2.1. Notations and agreements

2.2. Binomial distributions and beyond

The notation $\xi\sim B(n,p)$ means that the random variable ξ has binomial distribution with parameters n, p, that is $\mathbf{Pr}(\xi=k)=\binom nkp^k(1-p)^{n-k}$, $k=0,1,\ldots,n$. The archetypal example is the sum $\xi_1+\xi_2+\cdots+\xi_n$ of equidistributed Bernoulli random variables $\xi_j\sim B(1,p)$ with success parameter p.

The sum $\xi_1+\xi_2+\cdots+\xi_n$ of Bernoulli random variables $\xi_j\sim B(1,p_j)$ with different success parameters p_j has Poisson binomial distribution. It is used in Subsection 4.1.

So-called Markov binomial distribution corresponds the sum $\xi_1+\xi_2+\cdots+\xi_n$ where ξ_j form a Markov chain with two states 0 and 1. This idea was elaborated by A.A. Markov Sr. in his 1,907 paper, whose extended version is included in the 3rd edition of his textbook [Reference Markov16].

Another deformation of binomial distributions studied in Section 3 can be also described in terms of Markov chains. The additional positive integer (deterministic or random) parameter can be interpreted as a time delay.

2.3. Short description of Ouroboros Praos

We will use the next series of assumptions, which are the standard for the PoS model, in what follows. Thus, we assume that all epochs have the same duration, say T, and time interval T is divided into n equal intervals $[jT/n,(j+1)T/n]$ indexed by $j=0,1,\ldots,n-1$ and called timeslots. We say that some stakeholder S_i, $i\in I$ is a slot leader in jth timeslot if he was assigned for this timeslot, according to slot leader election procedure, described in Ouroboros Praos paper [Reference David, Gaži, Kiayias and Russell3]. The desirable properties of this procedure, achieved in Praos, are the following:

• • slot leaders are randomly selected, and the probability for stakeholder S_i with stake ratio α_i to become a slot leader in the jth time slot is proportional (with negligible deviation) to the ratio α_i and does not depend on j;

• • if S_i is a slot leader in the jth timeslot, nobody (except S_i) knows about this till the time when he creates and propagates the block;

• • after block creation, each participant can verify the validity of block creation (in particular, that the block was created by the assigned slot leader);

• • the probability of becoming a slot leader is indifferent w.r.t. stake union or splitting (no sense in uniting or dividing the stake, because it gives no extra profit).

The additional properties are:

• • depending on protocol parameters, some ratio of timeslots may be empty (without a slot leader), and some of them may have more than one slot leader;

• • in the case of multiple slot leaders in one timeslot, we necessary have a so-called orphan block(s), because only one block from one timeslot may be included in the chain.

Following the definitions and designations, introduced in [Reference David, Gaži, Kiayias and Russell3], we consider the function

(1)\begin{equation} \varphi_{f}(\alpha)=1-(1-f)^{\alpha}, \end{equation}

depending on the active slots coefficient $f\in(0,1)$. The exponential function $1-\varphi_{f}=(1-f)^\alpha$ is the solution of the Cauchy’s characteristic identity $E(x+y)=E(x)\cdot E(y)$ (see [Reference Kannappan9, Ch. 1]), which turns into the functional equation for φ_f:

(2)\begin{equation} \varphi_{f}\left(\alpha\right)+\varphi_{f}\left(\beta\right)-\varphi_{f}\left(\alpha\right)\varphi_{f}\left(\beta\right)=\varphi_{f}\left(\alpha+\beta\right), \end{equation}

whence by induction for a finite set $J\subseteq I$ we get the inequality:

(3)\begin{equation} \sum_{i\in J}\varphi_{f}\left(\alpha_{i}\right)\geqslant\varphi_{f}\left(\sum_{i\in J}\alpha_{i}\right). \end{equation}

We assume the existence of the finite set of stakeholders $(S_i)_{i \in I}$. Each stakeholder S_i owns corresponding stake ratio α_i. For $J\subseteq I$, denote $\alpha_J:=\sum_{i\in J}\alpha_i$. We assume that the total stake is taken as one:

(4)\begin{equation} \alpha_I=\sum_{i\in I}\alpha_i=1. \end{equation}

We consider the blockchain during an epoch, consisting of n timeslots indexed by integers $0,1,\ldots,n-1$. The whole slot leader election in each timeslot can be described by the family of independent Bernoulli random variables $\xi_{ij}\sim B(1,\varphi_f(\alpha_i))$ attributed to stakeholder S_i for $i\in I$ and to jth timeslot: $\xi_{ij}=1$ iff stakeholder S_i becomes jth slot leader.

For a subset $\Lambda\subseteq I$, denote $\chi_\Lambda:I\to\{0,1\}$ its characteristic function, that is $\chi_\Lambda(i)=1$ iff $i\in\Lambda$. So the set of slot leaders in the jth timeslot is $(S_i)_{i\in\Lambda}$ with the probability

(5)\begin{equation} \Pr\left(\bigcap_{i\in I}\xi_{ij}^{-1}(\chi_\Lambda(i))\right)=\prod_{i\in I\smallsetminus\Lambda}(1-f)^{\alpha_i}\cdot\prod_{i\in\Lambda}\varphi_f(\alpha_i) \underset{\text{by (4)}}=(1-f)^{1-\alpha_\Lambda}\prod_{i\in\Lambda}\varphi_f(\alpha_i), \end{equation}

which does not depend on j.

Usually, the set of stakeholders split into two classes of (H)onest and (M)alicious: $(S_i)_{i \in I}=(S_i)_{i \in I_{\text H}}\sqcup (S_i)_{i \in I_{\text M}}$. Here we assume that all stakeholders are honest, that is act according to block production rules:

• • create blocks in each corresponding timeslot and only in them;

• • in the case of a fork, support the longest chain.

In particular, they do not try to provide double spending or splitting attacks.

Note that in this case, forks may occur only due to two reasons—multiple slot leaders in one timeslot (each of them creates a block of the same height) or time delay in the network (two or more slot leaders refer to the same block).

3.1. Random chain $\gamma_{\delta,n}$

For our probabilistic model, $\delta,n,l$ are just integers which in the context of Ouroboros Praos can be interpreted as the time delay, the epoch length, and the chain length respectively. We define a $(\delta,n)$-chain as an increasing sequence of indexes of timeslots, such that two adjacent terms differ by at least the time delay:

During a random event that fixes the values of random variables $(\xi_j)_{0\leqslant j \lt n}$, for a given $(\delta,n)$-chain $c=(c_i)_{1\leqslant i\leqslant l}$, it is possible to construct a chain of blocks created exactly in the time intervals indexed by c_i if and only if $\xi_{c_i}=1$ for all c_i. This explains the following definition:

Definition 3.2. The random set $\Gamma_{\delta,n}\subseteq C_{\delta,n}$ of suitable chains is defined by the following formula:

\begin{equation*} \Gamma_{\delta,n}:=\{c\in C_{\delta,n}\,|\,\xi_{c_i}=1, 1\leqslant i\leqslant\ell(c)\}. \end{equation*}

The random set $\Gamma_{\delta,n}$ of suitable chains can contain several longest chains. A natural formalization of the longest chain rule assumes that we choose the longest chain c with the minimum possible values of c_i. To describe this precisely, we consider a total order $\succcurlyeq$ on $C_{\delta,n}$, which is a modification of the lexicographical order:

Definition 3.4. Let $(A,\geqslant)$ be a totally ordered set, and $(A^*,\cdot)$ be the free monoid of words $a_0a_1\cdots a_{n-1}$, $a_i\in A$, $n\geqslant0$ in the alphabet A. One can also identify a word $a_0a_1\cdots a_{n-1}$ with a sequence $(a_i)_{0\leqslant i \lt n}$. The structure binary operation is concatenation:

\begin{equation*} (a_0,a_1,\ldots a_{n-1})\cdot(b_0,b_1,\ldots b_{m-1}):=(a_0,a_1,\ldots a_{n-1},b_0,b_1,\ldots b_{m-1}); \end{equation*}

the neutral element is the empty string $()$.

