時間的隨機性，將此特性運用到泥砂傳輸，就可以進行泥砂顆粒在水中隨機的進入控制體積的行為模擬，其中泥砂顆粒的隨機進入包含泥砂隨機的到達以及每次到達含有隨機的泥砂顆粒數量。本研究以卜松過程(Poisson process)和二項分布(binominal distribution)分別模擬泥砂的隨機到達(random arrivals)以及每次到達的隨機泥砂數量(random magnitude)。
本研究含有三個不同的泥砂進入機制的模擬，分別是泥砂進入控制體積的時間是固定的，但是每次的量是隨機的(random magnitude, RM)、泥砂每次進入控制體積的量為固定的，但是進入控制體積的時間是隨機的(random arrivals, RA)以及泥砂進入控制體積的時間不固定而且每次的量也是隨機的(random-sized batch arrivals)。模擬結果以系綜統計(ensemble statistics)進行分析，統計的系綜包含泥砂濃度、泥沙傳輸速率的系綜平均(ensemble means)和系綜方差(ensemble variances)。再對三種不同的泥砂進入機制的模擬的結果進行比較，比較的結果顯示在相同的平均泥砂輸入速率下，不同於系綜平均，系綜方差的值可以反映出不同的泥砂進入機制。

關鍵字

排隊理論 ； 隨機泥砂進入 ； 隨機泥沙傳輸 ； 序率模式 ； 顆粒軌跡模型 ； 顆粒隨機被帶起 ； 卜松過程 ； 二項分布

並列摘要

Sediment transport is typically simulated using deterministic models. However, the intermittent and stochastic features of sediment transport make it suitable to be described by discrete random processes. This study attempts to apply queueing theory to develop a stochastic framework that could account for the random-sized batch arrivals of incoming sediment particles into receiving waters. Sediment particles, control volume, mechanics of sediment transport (such as mechanics of suspension, deposition and resuspension) are treated as the customers, service facility and server respectively in queueing theory. In the framework, the stochastic diffusion particle tracking model (SD-PTM) and resuspension of particles are included to simulate the random transport trajectories of suspended particles. The most distinguished characteristic of queueing theory is that customers come to the service facility in a random manner. In analogy to sediment transport, this characteristic is suitable to model the random-sized batch arrival process of sediment particles including the random occurrences and random magnitude of incoming sediment particles. The random occurrences of arrivals are simulated by Poisson process while the number of sediment particles in each arrival can be simulated by a binominal distribution. Simulations of random arrivals and random magnitude are proposed individually to compare with the random-sized batch arrival simulations. Simulation results are a probabilistic description for discrete sediment transport through ensemble statistics (i.e. ensemble means and ensemble variances) of sediment concentrations and transport rates. Results reveal the different mechanisms of incoming particles (RM, RA and BA) will result in differences in the ensemble variances of concentrations and transport rates under the same mean incoming rate of sediment particles.

並列關鍵字

queueing theory ； sediment transport ； random input ； SD-PTM ； stochastic method ； particle tracking model

參考文獻

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