Analytical Expressions of Maximum Throughput for Long ... - IEEE Xplore
Analytical Expressions of Maximum Throughput for Long-Frame Communications in One-way String Wireless Multihop Networks Hiroo Sekiya, Yoshihiro Tsuchiya, Nobuyoshi Komuro, and Shiro Sakata Graduate School of Advanced Integration Science, Chiba University, Chiba, 263–8522, JAPAN Email:sekiya/kmr/[email protected] Abstract—It is important to obtain analytical expressions of the maximum throughput in IEEE 802.11 Distributed Coordination Function (DCF) multi-hop networks. In the previous works, the analytical expressions of the maximum throughput for one-way string multi-hop networks taking into account the signal capture effect were obtained. In other researches, the analytical expressions of the maximum throughput for one-way string multi-hop networks were also obtained, which are, however, valid only for short-frame communications. There is no analytical expression for maximum throughput, which is valid for long-frame communications. This paper presents the analytical expressions of the maximum throughput for long-frame communications. For this analysis, we should make a different assumption from the previous-analyses. In the short-frame communication analyses, it is assumed that all nodes always have frames. In the presented analysis, however, we should assume that every equal to or more than three than three nodes in a string-topology network have frames. This is the most important progression in this paper. The assumptions and the analytical expressions are validated by the simulation results. Index Terms—Analytical expressions, maximum throughput, IEEE 802.11 DCF, frame length, multi-hop network, string topology, collision probability, carrea-sensing mechanism.
I. I NTRODUCTION Wireless ad-hoc networks have the flexibility and scalability. This is because they are multi-hop networks and require neither centralized control terminals nor existing infrastructure. The wireless ad-hoc networks are applicable for various networks such as military network, police network, temporal network in disaster, Intelligent Transport System (ITS), wireless home network, and Wireless Local Area Networks (WLANs). In wireless ad-hoc networks, the IEEE 802.11 Distributed Coordination Function (DCF) has commonly adopted to the Medium Access Control (MAC) layer protocol. In applying the IEEE 802.11 DCF to wireless ad-hoc networks, however, various problems occur such as the hidden terminal problem and the exposed terminal problem. The hidden terminal problem provides a high frame-drop rate and throughput degradations [1] - [3]. For avoiding these problems, various MAC protocols were proposed in recent years. Actually, many modified IEEE 802.11 DCFs for wireless ad-hoc networks have been presented. It is, however, undesirable to change and modify the fundamental functions of IEEE 802.11 DCF. It is also one of the approaches to analyze the network capacities
978-1-4244-8086-9/10/$26.00 ©2010 IEEE
for wireless ad-hoc networks. The analytical expressions are very helpful to understand the network behaviors intuitively and we can obtain the network performances much faster than the network simulations. The analytical expressions can be also applicable to the admission controls [3] - [6]. In the previous researches, the analytical expressions of the maximum throughput for one-way string multi-hop networks taking into account the signal capture effect were obtained [3]. However, the signal-capture effect cannot be usually observed in real hardware operations. Therefore, it is important to obtain the analytical expressions without signal-capture effect. Recently, the analytical expressions of the maximum throughput for one-way string multi-hop networks were presented, which are, however, valid only for short-frame communications such as a Voice over IP (VoIP) [4]. There is no analytical expression for maximum throughput, which is valid for long-frame communications. This means that the fundamental network behaviors for long-frame communications are different from those for short-frame communications. It is important that the differences between the long-frame and short-frame communications are clear and the analytical expressions for long-frame communications are given. This paper presents analytical expressions of the maximum throughput for one-way string multi-hop networks, which are valid for long-frame communications. For this analysis, we make new assumptions, which is different from the previous analyses. It is clarified that the network behaviors for long-frame communications are different from those for short-frame communications. In long-frame communications, every equal to or more than three than three nodes in a string-topology network have frames though all nodes have frames for short-frame communications. The indication of this difference is the most important progression in this paper. The assumptions and the analytical expressions are validated by the simulation results. II. R ELATED W ORKS A. IEEE 802.11 DCF Figure 1 shows an example of the channel access method of the IEEE 802.11 DCF based on Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA). In the IEEE 802.11 DCF, transmitters sense the channel before frame
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Not receiving ACK
Backoff
DIFS
FRAME
DIFS
Transmitter i
the transmission range and nodes within two-hop intervals are in the carrier-sensing range.
Retransmission
Backoff
FRAME
1) Maximum Throughput Analysis Taking into Account Signal Capture Effect: In [3], the analytical expression of maximum throughput in a one-way string multi-hop network is obtained, taking into account the signal-capture effect. The signal capture effect is: (1) if Node n starts a frame transmission during Node (n + 3) transmits a frame, Node (n + 1) cannot receive the frame from Node n but (2) if Node (n + 3) starts a frame transmission during Node n transmits a frame, Node (n+1) can receive the frame from Node n.
Backoff
SIFS ACK
Collision
Receiver
j th-frame transmission time
Transmitter i
si1
sij
siM
1st frame
jth frame
Mth frame
0
Time
Si Fig. 1.
Channel access process for IEEE 802.11 DCF.
