2.1. Notation

Section 2.2: $\epsilon$ denotes the time between arrivals, N denotes the number of vertices arriving at each time $t_n=n\epsilon$ , and $\lambda=N/\epsilon$ denotes the arrival rate of vertices. The values $h_1 < \cdots < h_M$ represent the possible durations of POW, and $N_i(t_n)$ denotes the number of Type i arrivals at time $t_n$ . The variable $L(t_n)$ denotes the number of tips at $t_n$ , while $F(t_n)$ and $W(t_n)$ , respectively, denote the number of free tips and the number of pending tips. The variable $F_i(t_n)$ counts the number of free tips selected as parents by Type i arrivals but not by Type j arrivals for any $j < i$ . The number of Type i pending tips at time $t_n$ with residual lifetime (RLT) u is recorded by $W_i(t_n,u)$ . The value $J_{i,j}(t_n,u)$ counts the number of Type i pending tips at time $t_n$ with RLT u that change to Type j pending tips. Sometimes we will use $F_\lambda(t)$ and $L_\lambda(t)$ to respectively denote $F(t)/\lambda$ and $L(t)/\lambda$ for simplicity of presentation.

Section 2.5: The values $l,f,w,w_i$ represent the fluid limit of variables $L,F,W,W_i$ .

Section 3: g(t) records the difference between the discrete process and its fluid limits, with T representing the time horizon. The values $\gamma,\xi,m$ are parameters describing the fluid limit, which are given by (3.3)–(3.5).

2.2. Preliminaries

Let $\epsilon$ and N be two constants, and let $t_n=n\epsilon$ for $n\in\mathbb{N}$ . We assume that N vertices arrive simultaneously at each time $t_n$ , with $\epsilon$ representing the inter-arrival time. We call a vertex that arrives at time $t_n$ an arrival at $t_n$ . We define $\lambda\,:\!=\,N/\epsilon$ , which represents the arrival rate of the vertices.

We assume that each new arriving vertex independently chooses its duration of POW from the set $\{h_1,h_2,\ldots,h_M\}\subseteq\mathbb{R}$ with probability $p_1,p_2,\ldots,p_M$ , respectively. Without loss of generality, we assume that $h_1 < h_2 < \cdots < h_M$ , which are not necessarily equidistant, and that any $h_i$ is an integer multiple of $\epsilon$ . Note that while arriving processes are commonly analyzed with $\epsilon = 1$ , where all time variables are normalized by the inter-arrival time, we take $\epsilon \rightarrow 0$ so that we can use fixed $\{h_i\}_{i=1,\ldots,M}$ to represent real-world POW durations (e.g. minutes). The alternative approach of fixing $\epsilon = 1$ and letting $\{h_i\}_{i=1,\ldots,M} \rightarrow \infty$ would be less natural for interpreting results in terms of actual timescales. This current setting captures scenarios where many blocks arrive rapidly but each requires substantial but bounded computation time. Throughout the paper, we call an arriving vertex with POW duration $h_i$ a Type i arrival and the corresponding POW a Type i POW. We define $N_i(t_n)$ to be the number of Type i arrivals that arrive at time $t_n$ ; their distribution will be provided in (2.1). Note that $\sum_{i=1}^{M}N_i(t_n)=N$ because at each time we have N arriving vertices. Figure 2 provides a graphical demonstration of the dynamics of the DAG used in this paper. It is worth noting that the assumption of a deterministic arrival process can be generalized to a binomial arrival process, a classical model closely related to Poisson arrivals. This can be done by assuming that each arrival is independently discarded with probability $p_0$ . While this generalization requires minor modifications to the proof, the main arguments remain intact, as will become clearer in Section 2.5.

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

We recall the terminology for tips and pending tips used in [Reference King14]. As in Figure 2, the solid vertices without any solid edge toward them are called tips, which represent the vertices that are accepted into the system and have not yet been attached by other vertices. For example, the tips in Figure 2 are $A_2,B_1,B_2,C_1,C_2$ . At time $t_n$ we let $L(t_n)$ denote the number of tips in the system. Among these, we distinguish between the pending tips and the free tips. A tip is a pending tip at time $t_n$ if it has been selected as a parent by some vertices that arrived at some time $t_j$ with $j < n$ . A pending tip appears as a solid vertex with a dashed edge toward it; for example, the pending tips in Figure 2 are vertices $B_1,A_2$ . A tip is a free tip at time $t_n$ if it is not a pending tip. A free tip appears as a solid vertex without any dashed edge toward it, for example, the free tips in Figure 2 are $B_2,C_1,C_2$ . We define $W(t_n)$ to be the number of pending tips at time $t_n$ and $F(t_n)$ to be the number of free tips at time $t_n$ . It follows that $L(t_n) = W(t_n) + F(t_n)$ . In order to analyze the evolution of random variables $L(t_n)$ and $F(t_n)$ , which will be respectively described in (2.8) and (2.7), we first introduce some other random variables.

Essentially, all DL protocols require each new vertex to have at least one parent. Vertices in blockchain typically have one parent, resulting in a linear graph. In addition to IOTA, there are other protocols where each vertex has multiple parents, including the work in [Reference Sompolinsky, Lewenberg and Zohar24, Reference Sompolinsky, Wyborski and Zohar25]. While this paper assumes that each vertex has two parents, the same procedure can be applied to protocols that require k parents.

To model the parent selection algorithm used in IOTA, we assume that each vertex arriving at time $t_n$ chooses two parents with replacement from the set of tips, whose population is recorded by $L(t_n)$ , and the tips are selected as parents with equal probability. The exact probability distribution will be given in (2.2). Note that it could be the case that the vertex makes the same choice for both selections, resulting in a situation where the vertex has only one parent. In fact, whether we consider replacement does not affect our results, as the difference only appears in lower-order terms. For example, (2.12) considers only the highest-order term, which suffices for our analysis. The selection of parents affects the dynamics of the system, and we assume that the effect resulting from the parent selection will be reflected at the next step. For example, after a free tip is selected as a parent at $t_n$ , it will become a pending tip at $t_{n+1}$ .

In order to analyze the dynamics of the DAG, we now introduce the variables used to record the number of free tips selected at each step. We first provide the definitions of these variables and then give an example for demonstration. We suppose that the Type 1 arrivals first make their selections, and we define $F_1(t_n)$ to be the number of free tips selected by Type 1 arrivals at time $t_n$ . Subsequently, the Type 2 arrivals make their selections, and we define $F_2(t_n)$ to be the number of free tips selected by Type 2 arrivals but not by any Type 1 arrivals at time $t_n$ . We then use the same idea to define $F_i(t_n)$ as the number of free tips that are selected by Type i arrivals but not by any Type j arrivals for all $j < i$ . An example is given in Figure 3. To model the situation where a free tip becomes a pending tip after it is selected as a parent, we require that all the tips counted in $\sum_{i=1}^M F_i(t_n)$ become pending tips at the next step; that is, they are counted as part of $W(t_{n+1})$ . Based on the description of $F_i(t_n)$ , we have the distribution of $F_i(t_n)$ as provided in (2.3)–(2.5).

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 $t_{k+1}$

Recall that a POW with duration $h_i$ is called a Type i POW, and a pending tip is a tip that has been selected as a parent. If vertex A is selected as a parent of a vertex B, we say that the POW corresponding to B is directed to vertex A. A pending tip will cease to be a tip when a POW directed to it is finished. We define the residual lifetime (RLT) of a pending tip at time $t_n$ as the remaining time (measured in units of $\epsilon$ ) required for finishing one of the ongoing POWs directed to the pending tip at time $t_n$ . For example, a pending tip A at time $t_7$ might have been selected by three vertices: an arrival at time $t_4$ with corresponding POW duration $h_2=7\epsilon$ whose POW will be finished at time $t_4+h_2=t_{11}$ ; an arrival at time $t_5$ with corresponding POW duration $h_3=8\epsilon$ whose POW will be finished at time $t_5+h_3=t_{13}$ ; and an arrival at time $t_6$ with corresponding POW duration $h_5=10\epsilon$ whose POW will be finished at $t_6+h_5=t_{16}$ . Since the three listed POWs will be finished at $t_{11}$ , $t_{13}$ , and $t_{16}$ , respectively, the RLT of this pending tip A at time $t_7$ is $(11-7)\epsilon=4\epsilon$ , which is determined by the POW with duration $h_2=7\epsilon$ . Suppose no arrivals at time $t_7$ select A as a parent. Then at time $t_8$ , the RLT of the pending tip A is $4\epsilon-\epsilon=3\epsilon$ because $\epsilon$ time has passed. However, notice that the RLT at $t_n$ only uses information up to time $t_n$ , and it can jump to a much lower value if the pending tip is selected as a parent by a new arrival. Using the same example, suppose that at time $t_8$ an arrival with POW duration $h_1=2\epsilon$ selects A as its parent. Then at time $t_9$ there are four POWs that are directed to A and the POW that starts at $t_8$ will finish at the earliest at time $t_8+2\epsilon=t_{10}$ , and hence the RLT of A at $t_9$ is $\epsilon$ . When the RLT of a pending tip becomes zero at $t_n$ , the pending tip will cease to be a tip at the next step; that is, the vertex will no longer be included in the set of tips at time $t_{n+1}$ .

Notice that there could be multiple POWs that determine the RLT of a pending tip at a given time. In this set of vertices whose corresponding POWs determine the RLT of a given pending tip at a given time, the vertices might be of different Types, i.e. they have different POW durations. For example, a pending tip A at time $t_9$ was selected by two vertices: a Type 2 arrival at time $t_6$ with POW duration $h_2=4\epsilon$ which will be finished at $t_{10}$ ; and a Type 1 arrival at $t_8$ with POW duration $h_1=2\epsilon$ which will be finished at $t_{10}$ . Then at time $t_9$ , the RLT of A is $\epsilon$ , which is determined by both the Type 1 POW and the Type 2 POW directed to A. If the smallest Type of the POWs that determine the RLT of the pending tip A at time $t_n$ is i, then we call the pending tip A a Type i pending tip. In the example, the smallest Type of POW that determines the RLT of A at $t_9$ is Type 1, and hence A is a Type 1 pending tip at $t_9$ . Note that a pending tip can still be selected as a parent by a new arrival before one of the POWs directed to it has been completed, so a pending tip may change from Type i to Type j for $i>j$ . However, notice that a pending tip cannot change from Type i to Type j if $i<j$ because being selected by a vertex with higher Type will not affect the related set of POWs that determine the RLT of the pending tip.

