On the packet delivery delay study for three-dimensional mobile ad hoc networks
Introduction
Three-dimensional mobile ad hoc networks (3D MANETs) are a class of flexible and distributed peer-to-peer networks, where mobile nodes move within 3D space and can communicate with each other via wireless link without any pre-existing infrastructure. As 3D MANETs can be rapidly deployed and flexibly reconfigured, they are appealing many critical applications: various military units communication (i.e., aircrafts, troops, and fleets) for modern combat, underwater vehicles communication for ocean surveillance, and unmanned aerial vehicles communication for disaster monitoring. To support these applications with different delay requirements in 3D MANETs, understanding the packet delivery delay performance in such networks is of fundamental importance.
The packet delivery delay performance for 2D MANETs has been extensively studied in the literature, in terms of its order sense scaling laws with network size or its closed-form analytical models (see Section 2 for related work). However, all the above work is conducted on 2D MANETs only. Recently, some initial work has focused on the study of performance for 3D MANETs, such as throughput capacity [1] and delivery rate [2], which are defined as the maximum packet input rate that the network can stably support and the probability that a packet is successfully transmitted to its destination, respectively.
It is notable that the packet delivery delay performance in 3D MANETs has not been investigated, which significantly hinders their applications. Different from the aforementioned work, this paper studies the packet delivery delay performance in 3D MANETs. This paper made a significant improvement on our previous work of Wang et al. [3]. In [3], we only studied the packet delivery delay. In this paper, we study the packet delivery delay as well as corresponding relative standard deviation. We also add new simulation results to validate the theoretical models on the packet delivery delay and corresponding relative standard deviation under the random walk and random waypoint mobility models, besides independent and identically distributed (i.i.d.) mobility model [4]. More simulated and numerical results with different parameters are further provided to do performance analysis and show the packet delivery performance in 3D MANET is different with that in 2D MANET. The main contributions of this paper are summarized as follows.
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First, a Markov chain theoretical framework is developed to model the packet delivery process under the two-hop relay algorithm with packet replication.
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Then, based on the developed theoretical framework, closed-form expressions are further derived for mean and relative standard deviation of packet delivery delay.
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Finally, simulation and numerical results are provided to validate our theoretical models and illustrate our findings.
The remainder of this paper is organized as follows. We review related work in Section 2. We introduce system models in Section 3. Section 4 presents two-hop relay algorithm with packet replication and corresponding transmission scheduling. In Section 5, we first develop a Markov chain theoretical framework and derive some related basic probabilities. Section 6 derives closed-form expressions for mean and relative standard deviation of packet delivery delay. Simulation and numerical results are presented in Section 7. Finally, Section 8 concludes this paper.
Section snippets
Related work
There have been many research efforts in the literature to study the packet delivery delay performance in MANETs. The packet delivery delay performance in two-hop relay MANETs is studied in [5], [6], [7], where Gamal et al. [5] consider random walk mobility model, Mammen and Shah [6] consider restricted mobility model, and Lin et al. [7] consider Brownian mobility model. Later, the packet delivery delay performance is explored in two-hop relay MANETs under discrete random direction model and
Network model
We consider a time-slotted network with n mobile nodes uniformly distributed in a unit cube area. The cube area is evenly divided into m × m × m equal-sized cells, as shown in Fig. 1. The mobile nodes roam from one cell to another following independent and identically distributed (i.i.d.) mobility model [4]. Under i.i.d. mobility model, at the beginning of each time slot, each node independently selects one from all m3 cells with the equal probability to move into, and stays at the selected
Two-hop relay algorithm with packet replication
We adopt two-hop relay algorithm for packet routing in 3D MANETs [13], as shown in Fig. 2. Under this algorithm, packet delivery process can be summarized as two phases. In phase 1, a packet is transmitted to an intermediate node (relay node) from its source node, and then in phase 2, the packet is transmitted to its destination node from the relay node. It is notable that the source node can directly transmit a packet to its destination node once such a transmission opportunity arises, and
Markov chain theoretical framework
In this section, we develop a Markov chain theoretical framework to depict the packet delivery process under two-hop relay algorithm with packet replication, and derive some related basic probability results.