1. (1) The lexicographical order (or dictionary order) $\succcurlyeq$ on $A^*$ is the total order uniquely determined by the following properties:

   1. (1) the empty string $()$ is the smallest element in $A^*$;

   2. (2) if a > b in A, then $(a,\ldots)\succ(b,\ldots)$ in $A^*$;

   3. (3) for $\alpha,\beta,\gamma\in A^*$, if $\beta\succ\gamma$ then $\alpha\cdot\beta\succ\alpha\cdot\gamma$.

2. (2) The Kleene–Brouwer order (or Lusin–Sierpiński order) $\succcurlyeq$ on $A^*$ is the total order uniquely determined by the following property

   1. (a) the empty string $()$ is the greatest element in $A^*$;

      and the above properties (b) and (c).

Now let take the alphabet $A=\{0 \lt 1 \lt \cdots \lt n-1\}$ and consider the restriction $\succcurlyeq$ of the Kleene–Brouwer order to the subset of $(\delta,n)$-chains $C_{\delta,n}\subset\{0 \lt 1 \lt \cdots \lt n-1\}^*$.

Example 3.5. The totally ordered set $(C_{2,5},\succcurlyeq)$ is the following:

\begin{equation*} (0,2,4)\prec (0,2)\prec (0,3)\prec (0,4)\prec (0)\prec (1,3)\prec (1,4)\prec (1)\prec (2,4)\prec (2)\prec (3)\prec (4)\prec (). \end{equation*}

In general, $\min_\succcurlyeq C_{\delta,n}=((i-1)\delta)_{1\leqslant i\leqslant \lceil n/\delta\rceil}$.

Definition 3.6. The optimal random chain $\gamma_{\delta,n}\in C_{\delta,n}$ is defined as the $\succcurlyeq$-minimal suitable chain:

\begin{equation*} \gamma_{\delta,n}:=\min\nolimits_\succcurlyeq\Gamma_{\delta,n}. \end{equation*}

Directly from the definition of the total order on $C_{\delta,n}$, we get the following lemma:

For each $c\in C_{\delta,n}^l$, let consider two subsets in $\{0,1,\ldots,n-1\}$:

(6)\begin{equation} \begin{aligned} J_1(c)&:=\{c_j\,|\,1\leqslant j\leqslant l\},\\ J_0(c)&:=[0,c_1)\cup[c_1+\delta,c_2)\cup\cdots\cup[c_i+\delta,c_{i+1})\cup\cdots\cup[c_{\ell}+\delta,n). \end{aligned} \end{equation}

Lemma 3.9. For $c\in C_{\delta,n}$, $\gamma_{\delta,n}=c$ iff $\xi_j=1$ for all $j\in J_1(c)$ and $\xi_j=0$ for all $j\in J_0(c)$. Hence,

(7)\begin{equation} \mathbf{Pr}\left(\gamma_{\delta,n}=c\right)=f^{\#J_1(c)}(1-f)^{\#J_0(c)}= \begin{cases} f^{\ell(c)}(1-f)^{n-\ell(c)\cdot\delta}, & \mbox{if } c_{\ell(c)}+\delta \lt n,\\ f^{\ell(c)}(1-f)^{c_{\ell(c)}}, & \mbox{otherwise.} \end{cases} \end{equation}

Definition 3.10. For $l\in\mathbb{Z}_{\geqslant0}$ and l-tuple $k=(k_1,\ldots,k_l)\in\mathbb{Z}^l_{\geqslant 0}$, we put $|k|:=k_1+\ldots +k_l$, and for m > 0, denote

(8)\begin{equation} \Delta_m^l:=\{k\in\mathbb{Z}^l_{\geqslant 0}\,|\,|k| \lt m\}. \end{equation}

Note that the cardinality of this set is the number of weak $(l+1)$-compositions of $(m-1)$ (see [Reference Stanley22, 1.2])

(9)\begin{equation} \#\Delta_m^l=\left(\!\!\binom{l+1}{m-1}\!\!\right)=\binom{m+l-1}l. \end{equation}

Lemma 3.11. For each $l\geqslant 0$, there is a bijection

\begin{equation*} \Delta^l_{n-(l-1)\delta}\ni k=(k_1,k_2,\ldots,k_l)\mapsto c(k)\in C^l_{\delta,n}, \qquad c(k)_i=(i-1)\delta+k_1+k_2+\cdots+ k_i. \end{equation*}

Lemma 3.12. For $k\in\Delta^l_{n-(l-1)\delta}$, (7) can be rewritten as follows

(10)\begin{equation} \mathbf{Pr}\left(\gamma_{\delta,n}=c(k)\right)= \begin{cases} f^l(1-f)^{|k|}, & \mbox{if } k\in\Delta^l_{n-(l-1)\delta}\smallsetminus\Delta^l_{n-l\delta}\\ f^l(1-f)^{n-l\delta}, & \mbox{if } k\in\Delta^l_{n-l\delta}. \end{cases} \end{equation}

In this section, we will study a random variable $\lambda_{\delta,n}:=\ell\left(\gamma_{\delta,n}\right)$, the length of the random chain $\gamma_{\delta,n}$, its expectation

(11)\begin{equation} L_n=L_{\delta,n}:= \mathbf{E}\lambda_{\delta,n}= \sum_{c\in C_{\delta,n}}\ell(c)\cdot\mathbf{Pr}\left(\gamma_{\delta,n}=c\right), \qquad n\geqslant 0, \end{equation}

and describe the sequence $(L_n)_{n\geqslant0}$ in terms of its ordinary generating series

(12)\begin{equation} L(t)=L^{(\delta)}(t):=\sum_{n\geqslant 0}L_nt^n. \end{equation}

Remark 3.13. In the case δ = 1,

\begin{equation*} \mathbf{Pr}\left(\gamma_{1,n}=c\right)=f^{\ell(c)}(1-f)^{n-\ell(c)},\qquad \#C_{1,n}^l=\binom nl. \end{equation*}

So we obtain binomial distribution for $\lambda_{1,n}\sim B(n,f)$ with well-known expectation

\begin{equation*} \mathbf{E}\lambda_{1,n}=nf, \qquad L^{(1)}(t):=\sum_{n\geqslant0}\mathbf{E}\lambda_{1,n}t^n=\frac{ft}{(1-t)^2}. \end{equation*}

3.2. Markov chains $X_\delta(n)$

In this subsection, we give an equivalent description of random chains $\gamma_{\delta,n}$ in terms of a family of discrete-time and time-homogeneous Markov chains X_δ.

Definition 3.14. For $\delta\in\mathbb{Z}_{ \gt 0}$, the Markov chain $X_\delta(n)\in\mathbb{Z}_{\geqslant0}$, $n\geqslant0$ is defined by a random mapping representation ([Reference Levin, Peres and Wilmer15, 1.2]):

(13)\begin{equation} X_\delta(0)=0, \qquad X_\delta(n+1)=X_\delta(n)+ \begin{cases} \xi_{n+1}, & \mbox{if } \delta|n \\ 1, & \mbox{otherwise}. \end{cases} \end{equation}

Hence, the transition matrix

\begin{equation*} A_{ij}:=\mathbf{Pr}\bigl(X_\delta(n+1)=i\,|\,X_\delta(n)=j\bigr), \qquad i,j\in\mathbb{Z}_{\geqslant0} \end{equation*}

has the following nonzero entries:

(14)\begin{equation} A_{\delta i,\delta i}=1-f, \qquad A_{i+1,i}= \begin{cases} f, & \mbox{if } \delta|i \\ 1, & \mbox{otherwise}, \end{cases} \qquad i\in\mathbb{Z}_{\geqslant0}. \end{equation}

The graph of this Markov chain is shown in Figure 1.

