Question

A packet switch receives a packet and determines the outbound link to which the packet should be forwarded. When the packet arrives, one other packet is halfway done being transmitted on this outbound link and four other packets are waiting to be transmitted. Packets are transmitted in order of arrival. Suppose all packets are 1,500 bytes and the link rate is 2 Mbps. What is the queuing delay for the packet? More generally, what is the queuing delay when all packets have length \(L\), the transmission rate is \(R, x\) bits of the currently-being-transmitted packet have been transmitted, and \(n\) packets are already in the queue?

Short answer

The queuing delay is 0.027 seconds. In general, \( t_{total} = \frac{L-x}{R} + n \times \frac{L}{R} \).

Step-by-step solution

Understand the Problem

We need to find the time it takes for a packet to be transmitted from the time it arrives until it begins transmission. This is the queuing delay. The scenario includes one packet currently being transmitted and four in the queue.

Calculate Transmission Time for One Packet

Transmission time for a packet of size 1,500 bytes (12,000 bits) on a 2 Mbps link is given by: \[ t_{transmission} = \frac{L}{R} = \frac{12,000 \text{ bits}}{2\times 10^6 \text{ bits per second}} = 0.006 \text{ seconds} \]

Calculate Remaining Transmission Time for Current Packet

Since half of the first packet has already been transmitted, it will take half of the transmission time calculated in Step 2 to finish. Thus, the remaining time is: \[ \frac{0.006}{2} = 0.003 \text{ seconds} \]

Calculate Total Queuing Time

The total queuing delay for the arriving packet is the sum of times for the remaining half of the first packet and the full transmission times for the four packets in the queue:\[ t_{queue} = 0.003 \text{ (current) + 4}\times 0.006 = 0.003 + 0.024 = 0.027 \text{ seconds} \]

Generalize for General Variables

For arbitrary packet length \(L\), transmission rate \(R\), \(x\) bits transmitted, and \(n\) packets:1. Remaining time for current packet: \( t_{remaining} = \frac{L-x}{R} \)2. Time for queued packets: \( t_{queue} = n \times \frac{L}{R} \)3. Total queuing delay: \[ t_{total} = \frac{L-x}{R} + n \times \frac{L}{R} \]

Key concepts

Packet Switching

Packet switching is an efficient method used in networks to transmit data where messages are broken down into smaller units called packets. This technique enables multiple packets from different sources to share the same network links.
When a packet arrives at a switch, the switch determines the best path for it to travel to its destination. Each packet can potentially take a different route, allowing for improved network utilization and fault tolerance.
Packet switching contrasts with circuit switching, where a dedicated communication path is established between the sender and receiver for the entire message. Packet switching is more dynamic and can better handle network congestion.

• Flexibility: Packets are routed based on current network conditions.
• Efficiency: Links are shared among multiple packets, maximizing resource use.
• Scalability: Network can grow without a linear increase in resources.

The primary challenge is managing the queue of packets waiting to be transmitted, which affects the queuing delay—an important metric in determining the network performance.

Transmission Time

Transmission time is the duration it takes to send a packet from one point to another in a network. It depends on the packet size and the link's transmission rate.
For example, if you have a packet size of 1,500 bytes (or 12,000 bits) and a transmission rate of 2 Mbps (2 million bits per second), you can calculate the transmission time using the formula:
\[ t_{transmission} = \frac{L}{R} \]
This formula tells us how long it will take to transmit the entire packet. In this case, it's:\[ t_{transmission} = \frac{12,000 \text{ bits}}{2 \times 10^6 \text{ bits per second}} = 0.006 \text{ seconds} \]

• Packet size \((L)\): Measured in bits or bytes.
• Transmission rate \((R)\): Measured in bits per second.

Understanding transmission time helps in network design and performance analysis by showcasing how fast data can travel across links.

Network Modeling

Network modeling is the practice of creating a representation of network operation. It allows network designers to predict how real networks will perform under different scenarios.
In this context, network modeling helps to estimate queuing delays, which occur when packets arrive at a network switch faster than they can be transmitted. The model often includes variables like packet size, transmission rate, and the state of ongoing transmissions.
By understanding these dynamics, network administrators can optimize configurations and resource allocation. It aids in visualizing the impact of different factors on network performance, such as increased traffic or changes in infrastructure.

• Simulation: Use models to simulate network performance and traffic.
• Analysis: Evaluate different strategies for network improvement.
• Optimization: Fine-tune networks to reduce delay and increase throughput.

Network modeling is crucial in making informed decisions about network expansion and management.

Packet Size

Packet size is a crucial factor that affects the performance of a network. It refers to the number of bits or bytes that each packet contains.
Larger packets may decrease the overhead for sending headers, as fewer packets are needed to transmit the same amount of data. However, they might also increase transmission time and queue lengths, potentially leading to increased delays.
On the other hand, smaller packets can lead to inefficiencies because more packets need to be sent, increasing header overhead, but they enable more flexible and responsive data handling, especially in networks with variable traffic.

• Overhead: Smaller packets increase overhead due to more frequent headers.
• Delay: Larger packet sizes might increase network delay due to longer transmission times.
• Efficiency: Finding an optimal packet size is critical for balanced network performance.

The choice of packet size is a trade-off that depends heavily on network conditions and the nature of the traffic being sent.