ABSTRACT The network of pipes in potable water distribution systems (WDS) are comprised of thousands of pipes made of various materials including PVC, concrete, cast iron, and steel, among several others. The pipes are subjected to internal and external conditions that lead to their failure. Stress conditions include, but are not limited to internal pressures, traffic loading, and corrosion. The deterioration of a pipe decreases its mechanical strength which results in an increase of its probability of failure. Failures lead to loss of service which translates to loss of money due to the cost of repairs and buildup of traffic caused by street closures. The focus of this study is the pipe network underneath cities that make it possible for communities to have access to potable water. The objective of this analysis is to evaluate the physical conditions of each pipe in a water distribution system in order to assess its probability of failure and ultimately calculate the risk associated with each pipe in the case that it were to fail. This model focuses only on the pipes of the WDS and does not take into consideration fittings, pumps, and other network components. This model assesses pipe age, material, diameter, internal pressure, traffic loading (industrial or residential), and length to determine the probability of failure. It then utilizes several economic factors such as material cost, customer criticality, demand, traffic impact, and land use to calculate the risk associated with each pipe. The risk associated with each pipe can then be used as a ranking system to identify the most vulnerable pipes, those with the highest economic impact upon failure. Identifying the pipes with the highest risk allows municipalities to better allocate funds for maintenance or replacement of pipes. It highlights the most critical pipes within a network of thousands. In order to check its functionality, this model applied to the WDS of the City of Arroyo Grande, California. Information on the City’s distribution system was analyzed using Bentley’s WaterCAD, ESRI’s ArcGIS, MathWorks’ MATLAB and Microsoft’s Excel software to perform the analysis. The risk analysis model provided 3 pipes within the distribution system made of cast iron as having a high probability of failure and a critical level of risk. A critical level of risk is defined as falling within the highest range of risk within this study. Considering that only 3 pipe segments were highlighted as having a Critical Risk, 4 as High Risk, and 6 as Medium Risk, in a system of 3572 pipes indicates that the model functions properly. This model was compared to a method developed by Jan C. Devera in his thesis “Risk Assessment Model for Pipe Rehabilitation and Replacement in a Water Distribution System” (2013), which was also applied to the City of Arroyo Grande’s distribution system. Results provided by this analysis prove that both models are functional due to similar results. The current study utilizes the concepts of random variables and conditional assessment to run various Monte Carlo Simulations as the means of calculating the probability of failure of a pipe. Mr. Devera’s model utilizes simplistic approach that does not involve intensive calculations, but results for both models turned out to be similar when looking at the Arroyo Grande distribution system. This risk assessment model demonstrates that a risk assessment model can provide a framework to prioritize pipes based on risk. The approach can help create a schedule for a city’s pipe distribution network for maintenance and repair. It is important to note that it is not a predictive model. This study may be employed to better allocate funds for the rehabilitation and replacement of a city’s existing pipe network to promote optimal operating conditions and service to the public.

Um dos principais problemas em testes de hipóteses para a homogeneidade de curvas de sobrevivência ocorre quando as taxas de falha (ou funções de intensidade) não são proporcionais. Apesar do teste de Log-rank ser o teste não paramétrico mais utilizado para se comparar duas ou mais populações sujeitas a dados censurados, este teste apresentada duas restrições. Primeiro, toda a teoria assintótica envolvida com o teste de Log-rank, tem como hipótese o fato das populações envolvidas terem distribuições contínuas ou no máximo mistas. Segundo, o teste de Log-rank não apresenta bom comportamento quando as funções intensidade cruzam. O ponto inicial para análise consiste em assumir que os dados são contínuos e neste caso processos Gaussianos apropriados podem ser utilizados para testar a hipótese de homogeneidade. Aqui, citamos o teste de Renyi e Cramér-von Mises para dados contínuos (CCVM), ver Klein e Moeschberger (1997) [15]. Apesar destes testes não paramétricos apresentar bons resultados para dados contínuos, esses podem ter problemas para dados discretos ou arredondados. Neste trabalho, fazemos um estudo simulação da estatística de Cramér von-Mises (CVM) proposto por Leão e Ohashi [16], que nos permite detectar taxas de