Question

Suppose four active nodes - nodes A, B, C and D-are competing for access to a channel using slotted ALOHA. Assume each node has an infinite number of packets to send. Each node attempts to transmit in each slot with probability \(p\). The first slot is numbered slot 1 , the second slot is numbered slot 2 , and so on. a. What is the probability that node A succeeds for the first time in slot \(5 ?\) b. What is the probability that some node (either A, B, C or D) succeeds in slot 4? c. What is the probability that the first success occurs in slot 3 ? d. What is the efficiency of this four-node system?

Short answer

a. \((1-p)^9 \times p\), b. \(4p(1-p)^3\), c. \((1-p)^8 \times 4p(1-p)^3\), d. Efficiency: \(4p(1-p)^3\)

Step-by-step solution

Probability of Node A's Success in Slot 5

For node A to succeed for the first time in slot 5, it means node A did not succeed in slots 1 to 4, and then succeeded in slot 5, while other nodes did not succeed in slot 5. The probability that a node does not succeed in a slot is \[(1-p)\]. Thus, the probability that node A does not succeed in slots 1 to 4 is \[(1-p)^4\]. Then, node A succeeds in slot 5 with probability \[p\]. Moreover, nodes B, C, and D should not succeed in slot 5, giving a probability of \[(1-p)^3\] for each to fail. Therefore, the overall probability is given by: \[((1-p)^4) \times p \times ((1-p)^3)^3 = (1-p)^9 \times p\].

Probability of Any Node's Success in Slot 4

For any node to succeed in slot 4, exactly one node - A, B, C, or D - must succeed, while the others don't. The probability that a specific node succeeds and the others don't is \[p \times (1-p)^3\]. As there are four nodes, by summing the probabilities, we get: \[4 \times p \times (1-p)^3\].

Probability of First Success in Slot 3

For the first success to occur in slot 3, no successes should occur in slots 1 and 2, but one success occurs in slot 3. The probability of no success in a slot is \[(1-p)^4\]. The probability that there is no success in the first two slots is: \[(1-p)^4 \times (1-p)^4\], and for a success in slot 3 for any node: \[4 \times p \times (1-p)^3\]. So, the total probability for this scenario is: \[((1-p)^8) \times (4p(1-p)^3)\].

Efficiency of the System

Efficiency in slotted ALOHA is defined as the probability that a slot is used for a successful transmission. The probability of a successful transmission by any one of the four nodes is: \[4 \times p \times (1-p)^3\]. This term already represents the system's efficiency.

Key concepts

Slotted ALOHA

Slotted ALOHA is a method used in computer networking to manage how nodes access a shared communication channel. This protocol operates in predefined time intervals called slots, where each node attempts to send data. If two or more nodes try to transmit during the same slot, a collision occurs, and the data transmission fails.

Slotted ALOHA offers a structured way of allowing nodes to send packets. It divides time into equal-sized slots, and a node can only send data at the beginning of a time slot. This reduces the chance of collisions compared to its predecessor, Pure ALOHA, improving the efficiency of channel access.

Using Slotted ALOHA, each active node decides whether to transmit based on a transmission probability, denoted by the variable \(p\). This probabilistic approach enables nodes to attempt transmission without prior consultation, allowing them to operate in a distributed manner without centralized control.

Channel Access Protocols

Channel access protocols like Slotted ALOHA are crucial in determining how multiple nodes interact over a shared channel to avoid data collisions. These protocols establish rules that manage how and when nodes can send data, thereby ensuring fair and efficient use of the channel.

There are various types of channel access protocols, each with unique methods to handle node transmissions and collison. They include:

• Time Division Multiple Access (TDMA)
• Carrier Sense Multiple Access (CSMA)
• Frequency Division Multiple Access (FDMA)

Slotted ALOHA is one such protocol that provides a balance between structure and flexibility by using time slots, which prevent nodes from transmitting data all the time, thereby reducing potential packet collisions. This protocol is especially useful in environments with many nodes competing for access to a single communication medium.

Network Efficiency

Network efficiency refers to how effectively a network manages resources to achieve successful data transmission. In the context of Slotted ALOHA, efficiency is defined as the fraction of time slots used for successful transmissions without collisions.

For a system using Slotted ALOHA, the probability that exactly one node sends its packet in a slot, and thereby achieves a successful transmission, is key to understanding efficiency. When multiple nodes are involved, like in the case of nodes A, B, C, and D, efficiency is calculated by summing the probability of success across all nodes.

In such a setup, efficiency is mathematically represented by the expression \(4 imes p imes (1-p)^3\), where the term \(4\) reflects the number of nodes, \(p\) is the probability of a node's transmission, and \((1-p)^3\) represents the probability that the other nodes do not transmit. This expression helps quantify how well the network makes use of the available slots to successfully transmit data.

Node Transmission Probability

Node transmission probability, denoted by \(p\), is a central concept in slotted ALOHA systems. It represents the likelihood that a given node will attempt to transmit a packet in a particular time slot. Managing this probability is essential in balancing transmission success and collision rates.

A higher transmission probability increases the chance that a node will send its packet, but also raises the collision risk when multiple nodes transmit simultaneously. Conversely, a lower \(p\) reduces the likelihood of collisions, but may lengthen the wait time before a packet is successfully transmitted.

In practice, choosing an optimal value for \(p\) requires considering the number of nodes and their activity levels. By adjusting \(p\), network designers strive to optimize the trade-off between maximizing slot usage for successful transmission and minimizing collisions.