On classes of Bitcoin-inspired infinite-server queueing systems

Abstract

We analyze the time-dependent behavior of various types of infinite-server queueing systems, where, within each system we consider, jobs interact with one another in ways that induce batch departures from the system. One example of such a queue was introduced in the recent paper of Frolkova and Mandjes (Stochastic Models, 2019) in order to model a type of one-sided communication between two users in the Bitcoin network: here we show that a time-dependent version of the distributional Little’s law can be used to study the time-dependent behavior of this model, as well as a related model where blocks are communicated to a user at a rate that is allowed to vary with time. We also show that the time-dependent behavior of analogous infinite-server queueing systems with batch arrivals and exponentially distributed services can be analyzed just as thoroughly.

Appendix

This section contains a slight rephrasing of Propositions 4.1 and 4.2 on page 17 of [15]. Throughout this appendix, we assume $\{X_{n}\}_{n \ge 1}$ is a sequence of i.i.d. nonnegative random variables having a finite, positive mean, and we set $\lambda := 1/\mathbb {E}[X_{1}]$. From these random variables, we define the partial sums $\{S_{n}\}_{n \ge 1}$ as

$$\begin{aligned} S_{n} := \sum _{k=1}^{n}X_{k}, ~~~~~~ n \ge 1. \end{aligned}$$

Proposition 4.1

As $n \rightarrow \infty$,

$$\begin{aligned} \frac{1}{n^{2}}\sum _{\ell = 1}^{n}S_{\ell } \rightarrow \frac{1}{2\lambda } \end{aligned}$$

with probability one.

Proof

This result can be proven through an application of the classical Strong Law of Large Numbers: we omit the details. $\square$

Proposition 4.2

Let $\{X_{n}\}_{n \ge 1}$ be a sequence of i.i.d random variables having finite mean, and fix two real numbers x, y satisfying $0< y < x$. Then, by setting $\rho := \lambda /\mu$, and letting $\mu \downarrow 0$ ($\lambda$ stays fixed) we find that

$$\begin{aligned} \frac{\alpha }{\sqrt{\rho }} \sum _{\ell = \lceil \sqrt{\rho } y \rceil }^{\lceil \sqrt{\rho } x \rceil }X_{\ell } + \sum _{\ell = \lceil \sqrt{\rho } y \rceil }^{\lceil \sqrt{\rho } x \rceil }\ell \mu X_{\ell } \Rightarrow \frac{\alpha (x-y)}{\lambda } + \frac{(x^{2} - y^{2})}{2} \end{aligned}$$

(36)

as $\rho \rightarrow \infty$. Furthermore,

$$\begin{aligned} \frac{\alpha }{\sqrt{\rho }}\sum _{\ell = 1}^{\lceil \sqrt{\rho } x \rceil }X_{\ell } + \sum _{\ell = 1}^{\lceil \sqrt{\rho } x \rceil }\ell \mu X_{\ell } \Rightarrow \frac{\alpha x}{\lambda } + \frac{x^{2}}{2} \end{aligned}$$

(37)

as $\rho \rightarrow \infty$.

Proof

First, note that for any two integers m, n satisfying $1 \le m \le n$, we have

$$\begin{aligned} \frac{\alpha }{\sqrt{\rho }}\sum _{\ell = m}^{n}X_{\ell } + \sum _{\ell = m}^{n}\mu \ell X_{\ell }&{\mathop {=}\limits ^{d}}\frac{\alpha }{\sqrt{\rho }}\sum _{\ell = m}^{n}X_{n - \ell + 1} + \frac{\lambda }{\rho }\sum _{\ell = m}^{n}\ell X_{n - \ell + 1} \\= & {} \frac{\alpha }{\sqrt{\rho }}\sum _{\ell = m}^{n}X_{n - \ell + 1}\\&+\, \frac{\lambda (m-1)}{\rho }\sum _{\ell = m}^{n}X_{n - \ell + 1} + \frac{\lambda }{\rho }\sum _{\ell = m}^{n}(\ell - (m-1))X_{n - \ell + 1} \\= & {} \frac{\alpha }{\sqrt{\rho }}S_{n - m + 1} + \frac{\lambda }{\rho }(m-1)S_{n - m + 1} + \frac{\lambda }{\rho }\sum _{\ell = m}^{n}\sum _{z = m}^{\ell }X_{n - \ell + 1} \\= & {} \frac{\alpha }{\sqrt{\rho }}S_{n - m + 1} + \frac{\lambda }{\rho }(m-1)S_{n - m + 1} + \frac{\lambda }{\rho }\sum _{z = m}^{n}\sum _{\ell = z}^{n}X_{n - \ell + 1} \\= & {} \frac{\alpha }{\sqrt{\rho }}S_{n - m + 1} + \frac{\lambda }{\rho }(m-1)S_{n - m + 1} + \frac{\lambda }{\rho }\sum _{z = m}^{n}S_{n - z + 1} \\= & {} \frac{\alpha }{\sqrt{\rho }}S_{n - m + 1} + \frac{\lambda }{\rho }(m-1)S_{n - m + 1} + \frac{\lambda }{\rho }\sum _{\ell = 1}^{n - m + 1}S_{\ell }. \end{aligned}$$

Using both the classical Strong Law of Large Numbers, as well as Proposition 4.1, we find

$$\begin{aligned}&\frac{\alpha }{\sqrt{\rho }} \sum _{\ell = \lceil \sqrt{\rho } y \rceil }^{\lceil \sqrt{\rho } x \rceil }X_{\ell } + \sum _{\ell = \lceil \sqrt{\rho } y \rceil }^{\lceil \sqrt{\rho } x \rceil }\ell \mu X_{\ell }\\&\quad {\mathop {=}\limits ^{d}} \frac{\alpha }{\sqrt{\rho }}S_{\lceil \sqrt{\rho } x \rceil - \lceil \sqrt{\rho } y \rceil + 1}\\&\qquad + \, \lambda \left[ \frac{\lceil \sqrt{\rho } y \rceil - 1}{\sqrt{\rho }}\right] \frac{1}{\sqrt{\rho }}S_{\lceil \sqrt{\rho } x \rceil - \lceil \sqrt{\rho } y \rceil + 1} + \frac{\lambda }{\rho }\sum _{\ell = 1}^{\lceil \sqrt{\rho } x \rceil - \lceil \sqrt{\rho } y \rceil + 1}S_{\ell } \\&\quad {\mathop {\rightarrow }\limits ^{a.s}} \frac{\alpha (x-y)}{\lambda } + y(x-y) + \frac{(x-y)^{2}}{2} \\&\quad = \frac{\alpha (x-y)}{\lambda } + \frac{(x^{2} - y^{2})}{2}, \end{aligned}$$

which proves that

$$\begin{aligned} \frac{\alpha }{\sqrt{\rho }} \sum _{\ell = \lceil \sqrt{\rho } y \rceil }^{\lceil \sqrt{\rho } x \rceil }X_{\ell } + \sum _{\ell = \lceil \sqrt{\rho } y \rceil }^{\lceil \sqrt{\rho } x \rceil }\ell \mu X_{\ell } \Rightarrow \frac{\alpha (x-y)}{\lambda } + \frac{(x^{2} - y^{2})}{2} \end{aligned}$$

as $n \rightarrow \infty$, thus proving (36). Finally, observe that statement (37) can be established with the same type of argument. $\square$