New Cross Layer Design Approach To Ad Hoc Networks ... - CiteSeerX

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New Cross Layer Design Approach To Ad Hoc Networks Under Rayleigh Fading † Peter P. Pham, Sylvie Perreau and Aruna Jayasuriya Institute for Telecommunications Research University of South Australia Mawson Lakes SA 5095 Australia Phone: 61-83025188. Fax: 61-83023873 Email:[email protected]

Abstract Interest in ad hoc networks has been significantly increased over the past few years. The IEEE 802.11 standard is the most mature technology for Wireless Local Area Networks (WLANs) and thus widely adopted as a Medium Access Control (MAC) mechanism in ad hoc networks. However, the research on the performance of ad hoc networks under Rayleigh fading channel is still at its early age. Some work have indicated that the network performance could be badly affected by Rayleigh fading channel [1]. In addition, the behavior of IEEE 802.11 standard equipped mobile nodes in a Rayleigh fading environment requires further thorough studies. In this paper, we identify the causes for the network performance degradation under Rayleigh fading channel. We then employ the predictability of Rayleigh fading channel to propose a new cross-layer design approach which improves the network throughput, decreases the unnecessary packet transmission, thus saving power and bandwidth resource, and reduces the packet loss due to network contention under IEEE 802.11 MAC. In addition, we also propose a Markovian model to study the dynamics of Rayleigh channel and then extend the existing Markov models for IEEE 802.11 to analyze the theoretical network throughput, packet processing rate, packet loss probability and delay under under Rayleigh channel in details. The simulation of the new cross-layer-design approach is carried out and shows an increase in network throughput, decrease in unnecessary packet retransmission and reduction in packet loss due to network contention. We also present our analytical results and confirm them by simulation.

† This work is partially supported by Australian Defence Science Technology Organisation (DSTO)

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Index Terms ad hoc networks, Rayleigh fading channel, cross layer design, IEEE 802.11

I. I NTRODUCTION Cross-layer design has attracted a lot of interests in ad hoc networks over the past few years. It relates to the sharing of information between various layers, as specified in the OSI layered architecture. A various number of research has been conducted on various aspects of crosslayer design principle [2], [3], [4], [5]. In [3], the “path-coupling” degree can be used by the routing layer as a criteria for route selection. This method was claimed to increase the network throughput and reduce the energy consumption. In [4], it is shown that the performance of the MAC is strongly dependent on the choice of the accompanying routing protocol. In [5], a spectrally efficient rate adaptation scheme for ad hoc network is proposed which is subsequently utilized by the routing layer to make better routing decisions. In this study, we make use of the channel state information from the physical layer, specifically the predictability of the slow Rayleigh fading channel, to improve the network performance. Specifically, we employ the predictability to predict when the channel will be in the “bad” state or “down” (i.e. when the receiver can not received the packets correctly due to low power). The channel uses this information to notify the upper layers in advance when the channel is expected to go “down”. The upper layers suspends the transmission until the channel is back to “good” state or “up” (i.e. when the packet could be received correctly). The paper is organized as the followings. In the next section, we propose an innovative cross-layer design approach employing the predictability of a slow Rayleigh fading channel. In section III, we propose a new mathematical model to analyze the IEEE 802.11 RTS/CTS access mechanism, and subsequently derive its performance under Rayleigh channel. We then analyze how our cross-layer design approach affects the performance of IEEE 802.11. The simulation results of our cross-layer design, which are compared against the normal approach and theoretical values, are presented in section IV and V. Finally, we summarize our achievements and propose the future research direction in section VI. It is important to note that we study our cross-layer design and its analytical model based on the usage of IEEE 802.11 RTS/CTS access mechanism though the mechanism can be applied for IEEE 802.11 basic access mechanism as well.

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II. P ROPOSED C ROSS - LAYER D ESIGN The principle of the proposed cross-layer design is based on sharing the channel status information with the upper layers. Having access to this, before sending a packet, the upper layers ideally know whether the channel is “good” enough for the successful transmission. This is based on the assumption that the physical layer knows the exact instance the channel changes from “good” to “bad” and vice versa. If the channel is “down”, the upper layers (e.g. Medium Access Control) can pause the transmission as it has a very low probability of being correctly received at the destination. This wastes power at the transmitter as well as the bandwidth resource in the vicinity at the transmitter. It could also increase packet delay waiting for the acknowledgements which would never arrives. If the channel is “good”, the upper layers proceed with the transmission. Therefore, as compared with the conventional layered approach where information on each network layers is not shared, our cross-layer design can save bandwidth, power resource and possibly reduces packet delay. In reality, the station does not know exactly whether the packet is transmitted correctly. Nevertheless, in Rayleigh channel, we can predict the next received power value based on the past values. Therefore, in this work, we use the predictability of the slow Rayleigh fading to predict the instance the channel is expected to go down and the duration of the “downtime”. The prediction is always carried out for the next packet transmission only to improve the prediction accuracy. It has been shown that the autocorrelation of the Rayleigh fading can be expressed as [6] ρβ = σ 2 J0 (2πfD υ)

(1)

Where fD is the maximum Doppler spread and upsilon is the time shift. Using the correlation function in (1), the predicted future can be expressed as a combination of the past V symbols with the range D, [β(i − D)β(i − D − 1) . . . β(i − D − V + 1)]. βpred (i) =

V −1 X

av (i)β(i − v − D)

(2)

v=0

where av (i), v = 0, 1, . . . , V − 1 are the linear prediction coefficients and can be derived as followings a(i) = R−1 (i)r(i)

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(3)

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Where R(i) is the V × V autocorrelation matrix of the input samples, whose elements are r(i)v,u = E[β(i − D − v)β ∗ (i − D − u)], u, v = 0, 1, . . . , V − 1. The elements of vector r(i) are r(i)v = E[β(i)β ∗ (i − D − v)]

v = 0, 1, . . . , V − 1

(4)

Readers can refer to the Appendix for a complete derivation of the autocorrelation function and the linear prediction coefficients. Using the prediction, the receiving station can “determine” whether the packet will be received correctly in the next transmission. If the packet will not be received correctly, the node performs the following actions: •

Stop the transmission of any reply packets to the sending node, freeze the Medium Access Control (MAC) until the channel is up again.

