[Paper Review] An Improved Analytical Expression for Write Amplification in NAND Flash

Xiang Luojie, Brian M. Kurkoski|arXiv (Cornell University)|Oct 19, 2011

Advanced Data Storage Technologies1 references19 citations

TL;DR

This paper proposes an improved analytical model for write amplification in NAND flash memory by modeling the number of invalid pages freed during garbage collection using a binomial distribution and the Lambert W function. The resulting expression, dependent only on the overprovisioning factor, provides significantly more accurate predictions than prior work, with a 2.36 predicted write amplification at ρ=0.30 versus the actual value of 2.35, outperforming the previous model's 2.17 estimate.

ABSTRACT

Agarwal et al. gave an closed-form expression for write amplification in NAND flash memory by finding the probability of a page being valid over the whole flash memory. This paper gives an improved analytic expression for write amplification in NAND flash memory by finding the probability of a page being invalid over the block selected for garbage collection. The improved expression uses Lambert W function. Through asymptotic analysis, write amplification is shown to depend on overprovisioning factor only, consistent with the previous work. Comparison with numerical simulations shows that the improved expression achieves a more accurate prediction of write amplification. For example, when the overprovisioning factor is 0.3, the expression proposed by this paper gives a write amplification of 2.36 whereas that of the previous work gives 2.17, when the actual value is 2.35.

Motivation & Objective

• To develop a more accurate analytical expression for write amplification in NAND flash memory.
• To model the number of invalid pages freed during garbage collection as a stationary binomial process.
• To improve upon Agarwal et al.'s closed-form expression by focusing on invalid page probability rather than valid page distribution.
• To validate the model through simulations and asymptotic analysis, showing write amplification depends only on the overprovisioning factor.

Proposed method

• The authors model the number of invalid pages freed per garbage collection as a binomially distributed random variable with probability $ p_{\mathsf{ie}} $.
• They derive a self-consistent equation for the expected number of freed invalid pages $ x $, leading to a transcendental equation involving the Lambert W function.
• The asymptotic write amplification is derived as $ \widehat{A} = \frac{-1-\rho}{-1-\rho - \mathsf{W}((-1-\rho)e^{-1-\rho})} $, a function of the overprovisioning factor $ \rho $ only.
• The model assumes stationarity in the number of invalid pages freed and justifies this via simulation and asymptotic analysis.
• Theoretical results are validated through extensive simulations with varying $ U $, $ N_{\mathsf{p}} $, and $ \rho $, showing convergence to the analytical value.
• The model is compared against Agarwal et al.'s prior expression, which assumes uniform distribution of valid pages and uses a different derivation.

Experimental results

Research questions

• RQ1 How can write amplification in NAND flash memory be more accurately modeled by focusing on invalid page recovery during garbage collection?
• RQ2 What is the asymptotic behavior of write amplification, and does it depend only on the overprovisioning factor?
• RQ3 Can a closed-form expression for write amplification be derived using the Lambert W function that better matches simulation results than prior models?
• RQ4 How do assumptions about page distribution and garbage collection dynamics affect the accuracy of write amplification predictions?

Key findings

• The improved analytical expression for write amplification depends solely on the overprovisioning factor $ \rho $, consistent with prior work but derived through a more accurate framework.
• At $ \rho = 0.30 $, the proposed model predicts a write amplification of 2.36, which closely matches the simulation result of 2.35, while the prior model predicts 2.17.
• The model's prediction error is significantly reduced compared to the previous work, with the largest deviation in the tested range being less than 0.02 for $ \rho \geq 0.25 $.
• Simulations confirm that the number of invalid pages freed per garbage collection converges to a stationary value independent of $ U $ and $ N_{\mathsf{p}} $, validating the model's assumptions.
• The Lambert W function enables a closed-form solution that captures the nonlinear behavior of write amplification more accurately than the previous quadratic approximation.
• The model justifies the use of $ T $, the number of blocks becoming full per cycle, as a good approximation of the actual number, supporting the analytical framework.