## 1 Introduction  
Cyber-Physical-Human Systems (CPHS) offer interesting perspectives for the comfort, health and safety of users [1; 2] with applications in building energy efficiency [3; 4] and emergency management [5]. While there has been extensive research regarding guidance systems, routing algorithms and efficient task assignment algorithms [6] for emergency evacuation applications, the operation of such systems in degraded conditions often associated with emergencies is generally overlooked. Recent work [7] shows that while a majority of sensors provide information of limited value, only a few strategically located sensors will critically affect the CPHS' performance and the outcome of an evacuation. While relevant analytical methods based on linear control exist to study systems in degraded conditions, in particular reliability and robustness [8, 9], redundancy [10, 11] and fault-tree analysis [12], these methods do not handle the complex and emergent aspects of a CPHS. On the other hand, much work in this area relies on experiments with purpose-built simulators such as the Distributed Building Evacuation Simulator (DBES) [13, 14], which provides an approach that is similar in many aspects to traditional tactical modelling and simulation in the military domain [15]. The work on the design and optimization of emergency management schemes based on discrete event simulation [16] is faced by the substantial amount of programming that is often needed when existing simulation tools are applied to a new problem, and by the scalability and large computing time needed to obtain useful performance estimates in realistic environments. Thus, this paper proposes a complementary approach that offers fast estimates based on queueing theoretic analytical models [17, 18, 19] as well as graph models and algorithms. The graph representations themselves, and in particular those used in this paper for our examples, will typically represent the space in a specific building. Since the space within modern buildings, or other built spaces such as areas within cities or towns, are typically represented with databases and graphical systems [20], going from the detailed building representation to a graph representation, which includes accurate distances and representative traversal times, can be carried out in an automatic and accurate manner. This is not too dissimilar (but generally simpler) than the construction of \"terrain models\" in simulation systems used in the military domain [21] where natural terrain features must also be included. The use of probability and analytical models in this context is not new [22, 23, 24, 25], but for a variety of reasons including the large computational costs of the early closed network (finite population) models, such models appear to have been abandoned in this area for the last decade or more. The approach we take here uses open models whose computational needs are very low and where useful formulas, for instance for average traverse times and throughput, can be obtained very rapidly [17, 19]. Such probability models are particularly useful in situations where there is a substantial amount of uncertainty [26], while graph models are useful when there is a substantial amount of topological structure such as the rooms, corridors, staircases in a building, or when area-related information will remain fixed for long periods of time. Combining the probability models for human movement and congestion with graph theory for locations offers a way forward to deal efficiently with the large and realistic representations that are needed in CPHS. Our work suggests that analytical models based on queueing theory can provide fast estimates for the location of points of congestion, for estimating the sensitivity of outcomes to the presence of hazards in particular areas, and for routing optimization. To illustrate the use of these approaches, we examine an emergency evacuation in the first three floors of a real-life building.