•

Notify the sender to stop the transmission and also indicate the expected downtime of the channel. This can be achieved in IEEE 802.11 by setting a special flag signifying the channel is going down in the packet header in Clear To Send (CTS) packet or Acknowlegement (ACK) packet. It will also store the expected downtime in a field in the header.

The sender receives the CTS or ACK, detect the message from the receiver. It performs the following actions •

Immediately halt the transmission to the subjected destination node in the MAC, routing layer, and application layer whichever applicable.

•

Obtain the expected downtime and schedule the transmission accordingly.

The neighboring nodes, upon hearing the CTS, or ACK and automatically set their network allocation vectors (NAV) accordingly. The channel can then be released for other nodes for transmission. The expected downtime of the channel is dependent on the average fade duration which can be expressed as [7]

2

eρ − 1 Fa = √ 2πfD ρ

(5)

Where

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•

ρ is the ratio between the power threshold and the Root Mean Square (RMS) value of the received power.

•

fD is the maximum Doppler frequency.

•

Fa is average duration of a fade.

The operation of the cross-layer design algorithm is illustrated in Fig 1.

predict wrongly

Fa Fa stop transmission

Fig. 1.

resume

stop transmission

retransmit

Cross-layer design using channel prediction

A. Performance Improvements Using the Cross-layer Design with Channel Prediction In the Rayleigh fading channel, the received signal can go into deep fades [7]. During these fades, the probability of the received signal strength dipping below the threshold, increases, which in turn increases the probability of packet error. This implies that the packet error rate is higher during these deep fades. This may lead to the reduction in network performance which was independently confirmed in [1]. If the cross-layer design is not used, the upper layers are not notified when the channel goes to “bad” state, hence the node keeps sending the packets. Nevertheless, these packets are discarded due to the low received power. Eventually, after waiting for a specified time to received a reply from the sender, destination node enters backoff followed by retries to resend the packets. After a specified number of retries, the sending node discards the packet permanently. In comparison, using the cross-layer design approach, we can obtain the following improvements. Firstly, it prevents the sender from starting and continuing with unnecessary packet transmissions. Therefore, it results in the reduction in power consumption for transmission as well as savings of the bandwidth resource which can be shared by other nodes. Therefore, this consequently increases the overall network throughput.

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Secondly, the cross-layer approach stops the senders from retransmissions which ultimately result in permanent packet loss. In fact, it is well documented that this type of packet drop has a significant effect on the system performance [8]. The prediction algorithm is dependent on the coefficients of the predictor which can be obtained from (1). However, the derivation of these values is computationally intensive, consume a lot of power which may not be viable during real-time operations. This problem can be solved by pre-computing the coefficients and “storing” them to be retrieved based on Doppler frequency and data rate. Obviously, this approach is fast and energy efficient compared to a real-time coefficient calculation. It can be noticed that, the improvements of the proposed cross-layer design is dependent on the accuracy of the prediction algorithm. Our prediction mechanism accuracy is dependent on the continuation of received power values which are obtained during packet receptions. In ad hoc network, the packet transmission is not continuous, leading to missing received power values. This can affect the accuracy of the prediction algorithm. However, this can be alleviated by performing the predictions to obtain these missing values. The basic principle behind the proposed channel state prediction-based cross-layer design is the communication is not viable between the source and the destination when the channel is “predicted” to be in the “down” state. Therefore, the source and the destination will suspend the transmission for a Fa period which depends on the Doppler frequency and the Root Mean Square (RMS) value of the received power. However, the downtime of the channel may be greater or less than the Fa value. If the downtime is larger than the Fa , the sender resumes the packet transmission which will not be received correctly by the receiver. This may affect the performance of the prediction algorithm. However, as shown in the simulation results, the algorithm improves the network performance as well as reduces the packet loss and unnecessary transmissions. If Fa is larger than the downtime, the packet is sent normally. To get a better understanding of the benefits of our cross-layer approach, we propose a new mathematical analysis to model the dynamics of the channel and the behaviors of a given mobile station using IEEE 802.11. Firstly, we present our analytical model for the Rayleigh channel.

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III. T HEORETICAL A NALYSIS A. Markovian Model for Rayleigh Fading Channel Various channel models have been proposed and studied. These include the well-known Gilbert-Elliott model [9], [10], [11]. These models focus on describing the channel error statistics. In terms of Rayleigh channel, several models have been proposed [12], [13], [14], [15]. In [15], a Markov process was derived for the path gain, which was defined as the ratio between the received signal power and transmitted signal power. In [14], a Markov model for the signal to noise ratio is proposed and then theoretically analysed. This work was later expanded for slowly fading channels [12]. Therefore, the Markov model for Rayleigh fading channel gained some acceptance and seems to find a growing number of applications. In this section, we aim at modelling the Rayleigh fading channel in a another context. We prove that we can use a Markov model consisting of two states taken by the received power: the first and second states respectively corresponding to a received signal below and above a given threshold. This new channel model is used to study the Performance of the IEEE 802.11. The following parameters are used in the analysis. The probability density function (p.d.f.) TABLE I S UMMARY OF PARAMETERS FOR R AYLEIGH C HANNELS

P

Received signal

Nr

Level Crossing Rate

p(r)

Prob of density function for P

Fa

Average fade duration

Pth

Received power threshold

fD

Maximum Doppler frequency

ρ

The ratio Pth /Prms

Prms

The RMS value for the received power

ti,j

Transition probability from state i to state

S0

“down” state

π0

The stationary probability at state S0

j S1

“good” state

π1

The stationary probability at state S1

of P can be expressed as

  p(r) =

r σ 2 exp

 0

³

2

r − 2σ 2

´ (0 ≤ r ≤ ∞) (r < 0)

(6)

Where σ is the rms value of the received signal before envelope detection [7].