Figure 1.

Transition digraph of the Markov chain $X_\delta(n)$.

We adopt Dirac bra-ket notation ([Reference Dirac4], [Reference Dirac5]) from linear algebra. A state $i\in\mathbb{Z}_{\geqslant0}$ is written as a ket $|i\rangle$. Nonnegative affine combinations of states are distributions. A bra $\langle f|$ is a linear form on linear combinations of states. $\langle f|x\rangle$ is a pairing, $|i\rangle\langle j|$ for $i,j\in\mathbb{Z}_{\geqslant0}$ is the matrix element. In this term, (14) takes form

\begin{equation*} A=\sum_{i\in\mathbb{Z}}\bigl(|i+1\rangle\langle i|+(1-f)(|\delta i\rangle\langle\delta i|-|\delta i+1\rangle\langle\delta i|)\bigr). \end{equation*}

Let S be a linear operator acting on states as a shift: $S|i\rangle=|i+1\rangle$. Note that our Markov chain is δ-periodic, that is:

(15)\begin{equation} AS^\delta=S^\delta A. \end{equation}

Proposition 3.15. The random chain $\gamma_{\delta,n}$ and the family of random variables $\bigl(X_\delta(k)\bigr)_{1\leqslant k\leqslant n}$ are recovered each from other:

(16)\begin{align} \lambda_{\delta,n}&=\lceil X_\delta(n)/\delta\rceil,\qquad\qquad\qquad \end{align}

\begin{align*} \nonumber \bigl(\gamma_{\delta,n}\bigr)_k&=\max\{m\,|\,X(m)=(k-1)\delta\},\qquad 1\leqslant k\leqslant\lambda_{\delta,n} \end{align*}

\begin{align*} \nonumber X_\delta(n)&=\delta\cdot\lambda_{\delta,n}-\max\bigl\{((\gamma_{\delta,n})_{i}+\delta-n)_{1\leqslant i\leqslant \lambda_{\delta,n}},0\bigr\}. \end{align*}

Expectations of random variables X_δ, $\lambda_{\delta,n}^m$ can be expressed with the help of the corresponding linear forms $\langle X_\delta|$ and $\langle\lambda_\delta^m|$:

\begin{align*} \nonumber \mathbf{E} X_\delta(n)&=\langle X_\delta|A^n|0\rangle, \qquad \langle X_\delta|i\rangle=i, \end{align*}

(17)\begin{align} \mathbf{E}\lambda_{\delta,n}^m&=\langle\lambda_\delta^m|A^n|0\rangle, \qquad \langle\lambda_\delta^m|i\rangle=\lceil i/\delta\rceil^m. \end{align}

Compatibility with shift for $\langle\lambda_\delta^m|$ and for formal series in x takes the form:

(18)\begin{equation} \langle\lambda_\delta^m|S^\delta=\sum_{k=0}^m\binom mk\langle\lambda_\delta^k|, \end{equation}

(19)\begin{equation} \langle e^{\lambda_\delta x}|S^\delta=\langle e^{(\lambda_\delta+1)x}|. \end{equation}

Lemma 3.16. The transition matrix A satisfies the following recurrent formula:

(20)\begin{equation} A^n|0\rangle=(1-f)^n|0\rangle +\sum_{k=1}^{\min\{\delta,n\}} f(1-f)^{n-k}|k\rangle +\sum_{k=0}^{n-\delta-1}f(1-f)^{k}A^{n-\delta-k}|\delta\rangle. \end{equation}

Or in terms of the generating function $\frac1{1-tA}=\sum_{n\geqslant0}(tA)^n$:

(21)\begin{equation} (1-tA)^{-1}|0\rangle=\frac1{1-(1-f)t}|0\rangle+\sum_{k=1}^{\delta-1}\frac{ft^k}{1-(1-f)t}|k\rangle+ \frac{ft^\delta}{(1-(1-f)t)(1-tA)}|\delta\rangle. \end{equation}

Negative binomial distribution

For $r\in\mathbb{Z}_{\geqslant0}$, let the random time $\tau_{\delta,r}$ is such that $X_\delta(\tau_{\delta,r})=r\delta$ and $X_\delta(\tau_{\delta,r}+1)=r\delta+1$. Then $\tau_{\delta,r}-r\delta$ has the negative binomial distribution:

(22)\begin{equation} \tau_{\delta,r}-r\delta\sim NB(r,f),\qquad \mathbf{Pr}(\tau_{\delta,r}-r\delta=k)=\binom{k+r-1}k(1-f)^kf^r, \quad k\in\mathbb{Z}_{\geqslant0}. \end{equation}

3.3. Generating functions for L_n

Proof. The formula (26) is an explicit solution of (25) with respect to L(t).

The formula (27) follows from (26) rewritten in the form

\begin{equation*} (1-(2-f)t+(1-f)t^2-ft^\delta+ft^{\delta+1})L(t)=ft. \end{equation*}

The formula (28) for initial conditions can be easily obtained if taken into attention that in this case the length of the chain $\leqslant1$.

Corollary 3.19. The sequence $(L_n)_{n\geqslant0}$ satisfies the recurrent relation:

(29)\begin{equation} L_n+f(L_{n-1}+L_{n-2}+\cdots+L_{\max\{n-\delta+1,0\}})=nf. \end{equation}

Proof. This follows from (26) rewritten in the form

\begin{equation*} (1+ft+ft^2+\cdots+ft^{\delta-1})L(t)=\sum_{n\geqslant0}nft^n. \end{equation*}

Theorem 3.18 and the following Theorem 3.20 agree with the scheme of usage of rational generating functions [Reference Stanley22, Thm. 4.1.1] and partial fractions [Reference Henrici8, Part 7].

Denote

(30)\begin{equation} g=g(\delta,f)=\frac{f}{1+(\delta-1)f}. \end{equation}

Let’s also consider two pairs of reciprocal adjoint polynomials (see Definition A.1)

(31)\begin{equation} \begin{aligned} p_{\delta,\,f}(t)&=t^{\delta-1}+f(t^{\delta-2}+t^{\delta-3}+\cdots+1), \\ p^*_{\delta,\,f}(t)&=1+f(t+t^2+\cdots+t^{\delta-1}); \end{aligned} \end{equation}

\begin{align*} r_{\delta,\,f}(t)&=\sum_{k=0}^{\delta-2}\left(\frac{(k+1)(\delta-1)(\delta-k-2)}2f-\binom{k+2}2\right)t^k, \\ r^*_{\delta,\,f}(t)&=\sum_{k=0}^{\delta-2}\left(\frac{k(\delta-1)(\delta-k-1)}2f-\binom{\delta-k}2\right)t^k. \end{align*}

Example 3.21. For δ = 2, $g=f/(1+f)$,

\begin{equation*} L_n=f\sum_{m=0}^{n-1}(n-m)(-f)^m=gn+g^2-g^2(-f)^n. \end{equation*}

Example 3.22. For δ = 3, $g=f/(1+2f)$,

\begin{align*} L(t)&=\frac{gt}{(1-t)^2}+\frac{3g^2}{1-t}+g^2\Re\frac{\alpha}{1-tq} \\ L_n&=ng+3g^2+g^2\Re(\alpha q^n), \end{align*}

where $q=-f/2+i\sqrt{f-f^2/4}$ and $\overline q$ are the roots of $p_{3,\,f}(z)=z^2+fz+f$, and

\begin{equation*} \alpha=2\frac{r_{3,\,f}(q)}{q-\overline q}=-3+i\frac{1-5f/2}{\sqrt{f-f^2/4}}. \end{equation*}

3.4. Asymptotic for L_n

Corollary 3.23. For each fixed $f\in(0,1)$ and $n\to\infty$,

(35)\begin{equation} L_n=ng+\binom\delta2g^2+o(1). \end{equation}

In Figure 2, the values of $L_n/n$ and their approximations according (35) (up to $O(1/n)$ and $O(1/n^2)$) are shown depending on f. In Figure 2(a), we can see that even $O(1/n)$-approximation of $L(n)/n$ with only one term of (35) is very close to the value. Taking the first two terms of (35) improves this approximation and is almost undistinguished from $L(n)/n$, especially for small values of f, till 0.62, usually in practice.

