Reliable Transport in Delay Tolerant Networks

Abstract

In this paper, we provide a holistic picture of the research efforts towards the design and development of transport protocols for DTN environments. In the first part, we provide an exhaustive and insightful survey of the literature on transport protocols and proposals aimed at DTNs. In the second part, we describe a new reliable transport protocol based on coding. Our proposed protocol is targeted at terrestrial DTN environments consisting of a large number of highly mobile nodes with random mobility. The key idea behind our proposal is that the average dynamics under such a network setting can be captured by a fluid-limit model and the protocol parameters can be optimized based on the fluid-limit model. Through simplified versions of our proposal, we guide the readers in a step-by-step manner through the intricacies of obtaining deterministic fluid-limit models for networks where the dynamics can be stochastically modeled by a continuous time Markov chain with a large state space. We also provide the relevant background material so as to help the readers clearly understand the methodology and enable him/her to apply the technique to their own research problems.

Appendices

Appendix A: Fluid-Limits and Fluid Approximations

In this appendix, we provide a brief background on fluid limits and fluid approximations. Please refer to [27, 56, 65, 66] and [97] for more details.

Intuitively speaking, a fluid-limit is a limit of a sequence of stochastic processes. The fluid approximation provides the first-order deterministic approximation to a stochastic process, and represents its average behavior.

Consider a sequence $\{Z^{(n)}(t), t \ge 0\}$, $n = 1, 2, \dots$, of stochastic processes. The index $n$ represents some quantity which is scaled up to infinity in order to study the sequence of processes at the limit, as ${n \uparrow \infty }$. For queuing systems, $n$ might represent “the number of servers” (as in infinite server approximations) or “a multiplying factor of one or more transition rates” (as in heavy-traffic approximations) or some other quantity w.r.t. which the scaling is performed. In our context of a mobile network, $n$ might represent the number of mobile nodes. For a chemical reaction, $n$ might represent the number of molecules and so on.

Recall that the Strong Law of Large Numbers (SLLN) (resp. Weak Law of Large Numbers (WLLN)) says that, under suitable conditions, the average of $n$ random variables with a common mean $\mu$, converges almost surely (resp. in probability) to $\mu$, as $n \uparrow \infty$. Consider the SLLN (or WLLN) type rescaling $z^{(n)}(t) := Z^{(n)}(t)/n$. Under certain conditions, as $n \uparrow \infty$, the sequence of rescaled processes $\{z^{(n)}(t), t \ge 0\}$, $n = 1, 2, \dots$, converges almost surely (or sometimes in probability) to a deterministic process $\{z(t), t \ge 0\}$ (see, for example, Theorem 4.1 of [66]). Then, the limit $\{z(t), t \ge 0\}$ is called the fluid limit associated with the sequence $\{Z^{(n)}(t), t \ge 0\}$, $n = 1, 2, \dots$, and the approximation

$$\begin{aligned} Z^{(n)}(t) \approx n z(t) , \quad \forall t \ge 0 , \end{aligned}$$

(10.12)

is called the fluid approximation for the $n$th system.

Appendix B: Density-Dependent Markov Chains

In this appendix, we recall a known fluid-limit result for the so-called density dependent Markov chains. First, we fix some notation. The set of integers (resp. real numbers) is denoted by $\mathbb{Z }$ (resp. $\mathbb{R }$). The space of $d$-dimensional vectors with integer (resp. real) components is denoted by $\mathbb{Z }^d$ (resp. $\mathbb{R }^d$). The absolute value of a scalar $b$ is denoted by $|b|$. The norm of a vector $z$ is denoted by $\Vert z\Vert$. The transpose of a vector $z$ (resp. a matrix $G$) is denoted by $z^T$ (resp. $G^T$).

Consider a one-parameter family of continuous time Markov chains $\{Z^{(N)}(t),$ ${t \ge 0}\}$, indexed by $N = 1, 2, \dots$, where $\{Z^{(N)}(t)\}$ has state space $\mathcal{S }^{(N)} \subset \mathbb{Z }^d$ and transition rate matrix $Q^{(N)} = [q^{(N)}(Z,Z^{\prime })]$, $Z,Z^{\prime } \in \mathcal{S }^{(N)}$.