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The RMS value of the received power can be derived as Prms =

sZ

p

E[r2 ]

∞

r2 p(r)dr =

=

√

2σ

(7)

0

Therefore Pth Pth =√ Prms 2σ

ρ=

(8)

With fD is the maximum Doppler frequency. The number of times per second the received power P crosses the level Pthresh in a positive direction is Nr = 2πfD ρeρ

2

(9)

As briefly mentioned previously, we divide the received power into two levels: 0 < P ≤ Pth

(10)

Pth < P < ∞

The Rayleigh channel is either in state S1 if Pth < P < ∞ or S0 if 0 < P ≤ Pth With the p.d.f of the received signal power as in (6) the steady state probabilities p1 and p2 of the state S1 and S0 are Z

∞

π1 = P r(r > Pth ) = Pth

Z

Pth

π0 = P r(0 < r ≤ Pth ) = 0

µ ¶ 2 r r2 exp − 2 dr = e−ρ σ2 2σ

(11)

µ ¶ 2 r2 r exp − 2 dr = 1 − e−ρ σ2 2σ

(12)

In order to calculate the transition probabilities ti,j

0 ≤ i, j ≤ 1, we can notice that during 1

second, there will be Nr crossings from state S0 to state S1 . The duration between each crossing will be 1/Nr and the channel remains in a fade (or in state S0 ) for Fa seconds. Therefore, the transition probability t0,1 and t1,0 can be approximated as: 2

2

t1,0 = Fa /(1/Nr ) = e−ρ (eρ − 1) = 1 − e−ρ t0,1 =

2

2 2 2 1/Nr − Fa = 1 − Nr Fa = 1 − e−ρ (eρ − 1) = e−ρ 1/Nr

(13) (14)

We can see that π0 .t0,1 = π1 .t1,0

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(15)

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Therefore, we can conclude that the state equilibrium equation holds. In summary, we can use the Markov model to model the status of the channel. The channel √ 2 is in state S1 with probability e−ρ with the expected duration 1/( 2πfD ρ)and in state S0 with √ 2 2 probability 1−eρ with the expected duration (eρ −1)/(ρfD 2π). Fig 2 illustrates the dynamics of the Rayleigh fading channel. B. IEEE 802.11 Markovian Model In this section, we study the performance of IEEE 802.11 under the Rayleigh fading channel with and without cross-layer design approach. In this paper, we use the results from [16] which is an enhancement to the work done by [17]. In [17], an Markov model was produce to represent the states of a mobile node under the “saturation” condition. In other words, it does not study the performance of IEEE 802.11 under finite load condition. In addition, it does not model the packet drop after a given retries as stated in the [18]. It is noted that the work in [16] provides a more comprehensive model. In this analysis, we only focus on the RTS/CTS access mechanism. In [16], the work is carried on assuming the perfect channel condition. We have to selectively adopt these results taking into account the characteristics of the Rayleigh fading channel. For convenience, the Markov model for the behavior of a given station is reproduced in Fig 3 with the parameters being summarized in Table II. Using the steady state condition of the Markov model, b(0, 0) can be obtained as 1

b(0, 0) = 1 2

h

W



2p (1−2p)

+

p(2p)m



+

p 1−p

i

+

2P 0 +W +1 q( 0 ,02 )

+

 qP

qP00 ,idle (Pidle,0 +1) Pidle,0 +Pidle,b

+

0

Pidle,b

(W +1) P 0 ,idle +(1−q) idle,0 +Pidle,b 2



(16)

Readers are referred to [16] for the values of the parameters, such as Pidle,0 , Pidle,b , . . . Therefore, the probability for transmitting a packet is τ=

m1 X i=0

b(i, 0) =

b(0, 0)(1 − pm+2 ) 1−p

(17)

Compared with the normal channel, the packet transmission under Rayleigh channel exhibits some differences.Under the Rayleigh channel, the packet can not be correctly received when there is a collision or when the channel is down. Therefore, the probability of being unable to receive a packet successfully in Rayleigh channel is higher. Furthermore, a successful RTS/CTS December 22, 2003

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TABLE II S UMMARY OF PARAMETERS

p

Prob of a packet not received correctly

b(i, j)

Stationary prob of the mobile node in backoff stage Si,j = P r(s(t) = i, b(t) = j)

b(idle)

Stationary prob at state (idle)

Wi

Backoff window size at stage s(t) = i

W

Min backoff window size

q

Prob of channel is empty after the finishing backoff

Ptr(n−1) Prob of at least one out of n − 1 node transmits

Ps(n−1)

Prob of one out of n − 1 transmit succesffully

τ

Prob of the packet to transmit

Si,j

State of mobile node when s(t) = i and b(t) = j

b(idle)

Stationary prob for the mobile node at idle state

P00 ,0

Transition prob from state (00 , 0) to (0, 0)

Pidle,0

Transition prob from state (idle) to (0, 0)