Carrier Sensing Range of Node i Transmission Range of Node i i-2
i-1
Fig. 2.
i
i +1
i +2
i +3
i +4
i +5
One-way string multi-hop network.
transmissions. If the channel is idle during the DIFS (Distributed Inter Frame Space) interval, the transmitter starts the frame-transmission process. When the channel is busy, the transmitter defers the transmission until the channel becomes idle. The first step to transmit the frame is the decrement of Backoff Timer (BT). The initial value of BT is randomly chosen from between 0 and Contention Window (CW). Only when the channel is idle, the BT decreases. When the channel is busy, the node stops the decrement of the BT. If the BT is equal to 0, the frame transmission is started. The Acknowledgement (ACK) frame is transmitted by the receiver after waiting for the Short Inter Frame Space (SIFS) interval if the data reception is successful. The transmitter confirms the successful transmission by receiving the ACK frame from the receiver. In case that the ACK frame is not received, it is recognized that the transmission is failed. Then a re-transmission is scheduled by the transmitter. B. Maximum Throughput Analyses for String Multi-hop Networks It is important and helpful to obtain the analytical expressions of maximum throughput for multi-hop networks. It is, however, not easy task. Therefore, the maximum throughput analyses for simple-topology networks were carried out as first step. Figure 2 shows a one-way string multi-hop network, which is one of the simplest topologies. In the previous researches [3]–[5], it is assumed that neighbor nodes are in
The bottleneck node, which is an important concept for the maximum throughput analysis, is defined as the node whose collision probability is the highest in the network. The maximum throughput is given as the throughput of the bottleneck node. In the string multi-hop network, the center node(s) becomes a bottleneck node, which is named as Node i. Additionally, it is assumed that the number of hops are sufficiently large. From this assumption, the Node-i has the same airtime as Nodes (i + 1), (i + 2) and (i + 3), where the airtime is defined as the time expended for frame transmissions. In detail, the airtime consists of the duration of the distributed interframe space (DIF S), the transmission time of data frame (F RAM E), the duration of the short interframe space (SIF S), the transmission time of acknowledgments (ACK) from the receiving node. The time used up for the re-transmissions is also included in the airtime. Of course, these assumptions are not strict. However, these assumptions provide not only a simple analysis but also sufficient accuracy. Additionally, it is assumed that all nodes always have at least one frame for the maximum-throughput situation. The signal capture effect is one of the functions included in the network simulator ns-2. However, the signal-capture effect cannot be usually observed in real hardware operations. Therefore, the analytical results do not agree with the experimentally results very well. It is important to obtain the analytical expressions without signal-capture effect. 2) Maximum Throughput Analysis Without Signal Capture Effect: In [4], the analytical expression of maximum throughput without signal-capture effect was presented. In stead of signal-capture effect, it is assumed that Node (n + 1) cannot receive the frame from Node n when Nodes n and (n+3) transmit frames simultaneously. Other assumptions are the same as those in [3]. The analytical maximum throughput shows good agreement not only simulation results but also experimental results, which indicate the validity of the change of the assumption. The analytical results in [4], however, show the good agreement only for short-frame communications such as VoIP application. Larger difference between the analytical results and the simulation and experimental results appears, Longer the frame length is. There is no analytical expression for long-frame communications until now.
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Number of frames
Number of frames
Number of frames
Number of frames
is presented. First, the network behaviors are investigated in detail. Figure 3 shows snap-shots of node frames. The 4 simulations are carried out by using the parameters given in 2 0 Table I. Figures 3(a) and (b) show snapshots of node frames, 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 taking into account the signal capture effect. It is seen from Node No. Figs. 3(a) and (b) that all the nodes around the bottleneck node, (a) 10 which is Node 10, has at least one frames regardless of frame 8 length. These results agree with the assumption. Because of 6 4 capture effect, Node n + 1 can receive a frame from Node n 2 with a half possibility even if Node n + 3 transmits a frame. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Therefore, frames are forwarded and all the nodes always have Node No. frames. Figure 3(c) and (d) shows the snapshots of node frames (b) 10 without signal capture effect. It is seen from Fig 3(c) that all 8 the nodes around the bottle neck node have at least one frame 6 for 100-byte frame length, which is the same as Figs. 3(a) 4 2 and (b). However, the snapshot for 1000-byte frame length 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 is different from the others. The frames appear every three Node No. nodes for the front nodes and there are nodes which have no (c) frame. This situation does not agree with the assumption that 10 8 all nodes have at least one frame. From these investigations, 6 it can be stated that we cannot apply the same assumption as 4 2 the previous researches and need to make a new assumption, 0 which agree with the situation in Fig. 3(c) well. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Node No. Figure 4 shows an example of the communication between (d) Node-i and Node-(i+3). In the maximum throughput situation, Fig. 3. Snapshots of node frames (a) taking into account signal capture effect most of Node i transmissions are collide with Node i + for 100-byte frame length, (b) taking into account signal capture effect for 3 transmissions. However, when the (F RAM E + SIF S) 1000-byte frame length, (c) without signal capture effect for 100-byte frame duration of Node i is smaller than the SIF S + ACK + length, and (d) without signal capture effect for 1000-byte frame length. DIF S + BACKOF F durations of Node i + 3 as shown in Fig. 4(a), the Node-i frames can forward to Node (i + 1). TABLE I For this behavior, however, the frame length should be short. S YSTEM PARAMETERS FOR S IMULATIONS If the frame length is long, it is impossible for Node i to transmit a frame during Node (i + 3) transmissions as Protocol IEEE 802.11a UDP/IP header 28 bytes shown in Fig. 4(b). Additionally, there is also no transmission MAC header 32 bytes opportunity for Node i during Nodes (i + 1) and (i + 2) PLCP preamble 16 µs transmissions. As a result, the frames appear every equal to PLCP header(signal) 4 µs PLCP header(service) 2 bytes or more than three nodes as shown in Fig. 3(d), which means FCS length 4 bytes the assumption that all nodes have at least one frame is not LLC length 8 bytes correct for long-frame communications. This is the reason why Tail length 6 bits ACK size 14 bytes the previous analytical results do not agree with the simulation Channel bit rate 18 Mbps and experimental results. ACK bit rate 12 Mbps The maximum throughput analysis for long-frame Slot time 9 µs SIFS time 16 µs communications is carried out with the following assumptions. DIFS time 34 µs 1) The diameter of the carrier-sensing range is twice as long ACK time 32 µs as that of the transmission range. The neighbor node is CWmin 15 CWmax 1023 in the transmission range and the nodes within two-hop Retransmission limit 7 intervals are in the carrier sensing range. 2) The frames appear equal to or more than three node around the bottle neck node. 3) The transmission probabilities and the airtime expended III. A NALYTICAL E XPRESSIONS FOR L ONG - FRAME frame transmissions around the bottleneck node are the C OMMUNICATIONS same as those of the bottleneck node. A. Assumption of throughput analysis 4) The airtime consists of the duration of backoff time decrement (BT ), DIF S, F RAM E, SIF S, and ACK. In this paper, an analytical expression of the maximum From the assumption 2, all nodes, which are in the throughput for an one-way flow in wireless string multi-hop carrier-sensing range of the bottleneck node, do not networks, which is valid for long-frame communications, 8
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Node i + 3
FRAME
SIFS ACK DIFS BACKOFF FRAME
Node i
FRAME
SIFS Time
(a) Node i + 3
FRAME
FRAME
SIFS ACK DIFS BACKOFF FRAME
Node i Collision
SIFS Collision
Time
(b) Fig. 4. Timing example of the frame transmissions between node i and node (i + 3) (a) for short-frame communications, (b) for long-frame communications
decrease the backoff timer simultaneously. Therefore, BACKOF F is included in the airtime in this analysis, which is different from the previous analyses. B. Expression for Throughput Now, a sufficient long time interval [0, T ime] is considered. The airtime for Node n is expressed as S n and the ratio of Sn to the entire time is defined as x = |S n |/T ime, where the airtime includes the durations for BT , DIF S, F RAM E, SIF S, and ACK from the assumption 4. The S n is expressed as the expended time to transmit M frames by Node i, namely, Sn =
M �
snj ,
(1)
j=1
where a = F RAM E/(DIF S + BACKOF F + F RAM E + SIF S + ACK) and [1 − x + x2 /(1 − x)] is the probability that Node (i + 3) has frames in the duration 1 − 2x. From the assumption 2, a node start a frame-transmission process continuously after the node receives a frame. Therefore, it is stated that the probability that Node i + 3 has a frame in the 1 − 2x duration is approximately equal to the time that Node (i + 4) does not transmit a frame. The Other case is the collision induced by the start of Node-(i+3) transmission. When Node (i+3) starts to transmit a frame during Node-i transmissions, Node i+1 cannot receive the frame. This type of a collision probability is expressed as � � x2 ax 1 − x + 1−x ρ2 = . (4) 1 − 2x
As shown in Fig. 4(b), There is a case that one frame transmitted by Node i collides twice with the Node-(i + 3)-transmission frames at both the front and rear parts of the frame. The front-frame-part collision is included in (3) and the rear-frame-part collision is included in (4). Therefore, two collisions are counted in the case of Fig. 4(b). The double counts occur only when when the frame length is F RAM E > SIF S + ACK + DIF S + BACKOF F . The event probability of the double count is �2 � x2 bx 1 − x + 1−x ρ3 = , (5) 1 − 2x
where snj is the transmission time of j -th frames by Node n. The time used up for the re-transmissions is also included in sij . The collision probability when a bottleneck node transmits a frame is defined as ρ. In this case, the network throughput T(x) is T(x) = x · (1 − ρ) · d · data rate, (2)
where, b = F RAM E − (SIF S + ACK + DIF S + BACKOF F )/(DIF S + BACKOF F + F RAM E + SIF S + ACK). From (3)–(5), the collision probability of the frame transmitted by Node i is
C. Collision probability
D. Capacity Limited by Carrier-sensing Mechanism
When Node i transmits a frame, there is possibility that the frame collides with the frame transmitted by Node (i + 3). These collisions can be classified into two types. One is the collision induced by the start of Node-i transmission. A collision occurs when Node i starts to transmit a frame during Node-(i + 3) frame transmissions. When the nodes in the carrier-sensing range of both Nodes i and (i+3), which are Nodes (i + 1) and (i + 2), transmit frames, Nodes i and i + 3 can transmit no frame. Therefore, Nodes i and (i + 3) should transmit frames in 1 − 2x duration. Therefore, this type of a collision probability is expressed as � � x2 ax 1 − x + 1−x ρ1 = , (3) 1 − 2x
The maximum throughput is limited by two factors. One is the collisions and the other is carrier-sensing mechanism. The nodes in the same carrier-sensing range never transmit frames simultaneously because of carrier-sensing mechanism. The normalized total airtimes of the nodes, which are in the carrier-sensing range of Node i, is
ρ
where d = DAT A/(DIF S + F RAM E + SIF S + ACK), DAT A is the transmission time of data payload, and data rate means the channel bit rate.