By the above definitions, we denote $\mathcal{W}_i(t_n, u)$ to be the set of Type i pending tips with RLT equal to u at time $t_n$ . And we denote $W_i(t_n, u)\,:\!=\,|\mathcal{W}_i(t_n, u)|$ as the number of Type i pending tips at time $t_n$ with RLT equal to u. For a Type i pending tip counted in $W_i(t_n,u)$ , it might jump to a lower Type j where $h_j\leq u$ because it gets selected by a Type j arrival at time $t_n$ . If this happens, the pending tip will be counted in $W_j(t_{n+1},h_j-\epsilon)$ as a Type j pending tip in the next moment. We hence define $J_{i,j}(t_n,u)$ to be the number of pending tips in the set ${\mathcal{W}}_i(t_n, u)$ that jump to Type j at the next step. The distribution of $J_{i,j}(t_n,u)$ will be defined in (2.6).

2.3. Distributions of random variables

Based on the intuition of the random variables described in Section 2.2, we first introduce the distribution of the variables, including $N_i(t_n)$ , $F_i(t_n)$ , and $J_{i,j}(t_n,u)$ . Since there are N arrivals at each time and each of them has probability $p_k$ of having POW duration $h_k$ for $k=1,\ldots,M$ , the term $N_i(t_n)$ satisfies a binomial distribution as it counts the number of arrivals at time $t_n$ that have POW duration $h_i$ . This means that

(2.1) \begin{align}N_i(t_n)\sim \text{Binomial}(N,p_i) \quad \text{and} \quad \sum_{i=1}^M N_i(t_n)=N.\end{align}

The random variables $F_i(t_n)$ and $J_{i,j}(t_n,u)$ are generated by selections of parents. To model the parent selection algorithm used in IOTA, we assume that each vertex arriving at time $t_n$ chooses two parents with replacement from the set of tips whose population is recorded by $L(t_n)$ and the tips are selected as parents with equal probability. Hence, for an arrival v at time $t_n$ and a tip v’ at time $t_n$ ,

(2.2) \begin{align}\mathbb{P}\big(v' \text{ is not selected as a parent of }v|L(t_n)\big)\,:\!=\, \left(1-\frac{1}{L(t_n)} \right)^2. \end{align}

The term $F_i(t_n)$ calculates how many free tips are selected by the arrivals with duration $h_i$ but are not selected by any arrival with POW duration less than $h_i$ at time $t_n$ . We suppose that the N arrivals first choose their duration for POW and then select their parents. By (2.2) and that the number of arrivals with POW duration $h_1$ is recorded by $N_1(t_n)$ , each free tip has probability $(1-1/L(t_n))^{2N_1(t_n)}$ of not getting selected by any Type 1 arrivals at $t_n$ . We use binary variables $X_{n,1,j}$ with $j=1,2,\ldots,F(t_n)$ to denote whether each free tip is selected as a parent by any Type 1 arrival at $t_n$ , where $X_{n,1,j}=1$ if the corresponding free tip is selected. The conditional probability that a free tip is not selected as a parent by any Type 1 arrival at $t_n$ is

\begin{align*}\mathbb{P}\big(X_{n,1,j}=0|L(t_n),N_1(t_n),F(t_n) \big)\,:\!=\,\left(1 - \frac{1}{L(t_n)} \right)^{2 N_1(t_n)} \quad\text{for } j=1,2,\ldots,F(t_n).\end{align*}

Therefore, we define

(2.3) \begin{align}F_1(t_n) \,:\!=\,\sum_{j=1}^{F(t_n)} X_{n,1,j} \end{align}

where the random variables $X_{n,1,1},X_{n,1,2},\ldots,X_{n,1,F(t_n)}$ are dependent.

Using the same idea, as well as the assumption that the Type i arrivals select their parents after the Type $i-1$ arrivals, we have that by the time the Type i arrivals select their parents, the number of tips that are possible to be counted toward $F_i(t_n)$ is $F(t_n)-\sum _{k=1}^{i-1}F_k(t_n)$ . Given n and i, we use $X_{n,i,j}$ with $j=1,\ldots,F(t_n)-\sum_{k=1}^{i-1}F_k(t_n)$ to denote whether each free tip that is possible to be counted toward $F_i(t_n)$ is selected by a Type i arrival, with $X_{n,i,j}=1$ if the free tip is selected and $X_{n,i,j}=0$ otherwise:

(2.4) \begin{align}\mathbb{P}\big(X_{n,i,j}=0|L(t_n),F(t_n),N_i(t_n), F_k(t_n)\colon k< i\big)\,:\!=\,\left(1 - \frac{1}{L(t_n)}\right)^{2 N_i(t_n)}. \end{align}

As a generalization of (2.3), we define

(2.5) \begin{align}F_i(t_n) \,:\!=\, \sum_{j=1}^{F(t_n)-\sum_{k=1}^{i-1}F_k(t_n)} X_{n,i,j} \quad\text{for } i=2,3,\ldots,M,\end{align}

where M is the total number of possible durations of POW.

Recall that $W_i(t_n,u)$ denotes the number of Type i pending tips whose RLT is u. For $i>j$ and $u\geq h_j$ , $J_{i,j}(t_n, u)$ counts how many pending tips with RLT u jump from Type i to Type j because they are selected by a Type j arrival. This number is also generated by random selection carried out by the N arrivals at time $t_n$ , and we use $X'_{n,i,u,j,k}$ to denote whether each Type i pending tip with RLT u is selected by a Type j arrival at $t_n$ :

\begin{align*}\mathbb{P}\big(X'_{n,i,u,j,k}=0|W_i(t_n, u),N_j(t_n)\big)\,:\!=\,\left(1 - \frac{1}{L(t_n)}\right)^{2 N_{j}(t_n)},\end{align*}

where $k=1,2,\ldots,W_i(t_n, u)$ . We then define

(2.6) \begin{align}J_{i,j}(t_n, u) \,:\!=\, \sum_{k=1}^{W_i(t_n, u)} X'_{n,i,u,j,k}.\end{align}

2.4. Update rules of the model

In this paper, our primary focus is to analyze the dynamics of the random variables $\{F(t_n),L(t_n), W_{i}(t_n,u)\colon u\in[0,\ h_i],\epsilon\text{ divides } u\}$ . While the initial condition will be introduced in Section 2.6, we now describe how the variables including $L(t_n), F(t_n)$ , and $W_i(t_n,u)$ are updated at each time step. First, consider the number of free tips $F(t_n)$ . This increases because some vertices complete their POWs, and it decreases because some free tips are selected as parents. Therefore we define

(2.7) \begin{align}F(t_{n+1}) - F(t_n) = \sum_{i=1}^{M} N_i(t_n-h_i)-\sum_{i=1}^{M}F_i(t_n).\end{align}

Next, consider the number of tips $L(t_n)$ , which increases because some vertices complete their POWs and decreases because some tips are attached by other vertices and cease to be tips. To model this mechanism, we define

(2.8) \begin{align}L(t_{n+1}) - L(t_n) = &\sum_{i=1}^M N_i(t_n-h_i) -\sum_{i=1}^M F_i(t_n-h_i)\nonumber\\&-\sum_{j < i}\sum_{u=h_j+\epsilon}^{h_i-\epsilon}J_{i,j}(t_n-h_j,u)+\sum_{j < i}\sum_{u=h_j+\epsilon}^{h_i-\epsilon}J_{i,j}(t_n-u,u)\end{align}

by considering the following scenarios: (1) $\sum_{i=1}^{M} N_i(t_n-h_i)$ counts how many vertices should finish their POWs at $t_n$ and become free tips at time $t_{n+1}$ ; (2) if a free tip was selected as a parent by a vertex whose POW will be finished at time $t_n$ , then it might be the case that this free tip ceases to be a tip due to the completion of the POW directed to it. We use $\sum_{i=1}^M F_i(t_n-h_i)$ to consider this situation; (3) some of the Type $i>1$ pending tips that are anticipated to finish POW after $t_{n}$ may have been selected by Type $j < i$ arrivals at time $t_n-h_j$ , thus changing to Type j pending tips and hence are validated at time $t_n$ . Such consideration corresponds to the first double summation in the second line of (2.8); (4) in scenarios (2) and (3), some of the Type i pending tips created at time $t_n - h_i$ may have been changed to Type $j < i$ at some time before $t_n-h_i$ , and thus have completed their POW prior to time $t_n$ . These tips should be removed from the count of the reduction of $L(t_n)$ and are taken into account through the last term in (2.8). Scenario (4) demonstrates how random POW durations complicate the model.

Next, we consider $W_i(t_{n}, u)$ , the number of Type i pending tips with RLT u at time $t_n$ . The value $W_i\big(t_{n+1}, h_i-\epsilon\big)$ consists of: (1) the free tips selected by Type i arrivals at $t_{n}$ and hence become Type i pending tips at time $t_{n+1}$ ; (2) the pending tips that just change to Type i at time $t_{n+1}$ . Hence we define

(2.9) \begin{align}W_i(t_{n+1}, h_i-\epsilon) \,:\!=\,F_i(t_n)+\sum_{j>i}\sum_{u=h_i}^{h_j-\epsilon} J_{j,i}(t_n,u).\end{align}

The Type i pending tips with RLT u will have a lower RLT at the next step but some of them will change to Type j for some $j < i$ . Hence we define

(2.10) \begin{align}W_i(t_{n+1}, u - \epsilon) \,:\!=\, W_i(t_{n}, u) - \sum_{h_j\leq u}J_{i,j}(t_n, u)\quad\text{for }{u\leq h_i-\epsilon}.\end{align}

If a pending tip has RLT equal to zero, it will no longer be a tip at the next step and hence ${W}_i(t_n, u)=0$ for $u<0$ . Furthermore, if the RLT of a Type i pending tip is less than $h_k$ then it cannot change to a Type k pending tip at any later time. In particular, if the RLT of a Type i pending tip is less than $h_1$ then it will always be Type i until it is no longer a tip. Therefore, we define $W_i(t_n + s, u-s) = W_i(t_n, u)$ for all $u<h_1$ , $s<u$ .