Pidle,b

Transition prob from state (idle) to back-off state

0

P00 ,idle

Transition prob from state (0 , 0) to (idle)

σ ¯

Average channel slot time

Ptr(n)

Prob at least one node out of n nodes transmit

Ps(n)

Prob successful transmission for n nodes

Ptr(n−1) Prob at least out of n − 1 nodes transmit

Ps(n−1)

Prob successful transmission for n − 1 nodes

U

Channel throughput

Ql

Queue length

Dave

Average throughput of the channel

Pl

Probability of packet lost

fD

Doppler frequency

Cc , Cnc Connection throughput for cross-layer and normal design

exchange does not guarantee a successful data packet transmission because the channel can go to “down” state after receiving the CTS. Therefore the value of Tc is described as followed: Tc =

p ∗ (RT S + DIF S + δ) + (1 − p) ∗ π0 ∗ (RT S + SIF S + δ + CT S + SIF S + δ + H + PL ) p + (1 − p) ∗ π0

(18)

Where π0 is the probability of the channel being in fade which can be obtained using 12. We denote this as Tcnc Using the cross-layer design approach, assuming the perfect prediction, the sending node suspends from transmitting packets till the channel is back to good state. Therefore, effectively, there is no packet loss due to low power. Therefore, the value of Tc is described as followed: Tc = RT S + DIF S + δ

(19)

It is noticed that this value of Tc is equivalent to the one described in [16] and [17] and is denoted as Tcc to differentiate from Tcnc in non-crosslayer design case. The average slot duration is σ ¯ = (1 − Ptr(n) )σ + Ptr(n) Ps(n) Ts + Ptr(n) (1 − Ps(n) )Tc

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(20)

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For better distinction between parameters for cross-layer approach and without cross-layer approach. Relevant parameters are appended with the subscript c (cross-layer) for cross-layer approach and nc (no-cross layer) for normal approach respectively unless its value is the same for both cases. C. Analysis without Cross-layer Design 1) Derivation of τ and p: The packet can be destroyed either due to collision or due to low received power, i.e. when the received power is lower than threshold. p = 1 − (1 − τ )n−1 + π0 (1 − τ )n−1

(21)

From (21), we can rewrite of the probability of packet transmission as µ τ =1−

1−p 1 − π0

¶1/(n−1) (22)

Where π0 is denoted as the probability of the channel in “down” state which are derived in III-A at the equation (12). Using (17) and (22), we can obtain τ and hence p which are denoted as τnc and pnc . 2) Channel and Connection Throughput Analysis: The probability for a successful transmission in a given channel slot is equal to the probability of exactly one mobile node transmitting on the channel, i.e. τnc (1 − τnc )n−1 . Since there are n combinations for n nodes, the probability of a packet transmission in a channel slot is: Psuccessnc = nτnc (1 − τnc )n−1

(23)

Let Unc is the normalized system throughput, defined as the amount of data bits transmitted on a slot time. We have: Unc =

E[payload information transmitted in a slot time] Psuccessnc PL = σ ¯nc σ ¯nc

(24)

The actual format of the data packet is shown in Fig 4. It consists of the actual useful payload PL and header information H. PL , H, σ, σ ¯c , σ ¯nc , Ts , and Tc are of the same unit. Therefore, the channel throughput can be express as Unc =

nτnc (1 − τnc )n−1 PL σ ¯nc

(25)

The probability of a packet of being retransmitted (due to fading channel) is Rnc =

m1 X i=1

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bi,0 =

m1 X

bi,0 − b0,0

(26)

i=0

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12

The connection throughput can then be expressed as Cnc = Unc ∗ (1 − Rnc )

(27)

3) Packet Processing Rate: In this analysis, we analyze the packet transmission rate with respect to the packet arrival rate. In order to facilitate this analysis, we assume that the incoming packets are of equal length and arriving exponentially with rate λ. Assuming µnc as the packet processing rate, and the average slot time as σ ¯nc . For a given node, the probability of successfully transmitting a packet in a slot time is τnc (1 − τnc )n−1 . Therefore the processing rate is: µnc = τnc (1 − τnc )n−1 /¯ σnc

(28)

4) Packet Loss Analysis: The packet can be dropped when the transmitting queue is full (queue loss) or after a given pre-define number of unsuccessful retries (contention loss). The illustration of queue losses is shown in Fig 5. Clearly seen from the Markov model in Fig 3, the state Sm1 ,0 is the state where the node drops the RTS packet and the DATA packet, reset the backoff window and enter a new backoff state. Therefore, the probability of packet loss due to contention is Plcnc = bnc (m1 , 0) = pm+1 nc bnc (0, 0)

(29)

Using the the arrival rate of λ, which is being dropped with probability Plcnc , the processing rate µnc and the queue length of Ql , the probability of the packet being destroyed by queue overflow is

µ Plqnc =

λ(1 − Plcnc ) µnc

¶Ql µ

1 − (λ(1 − Plcnc )/µnc ) 1 − (λ(1 − Plcnc )/µnc )Ql +1

¶ (30)

Therefore the total probability of packet loss is Plnc = Plqnc + Plcnc ¶Q µ ¶ µ 1 − (λ(1 − Plcnc )/µnc ) λ(1 − Plcnc ) l + pm+1 = nc bnc (0, 0) µnc 1 − (λ(1 − Plcnc )/µnc )Ql +1

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(31)

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5) Delay Analysis: We use the queueing theory to derive the average delay for a successfully transmitted packet. The effective arrival rate of the data packet is λef fnc = λ − Plnc λ "

