## 3 System Design
### System Scalability and Network Topology Extensions  
The chain-based architecture of the proposed system offers exceptional scalability characteristics that differentiate it from traditional centralized streaming solutions. Theoretically, the chain topology can be extended indefinitely, with each additional node contributing minimal overhead to the overall system while expanding its reach. This linear scaling property stands in stark contrast to centralized solutions, where increasing viewer capacity often incurs exponential cost increases in server infrastructure, bandwidth provisioning, and operational complexity.  
A particularly powerful aspect of the system's design is its inherent support for topology transformation and branching. Any node within an existing chain can serve as an origination point for a new chain by using its locally received stream as input, effectively creating a tree-structured network from the fundamental chain building blocks. This capability enables several significant advantages:  
\[T(n)=\begin{cases}O(1)&\text{for adding a new node to existing chain}\\ O(log(n))&\text{for establishing optimal tree depth with $n$ nodes}\end{cases} \tag{20}\]  
First, tree structures provide more efficient distribution paths compared with single long chains, reducing cumulative latency and enhancing viewership scalability. Second, this arrangement facilitates localized viewing clusters within LANs or geographically proximate networks, where a single entry point into a regional network can serve multiple downstream viewers through local branching. Third, network administrators can deliberately design the topology based on specific performance objectives, geographical constraints, or audience distribution patterns.  
Furthermore, this approach enables targeted optimization strategies. For instance, high-capacity nodes can be positioned at critical branch points to serve multiple downstream chains, while resource-constrained environments can be accommodated through careful branch placement and stream parameter adjustment. The system's flexibility allows implementers to minimize chain length where latency is critical while maximizing distribution breadth where coverage is the priority.  
This inherent extensibility presents a fundamental economic advantage: while traditional centralized solutions encounter bandwidth bottlenecks and processing limitations that require costly hardware upgrades or cloud service tier increases, our decentralized approach requires only the addition of standard nodes--which can be implemented on commodity hardware--to expand capacity. This results in near-linear cost scaling rather than the super-linear or exponential cost growth typical of centralized architectures under increasing load.  
Experimental validation confirms this scalability model, with successful deployments maintaining consistent performance characteristics across varied topologies ranging from simple 8-hop chains to complex tree structures with multiple branch points serving over 20 concurrent viewers. The architecture's ability to dynamically reorganize these topologies further enhances its adaptability to changing network conditions and audience requirements.