# Edge and Cloud Computing in Smart Cities
## 2 Architectural Models for Edge and Cloud Computing in Smart Cities
### Multi-Tier Hierarchical Architecture  
The multi-tier hierarchical architecture structures computational resources across multiple layers to optimize performance, latency, and resource utilization. This architecture balances centralized high-performance computing with distributed low-latency processing, ensuring intelligent service provisioning across urban environments [14, 15]. Formally, the architecture is modeled as a layered graph \(G=(\mathcal{N},\mathcal{L})\), where \(\mathcal{N}\) represents the set of computing nodes categorized into layers, and \(\mathcal{L}\) denotes the communication  
Figure 1: An overview of surveyed key topics: edge and cloud computing in smart cities. links between them. Each node \(n_{i}\in\mathcal{N}\) has a defined computational capacity, storage, and processing latency. The objective is to minimize cumulative service delay while optimizing resource allocation [16; 17]. The computational hierarchy, shown in Figure 2, consists of three basic layers unwrapped into \"Cloud, Edge, and Device\" [18]. Cloud layer \(C\) consists of \(M\) distinct nodes, the cloud servers, where each node provides centralized data storage, large-scale computational resources for data processing, AI model training, and global analytics. Computational resources at this layer are represented by  
\[C=\{C_{1},C_{2},\ldots,C_{M}\},\quad C_{m}=(P_{m},S_{m},L_{C_{m}}),\quad m=1,2, \ldots,M, \tag{1}\]  
where \(P_{m}\) denotes the processing power of cloud node \(C_{m}\), \(S_{m}\) its available storage, and \(L_{C_{m}}\) its inherent processing latency. The term \(L_{C_{m}}\) specifically represents the intrinsic processing delay at cloud node \(C_{m}\), which depends solely on the node's computational capacity and workload. However, tasks offloaded to the cloud experience an additional transmission delay, leading to the total execution latency at the cloud, denoted as \(T_{C}\). This total delay includes both network transmission time and processing latency in the cloud  
\[T_{C}=T_{D\to C}+L_{C_{m}}+T_{C\to D}, \tag{2}\]  
where \(T_{D\to C}\) is the time required to transmit data from the device or edge to the cloud, \(L_{C_{m}}\) is the actual processing delay at the cloud node, and \(T_{C\to D}\) is the response time for sending the processed results back to the device. The primary drawback of the cloud layer is the high transmission latency, \(T_{C}\), which varies with network conditions and task size [19; 20]. Edge layer \(E\) serves as an intermediary stratum that minimizes response times by processing tasks closer to data sources, alleviating network congestion and reducing latency. It is composed of \(K\) distinct nodes, called edge servers, each with its own computational and storage constraints. The edge layer is formally defined as  
Figure 2: Schematic representation of the three-tier architecture. \[E=\{E_{1},E_{2},\ldots,E_{K}\},\quad E_{k}=(P_{k},S_{k},L_{E_{k}}),\quad k=1,2, \ldots,K, \tag{3}\]  
where \(P_{k}\) represents the processing power, \(S_{k}\) the storage capacity, and \(L_{E_{k}}\) the inherent latency at edge node \(E_{k}\). The latency at the edge layer, \(L_{E_{k}}\), is significantly lower than at the cloud since tasks are processed closer to their sources. However, the edge is constrained by finite computing resources, which limits its capacity to handle computationally intensive tasks [21]. The device layer (\(D\)) comprises a set of IoT devices and user equipment that continuously generate data streams but possess limited computational and storage capabilities. These devices, typically operating in dynamic environments, rely on higher layers for processing-intensive tasks.