Question

Suppose that \(x\) bits of user data are to be transmitted over a \(k\)-hop path in a packet-switched network as a series of packets, each containing \(p\) data bits and \(h\) header bits, with \(x>p+h\). The bit rate of the lines is \(b\) bps and the propagation delay is negligible. What value of \(p\) minimizes the total delay?

Short answer

Maximize \(p\) such that \(p \approx x - h\) for minimum delay.

Step-by-step solution

Understand the Packetization Process

In a packet-switched network, user data is divided into packets. Each packet consists of payload data bits and header bits. For this problem, each packet contains \(p\) data bits and \(h\) header bits. Our goal is to minimize the total delay in sending \(x\) data bits over a \(k\)-hop network.

Express Total Number of Packets

The total number of packets needed is the number of times \(p\) data bits fit into \(x\) user bits. This can be calculated as \( \left\lceil \frac{x}{p} \right\rceil \). However, for simplicity and since we focus on minimizing delay, we assume \( \frac{x}{p} \) is an integer.

Calculate Transmission Delay per Packet

Each packet requires \(p + h\) bits to be sent. With a bit rate of \(b\) bps, the transmission delay per packet is \( \frac{p + h}{b} \) seconds.

Total Transmission Delay to Send All Packets

The total transmission delay is the transmission delay per packet times the total number of packets, given by \( \frac{x}{p} \times \frac{p + h}{b} \). This simplifies to \( \frac{x(p + h)}{pb} \) seconds.

Express Total Delay for Each Path Hop

With the packet making \(k\) hops, the total delay encountered will be multiplied by \(k\), yielding \( D(p) = k \times \frac{x(p + h)}{pb} \). Our task is to minimize this expression.

Simplify the Delay Expression and Optimize

Consider the function \( D(p) = \frac{khx}{bp} + \frac{kx}{b} \). This function decreases as \(p\) increases since the first term \(\frac{khx}{bp}\) decreases with increasing \(p\). Thus, to minimize \(D(p)\), maximize \(p\) such that \(p < x - h\). Since \(x > p + h\), \(p\) can be approximated to \(x - h\) to keep everything as one complete packet without leading the header to exceed total space.

Conclusion for p

Thus, the value of \(p\) that minimizes the total delay is just under \(x - h\), meaning \(p \approx x - h\). This ensures maximum data bits per packet while minimizing unnecessary header overhead.

Key concepts

Transmission Delay

Transmission delay refers to the time taken to push all the packet's bits into the transmission medium over a network. In a packet-switched network, the user data is divided into packets. Each packet consists of data bits and header bits.
Transmission delay per packet is determined by how many bits are contained in a packet and how fast these bits can be sent over the medium. This is further expressed mathematically as:

• Packet size: total bits per packet is given by the sum of payload, represented by data bits (data), and the header bits (hheader), leading to a total of \( p + h \) bits per packet.
• Transmission rate: the speed of transmission given by the bit rate (\( b \) bps).

The formula for transmission delay per packet is then \( \frac{p + h}{b} \) seconds, which tells us how long it takes to send one packet over a single path. Understanding this is essential in optimizing network efficiency.

Packetization Process

The packetization process involves breaking down the entire user data (x bits) into smaller units called packets. This process is crucial in a packet-switched network where the integrity and efficient delivery of data are paramount.
Each packet in this network has a payload and a header. While the payload carries the actual data, the header contains control information used for routing the packet across the network.
To calculate the number of packets needed for transmission, we typically use the formula:

• \( \left\lceil \frac{x}{p} \right\rceil \) - representing the ceiling function, which gives the smallest integer greater than or equal to the division result.

In our focus on delay optimization, assuming \( \frac{x}{p} \) is an integer simplifies calculations. Having more data in a single packet (i.e., a higher p value) reduces the overhead and, therefore, total delay, making the packetization process pivotal in optimization efforts.

Total Delay Optimization

Total delay optimization is centered around minimizing the total time taken for the data to traverse across multiple hops in a packet-switched network. Considering a network path with multiple hops (k), the aim is to optimize the packet size (p) to reduce delays.
The total transmission delay is computed as:

• \( D(p) = k \times \frac{x(p + h)}{pb} \) - which combines the delay per packet and the number of hops the packet must traverse.

By analyzing the function \( D(p) = \frac{khx}{bp} + \frac{kx}{b} \), it becomes apparent that increasing p reduces the first term, thereby lowering the total delay. The most effective way is aiming for p close to x minus the header size (h). This maximizes data per packet while minimizing header overhead, achieving optimal delay.