Question

Three packet-switching networks each contain \(n\) nodes. The first network has a star topology with a central switch, the second is a (bidirectional) ring, and the third is fully interconnected, with a wire from every node to every other node. What are the best-, average-, and worst-case transmission paths in hops?

Short answer

Star: best = 1, avg = 2, worst = 2; Ring: best = 1, avg ≈ n/4, worst = n/2; Fully connected: best, avg, worst = 1.

Step-by-step solution

Understand Specific Topologies

We are given three topologies: **star**, **ring**, and **fully interconnected network (complete graph)**. In the star configuration, there is a central node to which all other nodes are directly connected. In a ring, each node is connected to two other nodes, forming a circle. In a fully interconnected network, every node is connected to every other node directly.

Determine the Best-Case Path

The best-case path refers to the minimum number of hops required to send data from one node to another. For the **star topology**, the best-case path is 1 hop since any two nodes must communicate through the central switch. For the **ring topology**, the best-case path is also 1 hop if the destination node is the adjacent node. For the **fully interconnected network**, the best-case path is 1 hop because every node is directly connected to all others.

Determine the Average-Case Path

For each topology, the average-case path is the expected number of hops between any two nodes. In the **star topology**, since all communication goes through the central switch, the average is also 2 hops. In the **ring topology**, the average number of hops is approximately \( \frac{n}{4} \) because on average, a node is about a quarter way around the ring from another node.For the **fully interconnected network**, since all nodes are directly connected, the average number of hops is 1.

Determine the Worst-Case Path

The worst-case path indicates the maximum number of hops. In the **star topology**, any data must go through the central switch, resulting in a maximum of 2 hops.In the **ring topology**, the worst-case scenario happens when sending data between two nodes that are opposite each other in the ring, resulting in \( \frac{n}{2} \) hops.For the **fully interconnected network**, the worst-case path remains 1 hop as any node is directly connected to any other node.

Key concepts

Star Topology

In a star topology, all nodes connect directly to a central hub or switch. Think of it like a bicycle wheel, where the hub is the center, representing the central node, and each spoke is a direct connection to a surrounding node. This configuration is beneficial as every node has a single, simple connection path to the central hub. When a node wants to communicate with another node, the data is sent first to the central hub, and then to the target node.

• **Best-case transmission**: When information is sent from one node to another through the central hub, it takes 1 hop, as data always goes straight to the hub first.
• **Average-case transmission**: Since communication involves going to the hub and then to another node, the average transmission path is about 2 hops.
• **Worst-case transmission**: Similarly, even in the worst-case, it only takes 2 hops since all communications inevitably pass through the central node.

Star topologies are favored for ease of troubleshooting and management. If one connection fails, it doesn't impact other nodes, thanks to the individual direct connections to the hub.

Ring Topology

A ring topology connects each node to exactly two other nodes, creating a circular data path. Imagine a ring of people, where each person holds the hand of the person to their left and right. This configuration allows data to travel in one or both directions, forming a unidirectional or bidirectional ring.

• **Best-case transmission**: If two nodes that need to communicate are neighbors, data is transferred in just 1 hop.
• **Average-case transmission**: On average, a node is approximately a quarter-way around the ring, making the average number of hops about \( \frac{n}{4} \).
• **Worst-case transmission**: The most hops occur when nodes are directly opposite each other in the ring, resulting in \( \frac{n}{2} \) hops as data traverses half the ring.

Ring topology can become problematic if one connection breaks; however, using a bidirectional setup can mitigate this by rerouting data in the opposite direction.

Fully Connected Network

In a fully connected network, each node is linked to every other node directly. Picture every student in a class being personally connected to every other student through individual communication lines. This setup ensures the most direct and efficient communication possible.

• **Best-case transmission**: Every transmission is just 1 hop because each node has a direct line to any other node.
• **Average-case transmission**: As all nodes are interconnected directly, the average number of hops remains 1.
• **Worst-case transmission**: Even in the scenario considered as worst-case, it's still just 1 hop, since no intermediate nodes are involved.

Fully connected networks are optimal in terms of speed and robustness but are seldom used due to the impracticality of the vast number of connections required, especially as the number of nodes increases. This makes them more suitable for scenarios with a small number of nodes.