Question

Suppose that \(N\) Ethernet stations, all trying to send at the same time, require \(N / 2\) slot times to sort out who transmits next. Assuming the average packet size is 5 slot times, express the available bandwidth as a function of \(N\).

Short answer

\(\frac{5}{\frac{N}{2} + 5}\)

Step-by-step solution

Understanding the problem

Determine the time required for sorting out which station transmits next and the time taken for the average packet transmission.

Calculate sorting time

Given that it takes \(N/2\) slot times to sort out who transmits next.

Determine packet transmission time

Given that the average packet size equates to 5 slot times.

Compute total time per transmission

The total time for one complete cycle (sorting + transmission) is \(\frac{N}{2} + 5\) slot times.

Express the available bandwidth

The available bandwidth in terms of the number of stations \(N\) and the number of successful transmissions per slot time is given by \(\frac{5}{\frac{N}{2} + 5}\). This represents the fraction of time utilized for transmitting useful data.

Key concepts

network stations

Network stations are the computers or devices connected to a network. In an Ethernet network, these stations communicate with each other by transmitting data packets. Each station must wait for its turn to send data to avoid collisions, which can happen if two stations try to send data at the same time. The number of network stations, represented as \(N\), impacts the overall network performance and bandwidth.

slot time

Slot time is a critical concept in Ethernet networks. It represents the time window during which the network is checked for collisions. The length of a slot time is determined by the maximum network propagation delay. In this problem, sorting out which station will transmit next takes \( \frac{N}{2} \) slot times, where \(N\) is the number of stations.

packet transmission

Packet transmission refers to the sending of data packets from one network station to another. Each packet has a size, and in this problem, the average packet size is given as 5 slot times. This means it takes 5 slot times to send one packet completely without any interruptions.

available bandwidth

Available bandwidth is the measure of the network's ability to transmit data. It's affected by the time spent sorting out which station will send next and the actual transmission time of packets. It's expressed as the fraction of time used effectively for transmission. For \(N\) stations, the available bandwidth is represented as \( \frac{5}{\frac{N}{2} + 5} \). This formula shows that as the number of stations \(N\) increases, the time needed to sort out transmissions also increases, reducing the available bandwidth.

network efficiency

Network efficiency is about how well a network uses its available resources. In the context of Ethernet, it's about maximizing the amount of data successfully transmitted versus the time spent on managing transmissions and collisions. The available bandwidth formula \( \frac{5}{\frac{N}{2} + 5} \) indicates network efficiency. With fewer stations, sorting time is less, and efficiency is higher. As the number of network stations increases, more time is spent deciding who transmits next, lowering efficiency.