Optimization of transport capacity in wireless multihop networks
 SeungWoo Ko^{1} and
 SeongLyun Kim^{1}Email author
https://doi.org/10.1186/168714992013110
© Ko and Kim; licensee Springer. 2013
Received: 30 August 2012
Accepted: 27 March 2013
Published: 24 April 2013
Abstract
Gamal et al. (IEEE Trans. Inform. Theory 52:2568–2592, 2006) showed that the endtoend delay is n times the endtoend throughput under centralized time division multiple access scheduling. In our other work (IEEE Trans. Mobile Computing, in press), it was proved that the relationship between the endtoend throughput and the endtoend delay of Gamal et al. still holds under the IEEE 802.11 distributed coordination function (DCF) when the carrier sensing range and the packet generation rate are jointly optimized. The main purpose of this study is to determine whether the result in our other work is achievable when the transmission range is adjusted instead of the carrier sensing range. To this end, we revise the transport capacity by reflecting a queue at each node and optimize the revised transport capacity by jointly controlling the transmission distance and the packet generation rate. Under our system model, it is shown that the endtoend throughput and the endtoend delay scale are $\Theta \left(1/\sqrt{nlogn}\right)$ and $\Theta \left(\sqrt{n/logn}\right)$, respectively, where n is the number of nodes in the network. That is to say, the result that the endtoend delay is n times the endtoend throughput under the DCF mode is also estabilished while jointly optimizing the transmission range and packet generation rate.
Keywords
1 Introduction
Wireless multihop networks (WMNs) have received considerable attention because of their ability to enhance the spectrum and energy efficiency. To exploit such advantages, researchers have attempted to use various approaches at all levels of communication protocols. The key performance metric in a WMN is the endtoend throughput, referring to the number of packets that can be transported successfully from a given source to its destination. As the node density increases, each node transmits packets to its nearest node. In the seminal work in this area [1], Gupta and Kumar proved that the maximum endtoend throughput scales as $\Theta {\left(1/\sqrt{nlogn}\right)}^{\mathrm{a}}$, where n is the number of nodes in the network.
The endtoend delay, i.e., how fast a packet is transported from a given source to its destination, depends on the endtoend throughput. Because a node delivers its packet to the nearest node, the number of hops to the destination increases. Thus, more transmissions are needed to transport one packet from a source to the destination. In [2], Gamal et al. showed that the endtoend delay is n times the endtoend throughput under centralized time division multiple access (TDMA) scheduling. In our other work [3], we proved that the relationship between the endtoend throughput and the endtoend delay [2] still holds when the IEEE 802.11 distributed coordination function (DCF), a representative practical medium access control (MAC), is utilized. We defined the delayconstrained capacity as the maximum endtoend throughput satisfying the endtoend delay requirement. Given the endtoend delay requirement, we maximized the delayconstrained capacity by jointly controlling the carrier sensing range and the packet generation rate and proved that the delayconstrained capacity is the endtoend delay requirement divided by n.
When multiple IEEE 802.11 networks coexist, controlling the transmission range is more secure than controlling the carrier sensing range. Tuning the carrier sensing range of one network may significantly degrade the performance of the other networks [4]. Provided that the carrier sensing range of one network decreases, the nodes in the network have more chances to access the wireless medium whereas the nodes in the other networks do not have. These nodes receive excessive interference because the distance to the nearest interfering node becomes shorter than they estimated. On the other hand, adjusting the transmission range of one network does not increase the interference temperature significantly because the carrier sensing range, which determines the transmission opportunity, is not changed. The objective of our study is to determine whether the relationship between the endtoend throughput and the corresponding endtoend delay [3] is established when the transmission range is adjusted instead of the carrier sensing range. When controlling the transmission range, two different phenomena occur. As the transmission range increases, the number of transmissions to deliver one packet from a given source to its destination is reduced. On the other hand, the quality of each wireless link deteriorates. These phenomena can be explained by the transport capacity, which is defined as the product of the transmission rate and the transmission distance [1]. The original definition of the transport capacity does not contain the queuing delay at each node, which is a key factor to determine the performance of the WMN when the IEEE 802.11 DCF is used as a MAC protocol. We revise the transport capacity by incorporating a queuing delay at each node. This is referred to as the revised transport capacity throughout this paper. In [5], the authors defined the random access transport capacity to analyze the endtoend throughput and the endtoend delay of a WMN. However, the authors did not take into account the queue of each node.