Packet delivery delay modeling
With the help of the Markov chain theoretical framework and related basic probability results in 5, this section gives the derivation process of the closed-form expressions for expected value and relative standard deviation of packet delivery delay under the two-hop relay algorithm with packet replication. We first introduce the following definition of packet delivery delay.
Definition 1 For a tagged flow and a given packet, the delivery delay of a packet in considered 3D MANET is defined as time duration
Numerical results
In this section, we first provide simulation results to validate our theoretical models, and then illustrate the impact of network parameters on the packet delivery delay performance.
Conclusion
In this paper, we first develop a Markov chain theoretical framework to depict the packet delivery process under two-hop relay algorithm with packet replication. With the help of the Markov chain theoretical framework, we then derive closed-form expressions for mean and relative standard deviation of packet delivery delay. Simulation results indicate that our theoretical models can accurately predict packet delivery delay performance in 3D MANETs. Remarkably, our theoretical results indicate
Acknowledgment
This work is supported by the NSF of China Grants 61472057 and 61702068, Anhui Education Department Grants Kj2015A283, gxyqZD2016331 and KJ2017A425, Anhui Province’s Department of Human Resources and Social Security for the Returned Overseas Chinese Scholars, Talented Team of Computer System Architecture, Chuzhou University Grant 2015jsgg01, Kunming University Grant 2015CXTD04, and Reform and Innovation Project of Application-oriented Talents Training of Kunming University in 2015-Construction
Wu Wang received his B.S. and M.S. degrees both in information science and engineering from Yunnan University, Yunnan, China in 2003 and 2007, respectively. He is currently a Ph.D. candidate at the Graduate School of Systems Information Science at Future University Hakodate, Japan. His research interests include performance modeling and evaluation of wireless networks.
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Wu Wang received his B.S. and M.S. degrees both in information science and engineering from Yunnan University, Yunnan, China in 2003 and 2007, respectively. He is currently a Ph.D. candidate at the Graduate School of Systems Information Science at Future University Hakodate, Japan. His research interests include performance modeling and evaluation of wireless networks.
Bin Yang received his B.S. and M.S. degrees both in computer science from Shihezi University, China, in 2004 and from China University of Petroleum, Beijing Campus, in 2007, and Ph.D. degree in systems information science from Future University Hakodate, Japan in 2015, respectively. He is currently an associate professor at the School of Computer and Information Engineering, Chuzhou University, China, and is also a postdoctoral researcher at the School of Computer Science, Shaanxi Normal University, China. His research interests include performance modeling and evaluation, stochastic optimization and control in wireless networks.
Osamu Takahashi received his Ph.D. degree from Hokkaido University in 1975. He worked for NTT research laboratory and NTTDoCoMo research laboratory. He is currently a professor at the Department of System Information Science at Future University Hakodate. His research interest includes ad hoc networks, network security, and mobile computing. He is a fellow of IPSJ (Information Processing Society of Japan) and a member of IEEE and IEICE.
Xiaohong Jiang received his B.S., M.S. and Ph.D. degrees all from Xidian University, China. He is currently a full professor of Future University Hakodate, Japan. Dr. Jiang was an associate professor of Tohoku University, Japan, from February 2005 to March 2010, an assistant professor in Japan Advanced Institute of Science and Technology (JAIST), from October 2001 to January 2005. Dr. Jiang was a JSPS research fellow at JAIST from October1999 to October 2001. He was a research associate in the University of Edinburgh from March 1999 to October 1999. Dr. Jiang’s research interests include computer communications networks, mainly wireless networks, optical networks, etc. He has published over 250 technical papers at premium international journals and conferences, which include over 40 papers published in top IEEE journals and conferences like IEEE/ACM Transactions on Networking, IEEE Journal of Selected Areas on Communications, INFOCOM, etc. Dr. Jiang was the winner of the Best Paper Award and Outstanding Paper Award of IEEE HPCC 2014, IEEE WCNC 2012, IEEE WCNC 2008, IEEE ICC 2005-Optical Networking Symposium, and IEEE/IEICE HPSR 2002. He is a Senior Member of IEEE.
Shikai Shen received his B.S. and M.S. degrees from Yunnan Normal University in 1984 and from Yunnan University in 2003, respectively. He is currently a professor of Kunming University, China. His research interests include wireless sensor networks, network coding, internet of things, etc. He is a member of China Computer Federation.