falha não proporcionais (cruzamento das taxas de falha) sujeitas a censuras arbitrárias para dados discretos ou arredondados. Propomos também, uma modificação no teste de Log-rank clássico para dados dispostos em uma tabela de contingência. Ao aplicarmos as estatísticas propostas neste trabalho para dados discretos ou arredondados, o teste desenvolvido apresenta uma função poder melhor do que os testes usuais / One of the main problems in hypothesis testing for homogeneity of survival curves occurs when the failure rate (or intensity functions) are nonproportional. Although the Log-rank test is a nonparametric test most commonly used to compare two or more populations subject to censored data, this test presented two constraints. First, all the asymptotic theory involved with the Log-rank test, is the hypothesis that individuals and populations involved have continuous distributions or at best mixed. Second, the log-rank test does not show well when the intensity functions intersect. The starting point for the analysis is to assume that the data is continuous and in this case suitable Gaussian processes may be used to test the assumption of homogeneity. Here, we cite the Renyi test and Cramér-von Mises for continuous data (CCVM), and Moeschberger see Klein (1997) [15]. Despite these non-parametric tests show good results for continuous data, these may have trouble discrete data or rounded. In this work, we perform a simulation study of statistic Cramér-von Mises (CVM) proposed by Leão and Ohashi [16], which allows us to detect failure rates are nonproportional (crossing of failure rates) subject to censure for arbitrary data discrete or rounded. We also propose a modification of the test log-rank classic data arranged in a contingency table. By applying the statistics proposed in this paper for discrete or rounded data, developed the test shows a power function better than the usual testing

Um dos principais problemas em testes de hipóteses para a homogeneidade de curvas de sobrevivência ocorre quando as taxas de falha (ou funções de intensidade) não são proporcionais. Apesar do teste de Log-rank ser o teste não paramétrico mais utilizado para se comparar duas ou mais populações sujeitas a dados censurados, este teste apresentada duas restrições. Primeiro, toda a teoria assintótica envolvida com o teste de Log-rank, tem como hipótese o fato das populações envolvidas terem distribuições contínuas ou no máximo mistas. Segundo, o teste de Log-rank não apresenta bom comportamento quando as funções intensidade cruzam. O ponto inicial para análise consiste em assumir que os dados são contínuos e neste caso processos Gaussianos apropriados podem ser utilizados para testar a hipótese de homogeneidade. Aqui, citamos o teste de Renyi e Cramér-von Mises para dados contínuos (CCVM), ver Klein e Moeschberger (1997) [15]. Apesar destes testes não paramétricos apresentar bons resultados para dados contínuos, esses podem ter problemas para dados discretos ou arredondados. Neste trabalho, fazemos um estudo simulação da estatística de Cramér von-Mises (CVM) proposto por Leão e Ohashi [16], que nos permite detectar taxas de falha não proporcionais (cruzamento das taxas de falha) sujeitas a censuras arbitrárias para dados discretos ou arredondados. Propomos também, uma modificação no teste de Log-rank clássico para dados dispostos em uma tabela de contingência. Ao aplicarmos as estatísticas propostas neste trabalho para dados discretos ou arredondados, o teste desenvolvido apresenta uma função poder melhor do que os testes usuais / One of the main problems in hypothesis testing for homogeneity of survival curves occurs when the failure rate (or intensity functions) are nonproportional. Although the Log-rank test is a nonparametric test most commonly used to compare two or more populations subject to censored data, this test presented two constraints. First, all the asymptotic theory involved with the Log-rank test, is the hypothesis that individuals and populations involved have continuous distributions or at best mixed. Second, the log-rank test does not show well when the intensity functions intersect. The starting point for the analysis is to assume that the data is continuous and in this case suitable Gaussian processes may be used to test the assumption of homogeneity. Here, we cite the Renyi test and Cramér-von Mises for continuous data (CCVM), and Moeschberger see Klein (1997) [15]. Despite these non-parametric tests show good results for continuous data, these may have trouble discrete data or rounded. In this work, we perform a simulation study of statistic Cramér-von Mises (CVM) proposed by Leão and Ohashi [16], which allows us to detect failure rates are nonproportional (crossing of failure rates) subject to censure for arbitrary data discrete or rounded. We also propose a modification of the test log-rank classic data arranged in a contingency table. By applying the statistics proposed in this paper for discrete or rounded data, developed the test shows a power function better than the usual testing