Remark 3.24. Taking into account the negative binomial distribution (22) of $\tau_{\delta,r}-r\delta$ and asymptotic (35) for $\mathbf{E}\lambda$, we get the equality

(36)\begin{equation} \lim_{n\to\infty}\frac{\mathbf{E}\lambda_{\delta,n}}{n}=g=\frac{f}{1+(\delta-1)f}=\frac r{\mathbf{E}\tau_{\delta,r}}, \end{equation}

which is true for each $r\in\mathbb{Z}_{ \gt 0}$.

Figure 2.

The values of $L_n/n$ and their $O(1/n)-$, $O(1/n^2)$ approximations g, $g+\binom\delta2\frac{g^2}n$ depending on f. (a) $\frac{L_n}{n}$, g, and $g+\binom\delta2\frac{g^2}n$ for δ = 7 and n = 50. (b) g for $1\leqslant\delta\leqslant5$.

To pay more attention to the last member in the asymptotic (35), let’s consider the behavior of roots of $p_{\delta,\,f}(z)$ depending on f. For $\delta\geqslant 2$, the whole set $\bigcup_{f\in(0,1)}p_{\delta,\,f}^{-1}(0)$ of root is described in terms of variables $x=\Re z$ and $y=\Im z$.

Example 3.26. For δ = 3, Eq. (37) describes the circle $(x+1)^2+y^2=1$ or in polar form $\rho=-2\cos(\varphi)$. According to Example 3.22 for $f\in[0,4]$, roots of the polynomial $p_{3,\,f}(z)$ belong to this circle, moreover in the case $f\in(0,1)$, they are on its arc in the open unit disk $|z| \lt 1$.

For δ = 4, (37) turns into equation $x^4+2x^2y^2+y^4+2x^3+2xy^2+3x^2-y^2=0$ or in polar form $\rho^2+2\rho\cos(\varphi)+4cos^2(\varphi)-1=0$. So the pair of complex conjugate roots is described by the formulas

\begin{equation*} \rho=-\cos(\varphi)+\sqrt{1-3\cos^2(\varphi)}, \qquad \pi/3 \lt \varphi \lt \pi/2. \end{equation*}

In Figure 3, the roots of the polynomial $p_{\delta,\,f}(z)$ in the unit disk for $f\in(0,1)$ are shown in two cases δ = 4 and δ = 7.

Figure 3.

The roots of $p_{\delta,\,f}(z)$ in the unit disk $|z| \lt 1$ for $f\in(0,1)$ and $\delta=4,7$.

Note that for the roots q_k of the polynomial $p_{\delta,\,f}(t)$, the values $|q_k|/f^{\frac1{\delta-1}}$ are close to 1. Moreover, $|q_k|/f^{\frac1{\delta-1}}=1$ for $\delta=2,3$; and for any δ,

\begin{equation*} |q_k|/f^{\frac1{\delta-1}}\to 1 \qquad\text{whenever}\qquad f\to 0 \quad\text{or}\quad f\to 1. \end{equation*}

In Figure 4, the values $|q_k|/\sqrt[\delta-1]f$ for the roots q_k of the polynomial $p_{\delta,\,f}(t)$ are shown depending on f for δ = 4 and δ = 7.

Figure 4.

The values $|q_k|/\sqrt[\delta-1]f$ for the roots q_k of $p_{\delta,\,f}(t)$ depending on f for $\delta=4,7$.

Proposition 3.28. Coefficient of series (39) for L_n admits the asymptotic

(40)\begin{equation} \frac{a_{nm}}n\underset{n\to\infty}\nearrow(\delta-1)^{m}. \end{equation}

Proof.

\begin{equation*} \frac{a_{nm}}n\underset{n\to\infty}\nearrow \!\!\! \sum_{\substack{k_1,k_2,\ldots,k_{\delta-1}\geqslant 0\\k_1+k_2+\cdots+k_{\delta-1}=m}} \!\!\!\!\! \binom m{k_1,k_2,\ldots,k_{\delta-1}}=(\delta-1)^{m}. \end{equation*}

Remark 3.30. For small/large f, we have the following approximations:

(41)\begin{align} L_n&\approx fn, \qquad\;\; \text{whenever} \quad f\delta\ll 1. \end{align}

(42)\begin{align} L_n&\nearrow \lceil n/\delta\rceil, \quad \text{whenever} \quad f\nearrow 1. \end{align}

The fastest way to get (42) is to look on (45). The last formula (42) is illustrated in Figure 5 and the values of L_n and $L_n/n$ are shown depending on f for δ = 7 and $n=7\delta+r$, $r=0,1,2,3,4,5,6$.

Figure 5.

The values of L_n and $L_n/n$ depending on f for δ = 7 and $49\leqslant n\leqslant 55$. (a) L_n. (b) $L_n/n$.

3.5. Alternative formula for L_n

The rest of this section is devoted to two alternative proofs of another formula for L_n (45).

Lemma 3.31. L(t) is the product of two series:

(43)\begin{equation} L(t)=\left(\sum_{l\geqslant0}f^l\sum_{k\in\mathbb{Z}_{\geqslant0}^l}(1-f)^{l\delta+|k|}\cdot t^{|k|}\right)\cdot\sum_{k\geqslant0}\bigl(1-(1-f)^k\bigr)t^k. \end{equation}

Proof. Note that the ring $\mathbb{R}[[t]]$ of formal series is local whose maximal ideal $\frak m$ consists of the series $\sum_{n\geqslant0}a_nt^n$ with $a_0=0$. For each $a\in\frak m$, $(1-a)^{-1}=\sum_{l\geqslant 0}a^l$, where the infinite sum is defined because expression for each its coefficient at tⁿ turns into a finite sum.