The assumption of deterministic arrival and its generalization to binomial arrival lead to tractable update rules as described by (2.7)–(2.10), where each right-hand side of the update rules has finite terms. In contrast, Poisson arrivals introduce an unbounded number of possible arrivals in the update rules and continuous-time dynamics, both of which complicate the analysis.

Since a tip is either a pending tip or a free tip, we have $W(t_n)=L(t_n)-F(t_n)$ , and hence knowing (2.7)–(2.10) is enough. However, for the purpose of studying the fluid limit that will be introduced in Section 2.5, we introduce an alternative equation to calculate the number of pending tips, i.e. $W(t_n)$ . Note that a pending tip must be a free tip at a prior time. This implies that a pending tip at $t_n$ must correspond to a free tip that is selected as a parent by a Type i arrival in the time range $[t_n-h_i,\ t_n-\epsilon]$ for some i. Among the free tips that are selected as parents by any Type i arrivals in $[t_n-h_i,\ t_n-\epsilon]$ for some i, some of them might be selected again and get attached before or at time $t_n$ . So we need to subtract the number of pending tips that have anticipated attachment time after $t_n$ but changed to pending tips with lower Type and got attached before $t_n$ . These quantities that need to be subtracted are calculated by using $J_{i,j}$ . To illustrate, we first give an example using $M=2$ , i.e. there are two possible values of POW duration.

For the special case of $M=2$ , the number of pending tips is calculated as

\begin{align*}W(t_n) = \sum_{i=1}^2\sum_{s=\epsilon}^{h_i}F_i(t_n-s)-\sum_{s=h_1+\epsilon}^{h_2-\epsilon}\sum_{u=s}^{h_2-\epsilon} J_{2,1}(t_n-s,u)\!,\end{align*}

where the range of $J_{2,1}(t_n-s,u)$ should satisfy the following conditions: (1) we subtract $J_{2,1}(t_n-s,u)$ because the Type 2 pending tips were anticipated to finish POW at time $\geq t_n$ but changed to Type 1 and hence finished before $t_n$ . So $t_n-s+u\geq t_n$ and $t_n-s+h_1<t_n$ ; (2) since the greatest possible RLT is $h_2-\epsilon$ , we have $u\leq h_2-\epsilon$ ; (3) when $u=h_2-\epsilon$ , by (1) we need $t_n-s+h_2-\epsilon\geq t_n$ . Hence $s\leq h_2-\epsilon$ .

For $M>1$ , repeating the argument by generalizing the pair (2,1) to (i, j) yields

(2.11) \begin{align}W(t_n) = \sum_{i=1}^M\sum_{s=\epsilon}^{h_i}F_i(t_n-s)-\sum_{i=2}^M\sum_{j=1}^{i-1}\sum_{s=h_j+\epsilon}^{h_i-\epsilon}\sum_{u=s}^{h_i-\epsilon} J_{i,j}(t_n-s,u) .\end{align}

2.5. The fluid limit

We are interested in the model considering the random variables $L(t_n)$ , $F(t_n)$ , $W_i(t_n,u)$ in the limit where $\lambda, N, \epsilon^{-1} \rightarrow \infty$ . In particular, we will first make a reasonable conjecture for the fluid limit based on the expected values and then establish that the fluid limit is close to the random process. For the random variables, we focus on the leading-order term in their expected values and assume that the remainder terms are negligible in the limit $\lambda, N, \epsilon^{-1} \rightarrow \infty$ . Using the distributions from Sections 2.2 and 2.3, we obtain the following leading-order terms:

(2.12) \begin{align}N_i(t_n)& = N p_i + \cdots,\nonumber \\F_i(t_n) &= 2 \, N \, p_i \, \frac{F(t_n)}{L(t_n)} + \cdots,\nonumber \\J_{i,j}(t_n, u) &= 2 \, N \, p_j \, \frac{W_{i}(t_n, u)}{L(t_n)} + \cdots, \quad\quad \text{for } i > j,\ h_j\leq u.\end{align}

Note that the 2 multiplier appears in (2.12) and later derivations because two parents are selected for each vertex. If k parents are selected for each vertex instead, 2 will be replaced by k.

We expect that there are deterministic functions $l(t),f(t),w_i(t,u)$ such that $l(t_n)\approx L(t_n)/\lambda$ , $f(t_n) \approx F(t_n)/\lambda$ , and $w_i(t_n, u) \approx W_i(t_n, u)/N$ . By the update rules in (2.7)–(2.11), taking expected values on the right-hand side and only keeping the leading-order term in each expected value yields

(2.13) \begin{align}&f(t_{n+1}) - f(t_n) = \epsilon - 2 \, \epsilon \, \frac{f(t_n)}{l(t_n)} \nonumber,\\&l(t_{n+1}) - l(t_n) = \epsilon - 2 \, \epsilon \sum_{i=1}^M p_i\frac{f(t_n-h_i)}{l(t_n-h_i)} - 2 \, \epsilon \sum_{j < i}\sum_{u=h_j+\epsilon}^{h_i-\epsilon}\epsilon p_j\frac{w_i(t_n-h_i,u)}{l(t_n-h_i)} \nonumber \\&\quad\quad\quad\quad\quad\quad\quad\quad + 2 \, \epsilon \sum_{j < i}\sum_{u=h_j+\epsilon}^{h_i-\epsilon} \epsilon p_j\frac{w_i(t_n-u,u)}{l(t_n-u)}\nonumber,\\&w_i(t_{n+1}, u-\epsilon) - w_i(t_n, s) = - 2 \, \epsilon \,\sum_{h_j\leq u} p_j\frac{w_i(t_n,u)}{l(t_n)} \nonumber,\\&w_i(t_{n+1},h_i-\epsilon) = 2 \, p_i \, \frac{f(t_n)}{l(t_n)}+\sum_{j>i}\sum_{u= h_i}^{h_j-\epsilon}2p_i\frac{\epsilon w_j(t_n,u)}{l(t_n)}\nonumber,\\&w(t_n) =\sum_{i=1}^{M}\sum_{s=\epsilon}^{h_i}2\epsilon p_i\frac{f(t_n-s)}{l(t_n-s)}-2\sum_{i>j}\sum_{s=h_j+\epsilon}^{h_i-\epsilon}\sum_{u=s}^{h_i-\epsilon} \epsilon^2p_j\frac{w_i(t_n-s,u)}{l(t_n-s)}.\end{align}

We now formally take the limit $\lambda, N, \epsilon^{-1} \rightarrow \infty$ . The asymptotic equations (2.13) lead to the following delayed partial differential equations (PDEs), and we call the corresponding solution to the PDEs the fluid limit of the random DAG model:

(2.14) \begin{align}\frac{\mathrm{d}f}{\mathrm{d}t} & =1-2\frac{f(t)}{l(t)},\end{align}

(2.15) \begin{align}\frac{\mathrm{d}l}{\mathrm{d}t} & =1-2\sum_{i=1}^M p_i\frac{f(t-h_i)}{l(t-h_i)}-2\sum_{j < i}\int_{h_j}^{h_i}p_j\frac{w_i(t-h_i,u)}{l(t-h_i)}\,\mathrm{d}u\nonumber\\&\quad +2\sum_{j < i}\int_{h_j}^{h_i}p_j\frac{w_i(t-u,u)}{l(t-u)}\,\mathrm{d}u,\end{align}

(2.16) \begin{align}&\left(\frac{\partial}{\partial t}-\frac{\partial}{\partial u}\right)w_i(t,u)=-2\sum_{h_j\leq u} p_j\frac{w_i(t,u)}{l(t)},\\[-13pt]\nonumber\end{align}

(2.17)

(2.18)

As mentioned in Section 2.2, the assumption of deterministic arrivals can be generalized to binomial arrivals. This generalization can be carried out by multiplying the expected values in (2.12) by $1-p_0$ , similar to how $p_i$ appears in the calculations. The ansatz for the fluid limit and the proof of its approximation quality in Section 5 then proceed similarly.

In order to show the existence and uniqueness of the solution to the PDEs, we will first define the initial condition in Section 2.6.

2.6. Initial conditions

Both the random process of $\{F(t_n),L(t_n), W_{i}(t_n,u)\colon u\in[0,\ h_i], \epsilon \text{ {divides} } u\}$ defined in (2.7)–(2.10) and their corresponding PDEs that describe the fluid limit should be supplemented with an initial condition. The initial condition of the random process, which can be found by observing the update rules, is defined as follows:

\begin{align*}&F(t_j)=x_j,\quad\quad j=-2h_M\epsilon^{-1},\ldots,0,\\&L(t_j)=y_j,\quad\quad j=-2h_M\epsilon^{-1},\ldots,0,\\&W_i(t_0,k\epsilon)=\omega_k, \quad\quad k=1,2,\ldots, h_i\epsilon^{-1}-1,\\&N_i(t_j)=a_{i,j}, \quad\quad j=-h_M\epsilon^{-1},-h_M\epsilon^{-1}+1,\ldots,-1,\\&F_i(t_j)=b_{i,j}, \quad\quad j=-h_M\epsilon^{-1},-h_M\epsilon^{-1}+1,\ldots,-1,\\&J_{i,r}(t_j,k\epsilon)=c_{i,r,j,k}, \quad\quad j=-h_M\epsilon^{-1},-h_i\epsilon^{-1}+1,\ldots,-1,\\&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad k=h_i\epsilon^{-1}-1,h_i\epsilon^{-1}-2,\ldots,h_r\epsilon^{-1}.\end{align*}

Fixing these values is sufficient to generate the future of the model using (2.7)–(2.10).