(32) µ

= λ 1 − pm+1 bc (0, 0)π1 − π1 c

λ(1 − Plcc ) µc

¶Ql µ

1 − (λ(1 − Plcc )/µc ) 1 − (λ(1 − Plcc )/µc )Ql +1

¶#

From [19], using M/M/1/Ql queue results, the average number of packets in the system is Lnc =

Ql Ql +1 ηnc (1 − (Ql + 1)ηnc + Ql ηnc ηnc ) Ql +1 (1 − ηnc )(1 − ηnc )

where ηnc =

λef fnc µnc

(33)

Therefore using the Little theorem, the average delay for a packet is Davenc =

Lnc λef fnc

(34)

D. Analysis for Cross-layer Design 1) Derivation of τ and p: Assume that a packet drop is due to collisiononly. The packet is destroyed when two nodes out of n nodes transmit at the same time. In other words, the probability of collision is equal to the probability of at least one out of n−1 station transmitting. p = 1 − (1 − τ )n−1

(35)

From (35), we can deduct that the probability of a mobile node transmitting is τ = 1 − (1 − p)1/(n−1)

(36)

Using (17) and (36), the values of τ and hence p are readily obtained by using a numerical analysis. We denote these values as τc and pc . 2) Channel Throughput Analysis: Using our cross-layer design, when the channel is in a fade , the mobile node stops sending packets and only resumes the transmission when the channel gets out of the fade. It is recalled that, on average the channel will be in state S0 for with probability π0 and in good state (state S1 ) with probability π1 . Using the same derivation as in the previous section, the actual channel throughput is Uc =

nτc (1 − τc )n−1 PL × π1 σ ¯c

(37)

Where π1 can be obtained from (11). After a successful RTS/CTS exchange, it is guaranteed that the data packet gets transmitted successfully without any retransmission. December 22, 2003

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Therefore the connection throughput is nτc (1 − τc )n−1 PL × π1 σ ¯c

Cc = Uc =

(38)

3) Packet Processing Rate: Using the same argument as in the previous section, the packet processing rate is µc = τc (1 − τc )n−1 /¯ σc

(39)

4) Packet Loss Analysis: Here, we use a similar derivation as in the previous section, taking into account the fact that on average the total channel uptime is π1 of the total time. Therefore, the packet loss due to contention is bc (0, 0)π1 Plcc = pm+1 c

(40)

The packet loss due to queue overflow is µ Plqc = π1

λ(1 − Plcc ) µc

¶Ql µ

1 − (λ(1 − Plcc )/µc ) 1 − (λ(1 − Plcc )/µc )Ql +1

¶ (41)

Therefore, the total probability of packet loss is µ Plc = pm+1 bc (0, 0)π1 + π1 c

λ(1 − Plcc ) µc

¶Ql µ

1 − (λ(1 − Plcc )/µc ) 1 − (λ(1 − Plcc )/µc )Ql +1

¶ (42)

5) Delay Analysis: Since the mobile station suspends any transmission during the channel downtime, the delay must be scaled by 1/π1 to get the actual average packet delay. Davec =

Lc π1 λef fc

(43)

Where •

λef fc

= λ − Plc λ · ³ ´Ql ³ ´¸ λ(1−Plcc ) 1−(λ(1−Plcc )/µc ) m+1 = λ 1 − pc bc (0, 0)π1 − π1 µc 1−(λ(1−P )/µ )Ql +1 lcc

•

ηc

=

•

Lc

=

•

π1

c

λef fc µc Q Q +1 ηc (1−(Ql +1)ηc l +Ql ηc l ηc ) Ql +1

(1−ηc )(1−ηc

)

can be derived from (11)

IV. S IMULATION R ESULTS In the experiment, ns2 is chosen as the simulation environment. We used standard IEEE 802.11 using RTS/CTS access scheme in ns2 with the channel bandwidth of 1Mbps. We analyze our prediction algorithm and cross-layer design using several scenarios. December 22, 2003

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A. Channel Prediction Accuracy Firstly, we focus on analyzing how accurate is the prediction algorithm in the simulation environment. In the scenario, there are two mobile nodes A and B being separated by a distance 120 meters. Node B is moving away from node A with velocity 1 m/s. Node A transmits a constant number of packets to node B with the rate of 50 packets/second, each of size 512 bytes. The predicted received power of the next packet is then compared against actual received power. This result is shown in Fig 6. As shown, the algorithm does predict accurately future values of received power. This reinforces the validity of the derivation shown in the Appendix. B. Two-Node Scenario In this scenario, we focus on analyzing the improvements of cross-layer design approach when the channel goes to fades. By intuition, the improvement of cross-layer design is proportional to the probability of being in “fade”. It is noted that when two mobile nodes are moving away from each other, the RMS value of the received power decreases accordingly. This increases the probability of the channel being in a fade. Therefore, the benefits of using the cross-layer design are supposed to increase. We setup a scenario which consists of two nodes A and B with a given initial distance. Node B moves away from node A with the speed 1 m/s. Node A transmits packets to node B with a constant bit rate of 93440 bps. The simulation time is 50 seconds. The throughput of the connection between node A and node B as a function of the initial starting distance is shown in Fig 7. Clearly seen, the throughput is inversely proportional to the initial starting distance. Furthermore, we observe that the improvement brought out by the cross-layer approach is also proportional to the initial distance. Fig 8 shows that the amount of control packets is significantly decreased by adopting our mechanism. This implies a significant reduction in terms of bandwidth and power resources. The number of dropped packets due to contention and low power are shown in Fig 9 and Fig 10 respectively. As predicted, using the cross-layer design results in lower packet loss. C. Four-Node Scenario In this scenario, we focus on the performance of cross-layer mechanism in the presence of channel contention as well as fades. In the previous scenario, the channel contention is not examined because there are only two nodes in the transmission. As several nodes compete for December 22, 2003