Definition 10.1

(Density Dependent Markov Chains [27, 56]) The family of Markov chains $\{Z^{(N)}(t), t \ge 0\}$, $N = 1, 2, \dots$, is called density-dependent if there exist a subset $\mathcal{R }$ of $\mathbb{R }^d$ and continuous functions $f_l$, $l \in \mathbb{Z }^d$, with $f_l : \mathcal{R } \rightarrow \mathbb{R }$, such that

$$\begin{aligned} \quad \quad \quad \quad q^{(N)}(Z,Z+l) = N f_l \left( \frac{Z}{N}\right) , \quad \quad l \ne 0. \quad \quad \quad \quad \end{aligned}$$

In practice, instead of considering all possible $l \in \mathbb{Z }^d$, one only needs to consider the much smaller set

$$\begin{aligned} \mathcal{L } = \{l \in \mathbb{Z }^d : l \ne 0, q^{(N)}(Z,Z+l) \ne 0 \; \text{ for } \text{ some } \; Z \in \mathcal{S }^{(N)}\}, \end{aligned}$$

whose elements correspond to (actual) transitions of positive rate. For all $l \notin \mathcal{L }$, one can set $f_l$ identically equal to zero. Henceforth, we only consider transitions with positive rates and denote the total number of such transitions by $|\mathcal{L }|$.

Define the drift function $F(\cdot )$ by

$$\begin{aligned} F(w) := \sum _l l f_l(w), \quad w \in \mathcal{R }. \end{aligned}$$

(10.13)

Note that

$$\begin{aligned} F(w) = (F_1(w), \dots , F_d(w))^T \end{aligned}$$

is a $d$-dimensional column vector of functions, because $l$ is a $d$-dimensional column vector. Defining the density process $\{z^{(N)}(\cdot )\}$ by

$$\begin{aligned} z^{(N)}(t) = \frac{Z^{(N)}(t)}{N}, \end{aligned}$$

we recall the Functional Strong Law of Large Numbers (FSLLN) for density-dependent Markov chains (see [Chap. 11, Theorem 2.1][27]).

Theorem 10.1

(Kurtz [27]) Suppose that for each compact set $K \subset \mathcal{R }$,

$$\begin{aligned} \sum _l \Vert l\Vert \sup _{w \in K} f_l(w) < \infty , \end{aligned}$$

and there exists $M_K > 0$ such that

$$\begin{aligned} \Vert F(w) - F(w^{\prime })\Vert \le M_K \Vert w-w^{\prime }\Vert , \quad \forall w, w^{\prime } \in K. \end{aligned}$$

Suppose also that

$$\begin{aligned} \lim _{N \rightarrow \infty } z^{(N)}(0) = z_0, \end{aligned}$$

and $z(\cdot )$ satisfies

$$\begin{aligned} z(t) = z_0 + \int _{0}^t F(z(u)) {{d}}u, \quad t \ge 0. \end{aligned}$$

(10.14)

Then, for every $t$, $0 \le t < \infty$,

$$\begin{aligned} \lim _{N \rightarrow \infty } \sup _{0 \le s \le t} \Vert z^{(N)}(s)-z(s)| = 0, \end{aligned}$$

almost surely.

Remark 10.2

Theorem 1 says that, when the drift function $F(\cdot )$ is uniformly bounded and Lipschitz continuous over compact subsets of $\mathcal{R }$, then, as $N \rightarrow \infty$, the density process $\{z^{(N)}\}$ converges uniformly over compact subsets (u.o.c.) to a deterministic function $z(\cdot )$, almost surely (a.s.).

Appendix C: Derivation of Eq. (10.1)

In this appendix, we provide justifications for the ODE models which we have used as approximations for modeling the network dynamics when the number of relays $N$ is large. We formally derive only Eq. (10.1). Derivation of fluid-limit models is similar for the other cases.