= ρ 1 + ρ2 − ρ3 (6) � �2 � � 2 2 x x bx 1 − x + 2ax 1 − x + 1−x 1−x − (7) . = 1 − 2x 1 − 2x
y = |Si−2 ∪ Si−1 ∪ Si ∪ Si+1 ∪ Si+2 |/T ime,
(8)
where the durations of the overlap transmissions between non-career-sensing nodes are � � x2 x2 1 − x + 1−x (9) |Si−2 ∩ Si+1 | = 1 − 2x
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Maximum throughput (Mbps)
Maximum throughput (Mbps)
2.5 2 1.5 1
Simulation Analysis Conventional analysis
0.5 0
0
Fig. 5.
200
400
600 800 1000 1200 1400 Frame length (bytes)
|Si−2 ∩ Si+2 | = and
Therefore,
� x2 1 − 2x , 1−x
� � x2 x2 1 − x + 1−x . |Si−1 ∩ Si+2 | = 1 − 2x
y
0.95 0.9
Simulation Analysis Conventional analysis
0.85 0.8
200
220
240
260
IV. S IMULATION VERIFICATION
(10)
(11)
The sum of the total airtimes of all nodes, which are in the carrier-sensing range of Node i, should be (12)
Therefore, x should be in the region of x ≤ 0.271. E. Maximum Throughput
In this section, simulations are carried out to validate the analytical expressions. The network topology used for our simulations is the same as shown in Fig. 2, where the number of nodes is 20. The simulation parameters are given in Tab. I. Figure 5 shows the maximum throughput as a function of the frame length. Figure 6 is a expanded figure of Fig. 5 between 200 bytes and 270 bytes. It is seen from Fig 6 that the simulation results show a good agreement with the conventional analytical predictions when the frame length is less than 250 bytes. This is because that all nodes have frames to transmit when the frame length is short. When the frame length is long, however, the conventional analytical results are much different from the simulation results. This is because the assumption that all nodes have frames is invalid for long-frame communications. It is seen from Figs. 5 and 6 that the proposed analytical predictions agree with the simulation results quantitatively even when the frame length is long. This is because that the node transmits frames with equal to or more than three-hop intervals. From these results, we can show the validity of the new assumption and our analytical expressions.
From (2) and (7), T(x) is
T(x)
1
Fig. 6. the maximum throughput as a function of the frame length between 200bytes and 270 bytes.
� � x2 2x2 1 − x + 1−x = 5x − 1 − 2x � � x2 x2 1 − x + 1 − 2x . − 1−x
|Si−2 ∪ Si−1 ∪ Si ∪ Si+1 ∪ Si+2 | ≤ T ime.
1.05
Frame length (bytes)
the maximum throughput as a function of the frame length
� x2 1 − x +
1.1
� � x2 � 2ax 1 − x + 1−x = x· 1− 1 − 2x �2 � x2 � bx 1 − x + 1−x · d · data rate.(13) + 1 − 2x
Setting dT/dx = 0, the optimal value x ∗ which maximizes the throughput is obtained. Note that x ∗ ≤ 0.271. If x ∗ ≤ 0.271, the maximum throughput is limited by the collisions and it is T(x∗ ). If x∗ > 0.271, the maximum throughput is limited by the carrier-sensing mechanism, which is T(0.271).
V. F UTURE W ORKS This paper and previous papers show the analytical expressions in string-topology networks. The string topology is one of the simplest network topologies of multi-hop networks and the much more complicated topologies usual appear in real situations. We think that the analytical expressions both this and previous papers are the first step of the ”multi-hop network analysis” and there are many tasks, which should be addressed. This paper indicates that the network behaviors are strongly depends on the frame length. From the results, we can suppose that an admission control should be designed according to the frame length. Additionally, it is expected that these
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analytical expressions contribute to the development of Quality of Servece (QoS) control. We know that these analytical expressions never use some applications directly. We believe, however, the step-by-step progress of the network analysis helps many kinds of developments in this research field. VI. C ONCLUSION In this paper, analytical expressions of the maximum throughput for one-way string multi-hop networks, which is valid for long-frame communications, have been given. For this analysis, we make new assumptions, which is different from the previous analyses. It is clarified that the network behaviors for long-frame communications are different from those for short-frame communications. In long-frame communications, every more than three nodes have frames in the networks though all the nodes have frames in short-frame communications. The indication of this difference is the most important progression in this paper. The assumptions and the analytical expressions are validated by the simulation results. R EFERENCES [1] J. Li, C. Blake, D. S. J. D. Couto, H. I. Lee and R. Morris, “Capacity of ad hoc wireless networks,” ACM Mobile Computing and Networking, pp.61–69, July 2001. [2] P. C. Ng, S. C. Liew, K. C. Sha and W. T. To, “Experimental study of hidden-node problem in IEEE 802.11 wireless networks,” ACM Special Interest Group on Data Communications 2005, Aug. 2005. [3] P. C. Ng and S. C. Liew, “Throughput analysis of IEEE 802.11 multi-hop ad hoc networks,” IEEE/ACM Transactions on Networking, Volume 15, Issue 2, pp.309-322, Apr. 2007. [4] M. Inaba, Y. Tomita, M. Masaki, H. Sekiya, T. Yahagi, S. Sakata, and K. Yagyu, “Analysis and experiments of maximum throughput in wireless multi-hop networks,” IEICE Technical report, RCS 2007-121, p. 55-60, Dec. 2007. (In Japanese) [5] M. Inaba, Y. Tsuchiya, H. Sekiya, S. Sakata, and K. Yagyu, “Analysis and experiments of maximum throughput in wireless multi-hop networks for VoIP application,” IEICE Trans. on Communications, vol.E92-B, no.11, pp.3422-3431, Nov. 2009. [6] Y. Gao, D.-M. Chiu and J. C. S. Lui, “Determining the end-to-end throughput capacity in multi-hop networks: methodology and applications,” ACM SIGMetrics/Performance, vol. 34, Issue. 1, pp. 39–50, June 2006. [7] H.-S. Choi and J.-T. Lim, “Supervisory medium access control for multi-hop ad hoc networks,” IEEE SICE-ICASE ’06 International Joint Conference, pp.5973–5976, Oct. 2006. [8] K. Xu, M. Gerla and S. Bae, “How effective is the IEEE802.11 RTS-CTS handshake in ad hoc networks,” IEEE GLOBECOM, Volume 1, pp.72-76, Nov. 2002.