From (24), we get

\begin{align*} L(t)&=\left(1-ft^\delta\sum_{k\geqslant0}(1-f)^kt^k\right)^{-1}\cdot\sum_{k\geqslant0}\bigl(1-(1-f)^k\bigr)t^k \\ &=\left(\sum_{l\geqslant0}\left(ft^\delta\sum_{k\geqslant0}(1-f)^kt^k\right)^l\right)\cdot\sum_{k\geqslant0}\bigl(1-(1-f)^k\bigr)t^k. \end{align*}

Lemma 3.32. For $l\in\mathbb{Z}_{\geqslant0}$, $n\in\mathbb{Z}_{ \gt 0}$, the following identity between polynomials from $\mathbb{Z}[f]$ is true:

(44)\begin{equation} \sum_{k\in\Delta^l_{n}}f^l(1-f)^{|k|}=1-(1-f)^{n}\sum_{m=0}^{l-1}\binom{n+m-1}{m}f^m. \end{equation}

Proof. By induction on l, we have: For l = 0, (44) takes the from $\#\Delta^0_n=1$. For $l\geqslant 1$,

\begin{align*} \sum_{k\in\Delta^l_{n}}f^l(1-f)^{|k|}&=\sum_{k'\in\Delta^{l-1}_{n}}f^{l-1}(1-f)^{|k'|}\sum_{k''=0}^{n-|k'|-1}f(1-f)^{k''} \\ &=\sum_{k'\in\Delta^{l-1}_{n}}f^{l-1}(1-f)^{|k'|}\bigl(1-(1-f)^{n-|k'|}\bigr) \\ &=\sum_{k'\in\Delta^{l-1}_{n}}f^{l-1}(1-f)^{|k'|} -\binom{n+l-2}{l-1} f^{l-1}(1-f)^{n}. \end{align*}

Proposition 3.33. L_n admits the following presentations:

(45)\begin{equation} L_n=\lceil n/\delta\rceil-\sum_{l=0}^{\lceil n/\delta\rceil-1}(1-f)^{n-l\delta}\sum_{m=0}^{l}\binom{n-l\delta+m-1}{m}f^m. \end{equation}

Proof. The first way: In terms of coefficients (43) can be rewritten as

(46)\begin{equation} \begin{aligned} L_n&=\sum_{l=0}^{\lceil n/\delta\rceil-1}\sum_{k\in\Delta^l_{n-l\delta}}f^l(1-f)^{|k|}\bigl(1-(1-f)^{n-l\delta-|k|}\bigr) \\ &=\sum_{l=0}^{\lceil n/\delta\rceil-1}\sum_{k\in\Delta^l_{n-l\delta}}f^l(1-f)^{|k|} -\sum_{l=0}^{\lceil n/\delta\rceil-1}\binom{n-l\Delta+l-1}lf^l(1-f)^{n-l\delta}. \end{aligned} \end{equation}

Then we can apply (44) to the first summand of (46).

The second way: The explicit formula for the expectation (11) together with expressions for probabilities (10) gets

\begin{align*} L_n&=\sum_{l=1}^{\lceil n/\delta\rceil-1}l\sum_{k\in\Delta^l_{n-l\delta}}f^l(1-f)^{n-l\delta}+ \sum_{l=1}^{\lceil n/\delta\rceil}l\sum_{k\in\Delta^l_{n-(l-1)\delta}\smallsetminus\Delta^l_{n-l\delta}}f^l(1-f)^{|k|} \\ &=\sum_{l=1}^{\lceil n/\delta\rceil-1}l\binom{n-l\delta+l-1}lf^l(1-f)^{n-l\delta} \\ &\qquad\qquad+\sum_{l=1}^{\lceil n/\delta\rceil}l\sum_{k\in\Delta^l_{n-(l-1)\delta}}f^l(1-f)^{|k|}-\sum_{l=1}^{\lceil n/\delta\rceil-1}l\sum_{k\in\Delta^l_{n-l\delta}}f^l(1-f)^{|k|}. \end{align*}

Again we can apply (44) to the second and third summands.

3.6. Moments and variance

Let consider the mixed generating function for moments of $\lambda_{\delta,n}$:

\begin{equation*} L_\delta(t,x):= \sum_{n\geqslant0}t^n \sum_{m\geqslant0} \mathbf{E}\lambda_{\delta,n}^m\cdot\frac{x^m}{m!}. \end{equation*}

Theorem 3.34 The mixed generating function for moments of $\lambda_{\delta,n}$ is the following:

(47)\begin{equation} L_\delta(t,x)=\frac{1+fe^x(t+t^2+\cdots+t^{\delta-1})}{1-(1-f)t-fe^xt^\delta}. \end{equation}

Proof. We describe this generating function in terms of the transition matrix A of the Markov chain.

\begin{align*} L_\delta(t,x)&=\langle e^{\lambda_\delta^m}|(1-tA)^{-1}|0\rangle \\ &\underset{\text{by (21)}}=\frac1{1-(1-f)t}\langle e^{\lambda_\delta^m}|0\rangle+\sum_{k=1}^{\delta-1}\frac{ft^k}{1-(1-f)t}\langle e^{\lambda_\delta^m}|k\rangle+ \frac{ft^\delta}{1-(1-f)t}\langle e^{\lambda_\delta^m}|(1-tA)^{-1}|\delta\rangle \\ &\underset{\text{by} (15), (19)}=\frac{1+fe^x(t+t^2+\cdots+t^{\delta-1})}{1-(1-f)t}+\frac{fe^xt^\delta}{1-(1-f)t}L_\delta(t,x). \end{align*}

Finally, the obtained identity is solved with respect to $L_\delta(t,x)$.

Lemma 3.35. For $a,b,c,d$ independent on x,

(48)\begin{equation} \left.\frac{d^m}{dx^m}\frac{ae^x+b}{ce^x+d}\right|_{x=0}=\sum_{k=1}^n\frac{k!(-c)^{k-1}(ad-bc)\cdot S(n,k)}{(c+d)^{k+1}}. \end{equation}

Corollary 3.36. For $m\in\mathbb{Z}_{ \gt 0}$, the generating function for m-th moments of $\lambda_{\delta,n}$ is the following

(49)\begin{equation} \sum_{n\geqslant0}\lambda_{\delta,n}^mt^n=\sum_{k=1}^m\frac{(-1)^{k-1}k!S(m,k)f^kt^{(k-1)\delta+1}}{(1-t)^{k+1}\bigl(1+f(t+t^2+\cdots+t^{\delta-1})\bigr)^k}. \end{equation}

Example 3.37. The generating function for second moments of $\lambda_{\delta,n}$ is the following

(50)\begin{equation} \sum_{n\geqslant0}\lambda_{\delta,n}^2t^n==\frac{ft(1 - t + f t + f t^\delta)}{(1-t)^3\bigl(1+f(t+t^2+\cdots+t^{\delta-1})\bigr)^2}. \end{equation}

Theorem 3.38 The variance of $\lambda_{\delta,n}$ admits the following asymptotic for $n\to\infty$:

(51)\begin{equation} \mathbf{Var}\lambda_{\delta,n}=n\frac{f(1-f)}{(1+(\delta-1)f)^3}+o(n). \end{equation}

Proof. Let’s consider a partial fraction decomposition of (50) in the form

\begin{equation*} \frac{A}{(1-t)^3} +\frac{B}{(1-t)^2} +\frac{C}{1-t} +\frac{R(t)}{\bigl(1+f(t+t^2+\cdots+t^{\delta-1})\bigr)^2}. \end{equation*}

The values $A=\frac{2f^2}{(1+(\delta-1)f)^2}$ and $B=\frac{f-4f^2+(\delta-1)(\delta-3)f^3}{(1+(\delta-1)f)^3}$ allow to obtain the asymptotic $\mathbf{E}\lambda_{\delta,n}^2=n^2g^2+n\frac{f-f^2+\delta(\delta-1)f^3}{(1+(\delta-1)f)^3}+o(n)$. From (35), we get $(\mathbf{E}\lambda_{\delta,n})^2=n^2g^2+n\delta(\delta-1)g^3+o(n)$. Finally, $\mathbf{Var}\lambda_{\delta,n}=\mathbf{E}\lambda_{\delta,n}^2-(\mathbf{E}\lambda_{\delta,n})^2$.

3.7. Binomial distribution with random delay

In this subsection, we generalize the longest chain distribution to the case of random delay.