The requirements assumed in (5.2) , (5.3), 5.7, (5.8), (5.11), (5.12), and (5.14), which are stated in the lemmas where they are needed, are reasonable assumptions of initial conditions which require that some quantities in the initial condition are close to their expected values when they are generated. The validity of these conditions is stated in Proposition 2.1.

The proof of Proposition 2.1 is equivalent to the proof of the parallel statements that arise within the intervals $t\geq0$ instead of $t\leq 0$ . For illustration, see Lemmas 5.2 and 5.3, where Lemma 5.2 assumes the initial condition specified by (5.2) and Lemma 5.3 validates this assumption. The core idea is that the initial conditions are also randomly generated using the described dynamics, and therefore the properties that are established in $t\geq0$ can be applied to $t\leq0$ as well.

For the initial condition of the PDEs corresponding to the fluid limit, we first let $f(t_j)=F(t_j)/\lambda, l(t_j)=L(t_j)/\lambda$ for $t_j=-2h_M,\ldots,0$ . One can then use interpolations such as the Lagrange polynomial or cubic spline to construct f(t), l(t) for $t\in[{-}2h_M,\,0]$ so that both functions are continuously differentiable. By following (2.16) and (2.17), we can generate $w_i(t,u)$ for $t\in[{-}h_M,\,0]$ as part of the initial condition for the fluid limit.

Using the constructed initial condition for the fluid limit, we will use a similar approach as in [Reference King14] based on a method called the method of steps [Reference Driver5] to show the existence and uniqueness of the fluid limit. We first consider $f,l,w_i$ in the interval $t\in[0,\ h_1]$ . Note that l(t) for $t\in[0,\ h_1]$ can be directly computed using (2.15) because the right-hand side only considers initial conditions. Together with (2.14), the solution for f(t) within $t\in[0,\ h_1]$ can be calculated. Then l(t), f(t) within $t\in[0,\ h_1]$ along with the initial conditions can then be used to solve $w_i(t,h_i)$ within $t\in[0,\ h_1]$ using (2.16) and (2.17). Using the initial conditions and the solutions $f,l,w_i$ within the interval $t\in[0,\ h_1]$ , repeating the same procedure leads to the solution in $t\in[h_1,\ 2h_1]$ , and so on iteratively.

3.1. Discussion

While Theorem 3.1 assumes that if POW durations are independent, identically distributed with bounded support on M finite values, where M can be any finite number, the result can be extended to a distribution with unbounded support. One can construct a sequence of distributions with $M\rightarrow \infty$ so that the sequence converges to the unbounded distribution for POW durations. Each distribution with finite M can be seen as a truncated version of the unbounded distribution. By showing that the random process with bounded POW durations can approximate the process with unbounded POW durations within the fixed time horizon [0, T], Theorem 3.1 extends to unbounded distributions.

There are multiple future directions to extend this topic. For example, [Reference King14] has investigated how fast the fluid limit converges to a stable state when the POW has a fixed duration. Whether the convergence rate remains the same when POW is random is an interesting question. In addition, adversarial vertices are an important topic in DL systems, particularly in relation to security. This has been studied in [Reference Sankagiri, Gandlur and Hajek22] for blockchain systems but has not yet been explored in the context of IOTA and other similar protocols. For example, [Reference Dembo3, Reference Göbel, Keeler, Krzesinski and Taylor11] investigate the security of blockchain under adversarial attack by including the consideration of communication delay. Because of the differences between the attachment rules for blockchain and DAG-based protocols, many results from blockchain systems cannot be directly applied to protocols with DAG representation. Therefore, conducting deeper investigations into IOTA and similar systems by considering adversarial attack and communication delay provides valuable insights into the performance differences between DAG-based protocols and blockchain.

5.1. Difference in number of free tips

In this section, we establish the upper bound of $\big|\lambda^{-1}F(t_n)-f(t_n)\big|$ , which will be stated in (5.5). In order to do that, we will follow the idea introduced through (4.1)–(4.3). From now on, we assume that $n>0$ :

(5.1) \begin{align}\frac{1}{\lambda}F(t_n)-f(t_n) =&\frac{1}{\lambda}\sum_{j=0}^{n-1}F(t_{j+1})-F(t_j)-\lambda(f(t_{j+1})-f(t_j))\nonumber\\=&\frac{1}{\lambda}\sum_{j=0}^{n-1}\left(\sum_{i=1}^{M} N_i(t_j-h_i)-\sum_{i=1}^{M}F_i(t_j)-\lambda\int_{t_j}^{t_{j+1}} 1-2\frac{f(s)}{l(s)}\,\mathrm{d}s\right)\nonumber\\=&\frac{1}{\lambda}\sum_{j=0}^{n-1}\left(-N+\sum_{i=1}^{M} N_i(t_j-h_i)\right)+\sum_{i=1}^{M}\frac{1}{\lambda}\sum_{j=0}^{n-1}\big(\mathbb{E}\big[F_i(t_j)|\sigma_j\big]-F_i(t_j)\big)\nonumber\\&+\sum_{i=1}^{M}\frac{1}{\lambda}\sum_{j=0}^{n-1}\left(\lambda\int_{t_j}^{t_{j+1}} 2p_i\frac{f(s)}{l(s)}\,\mathrm{d}s-\mathbb{E}\big[F_i(t_j)|\sigma_j\big]\right)\!,\end{align}

which is the result following the idea in (4.1). We now establish Lemmas 5.2, 5.4, 5.5, and 5.6, which provide probability bounds on each summation in (5.1). All the lemmas from here on will be proved in Section 6.

Lemma 5.2. Let $D_n=\lambda^{-1}\sum_{j=0}^{n-1} (\!-N+\sum_{i=1}^M N_i(t_j-h_i))$ . Assume $\epsilon|h_i$ and let $n_i=h_i\epsilon^{-1}$ . If for the initial condition we have for all $i=1,2,\ldots,M$ , that

(5.2) \begin{align}\sup_{-n_i\leq k< 0}\Bigg|\frac{1}{\lambda}\sum_{j=-n_i}^{k} (N_i(t_j)-p_i N)\Bigg|<\theta_1 ,\end{align}

then

\begin{align*}\mathbb{P}(E_1)\,:\!=\,\mathbb{P}\Big(\sup_{0 < n\leq n_T}|D_n|\geq M\theta+M\theta_1\Big)\leq 2M\exp\left\{{-}\frac{\theta^2}{32T\epsilon}\right\}\!.\end{align*}

The validity of the assumption in (5.2) is given in Lemma 5.3, whose proof is analogous to the proof of Lemma 5.2. This similarity arises from the fact that both the random process after time $t=0$ and its initial condition are randomly generated by the same rules described in Section 2.3. Based on the illustration using Lemmas 5.2 and 5.3, the lemmas and proofs of other assumptions on the initial conditions will be omitted.

Lemma 5.3. Suppose the initial condition of the random process described in Section 2.6 is generated according to the distributions described in Section 2.3. Then

\begin{align*}\mathbb{P}\left(\bigcup_{i=1}^{M}\Bigg\{\sup_{-n_i\leq k<0}\Bigg|\frac{1}{\lambda}\sum_{j=-n_i}^{k} \left(N_i(t_j)-p_iN\right)\Bigg|\geq\theta_1\Bigg\}\right)\leq 2M\exp\left\{{-}\frac{\theta_1^2}{32h_M\epsilon}\right\}\!,\end{align*}

which vanishes for $\epsilon\rightarrow 0$ .

Lemma 5.4. For any i, we have

\begin{align*}\mathbb{P}\big(E_{2,i}\big)&\,:\!=\,\mathbb{P}\Bigg\{\sup_{0< n\leq n_T}\Bigg|\frac{1}{\lambda}\sum_{j=0}^{n-1} \mathbb{E}\big[F_i(t_j)|\sigma_j\big]-F_i(t_j)\Bigg|\geq \theta \Bigg\}\leq \exp\left\{{-}\frac{\theta^2}{32T\epsilon}\right\}\!,\\\mathbb{P}\big(E_2\big)&\,:\!=\,\mathbb{P}\Bigg(\bigcup_{i=1}^M E_{2,i}\Bigg)\leq 2M\exp\left\{{-}\frac{\theta^2}{32T\epsilon}\right\}\!.\end{align*}

Lemma 5.4 holds because $\mathbb{E}[F_i(t_j)|\sigma_j]-F_i(t_j)$ is a random variable that satisfies the requirements in Lemma 5.1.

Lemma 5.5. If for the initial condition we have

(5.3) \begin{align}\inf_{-h_M\epsilon^{-1}\leq n\leq \frac{3}{2}h_1\epsilon^{-1}}\sum_{j=n-2h_1\epsilon^{-1}}^{0} N_1(t_j)>1 ,\end{align}

\begin{align*}\text{then }\mathbb{P}\big(E_3\big)\,:\!=\,\mathbb{P}\Big(\inf_{-h_k\epsilon^{-1}\leq n\leq n_T}L_{\lambda}(t_n)\leq 1\Big)\leq T\epsilon^{-1} \exp\left\{{-}\frac{(1-p_1h_1/2)^2\lambda}{2h_1}\right\}\!.\end{align*}

Lemma 5.5 allows us to derive Lemma 5.6.

Lemma 5.6. Let

(5.4) \begin{align}H_{i,j+1}=-\mathbb{E}\big[F_i(t_j)|\sigma_j\big]+\lambda\int_{t_j}^{t_{j+1}} 2p_i\frac{f(s)}{l(s)}\,\mathrm{d}s.\end{align}

Then, on event $E_3^C$ , $\lambda^{-1}\sum_{j=0}^{n-1}|H_{i,j+1}|\leq 16\epsilon T+\sum_{j=0}^{n-1} 2p_i\epsilon g(t_{j+1})/m$ for all i.

