68 CHAPTER 3 An Analytical Framework for Cyber-Physical Networks B and that for a single sensor measurement is M . A set of messages with sum of their size S generates a total traffic T (S ) on the net- work when sent from sensor to fusion cen- ter. (a) Derive expressions for probability of detection and total network traffic generated for both methods. (b) Compare the physi- cal detection performance and cyber traffic levels generated by both methods. (c) Derive conditions for one method to outperform the other if traffic levels are not considered. 3. Optimization based on network size and bandwidth: Consider a variation of the sec- ond method described in Exercise 2, wherein each sensor sends only a fraction f of mea- surements to the fusion center. (a) Derive expressions for the detection probability and total network traffic T (fnM ) when a fraction f of measurements are sent from each sensor. (b) Derive conditions for optimal f and N would be needed to understand, design, analyze, deploy, and operate these systems. EXERCISES 1. Low-level radiation source detection: A mon- itoring system for a facility, such as a sta- dium or subway station, to detect low-level radiation sources consists of sensors that provide radiation measurements. The sen- sor measurements are sent over the net- work to a server that executes detection and fusion codes. (a) Identify and list the cyber and physical components of this sys- tem. (b) Develop a graph model for the sys- tem, where sensors are represented by nodes, and the communication links between the sensor and server are represented by edges. (c) Identify physical and cyber vulnerabili- ties of this system.