Let Γ be a transition digraph of a finite Markov chain. We assume that Γ is a tree with additional loops at the root and leaves. First, we will consider a special case of such a digraph, shown in Figure 6, where the root has $s\geqslant1$ children with weights $f_i \gt 0$, $i=1,\ldots,s$ and a loop with the weight $f_0=1-f_1-\cdots-f_s \gt 0$. Each other internal vertex has a single child with the weight 1 and the maximal subchains have lengths $\delta_i\in\mathbb{Z}_{ \gt 0}$, $i=1,\ldots,s$.

Figure 6.

Finite transition digraph Γ.

Let T_s be the infinite s-ary tree considered as a digraph with all edges oriented from the root as shown in Figure 7. Vertexes of this tree are labeled by elements of the free monoid $\{1,2\ldots,s\}^*$, that is by strings in the alphabet $\{1,2\ldots,s\}$. The monoid $\{1,2\ldots,s\}^*$ acts on T_s by endomorphisms: $\{1,2\ldots,s\}^*\ni w\mapsto S_w\in\operatorname{End}(T_s)$. Restricted to the vertexes, this action is isomorphic to the left regular action: $S_w(w')=ww'$.

Figure 7.

The infinite s-ary tree T_s.

Let $\widetilde\Gamma$ be an infinite transition digraph obtained if substitute each vertex in the infinite tree T_s by the digraph Γ as shown in Figure 8. The vertices of $\widetilde\Gamma$ are labeled by tuples $i_1,\ldots,i_k;j$, where $i_1,\ldots,i_k$ are elements of $\{1,2,\ldots,s\}$ and $j\in\mathbb{Z}/\delta_{i_k}\mathbb{Z}$. So $|i_1,\ldots,i_k;\delta_{i_k}\rangle=|i_1,\ldots,i_k;0\rangle$.

Figure 8.

Transition digraph of the infinite Markov chain with random delay.

Let $(\boldsymbol\iota_n)_{n\geqslant0}$ be the sequence of independent random variables taking values $0,1,2,\ldots,s$, respectively, with probabilities $f_0,f_1,\ldots,f_s$. The following random mapping representation generalizes Definition 3.14 and describes the Markov chain $(X(n))_{\geqslant0}$ corresponding to stochastic digraph $\widetilde\Gamma$.

Definition 3.39. $X(0)=|;0\rangle$, and if $X(n)=|i_1,\ldots,i_k;j\rangle$, then

\begin{equation*} \qquad X(n+1)= \begin{cases} X(n), & \mbox{if } j=0\land\boldsymbol\iota_n=0, \\ |i_1,\ldots,i_k,\boldsymbol\iota_n;1\rangle, & \mbox{if } j=0\land\boldsymbol\iota_n\ne 0,\\ |i_1,\ldots,i_k;j+1\rangle, & \mbox{if } j\ne 0. \end{cases} \end{equation*}

Negative binomial distribution

One can generalize (22): For $r\in\mathbb{Z}_{\geqslant0}$, let the random time τ_r is such that $X_\delta(\tau_{r})=|i_1,\ldots,i_r;0\rangle$ and $X_\delta(\tau_{r}+1)=|i_1,\ldots,i_r,\boldsymbol\iota_{\tau_r};1\rangle$. Then $\tau_{r}-\sum_{r'=0}^{r-1}\delta_{\boldsymbol\iota_{\tau_{r'}}}$ has the negative binomial distribution $NB(r,1-f_0)$.

Let A be the transition matrix corresponding to the stochastic digraph $\widetilde\Gamma$. The following generalization of (21) is true.

Lemma 3.40. The Green function $(1-tA)^{-1}:=\sum_{n\geqslant0}(tA)^n$ satisfies the following recurrent relation:

(52)\begin{equation} (1-tA)^{-1}|0\rangle= \frac1{1-f_0t}|0\rangle+ \sum_{i=1}^s\sum_{k=1}^{\delta_i-1}\frac{f_it^k}{1-f_0t}|i;k\rangle+ \sum_{i=1}^s\frac{f_it^{\delta_i}}{(1-f_0t)}(1-tA)^{-1}|i;\delta_i\rangle. \end{equation}

The action of the free monoid $\{1,2,\ldots,s\}^*$ on T_s is carried over to $\widetilde\Gamma$: In terms of generators, for each $i=1,2,\ldots,s$, there exists weight preserving endomorphism of the stochastic digraph $\widetilde\Gamma$

\begin{equation*} S_i:\widetilde\Gamma\to\widetilde\Gamma, \qquad|i_1,\ldots,i_k;j\rangle\mapsto|i,i_1,\ldots,i_k;j\rangle. \end{equation*}

Hence, the corresponding linear operators commute with the transition matrix:

(53)\begin{equation} AS_i=S_iA. \end{equation}

Let us consider a function on edges of $\widetilde\Gamma$, which extends to linear functional, given by the formula

(54)\begin{equation} \langle\lambda|i_1,\ldots,i_k;j\rangle=k. \end{equation}

The subject of our interest is the random variables

(55)\begin{equation} \lambda_n:=\langle\lambda|A^n|;0\rangle. \end{equation}

For $\langle e^{\lambda x}|:=\sum_{m\geqslant0}\frac{x^m}{m!}\langle\lambda^m|$, generalization of (19) is true:

(56)\begin{equation} \langle e^{\lambda x}|S_i=\langle e^{(\lambda+1)x}|. \end{equation}

Theorem 3.41 The mixed generating function $L_(t,x):=\sum_{n\geqslant0}t^n\sum_{m\geqslant0}\mathbf{E}\lambda_{n}^m\cdot\frac{x^m}{m!}$ is given by the identity

\begin{equation*} L(t,x)=\frac{1+e^x\sum_{i=1}^sf_i(t+t^2+\cdots+t^{\delta_i-1})}{1-f_0t-e^x\sum_{i=1}^sf_it^{\delta_i}}. \end{equation*}

Proof.

\begin{align*} L(t,x)&=\langle e^{\lambda x}|(1-tA)^{-1}|0\rangle \\ &\underset{\text{by (52)}}=\frac1{1-f_0t}\langle e^{\lambda x}|0\rangle+ \sum_{i=1}^s\sum_{k=1}^{\delta-1}\frac{f_it^k}{1-f_0t}\langle e^{\lambda x}|i;k\rangle+ \sum_{i=1}^s\frac{f_it^\delta}{1-f_0t}\langle e^{\lambda x}|(1-tA)^{-1}|i;\delta_i\rangle. \\ &\underset{\text{by (53), (19)}}=\frac{1+e^x\sum_{i=1}^sf_i(t+t^2+\cdots+t^{\delta_i-1})}{1-f_0t}+\frac{e^x\sum_{i=1}^sf_it^{\delta_i}}{1-f_0t}L(t,x). \end{align*}

Like in the proof of the special case, one can use the corollary (48) of Faá di Bruno formula to obtain expressions for generating function for mth moments of λ_n very close to (49).