In this paper, we start with maximizing the revised transport capacity of the WCM, where the IEEE 802.11 DCF MAC protocol is adopted. One way to increase the revised transport capacity is to minimize the number of hops by increasing the transmission distance. However, with a long transmission distance, the waiting time at the queue of each node will grow. When the transmission distance increases, fewer bits are transported to ensure reliable reception and more bits remain in the queue. Such a queuing delay affects the revised transport capacity as perceived by the receiver node. Reflecting such aspects, we optimize the revised transport capacity by jointly controlling the transmission distance and the packet generation rate. As a result, it is shown that under the IEEE 802.11 DCF, the endtoend throughput and the endtoend delay scale are $\Theta \left(1/\sqrt{nlogn}\right)$ and $\Theta \left(\sqrt{n/logn}\right)$, respectively. This indicates that adjusting the transmission range and choosing the proper packet generation rate can establish the tradeoff relationship between the endtoend throughput and the endtoend delay [2].
The rest of this paper is organized as follows: In Section 2, we summarize related works. In Section 3, our system model is described. In Section 4, we introduce the revised transport capacity and optimize it with respect to the transmission range and the packet generation rate. In Section 5, we derive the scaling laws of the throughput and the endtoend delay. Finally, we conclude the paper in Section 6.
2 Related works
In a random network, it is a critical problem as to how the endtoend throughput and the endtoend delay scale with the number of nodes, n. In [1], the authors showed that the endtoend throughput scales as $\Theta \left(1/\sqrt{n}\right)$ or $\Theta \left(1/\sqrt{nlogn}\right)$, depending on whether the network is arbitrary or random. The endtoend throughput gap between an arbitrary and a random network is caused by the network connectivity characteristics. In a random network, there is the additional cost of tuning the transmission range, which degrades the performance on the order of $\sqrt{logn}$ as compared with an arbitrary network. When the endtoend throughput is $\Theta \left(1/\sqrt{nlogn}\right)$, the corresponding endtoend delay becomes $\Theta \left(\sqrt{n/logn}\right)$[2]. On the other hand, a different study [6] derived the endtoend throughput of $\Theta (1/\sqrt{n})$ in a random network using what is known as percolation theory.
Supporting node mobility can improve the endtoend throughput. In [7], it was claimed that when all nodes move around the network, the endtoend throughput becomes independent of n, as described by the scaling law Θ(1). The mobile nodes communicate with each other only when they are very close. It was assumed that each node uniformly moves over the entire network area. In [8], the authors analyzed the endtoend throughput scaling laws under various mobility models. Scaling laws in [7, 8] are derived under a loose delay constraint, in which the speed of mobile movement is significantly slower than the delay requirement. In [2, 9–11], the authors explained the tradeoff relationship between the endtoend throughput and the endtoend delay of mobile networks. In [12], it was proved that the average delay scales with Θ(n), for both the stationary and mobile nodes. Seol and Kim [13, 14] claimed that controlling the mobility can decrease the endtoend delay while maintaining the constant throughput scaling law.
Most previous works [1, 2, 6–14] tend to simplify the MAC influencing the endtoend throughput and the endtoend delay. Some studies have attempted to verify the performance of a WMN when a practice MAC is utilized. Hwang and Kim [15] found that the result [1] is a reasonable throughput estimation for networks with ALOHAlike MAC. In [16, 17], it was proved that IEEE 802.11 DCF cannot achieve the $\Theta \left(1/\sqrt{nlogn}\right)$ endtoend throughput because of the randomness of the DCF. In [3], we jointly optimized the carrier sensing range and the packet generation rate and showed that the tradeoff relationship between the endtoend throughput and the endtoend delay occurs in spite of utilizing the DCF mode.