Combining Lemmas 5.2, 5.4, 5.5, and 5.6 yields that, on the event $(\bigcup_{i=1}^3 E_i)^c$ ,

\begin{align*}\bigg|\frac{1}{\lambda}F(t_n)-f(t_n)\bigg|\leq 2M\theta+M\theta_1+16M\epsilon T+\sum_{j=0}^{n-1} \frac{2\epsilon g(t_{j+1})}{m}.\end{align*}

We can extend this inequality to $t\in[t_{n},\ t_{n+1})$ by defining that $F(t)=F(t_k)$ for $t\in[t_k,\ t_{k+1})$ :

(5.5) \begin{align}\bigg|\frac{1}{\lambda}F(t)-f(t)\bigg| &\leq\bigg|\frac{1}{\lambda}F(t_n)-f(t_n)\bigg|+|f(t_n)-f(t)|\nonumber\\[5pt]&\leq\bigg|\frac{1}{\lambda}F(t_n)-f(t_n)\bigg|+\bigg|\int_{t_n}^{t}1-2\frac{f(s)}{l(s)}\,\mathrm{d}s\bigg|\leq\bigg|\frac{1}{\lambda}F(t_n)-f(t_n)\bigg|+\epsilon+2\epsilon\nonumber\\[5pt]&\leq3\epsilon+2M\theta+M\theta_1+16M\epsilon T+\sum_{j=0}^{n-1} \frac{2\epsilon g(t_{j+1})}{m},\end{align}

where we use the property that $f(s)\leq l(s)$ .

5.2. Difference in number of tips

Next, by following a similar procedure as in Section 5.1, we establish (5.9) that provides an upper bound to $|L(t_n)/\lambda-l(t_n)|$ :

(5.6) \begin{align}\frac{1}{\lambda}L(t_n)-l(t_n) =&\frac{1}{\lambda}(L(t_n)-L(t_0))-(f(t_n)-f(t_0))\nonumber\\[5pt]=&\frac{1}{\lambda}\sum_{j=0}^{n-1}\bigg[L(t_{j+1})-L(t_j)-\lambda\int_{t_j}^{t_{j+1}}f'(t)\,\mathrm{d}t\bigg]\nonumber\\[5pt]=&\frac{1}{\lambda}\sum_{j=0}^{n-1}\left(\sum_{i=1}^{M} N_i(t_j-h_i)-N\right)\nonumber\\[5pt]&+\frac{1}{\lambda}\sum_{j=0}^{n-1}\left(-\sum_{i=1}^M F_i(t_j-h_i)+2\lambda\sum_{i=1}^M p_i\int_{t_j}^{t_{j+1}}\frac{f(t-h_i)}{l(t-h_i)}\,\mathrm{d}t\right)\nonumber\\[5pt]&+\frac{1}{\lambda}\sum_{j=0}^{n-1}\left[2\lambda\sum_{r < i}\!\int_{t_j}^{t_{j+1}}\!\int_{h_r}^{h_i}p_r\frac{w_i(t-h_i,u)}{l(t-h_i)}\,\mathrm{d}u\,\mathrm{d}t-\!\sum_{r < i}\sum_{u=h_r+\epsilon}^{h_i-\epsilon}\!J_{i,r}(t_j-h_i,u)\right]\nonumber\\[5pt]&+\frac{1}{\lambda}\sum_{j=0}^{n-1}\left[2\lambda\sum_{r < i}\!\int_{t_j}^{t_{j+1}}\!\int_{h_r}^{h_i}p_r\frac{w_i(t-u,u)}{l(t-u)}\,\mathrm{d}u\,\mathrm{d}t-\sum_{r < i}\sum_{u=h_r+\epsilon}^{h_i-\epsilon}\!J_{i,r}(t_j-u,u)\right],\end{align}

where the last equality is derived from (2.8) and (2.15). In (5.6), we have decomposed the difference $L(t_n)/\lambda-l(t_n)$ into four parts, and we will analyze each part on the right-hand side of (5.6) by subtracting and adding conditional expectations as mentioned in (4.1).

The first part in (5.6) is $\lambda^{-1}\sum_{j=0}^{n-1} (\sum_{i=1}^M N_i(t_j-h_i) -N)$ , which is already bounded using Lemma 5.1. The second part in (5.6) is

\begin{align*}\frac{1}{\lambda}&\sum_{j=0}^{n-1}\left[\left(2\lambda\sum_{i=1}^M p_i\int_{t_j}^{t_{j+1}}\frac{f(t-h_i)}{l(t-h_i)}\,\mathrm{d}t\right)-\sum_{i=1}^M F_i(t_j-h_i)\right]\\[6pt]& \quad =\frac{1}{\lambda}\sum_{i=1}^M \sum_{j=0}^{n-1}\left[\left(2 p_i\lambda\int_{t_j}^{t_{j+1}}\frac{f(t-h_i)}{l(t-h_i)}\,\mathrm{d}t\right)-F_i(t_j-h_i)\right.\\[6pt]& \qquad-\mathbb{E}\big[F_i(t_j-h_i)|\sigma_{j-n_i}\big] +\mathbb{E}\big[F_i(t_j-h_i)|\sigma_{j-n_i}\big]\Biggr],\end{align*}

whose summands are very similar to the objects in Lemmas 5.4 and 5.6 except that the time is $t-h_i$ instead of t, and hence some of the terms are initial conditions. Therefore, we first make some modifications based on Lemma 5.4 to establish Corollary 5.1, and then we directly apply Lemma 5.6 to obtain Corollary 5.2.

Corollary 5.1. By Lemma 5.4, if for any i we have

(5.7) \begin{align}\sup_{0 < n\leq n_i}\Bigg|\frac{1}{\lambda}\sum_{j=0}^{n-1} \mathbb{E}\big[F_i(t_j-h_i)|\sigma_j\big]-F_i(t_j-h_i)\Bigg|\leq \theta_1,\end{align}

then

\begin{align*}\mathbb{P}\big(E_{4,i}\big)&\,:\!=\,\mathbb{P}\left\{\sup_{0< n\leq n_T}\Bigg|\frac{1}{\lambda}\sum_{j=0}^{n-1} \mathbb{E}\big[F_i(t_j-h_i)|\sigma_{j-n_i}\big]-F_i(t_j-h_i)\Bigg|\geq \theta_1+\theta \right\}\\[5pt]&\leq 2\exp\left\{{-}\frac{\theta^2}{32 T\epsilon}\right\}\\[5pt]\mathbb{P}(E_4)&\,:\!=\,\mathbb{P}\left(\bigcup_{i=1}^M E_{4,i}\right)\leq 2M \exp\left\{{-}\frac{\theta^2}{32T\epsilon}\right\}\!.\end{align*}

Note that the eligibility of 5.7 as an assumption for the initial conditions follows the same idea as shown in the proof of Lemma 5.2.

Corollary 5.2. By the proof of Lemma 5.6, on the event $E_3^c$ ,

\begin{align*}\frac{1}{\lambda} \sum_{j=0}^{n-1}\Bigg|\left(2 p_i\lambda\int_{t_j}^{t_{j+1}}\frac{f(t-h_i)}{l(t-h_i)}dt\right)-\mathbb{E}\big[F_i(t_j-h_i)|\sigma_{j-n_i}\big]\Bigg|\leq 16\epsilon T+\sum_{j=0}^{n-1} \frac{2p_i\epsilon g(t_{j+1})}{m}.\end{align*}

The third part in the right-hand side of (5.6) is

\begin{align*}\frac{1}{\lambda}\sum_{j=0}^{n-1}&\Bigg[2\lambda\sum_{r < i}\int_{t_j}^{t_{j+1}}\int_{h_r}^{h_i}p_r\frac{w_i(t-h_i,u)}{l(t-h_i)}\,\mathrm{d}u\,\mathrm{d}t-\sum_{r < i}\sum_{u=h_r+\epsilon}^{h_i-\epsilon}J_{i,r}(t_j-h_i,u)\Bigg]\\&=\sum_{r < i}\frac{1}{\lambda}\sum_{j=0}^{n-1}\Bigg[2\lambda\int_{t_j}^{t_{j+1}}\int_{h_r}^{h_i}p_r\frac{w_i(t-h_i,u)}{l(t-h_i)}\,\mathrm{d}u\,\mathrm{d}t-\sum_{u=h_r+\epsilon}^{h_i-\epsilon}\mathbb{E}\big[J_{i,r}(t_j-h_i,u)|\sigma_{j-n_i}\big]\\&+\sum_{u=h_r+\epsilon}^{h_i-\epsilon}\mathbb{E}\big[J_{i,r}(t_j-h_i,u)|\sigma_{j-n_i}\big]-\sum_{u=h_r+\epsilon}^{h_i-\epsilon}J_{i,r}(t_j-h_i,u)\Bigg],\end{align*}

which will be taken care of by Lemmas 5.7 and 5.8 with $h(u)=n_i$ .