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one channel, the probability of packet loss due to contention is higher. The cross-layer design alleviates this issue by minimizing the number of retransmissions. Therefore, it is predicted that the number of packet loss due to channel contention is lower using cross layer design. In addition, by using the cross-layer design, the receiving node releases the channel for other competing nodes when it predict an unsuccessful reception. Therefore, we can expect an improvement in the average connection throughput using cross-layer design. The scenario is shown in Fig 11. In this scenario, the mobile nodes 1, 2, 3, and 4 moves away from mobile node 0 with the velocity 1 m/s. Node 0 transmits packets to node 1, 2, 3 and 4 with the same data rate which is used as a variable. As predicted, an improvement in throughput is noted in Fig 12 when adopting our cross-layer design. Once again, we observe a significant reduction in control packets, in packet drops due to channel contention and in packet drops due to low power in Fig 13, Fig 14 and Fig 15. D. General Random Mobility Model We intent to measure the average improvement of using the cross-layer design in the scenario which is conventionally adopted by many researchers. In this scenario, there are 50 mobile nodes moving inside a square of 1000x1000 m according to the random waypoint mobility model. Each mobile node moves to a random destination with a speed of 1 m/s. After reaching the destination, the mobile node pauses for 10 seconds before moving to another random position. On average, there are 20 active data transmissions with the rate λ packets/second. The packet size is set to 512 bytes. Due to the space constraint, we only present the average throughput of the network which is shown in Fig 16. On average, we can notice an improvement of 8-10% in network throughput when using our cross layer design method. This is a significant result considering that this is an “all-win” design because we do not sacrifice any resource for the improvement. V. A NALYTICAL M ODEL R ESULTS In this section, we present the theoretical values from the analytical model. We then compare the theoretical results against the values obtained from the simulation experiments. The scenario is shown in Fig 17. In this scenario, there are 16 mobile nodes equally located on a circle with the radius 50 m. Each node moves away from the center with the velocity 0.5 m/s on the designated direction (i.e. relative speed is 1 m/s). The moving nodes transmit to the opposite December 22, 2003

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one with a constant rate λ packets/sec. The packet size is set to 1000 bytes. The transmitting queue length for each mobile node is set to 50 packets. The simulation time is 50 seconds. The theoretical connection throughput and simulation ones when using and not using crosslayer design approach is shown on Fig 18. Firstly, we notice a common trend of the theoretical and the simulation throughput values. However, the theoretical throughput values are higher than the simulation ones. This difference was also independently confirmed in [17], [20]. This may be due to the instability of the protocol itself [17]. However, we can also notice that the cross-layer design does achieve a higher theoretical connection throughput which is also confirmed by the simulation results. Furthermore, at low data rate, using cross-layer design is not recommended the prediction algorithm does not receive enough samples to achieve an accurate prediction. The theoretical and simulation average packet delay values are shown in Fig 19. Ones can notice that the theoretical average delays are similar to simulation values. This confirms the validity of our analytical model. Furthermore, theoretically, as presented in the figure, using crosslayer design achieves the same delay as normal approach. Theoretically, the packet delay reaches its maximum values when the average offered load for each node is around 8 packet/seconds. If the offered load is higher than this rate, the packets get dropped from the queue due to overflows. Using the simulation, the maximum packet delay is achieved when the packet rate is around 10 packets/second which is a only little bit higher than the theoretical value. This is due to higher packet drops rate from channel contention and queue overflows. VI. C ONCLUSION In summary, we have presented a new scheme that performs cross-layer design to improve the performance of ad hoc networks under the Rayleigh fading conditions. The approach, using the predictability of the channel, feeds the channel status back to upper network layers which can suspend packet transmission when the channel enters a fade. This minimizes unnecessary packet retransmissions, power and bandwidth usage while still increases the connection throughput by approximately 5-10% depending on the scenario. It is very significant considering that it is a “all-win” design. Furthermore, we have proposed a Markovian model for the uptime and downtime of the channel. This model is subsequently used to theoretically analyse the performance IEEE 802.11 DCF using the RTS/CTS hand shake when using or not using our cross layer approach. The December 22, 2003

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simulated channel throughput is lower than the theoretical one which is explained by a higher packet drop rate due to contention in simulations. In the future, it is interesting to explore the performance of this cross-layer mechanism for TCP traffic. A PPENDIX Predictability of Rayleigh Channel As shown in [6], in Rayleigh channel, there is a correlation between the received symbols, described by a correlation function. Therefore, ones can employ this property to predict the future symbols. In this section, the derivation of the autocorrelation function is summarized. We then illustrate how this function is used for the prediction. A. Correlation of Rayleigh Fading Channel We can express the transmitted signal x(t) as x(t) = s(t)ej(2πfc t)

(44)

The received signal is the superposition of L multipath components and expressed as r(t) =

L X

Cl s(t − τl )ej2π[(fc +fD cosφl )t−fc τl ]

(45)

l=1

Where •

Cl is the fraction of the lth path of the incoming signal amplitude.

•

τl is the lth path delay

•

fD = v/λ is the maximum Doppler spread

•

φl is the direction of lth scatterer with respect to the mobile velocity vector v.