3 Preliminaries
3.1 Transmission rate and range
Notations
Notations  Description 

θ  Path loss exponent 
n  Number of nodes 
D  Carrier sensing range 
R  Singlehop transmission range 
C(R,D)  Transmission rate 
m(R,D)  Maximum number of packets by singlehop transmission 
M  Unit packet size 
λ  Packet incoming rate 
E[H]  Average number of hops between a sourcedestination pair 
T  Time needed for one transmission 
E[d_{ q }]  Average queueing delay at each node 
v(R,Λ)  Packet velocity 
d_{ e }(R,Λ)  Endtoend delay 
λ  Packet generating rate (endtoend throughput when d_{ e }(R,Λ) is finite) 
Λ_{max}(R)  Throughput capacity 
C_{ T }(R,Λ)  Revised transport capacity 
R^{∗}(Λ)  Optimal transmission range when the packet generation rate is fixed to λ 
Λ^{∗}(R)  Optimal packet generation rate when the transmission range is fixed to R 
Λ _{opt}  Optimal packet generation rate (endtoend throughput) 
where M denotes the unit packet size (in bits), and the function ⌊x⌋ indicates the largest integer not greater than x. It is understood that $\u230a{log}_{2}\left(1+\frac{{\left(\frac{D}{R}\right)}^{\theta}}{6\zeta \left(\theta 1\right)}\right)/M\u230b$ packets are certainly delivered by a single transmission. As the transmission range R increases, m(R,D) decreases. There is a tradeoff between how many packets are transmitted and how far the packets are transported.
3.2 Queuing delay in each node
The average queuing delay (E[d_{ q }]) is dependent on the number of nodes (n), the sensing range (D), the transmission range (R), and the packet generation rate (λ). The average queuing delay E[d_{ q }] is a key factor that is used to express the performance of the IEEE 802.11 DCF. In the following section, we will define some performance metrics in terms of E[d_{ q }] and show how these metrics are affected by n, D, R, and λ.
4 Optimization of the transport capacity
4.1 Revised transport capacity of WMN
In [3], we defined the following terms:
Definition 1
Definition 2
Definition 3
(Feasible endtoend throughput) The endtoend throughput of λ packets per second for a node is feasible when the endtoend delay is finite.
Note that the packet generation rate λ is equivalent to the endtoend throughput when its corresponding endtoend delay is finite. In other words, the destination node can receive packets with rate λ.
Definition 4
Throughput capacity Λ_{max}(D/2) (14) is achieved by forcing the packet velocity to be very close to zero. However, this is accomplished at the cost of the infinite delay. In [1], the authors defined the transport capacity as the product of the transmission rate and the transmission distance of a single hop. The main motivation is to grasp the endtoend throughput and the endtoend delay within one framework. In a WMN, one node processes multiple sourcedestination pairs in a WMN and the corresponding queuing delay appears. Under the DCF mode, the queuing delay is an important factor to determine the endtoend throughput and the endtoend delay owing to its randomness property. The original transport capacity did not take into account the queuing delay. Therefore, we revise the transport capacity to consider the queuing delay in each node. We divide the transport capacity by the average queuing delay, E[d_{ q }] at each node, as follows:
Definition 5
4.2 Crosslayer optimization for maximizing the revised transport capacity
Our objective in this subsection is to determine the optimal transmission range and packet generation rate that maximize the revised transport capacity C_{ T }(R,Λ). We begin by finding the optimal transmission range R^{∗}(Λ) that maximizes C_{ T }(R,Λ) (maximizing the packet velocity v) when λ is fixed. The number of nodes that have at least one packet to transmit ($n{D}^{2}\lambda =n{D}^{2}\Lambda \frac{L}{R}$) depends on transmission range R, as given in (6). One way to decrease the transmission time T of (7) is to increase R and thus reduce the number of incoming packets λ (see also (6)). On the other hand, an excessive increase in the transmission range R may increase E[d_{ q }] by reducing the number of packets, m(R,D), as noted in (4) and (9).
Proposition 1
Proof 1
Appendix.
As λ increases, the optimal transmission range R^{∗}(Λ) should linearly increase to maintain the constraint (22).
Proposition 2
and c is an arbitrary constant value.
Proof 2
Appendix.