In the case of single packet transfer without buffer expiry, the positive transition rates of the Markov chain $\{(X^{(N)}(t), Y^{(N)}(t)), t \ge 0\}$ out of the state $(X,Y)$ are given by

$$\begin{aligned} \left( \begin{array}{c} X \\ Y \end{array} \right) \longrightarrow \left( \begin{array}{c} X+1 \\ Y \end{array} \right) \quad \text{ at } \text{ rate } \quad \left( \beta ^{(N)}_s n_s^{(N)} + \beta ^{(N)}_r X \right) (N-X-Y) \end{aligned}$$

$$\begin{aligned} \left( \begin{array}{c} X \\ Y \end{array} \right) \longrightarrow \left( \begin{array}{c} X \\ Y+1 \end{array} \right) \quad \text{ at } \text{ rate } \quad \beta ^{(N)}_r Y (N-X-Y) \end{aligned}$$

$$\begin{aligned} \left( \begin{array}{c} X \\ Y \end{array} \right) \longrightarrow \left( \begin{array}{c} X-1 \\ Y+1 \end{array} \right) \quad \text{ at } \text{ rate } \quad \beta ^{(N)}_d n_d^{(N)} X + \beta ^{(N)}_r XY \end{aligned}$$

Writing $Z = (X,Y)$, the positive transition rates out of the state $(X,Y)$ of the Markov chain $\{(X^{(N)}(t), Y^{(N)}(t)), t \ge 0\}$ can be written in the density-dependent form $N f_l\left( \frac{Z}{N}\right)$ given by (see Appendix B for a meaning of the density-dependent form)

$$\begin{aligned} N f_l\left( \frac{Z}{N}\right)&= \left\{ \begin{array}{l} N \left( \displaystyle \left( (N\beta ^{(N)}_s) \left( \frac{n_s^{(N)}}{N}\right) + (N\beta ^{(N)}_r) \left( \frac{X}{N}\right) \right) \left( 1-\frac{X}{N}-\frac{Y}{N}\right) \right) \\ \text{ for } \quad l = (1,0)^T, \\ \\ N \left( \displaystyle (N\beta ^{(N)}_r) \left( \frac{Y}{N}\right) \left( 1-\frac{X}{N}-\frac{Y}{N}\right) \right) , \quad \text{ for } \quad l = (0,1)^T, \\ \\ N \left( \displaystyle (N\beta ^{(N)}_d) \left( \frac{n^{(N)}_d}{N}\right) \left( \frac{X}{N}\right) + (N \beta ^{(N)}_r) \left( \frac{X}{N}\right) \left( \frac{Y}{N}\right) \right) , \\ \text{ for } \quad l = (-1,1)^T, \end{array} \right. \end{aligned}$$

provided that $\lambda _r := N \beta ^{(N)}_r$, $\lambda _s := N \beta ^{(N)}_s$, $\lambda _d := N \beta ^{(N)}_d$, $s := n^{(N)}_s/N$ and $d := n^{(N)}_d/N$ are constants, independent of $N$ (which we assume, as in [103]). Let $z=(x,y)$. Defining $f_l(z) = f_l(x,y)$ and $F(z) = F(x,y)$ by

$$\begin{aligned} f_l(x,y) = \left\{ \begin{array}{ll} (s \lambda _s + \lambda _r x) (1-x-y), &{} \quad l = (1,0)^T, \\ \lambda _r y (1-x-y), &{} \quad l = (0,1)^T, \\ {{d}} \lambda _d x + \lambda _r xy, &{} \quad l = (-1,1)^T, \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} F(x,y) = \sum _{l} l f(x,y,w) = \left[ \begin{array}{c} (s \lambda _s + \lambda _r x) (1-x-y) - \lambda _r xy - d \lambda _d x \\ {{d}} \lambda _d x + \lambda _r y (1-x-y) + \lambda _r xy \end{array} \right] , \end{aligned}$$

respectively, and applying Theorem 1 (see Appendix B), we observe that, as $N \rightarrow \infty$, the rescaled process $\left\{ \left( \frac{X^{(N)}(t)}{N}, \frac{Y^{(N)}(t)}{N}\right) , t \ge 0\right\}$ converges almost surely to the unique solution of the ODE

$$\begin{aligned} \frac{{{d}}z(t)}{{{d}}t} = F(z(t)), \end{aligned}$$

i.e., to the unique solutions $x(t)$ and $y(t)$ of the ODEs given by Eq. (10.1), with initial conditions $x(0) := \displaystyle \lim _{N \rightarrow \infty } \frac{X(0)}{N} = 0$ and $y(0) := \displaystyle \lim _{N \rightarrow \infty } \frac{Y(0)}{N} = 0$.