Question

Recall that AAL \(3 / 4\) has a CRC-10 checksum at the end of each cell, while AAL5 has a single CRC-32 checksum at the end of the PDU. If a PDU is carried in 12 AAL \(3 / 4\) cells, then AAL \(3 / 4\) devotes nearly four times as many bits to error detection as AAL-5. (a) Suppose errors are known to come in bursts, where each burst is small enough to be confined to a single cell. Find the probability that AAL3/4 fails to detect an error, given that it is known that exactly two cells are affected. Do the same for three cells. Under these conditions, is AAL \(3 / 4\) more or less reliable than AAL.5? Assume that an N-bit CRC fails to detect an error with probability \(1 / 2^{N}\) (which is strictly true only when all errors are equally likely). (b) Can you think of any error distribution in which AAL \(3 / 4\) would be more likely than AAL5 to detect an error? Do you think such circumstances are likely?

Short answer

AAL 3/4 is less reliable than AAL 5 in detecting errors for two and three affected cells. Certain error distributions may favor AAL 3/4, but these are unlikely.

Step-by-step solution

Understand the Problem

First, comprehend the given problem. We need to determine the error detection probability of AAL 3/4 and AAL 5 under specific conditions. AAL 3/4 uses a CRC-10 checksum for each cell, while AAL 5 uses a CRC-32 checksum at the end of the entire PDU. Each AAL 3/4 cell contains 10 bits for error checking, and the whole PDU (12 cells) of AAL 5 contains 32 bits for error checking.

Define Probabilities for AAL 3/4 and AAL 5

Given that the probability of a CRC detecting an error is defined by the formula: \[ P_{detection} = 1 - \frac{1}{2^N} \]The probability of failure is given by: \[ P_{fail} = \frac{1}{2^N} \]For a CRC-10 checksum: \[ P^{fail}_{10} = \frac{1}{2^{10}} \]For a CRC-32 checksum: \[ P^{fail}_{32} = \frac{1}{2^{32}} \]

Probability of AAL 3/4 Failing for Two Affected Cells

Since AAL 3/4 has two out of the 12 cells affected, the probability of AAL 3/4 failing to detect the errors in these cells is: \[ P_{fail}^{2 cells} = \frac{1}{2^{10}} \times \frac{1}{2^{10}} = \frac{1}{2^{20}} \]

Probability of AAL 3/4 Failing for Three Affected Cells

Similarly, for three cells affected, the probability of AAL 3/4 failing is: \[ P_{fail}^{3 cells} = \frac{1}{2^{10}} \times \frac{1}{2^{10}} \times \frac{1}{2^{10}} = \frac{1}{2^{30}} \]

Probability of AAL 5 Failing for Multiple Cells

AAL 5 has a single CRC-32 checksum for the whole PDU, and the error could be in any of the 12 cells. Thus, the probability of failure is: \[ P_{fail}^{AAL5} = \frac{1}{2^{32}} \]

Compare Reliability

Compare the probabilities calculated previously: For AAL 3/4 with two cells affected: \[ P_{fail}^{AAL 3/4, 2 cells} = \frac{1}{2^{20}} \] For AAL 3/4 with three cells affected: \[ P_{fail}^{AAL 3/4, 3 cells} = \frac{1}{2^{30}} \] For AAL 5: \[ P_{fail}^{AAL 5} = \frac{1}{2^{32}} \] Since for all the cases, the probabilities for AAL 3/4 are larger than AAL 5, it shows that AAL 3/4 is less reliable than AAL 5.

Consider Error Distribution Favoring AAL 3/4

Given CRC's mechanics, if errors are distributed across many cells, AAL 3/4 may detect more errors due to each cell having its own CRC. However, because bursts are usually confined within single cells, this exact error condition may not out-balance the larger probability of failure in AAL 3/4. Thus, such error distributions are unlikely to make AAL 3/4 more reliable than AAL 5.

Key concepts

CRC checksum

Cyclic Redundancy Check (CRC) is a widely used error-detecting code. It creates a checksum (a sequence of numbers and letters) based on the data. This checksum is added to the data before transmission. Upon reaching the destination, the checksum is recalculated to check for possible errors. If errors are detected, the data can be assumed corrupted, and corrective measures will be taken. CRC works by dividing the data into sections and performing arithmetic operations to create a checksum. Different types of CRC exist, such as CRC-10 and CRC-32, which indicate the length of the checksum in bits. For example, CRC-10 uses a 10-bit checksum, and CRC-32 uses a 32-bit checksum. In our problem, AAL 3/4 incorporates a CRC-10 checksum for each cell, while AAL 5 uses a single CRC-32 checksum for the entire PDU (Protocol Data Unit). The probability of failing to detect an error with an N-bit CRC is given by the formula: \[ P_{fail} = \frac{1}{2^N} \] For CRC-10, this failure probability becomes \[ P_{fail}^{CRC-10} = \frac{1}{2^{10}} \], and for CRC-32, it becomes \[ P_{fail}^{CRC-32} = \frac{1}{2^{32}} \]. These failure probabilities are critical when comparing different error detection mechanisms.

burst errors

Burst errors occur when multiple errors happen in a sequence within a small area of the data transmission. Such errors are usually due to physical disturbances or network issues. Burst errors are challenging to detect because they can corrupt more than one bit or cell in a data stream. In the context of our problem, if errors come in short bursts confined to a single cell, we need to consider each affected cell separately. For AAL 3/4, the probability of failing to detect an error affects each cell. If the burst affects two cells, the failure probability is compounded as: \[ P_{fail}^{AAL 3/4, 2 cells} = \frac{1}{2^{10}} \times \frac{1}{2^{10}} = \frac{1}{2^{20}} \] For three cells affected, it becomes: \[ P_{fail}^{AAL 3/4, 3 cells} = \frac{1}{2^{10}} \times \frac{1}{2^{10}} \times \frac{1}{2^{10}} = \frac{1}{2^{30}} \] Burst errors significantly impact calculations of failure probabilities, emphasizing why robust error-checking mechanisms are necessary for reliable data transmission.

AAL (ATM Adaptation Layer)

The ATM Adaptation Layer (AAL) is part of the Asynchronous Transfer Mode (ATM) networking. AAL adapts user traffic into ATM cells and provides essential services like error detection and correction. Different types of AAL serve particular applications, such as AAL 3/4 and AAL 5. AAL 3/4 provides error checking at the cell level using CRC-10, meaning each 48-byte cell has its own 10-bit checksum. This method allows each cell to be individually checked for errors, enhancing granularity in error detection. AAL 5, on the other hand, uses a CRC-32 checksum at the end of the entire PDU. A PDU comprises multiple ATM cells, making the CRC-32 checksum span the entire data block, resulting in a single comprehensive error check. Comparing the two, AAL 3/4 uses more bits in total for error checking (10 bits per cell across 12 cells = 120 bits) compared to AAL 5, which uses only 32 bits for the full PDU. The probability of detection failure is lower in AAL 5 due to the longer checksum, as seen in: \[ P_{fail}^{AAL 3/4, 2 cells} = \frac{1}{2^{20}} \] \[ P_{fail}^{AAL 3/4, 3 cells} = \frac{1}{2^{30}} \] \[ P_{fail}^{AAL 5} = \frac{1}{2^{32}} \] This comparison shows that AAL 5 generally provides more reliable error detection for burst errors affecting multiple cells.