Using Joint Probability Distribution f Reliability and Vulnerability to Develop a water System Performance Indexgo

Document Type



Assessing the long-term reliability and vulnerability of municipal water supply systems often employs system modeling to analyze performance. Generally, decision-makers have long been in search of a metric that presents a comprehensive assessment of water-supply systems. The system’s condition is often evaluated by a univariate measure of reliability, resiliency, or vulnerability, which individually cannot provide a comprehensive understanding of the system’s performance. In this study, instead of an individual measure, the joint probability distribution of reliability and vulnerability is used to assess the performance of water systems. To quantify the joint probability distribution between reliability and vulnerability, different copula functions are tested, and the most appropriate one is selected. The copula function couples one-dimensional marginal distributions of reliability and vulnerability to form the cumulative distribution function (CDF) of the joint probability. Then, a novel index, the Water System Performance Index (WSPI), is derived from the copula CDF based on exceedance and nonexceedance probability of these two metrics. WSPI presents simultaneous information about the frequency and magnitude of failures in water systems. The proposed WSPI increases with an increase in reliability and a decrease in vulnerability of the system and vice versa. The WSPI is demonstrated and tested using two reservoirs of the Salt Lake City Department of Public Utilities (SLCDPU) water system. WSPI, first, is employed to present the performance of the system during the historical period of 1981–2010. Then, WSPI values are estimated for the system under future climate projections. Results suggest the WSPI provides a useful tool for managers and stakeholders to represent simultaneous information about frequency, magnitude, and recovery period of a system under different failure conditions.

Digital Object Identifier (DOI)

APA Citation

Goharian, E., Burian, S. J., & Karamouz, M. (2018). Using Joint Probability Distribution of Reliability and Vulnerability to Develop a Water System Performance Index. Journal of Water Resources Planning and Management, 144(2), 04017081.


© ASCE, 2018