On Convergence Probability of Direct Acyclic Graph-Based Ledgers in Forking Blockchain Systems

Direct acyclic graph (DAG)-based ledger is a promising technology for the Internet of things (IoT). Compared with a single-chain topology, DAG and forking blockchain topology can solve some problems in IoT, such as high resource consumption, high transaction fee, low transaction throughput, and long confirmation delay. We propose the convergence probability to aid further analysis of the performance and security of DAG-based ledgers. Under unsteady load regimes, the convergence probability is the probability of each possible cumulative weight of the observed transaction when it is approved by all new arrival transactions. In this article, we derive a closed-form expression and an approximate expression of the convergence probability under the high-to-low regime (H2LR). Also, we verify the accuracy of the derived expressions through Markov chain Monte Carlo (MCMC) simulations. Numerical results shows that the simulation results match well with its analytical results, which indicates the accuracy of the exact expression and the approximate expression of the convergence probability.