Lemma 5.7. Given any pair $r < i$ , let h(u) be either $n_i\,:\!=\,h_i/\epsilon$ (so the function is constant with respect to u) or $u\epsilon^{-1}$ . If for the initial condition we have

(5.8) \begin{align}\sup_{0 < n\leq n_T}\Bigg|\frac{1}{\lambda}\sum_{j=0}^{n-1}\sum_{u=h_r+\epsilon}^{h_i-\epsilon}(\mathbb{E}[J_{i,r}(t_{j-h(u)},u)|\sigma_{j-h(u)}]-J_{i,r}(t_{j-h(u)},u))I_{\{j-h(u)\leq 0\}}\Bigg|<\theta_1 ,\end{align}

and we let $E_{5,i,r}(h(u)\epsilon)\,:\!=\,$

\begin{align*}\Bigg\{\sup_{0 < n\leq n_T}\Bigg|\frac{1}{\lambda}\sum_{j=0}^{n-1}\Bigg(\sum_{u=h_r+\epsilon}^{h_i-\epsilon}\mathbb{E}[J_{i,r}\big(t_{j-h(u)},u\big)|\sigma_{j-h(u)}]-\sum_{u=h_r+\epsilon}^{h_i-\epsilon}J_{i,r}\big(t_{j-h(u)},u\big)\Bigg)\Bigg|\geq\theta_1+\theta\Bigg\}\!,\end{align*}

then

\begin{align*}&\mathbb{P}(E_{5,i,r}(h(u)\epsilon))\leq 2T\epsilon^{-1}\exp\left\{{-}\frac{\theta^2}{32T\epsilon}\right\}\!,\\&\mathbb{P}(E_{5,1})\,:\!=\,\mathbb{P}\left(\bigcup_{r < i}E_{5,i,r}(h_i)\right)\leq \frac{M^2}{2}\cdot2T\epsilon^{-1}\exp\left\{{-}\frac{\theta^2}{32T\epsilon}\right\}\!,\\&\mathbb{P}(E_{5,2})\,:\!=\,\mathbb{P}\left(\bigcup_{r < i}E_{5,i,r}(u)\right)\leq \frac{M^2}{2}\cdot2T\epsilon^{-1}\exp\left\{{-}\frac{\theta^2}{32T\epsilon}\right\}\!,\\&\mathbb{P}(E_5)\,:\!=\,\mathbb{P}\big(E_{5,1}\cup E_{5,2}\big)\leq M^2T\epsilon^{-1}\exp\left\{{-}\frac{\theta^2}{32T\epsilon}\right\}\!.\end{align*}

Lemma 5.8. Using the same setup for h(u) from Lemma 5.7, we get

\begin{align*}\Bigg| \frac{1}{\lambda} \sum_{j=0}^{n-1} & \Bigg[2\lambda\int_{t_j}^{t_{j+1}}\int_{h_r}^{h_i} p_r \frac{w_i(t-h(u)\epsilon,u)}{l(t-h(u)\epsilon)} \,\mathrm{d}u \,\mathrm{d}t-\sum_{u=h_r+\epsilon}^{h_i-\epsilon} \mathbb{E}\big[J_{i,r}(t_{j-h(u)},u)|\sigma_{j-h(u)}\big]\Bigg]\Bigg| \\{}& \leq 16\epsilon T+2Tp_r\epsilon(\xi+(h_i-h_r)^2\gamma)+6p_r(h_i-h_r)\epsilon \sum_{j=0}^{n-1}\frac{g(t_{j+1})}{m}.\end{align*}

For the fourth part of (5.6), we apply Lemmas 5.7 and 5.8 with $h(u)=u\epsilon^{-1}$ . Lemmas 5.2, 5.7, and 5.8 and Corollaries 5.1 and 5.2 cover all terms in (5.6). Putting them together, we get that, on the event $(E_{4}\cup E_{5})^c$ ,

\begin{align*}&\bigg|\frac{1}{\lambda}L(t_n)-l(t_n)\bigg|\leq (M^2+2M)(\theta_1+\theta)+16M\epsilon T+\sum_{j=0}^{n-1} \frac{2\epsilon g(t_{j+1})}{m}\\\leq& (M^2+2M)(\theta_1+\theta)+16(M^2+M)\epsilon T+4 M T\epsilon(\xi+\big(h_M - h_1\big)^2\gamma)\\&+\left[12M(h_M-h_1)\epsilon+2\epsilon\right]\sum_{j=0}^{n-1} \frac{ g(t_{j+1})}{m}.\end{align*}

We can extend this inequality to $t\in[t_n,\ t_{n+1})$ by setting $L(t)\,:\!=\,L(t_n)$ in this interval, yielding

(5.9) \begin{align}\bigg|\frac{L(t)}{\lambda}-l(t)\bigg|&\leq \bigg|\frac{L(t_n)}{\lambda}-l(t_n)\bigg|+\bigg|\int_{t_n}^t\frac{\mathrm{d}l}{\mathrm{d}s}\,\mathrm{d}s\bigg|\leq \bigg|\frac{L(t_n)}{\lambda}-l(t_n)\bigg|+\epsilon(1+2+4M(h_M-h_1)\xi)\nonumber\\&\leq 2M^2(\theta_1+\theta)+38M^2\epsilon T+4 M T\epsilon((h_M+1)\xi+(h_M)^2\gamma)\\\nonumber&\quad +\left[12 M h_M + 2\right]\epsilon\sum_{j=0}^{n-1} \frac{ g(t_{j+1})}{m}\\\nonumber&\leq 2M^2(\theta_1+\theta)+38M^2 T \epsilon +8 M T (h_M)^2(\xi+\gamma)\epsilon+13 M h_M \epsilon\sum_{j=0}^{n-1} \frac{ g(t_{j+1})}{m},\end{align}

where for simplicity of expression we assume $T\geq 1$ and $h_M>2$ .

5.3. Difference in number of pending tips

In this section, we will establish (5.20) as an upper bound of $|{W_i({\cdot})}/N-w_i({\cdot})|$ . As usual, we first rewrite the difference using a telescoping sum using (2.9)–(2.18):

\begin{align*}&\left[\frac{W_i(t_n,h_i-k\epsilon)}{N}-w_i(t_n,h_i-k\epsilon)\right]\nonumber\\[3pt]=& \frac{1}{N} \big\{W_i(t_n,h_i-k\epsilon)-W_i(t_n-(k-1)\epsilon,h_i-\epsilon) \nonumber\\[3pt]&-N \left[w_i(t_n,h_i-k\epsilon)- w_i(t_n-(k-1)\epsilon,h_i-\epsilon)\right]\big\}\nonumber\\[3pt]&+\frac{1}{N}\left[W_i(t_n-(k-1)\epsilon,h_i-\epsilon)-N w_i(t_n-(k-1)\epsilon,h_i-\epsilon)\right]\nonumber\\[3pt]=& \frac{1}{N} \sum_{j=1}^{k-1}\{W_i(t_{n-k+j+1},h_i- (j+1) \epsilon)-W_i (t_{n-k+j},h_i-j \epsilon)\nonumber\\[3pt]&-N[w_i(t_{n-k+j+1},h_i- (j+1) \epsilon)-w_i(t_{n-k+j},h_i-j \epsilon)]\}\\&+ \frac{1}{N}\left[F_i(t_{n-k})+\sum_{j>i} \sum_{u=h_i}^{h_j-\epsilon}J_{j,i}(t_{n-k},u)-N\left(2p_i\frac{f(t_{n-k})}{l(t_{n-k})}+\sum_{j>i} \int_{h_i}^{h_j}2p_i\frac{w_j(t_{n-k},u)}{l(t_{n-k})}\,\mathrm{d}u\right)\right.\nonumber\\[3pt]&\left.+2N \int_{0}^{\epsilon} \sum_{h_j\leq u} p_j\frac{w_i(t_{n-k}+u,h_i-u)}{l(t_{n-k}+u)}\,\mathrm{d}u\right]\nonumber\end{align*}

(5.10) \begin{align}=& \frac{1}{N} \sum_{j=1}^{k-1} \sum_{h_r\leq h_i-j\epsilon}\left\{{-}J_{i,r}(t_{n-k+j},h_i-j\epsilon)+N \int_{0}^{\epsilon}2 p_r\frac{w_i(t_{n-k+j}+u,h_i-j\epsilon-u)}{l(t_{n-k+j}+u)}\,\mathrm{d}u\right\}\nonumber\\[3pt]&+ \frac{1}{N}\left(F_i(t_{n-k})-2p_i N\frac{f(t_{n-k})}{l(t_{n-k})}\right)+ \frac{1}{N}\left(2N\int_{0}^{\epsilon} \sum_{h_j\leq u} p_j\frac{w_i(t_{n-k}+u,h_i-u)}{l(t_{n-k}+u)}\,\mathrm{d}u\right)\nonumber\\[3pt]&+\frac{1}{N}\left(\sum_{j>i} \sum_{u=h_i}^{h_j-\epsilon}J_{j,i}(t_{n-k},u)-N\sum_{j>i} \int_{h_i}^{h_j}2p_i\frac{w_j(t_{n-k},u)}{l(t_{n-k})}\,\mathrm{d}u\right)\!.\end{align}

In the last equality of (5.10), we have separated the difference into four terms, each of which is scaled by N. Using the separation technique to add and subtract conditional expectations, we will establish Lemmas 5.9 and 5.10 for the first scaled term and establish Lemmas 5.11 and 5.12 for the second scaled term. The fourth term will be covered by Lemma 5.15, which is derived from Lemmas 5.13 and 5.14. Last, we will establish an upper bound for the third scaled term in (5.15). Combining these results leads to the upper bound for (5.10) as given in (5.16). With (5.16), we are then able to derive an upper bound on the difference between the number of pending tips and its fluid limit as given in (5.20). Let

\begin{align*} G_{i,n,k,j}\,:\!=\,\sum_{h_r\leq h_i-j\epsilon}\big\{{-}J_{i,r}(t_{n-k+j},h_i-j\epsilon)+\mathbb{E}[J_{i,r}(t_{n-k+j},h_i-j\epsilon)|\sigma_{n-k+j}]\big\}\!.\end{align*}

Lemma 5.9. If for the initial condition we have

(5.11) \begin{align}&\sup_{i\leq M}\sup_{0 < n\leq n_T}\sup_{1\leq k\leq n_i}\Bigg|\frac{1}{N}\sum_{j=1}^{k-1} G_{i,n,k,j}\mathbf{1}_{\{n-k+j < 0\}}\Bigg|\leq\theta_1,\\[5pt]{then}\ \mathbb{P}\big(E_6\big)&\,:\!=\,\mathbb{P}\left(\sup_{i\leq M}\sup_{0 < n\leq n_T}\sup_{1\leq k\leq n_i}\Bigg| \frac{1}{N} \sum_{j=1}^{k-1} G_{i,n,k,j}\Bigg|\geq \theta_1+\theta\right)\nonumber\\[5pt]&\leq M T h_M^2 \frac{\epsilon^{-3}}{N}\left(512\frac{h_M^2}{\theta^2}+\exp\left\{{-}\frac{\theta^2 \epsilon^2 N}{8h^2_M}\right\}\right)\!.\nonumber\end{align}

Lemma 5.10.

