This article proposes a stochastic model to obtain the end-to-end delay law between two nodes of a Delay Tolerant Network (DTN). We focus on the commonly used Binary Spray and Wait (BSW) routing protocol and propose a model that can be applied to homogeneous or heterogeneous networks (i.e. when the inter-contact law parameter takes one or several values). To the best of our knowledge, this is the first model allowing to estimate the delay distribution of Binary Spray and Wait DTN protocol in heterogeneous networks. We first detail the model and propose a set of simulations to validate the theoretical results.
Delay Tolerant Networks (DTN) is a concept initially created for interplanetary networks [6]. However, it also receives a great success for intermittently connected networks and particularly for opportunistic networks [5]. In these networks, a node can send data to another if both are in the same transmission range. Due to the dynamic character of these networks, there is no guarantee that a direct connected path from a given source to a given destination exists at any time. As a result, routing protocols using relay nodes and replication such as MaxProp [3], Spray and Wait [13], PRoPHET [12] and RAPID [2] have been proposed to increase the message delivery ratio over such intermittently connected networks.
The performance evaluation of relay protocols in terms of message delivery ratio, end-to-end delivery delay or throughput is a difficult task due to the complexity to drive mobile network simulations. Several efforts have been done in order to assess the performance of routing schemes with simulations. Today, The ONE simulator became a referent tool in this area [1]. Other approaches have proposed Markovian and ordinary differential equations (ODEs) models to study the performance of some basic routing protocols such as Epidemic, Epidemic limited, 2-hop routing and 2-hop limited routing protocols [14], [7] while others focus on the ressource constraints issues in these networks [15], [10]. However, all these models do not consider both Binary Spray and Wait (BSW) routing protocol and different inter-contact law parameters (called in this study heterogeneous case).
In this paper, we introduce a Markovian model to obtain the end-to-end delivery delay law and the average delivery ratio of an intermittently connected network. Compare to previous existing works, we propose to fill a gap by introducing a model of the commonly-used Binary Spray and Wait routing protocol in both homogeneous and heterogeneous cases. Indeed, in most DTN routing studies, this protocol is used as a reference for comparison purpose as BSW has been proved to be optimal in a fully random network [13]. To the best of our knowledge this is the first model proposed for BSW performances. Section 2.1 presents and justifies the assumptions chosen and sums up the notations used inside this paper. In Section 3, we first propose a BSW model for the homogeneous case. This model is then extended to handle heterogeneous networks in Section 4. In each section we provide examples to assess the consistency and efficiency of the developed model and compare the results obtained with The ONE simulator. Section 5 concludes this work and details the future work.
Before presenting the assumptions used to build our model, we first recall how the BSW routing protocol operates.
The source node of a message initially starts with a fixed number of copies denoted L. This number is called the replication factor. Then, the spray phase is directed by the following rule: any node that has strictly more than one message copy (source or relay) gives half of its copies to the first node (without copies) encountered. When a node has only one copy, it switches to the wait phase and give its copy to and only to the destination.
Our model is based on two main assumptions:
the model does not consider buffer constraints (i.e. losses resulting from congestion) and losses due to link failure. That means that we model a case where each contact is long enough to send and/or receive all required messages. Note that the case of congestion is discussed later in Section 5;
we consider all inter-contact laws as exponential. Following [9], the authors show that the time scale of interest for opportunistic forwarding may be of the same order as the characteristic time, and thus the exponential tail is important. As a result, the exponential distribution of inter-contact is meaningful and justifies a Markovian model. In this paper, the authors also claim that the choice of a power law (as proposed in [4]) in these cases leads to pessimistic results. The use of exponential laws is clearly justified, however it would be interesting to qualify and quantify the error done with such an assumption in a case of network characterized by different inter-contact laws. This problem will be tackled in a future work.
We consider a network with N nodes, noted ni, i ∈ {1, .., N }. ∀(i, j) ∈ {1, .., N } × {1, .., N }, i = j, the inter-contact law between ni and nj is an exponential law of parameter λi,j = λj,i. In our study, we also consider homogeneous networks that means ∀(i, j) ∈ {1, .., N } × {1, .., N }, i = j, λi,j = λ. Thus, there is only one parameter: λ. Previous notations are summed up in Table 1.
The model is done in two parts. First, we build a Markov chain representing the dissemination of copies in the network with an absorbing state corresponding to the delivery of the message. Then, we apply the first hitting time theorem [11] between the initial state representing the cr
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