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I. I NTRODUCTION Wireless ad-hoc networks have the flexibility and scalability. This is because they are multi-hop networks and require neither centralized control terminals nor existing infrastructure. The wireless ad-hoc networks are applicable for various networks such as military network, police network, temporal network in disaster, Intelligent Transport System (ITS), wireless home network, and Wireless Local Area Networks (WLANs). In wireless ad-hoc networks, the IEEE 802.11 Distributed Coordination Function (DCF) has commonly adopted to the Medium Access Control (MAC) layer protocol. In applying the IEEE 802.11 DCF to wireless ad-hoc networks, however, various problems occur such as the hidden terminal problem and the exposed terminal problem. The hidden terminal problem provides a high frame-drop rate and throughput degradations [1] - [3]. For avoiding these problems, various MAC protocols were proposed in recent years. Actually, many modified IEEE 802.11 DCFs for wireless ad-hoc networks have been presented. It is, however, undesirable to change and modify the fundamental functions of IEEE 802.11 DCF. It is also one of the approaches to analyze the network capacities
978-1-4244-8086-9/10/$26.00 ©2010 IEEE
for wireless ad-hoc networks. The analytical expressions are very helpful to understand the network behaviors intuitively and we can obtain the network performances much faster than the network simulations. The analytical expressions can be also applicable to the admission controls [3] - [6]. In the previous researches, the analytical expressions of the maximum throughput for one-way string multi-hop networks taking into account the signal capture effect were obtained [3]. However, the signal-capture effect cannot be usually observed in real hardware operations. Therefore, it is important to obtain the analytical expressions without signal-capture effect. Recently, the analytical expressions of the maximum throughput for one-way string multi-hop networks were presented, which are, however, valid only for short-frame communications such as a Voice over IP (VoIP) [4]. There is no analytical expression for maximum throughput, which is valid for long-frame communications. This means that the fundamental network behaviors for long-frame communications are different from those for short-frame communications. It is important that the differences between the long-frame and short-frame communications are clear and the analytical expressions for long-frame communications are given. This paper presents analytical expressions of the maximum throughput for one-way string multi-hop networks, which are valid for long-frame communications. For this analysis, we make new assumptions, which is different from the previous analyses. It is clarified that the network behaviors for long-frame communications are different from those for short-frame communications. In long-frame communications, every equal to or more than three than three nodes in a string-topology network have frames though all nodes have frames for short-frame communications. The indication of this difference is the most important progression in this paper. The assumptions and the analytical expressions are validated by the simulation results. II. R ELATED W ORKS A. IEEE 802.11 DCF Figure 1 shows an example of the channel access method of the IEEE 802.11 DCF based on Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA). In the IEEE 802.11 DCF, transmitters sense the channel before frame
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Not receiving ACK
Backoff
DIFS
FRAME
DIFS
Transmitter i
the transmission range and nodes within two-hop intervals are in the carrier-sensing range.
Retransmission
Backoff
FRAME
1) Maximum Throughput Analysis Taking into Account Signal Capture Effect: In [3], the analytical expression of maximum throughput in a one-way string multi-hop network is obtained, taking into account the signal-capture effect. The signal capture effect is: (1) if Node n starts a frame transmission during Node (n + 3) transmits a frame, Node (n + 1) cannot receive the frame from Node n but (2) if Node (n + 3) starts a frame transmission during Node n transmits a frame, Node (n+1) can receive the frame from Node n.
Backoff
SIFS ACK
Collision
Receiver
j th-frame transmission time
Transmitter i
si1
sij
siM
1st frame
jth frame
Mth frame
0
Time
Si Fig. 1.
Channel access process for IEEE 802.11 DCF.
Carrier Sensing Range of Node i Transmission Range of Node i i-2
i-1
Fig. 2.
i
i +1
i +2
i +3
i +4
i +5
One-way string multi-hop network.