In the time domain, for a Rayleigh fading channel, the time delay is much less than symbol duration. Therefore, the received signal can be expressed as r(t) = s(t − τ0 )

à L X

! Cl e

jφl (t)

ej2πfc t

(46)

l=1

Where φl (t) = 2π(fD cosψl t − fc τl ) and τ0 ∈ [minτl , maxτl ]. The phase φl (t) is an independent and identically distributed (i.i.d) random variable [21] and uniformly distributed over [0, 2π]. We can see that in (46) the first two terms are the equivalent low pass received signal [6]. The first term shows that the transmitted baseband signal is delayed due to propagation time, and the second term reflects the amplitude fluctuation of the baseband signal by β(ω, t) =

L X

Cl ejφl (t) = α(t)ejφ(t)

(47)

l=1

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We also assume that the time delay τl is an i.i.d variable with the probability density function fT (τ ) where fT (τ ) is nonzero for 0 ≤ τ < ∞ and zero otherwise. The time frequency correlation of the fading factor β(ω, t) is then ρβ (ω1 , ω2 , t, t + υ) = E [β(ω1 , t)β ∗ (ω2 , t + υ)] # " L L XX j(φi (ω1 ,t)−φl (ω2 ,t+υ)) =E Ci Cl e

(48)

i=1 l=1

(49)

Take ∆ω = ω1 − ω2 , we have ρβ (ω1 , ω2 , t, t + υ) = ρβ (∆ω, υ) X £ ¤ = E Ci2 ej(ωD cosφi υ−∆ωτi )

(50)

i

Where E[Ci2 ] is the average value of the fraction of the incoming power in the ith path that can be expressed as E[Ci2 ] = σ 2 fψ (ψi )fT (τi )dψi dτi

(51)

Where σ 2 is the radiated power from the mobile and fψ (ψi )ft (τi )dψi dτi is the average fraction of incoming power within dψi of the time τi For large L, (50) can be rewritten as ρβ (∆, υ) =

σ2 2π

Z

2π

0

Z

∞

ej(wD cosψυ−∆ωτ ) fT (τ )dψdτ

(52)

0

= σ 2 Jo (ωD υ)FT (j∆ω)

Where Jo is the zero-th Bessel function of the first kind and FT (s) is the characteristic function of the time delay τ . For frequency-nonselective Rayleigh fading channel, only the time correlation is considered, the correlation of the Rayleigh fading can be derived as ρβ = σ 2 Jo (2πfD υ)

(53)

Where fD is the maximum Doppler spread and υ is the time shift.

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B. Channel Predictor Using the correlation function in (53), the predicted value can be expressed as a combination of the past V symbols with the range D, [β(i − D)β(i − D − 1) . . . β(i − D − V + 1)]. βpred (i) =

V −1 X

av (i)β(i − v − D)

(54)

v=0

where av (i), v = 0, 1, . . . , V − 1 are the linear prediction coefficients and can be derived as followings a(i) = R−1 (i)r(i)

(55)

Where R(i) is the V × V autocorrelation matrix of the input samples, whose elements are r(i)v,u = E[β(i − D − v)β ∗ (i − D − u)], u, v = 0, 1, . . . , V − 1. The elements of vector r(i) are r(i)v = E[β(i)β ∗ (i − D − v)]

v = 0, 1, . . . , V − 1

(56)

The coefficients vector a(i) is readily known from (53). Therefore, if we know the Doppler spread of the fading channel, the correlation matrix R(i) is also known. R EFERENCES [1] M. Takai and J. Martin, “Effects of wireless physical layer modeling in mobile ad hoc networks.” [Online]. Available: citeseer.nj.nec.com/460094.html [2] A. J. Goldsmith and S. B. Wicker, “Design challenges for energy-constrained ad hoc wireless networks,” IEEE Wireless Communications, vol. 02, 2002. [3] Y. Fang and A. B. McDonald, “Cross-layer performance effects of path-coupling in wireless ad hoc networks: Implications for throughput, power and scalability,” in IEEE IPCCC, 2002. [4] S. Toumpis and A. Goldsmith, “Performance, optimization, and cross-layer design of media access protocols for wireless ad hoc networks,” in IEEE ICC, 2003. [5] W. Yuen, H. Lee, and T. Andersen, “A simple and effective cross layer networking system for mobile ad hoc networks,” 2002. [Online]. Available: citeseer.nj.nec.com/yuen02simple.html [6] A. Kurniawan, “Predictive power control in cdma systems,” Ph.D. dissertation, University of South Australia, 2003. [7] T. S. Rappaport, Wireless Communication: Principles and Practice. Printice Hall, 1996. [8] Z. F. et al, “On tcp performance in multihop wireless networks,” in UCLA WiNG Technical Report, 2002. [9] E. Elliot, “Estimates on error rates for codes in burst-noise channels,” Bell Syst. Tech. Journals, vol. 42, 1963. [10] B. Fritchman, “A binary channel characterization using partitioned markov chains,” IEEE Trans. Inform. Theory, vol. 13, 1967. [11] R. McCullough, “The binary regenerative channel,” Bell Syst. Tech. Journal, vol. 47, 1968.

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[12] C. C. Tan and N. C. Beaulieu, “On first-order markov modeling for the rayleigh fading channel,” IEEE Transactions on Communications, 2000. [13] H. S. Wang and P. Chang, “On verifying the first-order markovian assumption for a rayleigh fading channels,” IEEE Transaction on Vehicular Technology, 1996. [14] H. S. Wang and N. Moayeri, “Modeling, capacity, and joint source/channel coding for rayleigh fading channels,” in IEEE Vehicular Technology Conference (VTC), 1993. [15] R. Chen, K. C. Chua, B. T. Tan, and C. S. Ng, “Adaptive error coding using channel prediction,” Wireless Networks, vol. 5, no. 1, pp. 23–32, 1999. [16] P. P. Pham, “Comprihensive analysis of ieee 802.11,” Institute for Telecommunications Research Internal Report, October 2003. [Online]. Available: http://www.itr.unisa.edu.au/ ppham/ITRreport/report.pdf [17] G. Bianchi, “Performance analysis of the ieee 802.11 dcf,” JSAC, vol. 18, no. 3, pp. 535–547, March 2000. [18] I. C. S. L. M. S. Committee, “Wireless lan medium access control MAC and physical layer PHY specifications,” The Institute of Electrical and Electronics Engineer, 1997. [19] L. Kleinrock, Queueing systems.