The total gain G of point (v(R^{∗}(Λ_{opt}),Λ_{opt}),Λ_{opt}) is always higher than any reference coordinate in the feasible region.
5 Scaling laws of endtoend throughput and delay
In the case of a WMN, we expect that the endtoend throughput decreases and that the endtoend delay increases as the node density increases. An important problem is how to quantify these relationships.
Proposition 3
Proof 3
Appendix.
6 Conclusions
In [2], Gamal et al. proved that the endtoend delay is n times the endtoend throughput by controlling the carrier sensing range and the packet generation rate under centralized TDMA scheduling. In this paper, we verified whether IEEE 802.11 DCF can establish this tradeoff relationship by jointly controlling the transmission range and the packet generation rate. To this end, we redefined the transport capacity in [1] by considering the average queuing time at each node. There are interdependencies among the transmission range, the maximum data rate by a singlehop transmission, and the average waiting time in the node queue. Increasing the transmission range plays a key role in decreasing the hop count, which in turn shortens the packet delay. On the other hand, increasing the transmission range reduces the number of packets that can be transmitted by a singlehop transmission, which therefore increases the average packet waiting time. Therefore, there exists an optimal transmission range suitable to minimizing the endtoend delay. In this study, we analyzed the optimal revised transport capacity by controlling the transmission range and by choosing the appropriate packet generating rate. Finally, it is proved that the scaling laws of the endtoend throughput and endtoend delay are $\Theta \left(1/\sqrt{nlogn}\right)$ and $\Theta \left(\sqrt{n/logn}\right)$, respectively, which are equivalent to the results in [2]. Note that when the transmission range is fixed, the scaling law of the endtoend throughput is $o\left(1/\sqrt{nlogn}\right)$. This means that the performance degradation by using IEEE 802.11 DCF instead of centralized TDMA scheduling is mitigated by adjusting the transmission range and the corresponding packet generating rate.
Endnotes
^{a}We recall the following notations:

f(n)=Θ(g(n)) ⇒ ∃ c_{1}, c_{2}, n_{0}>0 s.t. c_{1}g(n)≤f(n)≤c_{2}g(n), ∀n≥n_{0}.

f(n)=O(g(n)) ⇒ ∃c, n_{0}>0 s.t. 0≤f(n)≤c g(n), ∀n≥n_{0}.

f(n)=O(g(n)) ⇒ ∃c, n_{0}>0 s.t. 0≤f(n)<c g(n), ∀n≥n_{0}.
^{b}The transmission range (R) should be less than D/2 to avoid the hidden node problem [16]. ^{c}The backoff timer stops if any node in the sensing range is transmitting. ^{d} D cannot shrink faster than $\sqrt{\frac{logn}{n}}$ to maintain the connectivity of the nodes [1].
Appendix
Proof of Proposition 1
which is equivalent to the result of Proposition 1.
Proof of Proposition 2
Here, c is a parameter that minimizes the mean square error (we set c to 0.3 when θ is 4). We rearrange the revised transport capacity by inserting $\widehat{m\left({R}^{\ast}\right(\Lambda ),D)}$ instead of m(R^{∗}(Λ),D):
which is a quadratic equation. The solution of Equation 39 is identical to the result of Proposition 2.
Proof of Proposition 3
Network connectivity is a critical problem in a WMN because nodes are randomly distributed. In our system model, sensing range D is a critical parameter in the determination of the network connectivity. Let us define D_{ c }(n) as the sensing range that guarantees network connectivity. We use D_{ c }(n) instead of D.
Because D_{ c }(n) cannot shrink faster than $\sqrt{\frac{logn}{n}}$ to maintain the connectivity of the nodes [1], (40) and (41) are reduced to Proposition 3.
Declarations
Acknowledgements
This research was supported by the International Research & Development Program of the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MEST) of Korea (grant number: 2012K1A3A1A26034281, FY 2012).