Theorem 3.42 The generating function $L(t)=\sum_{n\geqslant0}\mathbf{E}\lambda_nt^n$ is

\begin{equation*} L(t)=\frac{d}{dx}L(t,x)|_{x=0}=\frac{t\sum_{i=1}^sf_i}{(1-t)^2(1+\sum_{i=1}^sf_i(t+t^2+\cdots+t^{\delta_i-1}))}. \end{equation*}

Its partial fraction decomposition takes the form

\begin{align*} L(t)&=\frac{t\sum_{i=1}^sf_i}{\left(1+\sum_{i=1}^sf_i(\delta_i-1)\right)(1-t)^2} +\frac{\left(\sum_{i=1}^sf_i\right)\left(\sum_{i=1}^s\binom{\delta_i}2f_i\right)}{\left(1+\sum_{i=1}^sf_i(\delta_i-1)\right)^2(1-t)} \\ &+\frac{R(t)}{1+\sum_{i=1}^sf_i(t+t^2+\cdots+t^{\delta_i-1})}, \end{align*}

where

\begin{equation*} R(t)=\frac{\left(\sum_{i=1}^sf_i\right)\left(\sum_{i=1}^s\binom{\delta_i}2f_i\right)\sum_{i=1}^sf_i\sum_{k=0}^{\delta_i-2}(\delta_i-k-1)t^k} {\left(1+\sum_{i=1}^sf_i(\delta_i-1)\right)^2} -\frac{\left(\sum_{i=1}^sf_i\right)\sum_{i=1}^sf_i\sum_{k=0}^{\delta_i-2}\binom{\delta_i-k}2t^k} {\left(1+\sum_{i=1}^sf_i(\delta_i-1)\right)}. \end{equation*}

Similarly to (33), one can get the full fraction decomposition of L(t) and generalization of (34) for expectations $\mathbf{E}\lambda_n$. According to Example A.3, all roots of the denominator $1+\sum_{i=1}^sf_i(t+t^2+\cdots+t^{\delta_i-1})$ are out of the closed unit disc $|t| \gt 1$. So finally we get the asymptotic formula generalizing (35).

Theorem 3.43 For each fixed $f\in(0,1)$ and $n\to\infty$,

(57)\begin{equation} \mathbf{E}\lambda_{n}=\frac{n\sum_{i=1}^sf_i}{1+\sum_{i=1}^sf_i(\delta_i-1)} +\frac{\left(\sum_{i=1}^sf_i\right)\left(\sum_{i=1}^s\binom{\delta_i}2f_i\right)}{\left(1+\sum_{i=1}^sf_i(\delta_i-1)\right)^2}+o(1). \end{equation}

Let δ be a random variable taking values δ_i with probabilities $f_i/f$ for $i=1,2,\ldots,s$, where $f=f_1+f_2+\cdots+f_s$. Then the expression (57) can be rewritten in the form very close to (35)

(58)\begin{equation} L_n=ng+g^2\mathbf{E}\binom{\boldsymbol\delta}2+o(1), \qquad g=\frac{f}{1+f\mathbf{E}(\boldsymbol\delta-1)}. \end{equation}

Remark 3.44. We consider special stochastic digraphs: finite trees with loops allowed only at the root and leaves, and where all edges are oriented from the root to the leaves. We say that two such stochastic digraphs Γ and $\Gamma'$ are equivalent if

• • the bijection between leaves of Γ and leaves of $\Gamma'$ is given;

• • for each path π in Γ from the root to a leaf and the path $\pi'$ in $\Gamma'$ from the root to the corresponding leaf

  • • π and $\pi'$ have the same length,

  • • the products of the weights of all edges in π and $\pi'$ are equal.

For each such stochastic digraph Γ, there exists a unique equivalent stochastic digraph $\Gamma'$ where each internal vertex has a single child with the weight 1 (i.e., similar to the one described at the beginning of this section and shown in Figure 6).

If we apply the construction from this subsection to such a stochastic digraph Γ, then the distributions of the resulting random variables λ_n will be independent of the representative Γ of the equivalence class.

4.1. The expected number of slot leaders in a timeslot

The random number of slot leaders in the fixed jth timeslot is the sum $\nu_{j}=\sum_{i\in I}\xi_{ij}$ of independent Bernoulli random variables $\xi_{ij}\sim B(1,\varphi_f(\alpha_i))$. Thus, all ν_j are independent, and regardless of the slot index j, have the same Poisson binomial distribution:

(59)\begin{equation} \mathbf{Pr}(\nu_{j}=k)=\sum_{I'\in\binom Ik}(1-f)^{1-\sum_{i\in I'}\alpha_i}\prod_{i\in I'}\varphi_{\alpha_i}(f), \end{equation}

and (as the expected value of the sum of independent random variables)

(60)\begin{equation} \mathbf{E}\nu_j=\sum_{i\in I}\mathbf{E}\xi_{ij}=\sum_{i\in I}\varphi_f(\alpha_i). \end{equation}

Lemma 4.1. For the above random variables ν_j with Poisson binomial distribution (59), the expectation is represented by the polynomial series

(61)\begin{equation} \mathbf{E}\nu_j=f+\sum_{k\geqslant2}a_kf^k, \qquad a_k=\sum_{i\in I}\frac{\alpha_i}{k}\cdot\frac{(1-\alpha_i)(2-\alpha_i)\cdots(k-1-\alpha_i)}{(k-1)!} \end{equation}

convergent for $|f| \lt 1$.

For $\alpha\in(0,1]$, let consider the function:

(62)\begin{equation} \Phi_f(\alpha):=\frac{\varphi_f(\alpha)}\alpha=\frac{1-(1-f)^\alpha}\alpha. \end{equation}

It can be continued continuously:

(63)\begin{equation} \Phi_f(0):=\lim_{\alpha\searrow 0}\Phi_f(\alpha)=-\log(1-f). \end{equation}

One can rewrite (60) as

(64)\begin{equation} \mathbf{E}\nu_j=\sum_{i\in I}\alpha_i\Phi_f(\alpha_i). \end{equation}

In the case when all stakeholders have the stake ratio $\alpha_j=1/|I|$ and each ν_j has the binomial distribution $B(|I|,\varphi_f(1/|I|))$,

(65)\begin{equation} \mathbf{E}\nu_j=\Phi_f(1/|I|). \end{equation}

More generally, one can group stakeholders in (64) by its stake ratios:

Proposition 4.2. Suppose that I is the disjoint union $\coprod_{k\in K} I_k$ and for each $i\in I_k$ the stake ratio $\alpha_i=\beta_k/|I_k|$ depends only on k, with $\sum_k\beta_k=1$. In this case:

(66)\begin{equation} \mathbf{E}\nu_j =\sum_{k\in K}\beta_k\Phi_f(\beta_k/|I_k|). \end{equation}

Proposition 4.3. For the above random variables ν_j with Poisson binomial distribution (59), the following inequalities hold:

(67)\begin{equation} \Phi_f(1)=f\leqslant\mathbf{E}\nu_j\leqslant \Phi_f(1/|I|). \end{equation}

The lower bound is reached when a single stakeholder owns the whole stake. The upper bound is reached in the case (65) of equal stake ratios.

On Figure 9, the values of $\mathbf{E}\nu_j=\Phi_f(1/|I|)$ are shown depending on f in the case of equal stakes $\alpha_j=1/|I|$ for $|I|=1,2,3,5,10,\infty$. Here we can see that, in the case of equally distributed stake, the average number of slot leaders in a timeslot for small f (less than 0.2) does not depend on the number of stakeholders. But the difference between charts increases dramatically when f tends to 1.

Figure 9.

The values of $\mathbf{E}\nu_j=\Phi_f(1/|I|)$ depending on f for equal stakes $\alpha_j=1/|I|$. (a) $|I|=1,2,3$; $f\in[0,1]$. (b) $|I|=1,2,3,5,10,\infty$; $f\in[0,.95]$.

4.2. Efficiency

Definition 4.5. The efficiency is the ratio of the expected number of useful blocks to the expected number of all produced blocks during the epoch:

\begin{equation*} \operatorname{Eff}:=\frac{\mathbf{E}\lambda_{\delta,n}}{\sum_{0\leqslant j \lt n}\mathbf{E}\nu_j}=\frac{L_{\delta,n}}{n\cdot\sum_{i\in I}\varphi(\alpha_i)}. \end{equation*}

Note that the number of orphan blocks is $\sum_{0\leqslant j \lt n}\mathbf{E}\nu_j-\mathbf{E}\lambda_{\delta,n}$ and so the rate of orphan blocks is $1-\operatorname{Eff}$.