New York : Wiley, 1975-1976, ch. 3, p. 120.

[20] H. S. Chhaya and S. Gupta, “Performance modeling of asynchronous data transfer methods of ieee 802.11 mac protocol,” Wireless Networks, vol. 3, no. 3, pp. 217–234, 1997. [21] W. R. Braun and U. Dersh, “A physical mobile radio channel model,” IEEE Transaction on Vehicular Technology, vol. VT-40, no. 2, pp. 472–482, May 1991.

t 1,0 S1 π1 Fig. 2.

S0 t 0,1

π0

The Channel Model

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1

.......

0’,W −1 0

q

P0’,idle

0’,0

idle

q

Pidle,b P0’,0

Pidle,0

1−q

Pidle,b /W0

(1−p)q

Pidle,b /W0 (1−p)(1−q)

0,0

1−p

p −−−−−−−

1−p

−−−−−−−

0,1

1

0,W 0 −2

1

0,W 0−1

p −−−−−−−

i−1,0 p p

1−p

i,0

i,1

1

−−−−−−−

i,Wi −2

1

−−−−−−−

−−−−−−−

m0 ,0

Fig. 3.

−−−−−−−

m0 ,W m−2

1

p

1

m1,0

m1,1

−−−−−−−

m0 ,W m−1 −−−−−−−

p

m1,Wm−2

1

m1 W m−1

Markov Model for Backoff of IEEE 802.11

PL

H PHY HDR

Fig. 4.

m0 ,1

1

p

1

p

p

p

1−p

i,Wi −1

MAC HDR

ACTUAL PAYLOAD

Data Packet Format

December 22, 2003

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λ Plq

Pl

Plc

µ

Fig. 5.

Packet Loss in the Queue

11.3

11.1

10.9

10.7

10.6

10

10.4

10.2

9.82

9.63

9.43

9.26

9.07

8.91

8.73

8.2

8.56

8.37

8.02

7.84

7.67

7.49

7.31

7.14

6.96

6.79

6.62

6.1

6.45

6.27

5.93

5.75

5.58

5.42

5.23

5.05

4.87

4.71

4.51

4.35

-60

-70

Power (dB)

-80

-90

-100

-110 Actual Received Power Predicted Received Power Threshold -120 Time (secs)

Fig. 6.

The Comparison Between Predicted and Actual Received Power

90000

80000

70000

Throughput (kbps)

60000

50000

40000

30000

20000

10000

0 85

92

101

106

113

distance

Fig. 7.

120

127.3

134.3

crosslayer nocrosslayer

The Throughput Comparison under Different Initial Starting Position

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5000 4500 4000 3500

packets

3000 2500 2000 1500 1000 500 0 85

92

101

106

113

120

127.3

Fig. 8.

134.3

crosslayer nocrosslayer

distance

The Control Packets Comparison under Different Starting Position

180 160 140

Packets

120 100 80 60 40 20 0 85

92

101

106

113

120

127.3

Distance

Fig. 9.

134.3 crosslayer nocrosslayer

The Number of Packet Dropped due to Retries

800

700

600

Packets

500

400

300

200

100

0 85

92

101

106

113

distance

Fig. 10.

120

127.3

134.3

crosslayer nocrosslayer

The Number of Packet Dropped due to Low Power

December 22, 2003

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1m/s

1m/s

mobide node 2

mobide node 1

mobide node 0

mobide node 3

mobide node 4

1m/s

Fig. 11.

1m/s

Diagram of Second Scenario

90000

80000

Throughput (kbps)

70000

60000

50000

40000

30000

20000

10000

0 20480

40960

51200

58491

Data Rate (pkts/sec)

Fig. 12.

68268

81920

102400 cross-layer no-cross

Average Throughput

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25000

Number of packets

20000

15000

10000

5000

0 20480

40960

51200

58491

68268

81920

Fig. 13.

102400 cross-layer no cross

Transmission rate

Number of Control Packets

1000 900 800 700

Packets

600 500 400 300 200 100 0 20480

40960

51200

58491

68268

81920

Data rate

Fig. 14.

102400 cross-layer no-cross

Packets Drop due to Retries

12000

10000

Packets

8000

6000

4000

2000

0 20480

40960

51200

58491

Data rate

Fig. 15.

68268

81920

102400

cross-layer no-cross

Packets Drop due to Low Power

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45000 40000 35000

throughput

30000 25000 20000 15000 10000 5000 0 373

9600

24570

52100

104567

offered load

Fig. 16.

cross-layer no-cross

Throughput for 50 node simulation

node4

node3

node5 node2 node6 node1 λ node7

node0

60m node8 λ

node15

node9

node14 node10 node13

Fig. 17.

node11

node12

Scenario for Analytical Model Validation

0.9000 0.8000

Connection Throughput

0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000 0.0000 1

2

3

4

5

8

Data Rate (packets/sec)

Fig. 18.

10

20

30 real cross-layer real no-cross theo-cross-layer theo-no-cross

Network Throughput

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4.5 4

average packet delay

3.5 3 2.5 2 1.5 1 0.5 0 1

2

3

4

5

offered load

Fig. 19.

8

10

20 theo-cross-layer theo-no-cross real-cross-layer real-nocross

Average Packet Delay

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