transmissions. If the channel is idle during the DIFS (Distributed Inter Frame Space) interval, the transmitter starts the frame-transmission process. When the channel is busy, the transmitter defers the transmission until the channel becomes idle. The first step to transmit the frame is the decrement of Backoff Timer (BT). The initial value of BT is randomly chosen from between 0 and Contention Window (CW). Only when the channel is idle, the BT decreases. When the channel is busy, the node stops the decrement of the BT. If the BT is equal to 0, the frame transmission is started. The Acknowledgement (ACK) frame is transmitted by the receiver after waiting for the Short Inter Frame Space (SIFS) interval if the data reception is successful. The transmitter confirms the successful transmission by receiving the ACK frame from the receiver. In case that the ACK frame is not received, it is recognized that the transmission is failed. Then a re-transmission is scheduled by the transmitter. B. Maximum Throughput Analyses for String Multi-hop Networks It is important and helpful to obtain the analytical expressions of maximum throughput for multi-hop networks. It is, however, not easy task. Therefore, the maximum throughput analyses for simple-topology networks were carried out as first step. Figure 2 shows a one-way string multi-hop network, which is one of the simplest topologies. In the previous researches [3]–[5], it is assumed that neighbor nodes are in
The bottleneck node, which is an important concept for the maximum throughput analysis, is defined as the node whose collision probability is the highest in the network. The maximum throughput is given as the throughput of the bottleneck node. In the string multi-hop network, the center node(s) becomes a bottleneck node, which is named as Node i. Additionally, it is assumed that the number of hops are sufficiently large. From this assumption, the Node-i has the same airtime as Nodes (i + 1), (i + 2) and (i + 3), where the airtime is defined as the time expended for frame transmissions. In detail, the airtime consists of the duration of the distributed interframe space (DIF S), the transmission time of data frame (F RAM E), the duration of the short interframe space (SIF S), the transmission time of acknowledgments (ACK) from the receiving node. The time used up for the re-transmissions is also included in the airtime. Of course, these assumptions are not strict. However, these assumptions provide not only a simple analysis but also sufficient accuracy. Additionally, it is assumed that all nodes always have at least one frame for the maximum-throughput situation. The signal capture effect is one of the functions included in the network simulator ns-2. However, the signal-capture effect cannot be usually observed in real hardware operations. Therefore, the analytical results do not agree with the experimentally results very well. It is important to obtain the analytical expressions without signal-capture effect. 2) Maximum Throughput Analysis Without Signal Capture Effect: In [4], the analytical expression of maximum throughput without signal-capture effect was presented. In stead of signal-capture effect, it is assumed that Node (n + 1) cannot receive the frame from Node n when Nodes n and (n+3) transmit frames simultaneously. Other assumptions are the same as those in [3]. The analytical maximum throughput shows good agreement not only simulation results but also experimental results, which indicate the validity of the change of the assumption. The analytical results in [4], however, show the good agreement only for short-frame communications such as VoIP application. Larger difference between the analytical results and the simulation and experimental results appears, Longer the frame length is. There is no analytical expression for long-frame communications until now.
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Number of frames
Number of frames
Number of frames
Number of frames
is presented. First, the network behaviors are investigated in detail. Figure 3 shows snap-shots of node frames. The 4 simulations are carried out by using the parameters given in 2 0 Table I. Figures 3(a) and (b) show snapshots of node frames, 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 taking into account the signal capture effect. It is seen from Node No. Figs. 3(a) and (b) that all the nodes around the bottleneck node, (a) 10 which is Node 10, has at least one frames regardless of frame 8 length. These results agree with the assumption. Because of 6 4 capture effect, Node n + 1 can receive a frame from Node n 2 with a half possibility even if Node n + 3 transmits a frame. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Therefore, frames are forwarded and all the nodes always have Node No. frames. Figure 3(c) and (d) shows the snapshots of node frames (b) 10 without signal capture effect. It is seen from Fig 3(c) that all 8 the nodes around the bottle neck node have at least one frame 6 for 100-byte frame length, which is the same as Figs. 3(a) 4 2 and (b). However, the snapshot for 1000-byte frame length 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 is different from the others. The frames appear every three Node No. nodes for the front nodes and there are nodes which have no (c) frame. This situation does not agree with the assumption that 10 8 all nodes have at least one frame. From these investigations, 6 it can be stated that we cannot apply the same assumption as 4 2 the previous researches and need to make a new assumption, 0 which agree with the situation in Fig. 3(c) well. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Node No. Figure 4 shows an example of the communication between (d) Node-i and Node-(i+3). In the maximum throughput situation, Fig. 3. Snapshots of node frames (a) taking into account signal capture effect most of Node i transmissions are collide with Node i + for 100-byte frame length, (b) taking into account signal capture effect for 3 transmissions. However, when the (F RAM E + SIF S) 1000-byte frame length, (c) without signal capture effect for 100-byte frame duration of Node i is smaller than the SIF S + ACK + length, and (d) without signal capture effect for 1000-byte frame length. DIF S + BACKOF F durations of Node i + 3 as shown in Fig. 4(a), the Node-i frames can forward to Node (i + 1). TABLE I For this behavior, however, the frame length should be short. S YSTEM PARAMETERS FOR S IMULATIONS If the frame length is long, it is impossible for Node i to transmit a frame during Node (i + 3) transmissions as Protocol IEEE 802.11a UDP/IP header 28 bytes shown in Fig. 4(b). Additionally, there is also no transmission MAC header 32 bytes opportunity for Node i during Nodes (i + 1) and (i + 2) PLCP preamble 16 µs transmissions. As a result, the frames appear every equal to PLCP header(signal) 4 µs PLCP header(service) 2 bytes or more than three nodes as shown in Fig. 3(d), which means FCS length 4 bytes the assumption that all nodes have at least one frame is not LLC length 8 bytes correct for long-frame communications. This is the reason why Tail length 6 bits ACK size 14 bytes the previous analytical results do not agree with the simulation Channel bit rate 18 Mbps and experimental results. ACK bit rate 12 Mbps The maximum throughput analysis for long-frame Slot time 9 µs SIFS time 16 µs communications is carried out with the following assumptions. DIFS time 34 µs 1) The diameter of the carrier-sensing range is twice as long ACK time 32 µs as that of the transmission range. The neighbor node is CWmin 15 CWmax 1023 in the transmission range and the nodes within two-hop Retransmission limit 7 intervals are in the carrier sensing range. 2) The frames appear equal to or more than three node around the bottle neck node. 3) The transmission probabilities and the airtime expended III. A NALYTICAL E XPRESSIONS FOR L ONG - FRAME frame transmissions around the bottleneck node are the C OMMUNICATIONS same as those of the bottleneck node. A. Assumption of throughput analysis 4) The airtime consists of the duration of backoff time decrement (BT ), DIF S, F RAM E, SIF S, and ACK. In this paper, an analytical expression of the maximum From the assumption 2, all nodes, which are in the throughput for an one-way flow in wireless string multi-hop carrier-sensing range of the bottleneck node, do not networks, which is valid for long-frame communications, 8
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Node i + 3
FRAME
SIFS ACK DIFS BACKOFF FRAME
Node i
FRAME
SIFS Time
(a) Node i + 3
FRAME
FRAME
SIFS ACK DIFS BACKOFF FRAME
Node i Collision
SIFS Collision
Time
(b) Fig. 4. Timing example of the frame transmissions between node i and node (i + 3) (a) for short-frame communications, (b) for long-frame communications
decrease the backoff timer simultaneously. Therefore, BACKOF F is included in the airtime in this analysis, which is different from the previous analyses. B. Expression for Throughput Now, a sufficient long time interval [0, T ime] is considered. The airtime for Node n is expressed as S n and the ratio of Sn to the entire time is defined as x = |S n |/T ime, where the airtime includes the durations for BT , DIF S, F RAM E, SIF S, and ACK from the assumption 4. The S n is expressed as the expended time to transmit M frames by Node i, namely, Sn =
M �
snj ,
(1)
j=1
where a = F RAM E/(DIF S + BACKOF F + F RAM E + SIF S + ACK) and [1 − x + x2 /(1 − x)] is the probability that Node (i + 3) has frames in the duration 1 − 2x. From the assumption 2, a node start a frame-transmission process continuously after the node receives a frame. Therefore, it is stated that the probability that Node i + 3 has a frame in the 1 − 2x duration is approximately equal to the time that Node (i + 4) does not transmit a frame. The Other case is the collision induced by the start of Node-(i+3) transmission. When Node (i+3) starts to transmit a frame during Node-i transmissions, Node i+1 cannot receive the frame. This type of a collision probability is expressed as � � x2 ax 1 − x + 1−x ρ2 = . (4) 1 − 2x
As shown in Fig. 4(b), There is a case that one frame transmitted by Node i collides twice with the Node-(i + 3)-transmission frames at both the front and rear parts of the frame. The front-frame-part collision is included in (3) and the rear-frame-part collision is included in (4). Therefore, two collisions are counted in the case of Fig. 4(b). The double counts occur only when when the frame length is F RAM E > SIF S + ACK + DIF S + BACKOF F . The event probability of the double count is �2 � x2 bx 1 − x + 1−x ρ3 = , (5) 1 − 2x
where snj is the transmission time of j -th frames by Node n. The time used up for the re-transmissions is also included in sij . The collision probability when a bottleneck node transmits a frame is defined as ρ. In this case, the network throughput T(x) is T(x) = x · (1 − ρ) · d · data rate, (2)
where, b = F RAM E − (SIF S + ACK + DIF S + BACKOF F )/(DIF S + BACKOF F + F RAM E + SIF S + ACK). From (3)–(5), the collision probability of the frame transmitted by Node i is
C. Collision probability
D. Capacity Limited by Carrier-sensing Mechanism
When Node i transmits a frame, there is possibility that the frame collides with the frame transmitted by Node (i + 3). These collisions can be classified into two types. One is the collision induced by the start of Node-i transmission. A collision occurs when Node i starts to transmit a frame during Node-(i + 3) frame transmissions. When the nodes in the carrier-sensing range of both Nodes i and (i+3), which are Nodes (i + 1) and (i + 2), transmit frames, Nodes i and i + 3 can transmit no frame. Therefore, Nodes i and (i + 3) should transmit frames in 1 − 2x duration. Therefore, this type of a collision probability is expressed as � � x2 ax 1 − x + 1−x ρ1 = , (3) 1 − 2x
The maximum throughput is limited by two factors. One is the collisions and the other is carrier-sensing mechanism. The nodes in the same carrier-sensing range never transmit frames simultaneously because of carrier-sensing mechanism. The normalized total airtimes of the nodes, which are in the carrier-sensing range of Node i, is
ρ
where d = DAT A/(DIF S + F RAM E + SIF S + ACK), DAT A is the transmission time of data payload, and data rate means the channel bit rate.
= ρ 1 + ρ2 − ρ3 (6) � �2 � � 2 2 x x bx 1 − x + 2ax 1 − x + 1−x 1−x − (7) . = 1 − 2x 1 − 2x
y = |Si−2 ∪ Si−1 ∪ Si ∪ Si+1 ∪ Si+2 |/T ime,
(8)
where the durations of the overlap transmissions between non-career-sensing nodes are � � x2 x2 1 − x + 1−x (9) |Si−2 ∩ Si+1 | = 1 − 2x
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Maximum throughput (Mbps)
Maximum throughput (Mbps)
2.5 2 1.5 1
Simulation Analysis Conventional analysis
0.5 0
0
Fig. 5.