\begin{align*}&\Bigg| \frac{1}{N} \sum_{j=1}^{k-1} \sum_{h_r\leq h_i-j\epsilon}\Bigg(-\mathbb{E}[J_{i,r}(t_{n-k+j},h_i-j\epsilon)|\sigma_{n-k+j}]\\&\quad\quad\quad\quad\quad\quad\quad\quad+ \int_{0}^{\epsilon}2 Np_r\frac{w_i(t_{n-k+j}+u,h_i-j\epsilon-u)}{l(t_{n-k+j}+u)}\,\mathrm{d}u\Bigg)\Bigg|\\\leq&16 T M \epsilon+2p_r N \epsilon^2 \gamma +6\epsilon\sum_{j=1}^{n-1} \frac{g(t_{j})}{m}.\end{align*}

Lemma 5.11. If for the initial condition we have

(5.12) \begin{align}\sup_{i\leq M} \sup_{n\in[{-}h_M\epsilon^{-1},0]} \bigg| \frac{1}{N} (F_i(t_{n})-\mathbb{E}[F_i(t_{n})|\sigma_{n}])\bigg|\leq \theta_1,\end{align}

then using the same proof as in Lemma 5.9 we get

\begin{align*}\mathbb{P}(F_i(t_n)-\mathbb{E}[F_i(t_n)|\sigma_{n}]> N\theta)\leq 256\epsilon\frac{1}{N\theta^2}+\exp{\left\{{-}\frac{\theta^2 N}{8}\right\}} \quad\text{for }n> 0,\end{align*}

and hence

\begin{align*}\mathbb{P}(E_7)\,:\!=\,\mathbb{P}\left(\sup_{i\leq M} \sup_{n\leq n_T} \sup_{k\leq h_M\epsilon^{-1}} \frac{1}{N} \left(F_i(t_{n-k})-\mathbb{E}[F_i(t_{n-k})|\sigma_{n-k}]\right)> \theta+\theta_1\right)\\\leq 256\frac{1}{N\theta^2}+\exp{\left\{{-}\frac{\theta^2 N}{8}\right\}}.\end{align*}

Lemma 5.12. By the proof of Lemma 5.6, we have

\begin{align*}\frac{1}{N}\bigg|\mathbb{E}\big[F_i(t_{n-k})|\sigma_{n-k}\big]-2p_i\frac{f(t_n-k)}{l(t_{n-k})}\bigg|\leq \frac{2p_i}{N} \frac{\big|f(t_{n-k})-F_\lambda(t_{n-k})\big|+ \big|(t_{n-k})-L_\lambda(t_{n-k})\big|}{m}.\end{align*}

We now establish Lemmas 5.13, 5.14, and 5.15 to cover the fourth scaled term in the last equality of (5.10). The fourth scaled term is

(5.13) \begin{align}\frac{1}{N}&\Bigg|\sum_{j>i} \sum_{u=h_i}^{h_j-\epsilon}J_{j,i}(t_{n-k},u)-N\sum_{j>i} \int_{h_i}^{h_j}2p_i\frac{w_j(t_{n-k},u)}{l(t_{n-k})}\,\mathrm{d}u\Bigg|\nonumber\\[5pt]\leq& \frac{1}{N}\Bigg|\sum_{j>i} \sum_{u=h_i}^{h_j-\epsilon}\left( J_{j,i}(t_{n-k},u)-\mathbb{E}\big[J_{j,i}(t_{n-k},u)|\sigma_{n-k}\big] \right)\Bigg|\nonumber\\[5pt] &+\frac{1}{N}\Bigg|\sum_{j>i} \sum_{u=h_i}^{h_j-\epsilon} \mathbb{E}[J_{j,i}(t_{n-k},u)|\sigma_{n-k}]-N\sum_{j>i} \int_{h_i}^{h_j}2p_i\frac{w_j(t_{n-k},u)}{l(t_{n-k})}\,\mathrm{d}u\Bigg|.\end{align}

Lemma 5.13. If for the initial condition we have

(5.14) \begin{align}\sup_{n\in[{-}h_M\epsilon^{-1},\ 0]} \Bigg|\frac{1}{N}\sum_{j>i}\sum_{u=h_i}^{h_j-\epsilon} ( J_{j,i}(t_{n},u)-\mathbb{E}[J_{j,i}(t_{n},u)|\sigma_{n}])\Bigg| \leq \theta_1,\end{align}

then $\mathbb{P}\big(E_8\big)\,:\!=\,$

\begin{align*}&\mathbb{P}\left(\sup_{i}\sup_{n-k\leq n_T}\frac{1}{N}\Bigg|\sum_{j>i}\sum_{u=h_i}^{h_j-\epsilon} ( J_{j,i}(t_{n-k},u)-\mathbb{E}[J_{j,i}(t_{n-k},u)|\sigma_{n-k}])\Bigg|>\theta+\theta_1\right)\\&\leq 512 M^2 h_M\frac{1}{N\theta^2}+\exp\left\{{-}\frac{\theta^2N}{8}\right\}\!.\end{align*}

Next, we establish Lemma 5.14, which will be used in the proof of Lemma 5.15 and in the argument that extends (5.18)–(5.19).

Lemma 5.14. Consider $s_1,s_2\in[k\epsilon,\ (k+1)\epsilon)$ and $s_2>s_1$ . Then for $i=1,2,\ldots,M$ , $\bigg|w_i(t,s_2)-w_i(t,s_1)|\leq \alpha_{0,1}M(h_M)^M\exp\{2Mh_M/m\}\epsilon$ , where

\begin{align*}\alpha_{0,1}=(2+2Mh_M)\xi+\frac{6}{m}+\frac{4(3+4M h_M \xi)}{m}+2h_M\xi \frac{2(3+4 M h_M \xi)}{m}.\end{align*}

Applying Lemmas 5.13 and 5.14 on (5.13), we get the following Lemma 5.15 as an upper bound of (5.13).

Lemma 5.15. In the event of $E_8^c$ ,

\begin{align*}\frac{1}{N}&\Bigg|\sum_{j>i} \sum_{u=h_i}^{h_j-\epsilon}J_{j,i}(t_{n-k},u)-N\sum_{j>i} \int_{h_i}^{h_j}2p_i\frac{w_j(t_{n-k},u)}{l(t_{n-k})}\,\mathrm{d}u\Bigg|\\\leq& \theta+\theta_1+32Mh_M\frac{\epsilon^{2}}{N}+2\alpha_{0,1}M^2(h_M)^{M+1}\exp\left\{\frac{2Mh_M}{m}\right\}\epsilon\\&+2p_i\epsilon \sum_{j>i} \sum_{u=h_i}^{h_j-\epsilon}\left(\frac{ \big|W_j(t_{n-k},u)/N- w_j(t_{n-k},u)\big|}{m}\right)+\frac{4p_i}{m}Mh_M\big|L_{\lambda}(t_{n-k})-l(t_{n-k})\big|.\end{align*}

For the third term in last equality of (5.10), recall that $|w_i(t,u)/l(t)|<\xi$ as stated in the conditions of Theorem 3.1, so we have

(5.15) \begin{align}\Bigg|2\int_{0}^{\epsilon} \sum_{h_j\leq u} p_j\frac{w_i(t_{n-k}+u,h_i-u)}{l(t_{n-k}+u)}\,\mathrm{d}u\Bigg|&\leq2 \int_{0}^{\epsilon} \sum_{h_j\leq u} p_j \bigg|\frac{w_i(t_{n-k}+u,h_i-u)}{l(t_{n-k}+u)}\bigg|\,\mathrm{d}u\nonumber\\&\leq 2\int_{0}^{\epsilon} \sum_{h_j\leq u} p_j \xi \,\mathrm{d}u\leq 2\epsilon \xi .\end{align}

By applying Lemmas 5.3–5.3, and 5.3 as well as (5.15)–(5.10), we conclude that there exists a constant $\alpha_1$ independent of $\lambda,\epsilon$ such that

(5.16) \begin{align}\bigg|&\frac{W_i(t_n,h_i-k\epsilon)}{N}-w_i(t_n,h_i-k\epsilon)\bigg|\nonumber\\\leq& 3\theta_1+3\theta +16 T M \epsilon+2p_r N \epsilon^2 \gamma +2\epsilon \xi+ 32Mh_M\frac{\epsilon^{2}}{N}\nonumber\\&+2\alpha_{0,1}M^2(h_M)^{M+1}\exp\left\{\frac{2Mh_M}{m}\right\}\epsilon +6\epsilon\sum_{j=1}^{n-1} \frac{g(t_{j})}{m}\nonumber\\&+2 \frac{p_i}{N}\frac{\big|f(t_{n-k})-F_\lambda(t_{n-k})\big|+ \big|l(t_{n-k})-L_\lambda(t_{n-k})\big|}{m}\nonumber\\&+\frac{4p_i}{m}Mh_M\big|L_{\lambda}(t_{n-k})-l(t_{n-k})\big|+2p_i\epsilon \sum_{j>i} \sum_{u=h_i}^{h_j-\epsilon}\left(\frac{ \big|W_j(t_{n-k},u)/N- w_j(t_{n-k},u)\big|}{m}\right)\nonumber\\\leq&3\theta_1+3\theta+\alpha_1\epsilon+6\epsilon\sum_{j=1}^{n-1} \frac{g(t_{j})}{m}+2 \frac{1}{N}\frac{\big|f(t_{n-k})-F_\lambda(t_{n-k})\big|+ \big|l(t_{n-k})-L_\lambda(t_{n-k})\big|}{m}\nonumber\\&+\frac{4}{m}Mh_M\big|L_{\lambda}(t_{n-k})-l(t_{n-k})\big|+2p_i\epsilon \sum_{j>i} \sum_{u=h_i}^{h_j-\epsilon}\left(\frac{ \big|W_j(t_{n-k},u)/N- w_j(t_{n-k},u)\big|}{m}\right)\nonumber\\=\!:\,&y(n,k)+2p_i\epsilon \sum_{j>i} \sum_{u=h_i}^{h_j-\epsilon}\left(\frac{ \big|W_j(t_{n-k},u)/N- w_j(t_{n-k},u)\big|}{m}\right) ,\end{align}

where

(5.17) \begin{align}y(n,k)\,:\!&=\,3\theta_1+3\theta+\alpha_1\epsilon+6\epsilon\sum_{j=1}^{n-1} \frac{g(t_{j})}{m} +\frac{4}{m}Mh_M\big|L_{\lambda}(t_{n-k})-l(t_{n-k})\big|\nonumber\\&\quad +2 \frac{\big|f(t_{n-k})-F_\lambda(t_{n-k})\big|+ \big|l(t_{n-k})-L_\lambda(t_{n-k})\big|}{mN}.\end{align}