If $n\gg 1$, one can use the asymptotic $\mathbf{E}\lambda_{\delta,n}\approx ng$ from (35) or (58); for the case of small stakes $\max_{i\in I}\alpha_i\ll 1$, we have $\mathbf{E}\nu_j\approx\Phi_f(0)=-\log(1-f)$. So in this case, one can use the approximation

(68)\begin{equation} \operatorname{Eff}\approx\frac{g}{\Phi_f(0)}=\frac{-f/\log(1-f)}{1+(\delta-1)f}. \end{equation}

If additionally $f\ll 1$, one can replace the numerator in (68) by 1:

\begin{equation*} \operatorname{Eff}\approx\frac{1}{1+(\delta-1)f}. \end{equation*}

In both cases for the random delay presented by the data $(\delta_i,f_i)_{i=1,2,\ldots,s}$, one should put $f=f_1+f_2+\cdots+f_s$ and $\delta=\mathbf{E}\boldsymbol\delta=\frac{f_1\delta_1+f_2\delta_2+\cdots+f_s\delta_s}{f}$. As was expected, the large δ causes the smaller efficiency, but for all δ efficiency tends to 0 when f tends to 1.

One can inverse the identity (30) as

\begin{equation*} f=\frac{g}{1-(\delta-1)g}, \qquad g\in[0,1/\delta], \end{equation*}

and substituting this value in (68), we express the efficiency depending on the new dimensionless quantity $h=\delta\cdot g$, the expected length of chain produced during the propagation time δ

(69)\begin{equation} \operatorname{Eff}\approx\operatorname{Eff}_\delta(h):=\frac h{\delta\log\left(1+\frac h{\delta(1-h)}\right)}. \end{equation}

Note that $\operatorname{Eff}_\infty(h):=\lim_{\delta\to\infty}\operatorname{Eff}_\delta(h)\to 1-h$. So we get an approximative conservation law:

(70)\begin{equation} \operatorname{Eff}+h\approx 1 \qquad \text{for } \delta\gg1. \end{equation}

The deviation $\operatorname{Eff}_\delta(h)-\operatorname{Eff}_\infty(h)$ from the linear law admits the following series expansion at h = 0

\begin{equation*} \operatorname{Eff}_\delta(h)-\operatorname{Eff}_\infty(h)=\frac h{\delta\log\left(1+\frac h{\delta(1-h)}\right)}+h-1 = \frac{h}{2\delta}-\frac{h^2}{12\delta^2}+\frac{(1-2\delta) h^3}{24\delta^3}+O(h^4). \end{equation*}

On Figure 10 for $\delta=1,2,3,5,10$ is shown (a) the asymptotic of $\operatorname{Eff}$ according to (68) depending on f, (b) the deviation $\operatorname{Eff}_\delta(h)-\operatorname{Eff}_\infty(h)$.

Figure 10.

The asymptotic of efficiency. (a) $\operatorname{Eff}(f)$ for $\delta=1,2,3,5,10$. (b) $\operatorname{Eff}_\delta(h)-\operatorname{Eff}_\infty(h)$ for $\delta=1,2,3,5,10$.

4.3. About length of forks

Forks are much more dangerous in the PoS consensus protocol than in PoW. It is connected with the procedure of slot leader election, which is strictly bound to the epoch. In this case, if the fork occurs with a length more than epoch length, there will be many problems not only with canceled transactions but also with protocol operation. So the probability distribution of fork length is also of significant importance.

The notion of a fork is widely used but still lacks formalization, though intuitively one understands its meaning. It causes additional difficulties in fork length estimation.

In our model, a fork may occur because of two reasons:

1. (1) two (or more) blocks were created by different slot leaders in different timeslots during the time which is less than block propagation time;

2. (2) there are two (or more) slot leaders at the same timeslot.

Note that the influence of the first reason may be reduced, if we add the new rule for choosing a valid branch:

In this case, the forks that occur for the first reason may have only the length 1. If we consider the second reason, we note that the length of the current fork may increase only in the case when the next nonempty timeslot has more than 1 slot leader. In other words, the timeslot with only one slot leader stops the fork.

For a fixed timeslot, let $\Lambda\subseteq I$ be the random set such that $(S_i)_{i\in\Lambda}$ is the set of slot leaders in the fixed timeslot. As the special cases of (5), we get:

(71)\begin{equation} \mathbf{Pr}(\Lambda=\varnothing)=1-f, \end{equation}

and for each $i\in I$,

(72)\begin{equation} \mathbf{Pr}(\Lambda=\{i\})=(1-f)^{1-\alpha_{i}}\varphi_f(\alpha_i) \gt rapprox(1-f)\varphi_f(\alpha_i). \end{equation}

The symbol “ $ \gt rapprox$” in (72) means inequality “ $\geqslant$” and approximation “≈” whenever all $\alpha_i\ll1$. From (71) and (72), we get

\begin{align*} &\mathbf{Pr}(\Lambda\ne\varnothing)=f, \\ \Pr\left(\#\Lambda=1\right)=\sum_{i\in I}\mathbf{Pr}(\Lambda=\{i\})& \gt rapprox(1-f)\sum_{i\in I}\varphi_f(\alpha_i) \underset{\text{by (3)}}\geqslant(1-f)\varphi_f(\alpha_{I})=f(1-f). \end{align*}

Finally, we get the inequality for the conditional probability

\begin{equation*} \Pr\left(\#\Lambda \gt 1\,|\,\Lambda\ne\varnothing\right)= \frac{\Pr\left(\Lambda\ne\varnothing\right)-\Pr\left(\#\Lambda=1\right)}{\Pr\left(\Lambda\ne\varnothing\right)} \leqslant f. \end{equation*}

The value f^r from the lemma above can be considered an upper bound on the probability that a new fork of length r will begin in a fixed habitable timeslot.

4.4. Conclusion

The paper’s results allow us to estimate different important parameters, connected with the operation of Ouroboros-based blockchains. Within the framework of our mathematical model, explicit analytical expressions for the quantities of interest to us are obtained, allowing convenient approximation. They were nevertheless obtained under the following simplified assumptions:

• • all slot leaders in the current epoch are honest and act according to the consensus protocol;

• • block propagation time is either a constant or random variable with finite support;

• • epoch length is sufficiently large to use the asymptotics.

Deviations from at least one of these assumptions cause additional analytical problems, which, in turn, essentially complicate research. The most interesting and promising direction of the next research may be an estimation of the longest chain length under the presence of an adversary, which tries to split the blockchain. Note that this problem can’t be solved by simply reducing the value of the active slot coefficient proportionally to the stake of honest slot leaders, because Adversary may try to use honest stakeholders’ potential for splitting, by supporting the creation of chains of equal length. We also have a conjecture that the presence of an adversary cannot decrease the speed of chain growing by honest participants, though adversarial activity increases the length of forks and block stabilization time together with the share of orphan blocks, as well as decreases chain quality.

Note that results characterizing blockchain behavior in a model without adversary are also of great interest. In reality, most of the time the blockchain operates without any visible attacks. For example, during Cardano’s work (since 2018), no double spending or splitting attacks were fixed—maybe because of protocol security, which makes such attacks impractical. Moreover, knowing the expected characteristics of the chain parameters (length of the longest fork, growth of the chain, number of lost blocks, etc.) allows an observer to distinguish some deviations in the behavior of the chain that may be caused by the activity of an attacker, and to be aware of his malicious actions.

The obtained results may also be used by developers to choose blockchain parameters, first of all, active slot coefficient f, according to their preferences, such as to increase the speed of chain growing or to minimize the orphan block ratio.