200
400
600 800 1000 1200 1400 Frame length (bytes)
|Si−2 ∩ Si+2 | = and
Therefore,
� x2 1 − 2x , 1−x
� � x2 x2 1 − x + 1−x . |Si−1 ∩ Si+2 | = 1 − 2x
y
0.95 0.9
Simulation Analysis Conventional analysis
0.85 0.8
200
220
240
260
IV. S IMULATION VERIFICATION
(10)
(11)
The sum of the total airtimes of all nodes, which are in the carrier-sensing range of Node i, should be (12)
Therefore, x should be in the region of x ≤ 0.271. E. Maximum Throughput
In this section, simulations are carried out to validate the analytical expressions. The network topology used for our simulations is the same as shown in Fig. 2, where the number of nodes is 20. The simulation parameters are given in Tab. I. Figure 5 shows the maximum throughput as a function of the frame length. Figure 6 is a expanded figure of Fig. 5 between 200 bytes and 270 bytes. It is seen from Fig 6 that the simulation results show a good agreement with the conventional analytical predictions when the frame length is less than 250 bytes. This is because that all nodes have frames to transmit when the frame length is short. When the frame length is long, however, the conventional analytical results are much different from the simulation results. This is because the assumption that all nodes have frames is invalid for long-frame communications. It is seen from Figs. 5 and 6 that the proposed analytical predictions agree with the simulation results quantitatively even when the frame length is long. This is because that the node transmits frames with equal to or more than three-hop intervals. From these results, we can show the validity of the new assumption and our analytical expressions.
From (2) and (7), T(x) is
T(x)
1
Fig. 6. the maximum throughput as a function of the frame length between 200bytes and 270 bytes.
� � x2 2x2 1 − x + 1−x = 5x − 1 − 2x � � x2 x2 1 − x + 1 − 2x . − 1−x
|Si−2 ∪ Si−1 ∪ Si ∪ Si+1 ∪ Si+2 | ≤ T ime.
1.05
Frame length (bytes)
the maximum throughput as a function of the frame length
� x2 1 − x +
1.1
� � x2 � 2ax 1 − x + 1−x = x· 1− 1 − 2x �2 � x2 � bx 1 − x + 1−x · d · data rate.(13) + 1 − 2x
Setting dT/dx = 0, the optimal value x ∗ which maximizes the throughput is obtained. Note that x ∗ ≤ 0.271. If x ∗ ≤ 0.271, the maximum throughput is limited by the collisions and it is T(x∗ ). If x∗ > 0.271, the maximum throughput is limited by the carrier-sensing mechanism, which is T(0.271).
V. F UTURE W ORKS This paper and previous papers show the analytical expressions in string-topology networks. The string topology is one of the simplest network topologies of multi-hop networks and the much more complicated topologies usual appear in real situations. We think that the analytical expressions both this and previous papers are the first step of the ”multi-hop network analysis” and there are many tasks, which should be addressed. This paper indicates that the network behaviors are strongly depends on the frame length. From the results, we can suppose that an admission control should be designed according to the frame length. Additionally, it is expected that these
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analytical expressions contribute to the development of Quality of Servece (QoS) control. We know that these analytical expressions never use some applications directly. We believe, however, the step-by-step progress of the network analysis helps many kinds of developments in this research field. VI. C ONCLUSION In this paper, analytical expressions of the maximum throughput for one-way string multi-hop networks, which is valid for long-frame communications, have been given. For this analysis, we make new assumptions, which is different from the previous analyses. It is clarified that the network behaviors for long-frame communications are different from those for short-frame communications. In long-frame communications, every more than three nodes have frames in the networks though all the nodes have frames in short-frame communications. The indication of this difference is the most important progression in this paper. The assumptions and the analytical expressions are validated by the simulation results. R EFERENCES [1] J. Li, C. Blake, D. S. J. D. Couto, H. I. Lee and R. Morris, “Capacity of ad hoc wireless networks,” ACM Mobile Computing and Networking, pp.61–69, July 2001. [2] P. C. Ng, S. C. Liew, K. C. Sha and W. T. To, “Experimental study of hidden-node problem in IEEE 802.11 wireless networks,” ACM Special Interest Group on Data Communications 2005, Aug. 2005. [3] P. C. Ng and S. C. Liew, “Throughput analysis of IEEE 802.11 multi-hop ad hoc networks,” IEEE/ACM Transactions on Networking, Volume 15, Issue 2, pp.309-322, Apr. 2007. [4] M. Inaba, Y. Tomita, M. Masaki, H. Sekiya, T. Yahagi, S. Sakata, and K. Yagyu, “Analysis and experiments of maximum throughput in wireless multi-hop networks,” IEICE Technical report, RCS 2007-121, p. 55-60, Dec. 2007. (In Japanese) [5] M. Inaba, Y. Tsuchiya, H. Sekiya, S. Sakata, and K. Yagyu, “Analysis and experiments of maximum throughput in wireless multi-hop networks for VoIP application,” IEICE Trans. on Communications, vol.E92-B, no.11, pp.3422-3431, Nov. 2009. [6] Y. Gao, D.-M. Chiu and J. C. S. Lui, “Determining the end-to-end throughput capacity in multi-hop networks: methodology and applications,” ACM SIGMetrics/Performance, vol. 34, Issue. 1, pp. 39–50, June 2006. [7] H.-S. Choi and J.-T. Lim, “Supervisory medium access control for multi-hop ad hoc networks,” IEEE SICE-ICASE ’06 International Joint Conference, pp.5973–5976, Oct. 2006. [8] K. Xu, M. Gerla and S. Bae, “How effective is the IEEE802.11 RTS-CTS handshake in ad hoc networks,” IEEE GLOBECOM, Volume 1, pp.72-76, Nov. 2002.
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