Equation (5.16) will be bounded above using induction as follows:

Base case: $i=M$ . In such a case there is nothing in the term $\sum_{j>i}$ , and hence

\begin{align*} \bigg|\frac{W_i(t_n,h_i-k\epsilon)}{N}-w_i(t_n,h_i-k\epsilon)\bigg|\leq y(n,k).\end{align*}

Inductive case: $i<M$ . First, consider $i=M-1$ and use the base case to get

\begin{align*}\bigg|&\frac{W_i(t_n,h_i-k\epsilon)}{N}-w_i(t_n,h_i-k\epsilon)\bigg|\\&\leq y(n,k)+2p_i\epsilon \sum_{u=h_i}^{h_M-\epsilon}\left(\frac{ \big|W_M(t_{n-k},u)/N- w_M(t_{n-k},u)\big|}{m}\right)\\&\leq \left(2\frac{h_M}{m}+1\right)y(n,k).\end{align*}

Iterating on $i\leq M$ , we have that there exists an $\alpha_2$ independent of $\lambda,\epsilon$ such that for all i,

(5.18) \begin{align} \bigg|\frac{W_i(t_n,h_i-k\epsilon)}{N}-w_i(t_n,h_i-k\epsilon)\bigg|\leq \alpha_2 y(n,k). \end{align}

Now we extend the inequality to $t\in [t_n,\ t_{n+1})$ and $s\in((k-1)\epsilon,k\epsilon]$ :

(5.19) \begin{align}\bigg|&\frac{W_i(t,s)}{N}-w_i(t,s) \bigg|\nonumber\leq \bigg|\frac{W_2(t_n,k\epsilon)}{N}-w_2(t_n,k\epsilon) \bigg|+ |w_2(t,s)-w_2(t_n,k\epsilon)|\nonumber\\[6pt]\leq& \bigg|\frac{W_2(t_n,k\epsilon)}{N}-w_2(t_n,k\epsilon) \bigg|+|w_2(t,k\epsilon-(t-t_n))-w_2(t_n,k\epsilon)|\nonumber\\[6pt]&+|w_2(t,k\epsilon-(t-t_n))-w_2(t,s)|\nonumber\\[6pt]\leq&\bigg|\frac{W_2(t_n,k\epsilon)}{N}-w_2(t_n,k\epsilon) \bigg|+\int_{0}^{t-t_n}\Bigg|-2\sum_{h_j\leq u}p_j\frac{w_i(t_n+u,k\epsilon-u)}{l(t_n+u)}\Bigg|\,\mathrm{d}u\nonumber\\[6pt]&+\alpha_{0,1}M(h_M)^M\exp\left\{\frac{2Mh_M}{m}\right\}\epsilon \nonumber\\[6pt]\leq& \alpha_2 y(n,k)+2\xi\epsilon+\alpha_{0,1}M(h_M)^M\exp\left\{\frac{2Mh_M}{m}\right\}\epsilon,\end{align}

where $\alpha_{0,1}$ is defined in Lemma 5.14.

Recall (5.17). Then there exist $\alpha_3$ independent of $\lambda, \epsilon$ such that in the event $(E_6\cup E_7\cup E_8)^c$ ,

(5.20) \begin{align}&\bigg|\frac{W_i(t,s)}{N}-w_i(t,s) \bigg|\nonumber\\&\leq \alpha_3\left(\theta_1+\theta+\epsilon+\epsilon\sum_{j=1}^{n-1} \frac{g(t_{j})}{m}+ \big|f(t_{n-k})-F_\lambda(t_{n-k})\big| +\big|L_{\lambda}(t_{n-k})-l(t_{n-k})\big|\right)\!.\end{align}

5.4. Applying Gronwall’s inequality

To summarize, we have (5.5), (5.9), and (5.20) hold on the event $(\bigcup_{i=1}^8 E_i)^c$ . Recall that

\begin{align*}g(t)=\sup_{0 < s\leq t}\left\{\bigg|\frac{F(s)}{\lambda}-f(s)\bigg|+\bigg|\frac{L(s)}{\lambda}-l(s)\bigg|+\sum_{i=1}^M \sup_{u\leq h_i} \bigg|\frac{W_i(s,u)}{N}-w_i(s,u) \bigg| \right\}\!.\end{align*}

Since g is non-decreasing, (5.5), (5.9), and (5.20) imply

\begin{align*}g(t_n)\leq& \sup_{t\leq t_n} \big|F_\lambda(t)-f(t)\big|+\sup_{t\leq t_n} \big|L_\lambda(t)-l(t)\big|+M\sup_{t\leq t_n}\sup_{u\leq h_i} \bigg|\frac{W_i(t,u)}{N}-w_i(s,u)\bigg|\\\leq& (2M+2M^2+\alpha_3)(\theta_1+\theta)+(16MT+38M^2 T+8 M T (h_M)^2(\xi+\gamma)+\alpha_3)\ \epsilon\\&+(2+13 M h_M+\alpha_3)\epsilon\! \sum_{j=0}^{n-1} \frac{ g(t_{j+1})}{m}+\alpha_3(|f(t_{n-1})-F_\lambda(t_{n-1})|+|L_{\lambda}(t_{n-1})-l(t_{n-1})|)\\\leq& \big[2M+2M^2+\alpha_3(1+2M+2M^2)\big](\theta_1+\theta)\\&+\big[16MT+38M^2 T+8 M T h_M^2(\xi+\gamma)+\alpha_3(1+54M^2 T+8 M T h_M^2(\xi+\gamma))\big]\ \epsilon\\&+\left[2+13 M h_M+\alpha_3\big(3+13 M h_M\big)\right]\epsilon \sum_{j=0}^{n-1} \frac{ g(t_{j})}{m}+(2+13 M h_M+\alpha_3)\ \epsilon \frac{ g(t_n)}{m}.\end{align*}

Hence, there exists a constant $\alpha$ independent of $\lambda,\epsilon$ such that

\begin{align*}&g(t_n)\leq \frac{1}{2}\alpha\Bigg(\theta_1+\theta+\epsilon+\epsilon \sum_{j=0}^{n-1} g(t_{j})\Bigg)+\big(2+13 M h_M+\alpha_3\big)\epsilon \frac{ g(t_n)}{m}.\end{align*}

If we assume $\epsilon$ small enough such that $\big(2+13 M h_M+\alpha_3\big)\epsilon <\frac{1}{2}$ , then we get $g(t_n)\leq \alpha(\theta_1+\theta+\epsilon+\epsilon \sum_{j=0}^{n-1} g(t_{j}))$ . By Gronwall’s lemma,

\begin{align*}g(t_n) &\leq \alpha \big(\theta_1+\theta+\epsilon\big)\exp{\left\{\sum_{j=0}^{n-1} \alpha \epsilon\right\}}\leq \alpha \big(\theta_1+\theta+\epsilon\big)e^{\alpha T},\\g(T)&\leq \alpha (\theta_1+\theta+\epsilon)e^{\alpha T}.\end{align*}

Given $\delta>0$ , choose $\theta=\theta_1=\delta e^{-\alpha T}/(3\alpha)$ and $\epsilon_0=\theta$ . Then for $\epsilon\leq \epsilon_0$ , we have $\alpha (\theta_1+\theta+\epsilon)e^{\alpha T}\leq \delta$ . Therefore, on the event $\big(\bigcup_{i=1}^8 E_i\big)^c$ , $g(T)\leq \delta$ . Hence

\begin{align*}\mathbb{P}\big(g(T)>\delta\big)&\leq \mathbb{P}\left(\bigcup_{i=1}^8 E_i\right) \leq \sum_{i=1}^8 \mathbb{P}\big(E_i\big)\\\leq& 2M\exp\left\{{-}\frac{\theta^2}{32T\epsilon}\right\} +2M\exp\left\{{-}\frac{\theta^2}{32T\epsilon}\right\} +T\epsilon^{-1} \exp\left\{{-}\frac{(1-p_1h_1/2)^2\lambda}{2h_1}\right\}\\& +2M \exp\left\{{-}\frac{\theta^2}{32 T \epsilon}\right\} +M^2T\epsilon^{-1}\exp\left\{{-}\frac{\theta^2}{32T\epsilon}\right\}+ 256\frac{1}{N\theta^2}+\exp{\left\{{-}\frac{\theta^2 N}{8}\right\}}\\& +M T h_M^2 \frac{\epsilon^{-3}}{N}\left(512\frac{h_M^2}{\theta^2}+\exp\left\{{-}\frac{\theta^2 \epsilon^2 N}{8h^2_M}\right\}\right)+ 512 M^2 h_M\frac{1}{N\theta^2}+\exp\left\{{-}\frac{\theta^2N}{8}\right\}\!.\end{align*}

Combining similar terms shows that there exists positive constant $\beta$ such that

\begin{align*}\mathbb{P}\big(g(T)>\delta\big)\leq& \beta\epsilon^{-1}\exp\left\{{-}\beta \frac{\theta^2}{T\epsilon} \right\}+\beta T\epsilon^{-1}e^{-\beta\lambda}\\&+\beta \exp\big\{{-}\beta\theta^2N\big\}+\beta T\frac{1}{\epsilon^3N\theta^2}+\beta T\frac{1}{\epsilon^3N}\exp\big\{{-}\beta\theta^2\epsilon^2N\big\}\!.\end{align*}

Hence as $\lambda,\epsilon^{-1},N,\epsilon^3N\rightarrow \infty$ the probability goes to 0.