Authors’ Affiliations
References
 Gupta P, Kumar PR: The capacity of wireless networks. IEEE Trans. Inform. Theory 2000, 46: 388404. 10.1109/18.825799MathSciNetView ArticleGoogle Scholar
 Gamal AE, Mammen J, Prabhakar B, Shah D: Optimal throughputdelay scaling in wireless networks  part I: the fluid model. IEEE Trans. Inform. Theory 2006, 52: 25682592.MathSciNetView ArticleGoogle Scholar
 Ko SW, Kim SL: Delayconstrained capacity of the IEEE 802.11 DCF in wireless multihop networks. IEEE Trans. Mobile Computing submittedGoogle Scholar
 Park KJ, Choi J, Hou JC, Hu YC, Lim H: Optimal physical carrier sense in wireless networks. Ad Hoc Netw 2011, 9: 1627. 10.1016/j.adhoc.2010.04.006View ArticleGoogle Scholar
 Andrews JG, Baccelli F, Kountouris M, Haenggi M: Random access transport capacity. IEEE Trans. Wireless Commun 2010, 9: 21012111.View ArticleGoogle Scholar
 Franceschetti M, Douse O, Tse DNC, Thiran P: Closing the gap in the capacity of wireless networks via percolation theory. IEEE/ACM Trans. Netw 2007, 53: 10091018.Google Scholar
 Grossglauser M, Tse DNC: Mobility increases the capacity of ad hoc wireless networks. IEEE/ACM Trans. Netw 2002, 10: 477486. 10.1109/TNET.2002.801403View ArticleGoogle Scholar
 Garetto M, Giancone P, Leonardi E: Capacity scaling in ad hoc networks with heterogenous mobile nodes: the super critical regime. IEEE/ACM Trans. Netw 2009., 17:Google Scholar
 Neely MJ, Modiano E: Capacity and delay tradeoffs for adhoc mobile networks. IEEE Trans. Inform. Theory 2005, 51: 19171936. 10.1109/TIT.2005.847717MathSciNetView ArticleGoogle Scholar
 Sharma G, Mazumdar RR: Scaling laws for capacity and delay in wireless ad hoc networks with random mobility. Proceedings of the 2004 IEEE International Conference on Communications Paris, 20–24 June 2004, 38693873.Google Scholar
 Sharma G, Mazumdar RR, Shroff NB: Delay and capacity tradeoffs in mobile ad hoc networks: a global perspective. IEEE Trans. Netw 2007, 15: 981992.View ArticleGoogle Scholar
 Yu SM, Kim SL: Endtoend delay in wireless random networks. IEEE Commun. Lett 2010, 14: 109111.View ArticleGoogle Scholar
 Seol JY, Kim SL: Node mobility control and capacity in wireless ad hoc networks. Proceedings of the 2009 IFAC/IEEE Workshop on Networked Robotics Golden, 6–8 Oct 2009.Google Scholar
 Seol JY, Kim SL: Node mobility and capacity in wireless controllable ad hoc networks. Comput. Commun 2012, 35: 13451354. 10.1016/j.comcom.2012.03.012View ArticleGoogle Scholar
 Hwang YJ, Kim SL: The capacity of random wireless networks. IEEE Trans. Wireless Comm 2008, 7: 49684975.View ArticleGoogle Scholar
 Bisnik N, Abouzeid AA: Queueing network models for delay analysis of multihop wireless ad hoc networks. Ad Hoc Netw 2009, 10: 7997.View ArticleGoogle Scholar
 Chau CK, Chen M, Liew SC: Capacity of largescale CSMA wireless networks. Proceedings of the ACM 15th Annual International Conference on Mobile Computing and Networking Beijing, 20–25, Sept 2009, 97108.Google Scholar
 IEEE Standards Association: IEEE Standard for Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications, P802.11. Piscataway: IEEE; 1997.Google Scholar
 Hekmat R, Mieghem PV: Interference in wireless multihop adhoc networks and its effect on network capacity. Wireless Netw 2004, 10: 389399.View ArticleGoogle Scholar
 Kim TS, Lim H, Hou JC: Improving spatial reuse through transmit power, carrier sense threshold and data rate in multihop wireless networks. Proceedings of the ACM 12th Annual International Conference on Mobile Computing and Networking Los Angeles, 24–29 Sept 2006, 366377.Google Scholar
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