Fluid limit of a distributed ledger model with random delay

TL;DR

The paper analyzes a DAG-based distributed ledger model with batch arrivals and random proof-of-work (POW) durations. It derives a fluid-limit description in the high-traffic limit, yielding a system of delayed partial differential equations governing the evolution of the scaled numbers of tips and their classifications. The main result provides a probabilistic convergence bound showing that the stochastic process remains close to its fluid limit over a finite horizon, with explicit dependence on parameters such as the arrival rate, batch size, and POW duration distribution; the bound implies convergence in probability as $\epsilon^{-1}$, $\lambda$, $N$, and $\epsilon^3N$ grow. The work also characterizes stable states (e.g., for $M=2$, $f=w=l/2$ and explicit forms for $w_i$) and discusses extensions to unbounded POW distributions, with simulations validating the fluid-limit predictions. Overall, the results offer a rigorous, quantitative bridge between random DAG dynamics and deterministic PDE-based fluid descriptions, enabling prediction of tip behavior and verification speed in IOTA-like distributed ledgers.

Abstract

Distributed ledgers, including blockchain and other decentralized databases, are designed to store information online where all trusted network members can update the data with transparency. The dynamics of ledger's development can be mathematically represented by a directed acyclic graph (DAG). In this paper, we study a DAG model which considers batch arrivals and random delay of attachment. We analyze the asymptotic behavior of this model by letting the arrival rate go to infinity and the inter-arrival time go to zero. We establish that the number of leaves in the DAG and various random variables characterizing the vertices in the DAG can be approximated by its fluid limit, represented as the solution to a set of delayed partial differential equations. Furthermore, we establish the stable state of this fluid limit and validate our findings through simulations.

Figures (4)

• Figure 1: Example of a DAG. A solid directed edge implies that the associated POW has been completed and the data has been accepted into the ledger. For example, vertex 3 has selected 1 as its parent and finished its POW. A dashed vertex with outgoing dashed edge implies the POW has started but not yet been finished. For example, vertex 5 has selected 2 and 3 as its parents but its POW has not yet been finished. We generally refer to the solid vertices that have no solid edge pointing toward them as tips, which means the vertices have been accepted to the distributed ledger but has not yet been attached by any other vertices. For example, vertices $2$ and $3$ are tips because there is no solid edge toward them, while vertices $0$ and $1$ are not tips because they have been attached by other solid vertices. Also, vertices $4$ and $5$ are not considered as tips because they have not been accepted to the system by the fact that their POW have not been finished.
• Figure 2: A demonstration of a DAG modeling IOTA at some time $t$. Each time there are $N=2$ arriving vertices, for example, $C_1,C_2$ arrived at the same time while $D_1,D_2$ arrived simultaneously $\epsilon$ time after $C_1,C_2$ arrived. A solid vertex with outgoing solid edge(s) implies that its corresponding POW has been completed and the vertex has been accepted into the system. For example, vertex $C_1$ has selected $A_1$ as its parent and finished its POW, hence it has been accepted into the system. A dashed vertex with outgoing dashed edge(s) implies the POW has not yet been finished. For example, vertex $D_2$ has selected $B_1$ and $A_2$ as its parents but the POW corresponding to $D_2$ has not yet been finished. The dashed vertices are the ones that have not been accepted into the system because their POWs are still in process. A dashed vertex will be included in the system once its corresponding POW is finished, then the dashed vertex and the dashed edge(s) originating from it will become solid.
• Figure 3: Example of the random variables $F_i(t_k)$. In this example, $N=3$ which means there are 3 arrivals at each time. At some time $t_k$ the vertices $D_1,D_2,D_3$ arrive and they have POW durations $h_1,h_2,h_3$ respectively. The graph on the left represents the graph at $t_k$ when $D_1,D_2,D_3$ arrive while the right graph represents the graph at $t_{k+1}$. At time $t_k$, the vertices $A_1,B_1,C_1,C_2$ are free tips since their corresponding POWs have been finished but they have not yet been selected as parents. The free tip $A_1$ is selected as a parent by a $Type\ 1$ arrival $D_1$. The free tip $B_1$ is selected by a $Type\ 1$ arrival $D_1$ and a $Type\ 2$ arrival $D_2$. The free tip $C_2$ is selected by a $Type\ 2$ arrival $D_2$ and a $Type\ 3$ arrival $D_3$. Then $F_1(t_k)=2$ because both $A_1,B_1$ are selected by a $Type\ 1$ arrival. Because $B_1$ is already included in $F_1(t_k)$, we have $F_2(t_k)=1$ although both $B_1,C_2$ are selected by a $Type\ 2$ arrival. Finally, $F_3(t_k)=0$ because $C_2$ is the only free tip that is selected by a $Type\ 3$ arrival but $C_2$ has been counted in $F_2(t_k)$. The selection will be reflected at $t_{k+1}$ and hence $A_1,B_1,C_2$ become pending tips at $t_{k+1}$.
• Figure 4: Simulations with $\lambda=400, N=20, \epsilon=0.05$ where the vertical axis displays values for $L(t)/\lambda,F(t)/\lambda$ and the horizontal axis represents time $t_n$. $M$ is assumed to be two corresponding to Proposition \ref{['k_equili_2']}. The left figure corresponds to the simulation with $p_1=0.8$ and $p_2=0.2$ while the right figure corresponds to the simulation with $p_1=0.3$ and $p_2=0.7$. Both simulations assume $h_1=3$ and $h_2=5$. Multiple simulations are performed with the same parameters and we can see that the scaled random processes behave almost deterministically with minor deviation.

Theorems & Definitions (32)

• Definition 2.1
• Proposition 2.2
• Theorem 3.1
• Proposition 3.2
• Lemma 5.1
• Lemma 5.2
• Lemma 5.3
• Lemma 5.4
• Lemma 5.5
• Lemma 5.6