We consider a one Source (S), N relays (Ri, i = 1, 2, ..., N ), one destination (D) network, with only one relay chosen to assist the source to transmit the desired packet to the destination. Decode-and-Forward (DF) is assumed. We also make the assumption that the distance between the source and the destination is too great to have a significant direct link, and thus it is ignored. An illustration of a relay network can be found in Fig 2.1. For simplicity, we assume a fixed-power transmission policy for both the source and the relays across all time slots represented by PS and PRi, respectively. Note that every relay is going to use the same transmitting power PR no matter what the channel condition is. That is, PRi=PR, ∀ i = 1, 2, 3, · · · , N .
We denote the distance between the source and relay i by dS,iand that between relay i and destination by di,D. With these, we introduce the path loss model
Lab = n × 10 log10(dab) , (2.1)
Figure 2.1: System model of a relay network.
where Lab is the path loss from node a to node b in dB, dab is the distance from node a to node b. n is the path loss exponent.
Single carrier scheme is adopted, and the energy required for the relays to decode and process the packet is neglected. We assume that it is much smaller than the energy used for transmission. Perfect CSI is assumed unless otherwise specified.
The relay network adopts a two-stage transmission scheme[8]. In the first time slot, the source will first do a CSI acquisition and decide the selected relay according to different algorithms. Then it broadcasts the data to all relays, and every relay tries to decode the message transmitted from the source. In the second time slot, the selected relay transmit the received packet to the destination. Such transmission process is demonstrated in Table 2.1.
We would like to mention here that Table 2.1 is not representing the real time spend-ing ratio between the CSI acquisition and the actual transmission. The former takes much less time than the later.
Table 2.1: An example of how a two-stage transmission takes place.
Timeslot 1 2 3
Source Getting CSIBroadcast X Getting CSIBroadcast
Selected Relay X Send to Destination X
The received signal at the relay Ri and the destination D can be written as rRi = q P
s
10LSi10
hSRix + vi, rD = q P
R
10Li∗D10
hRi∗Dx + vD (2.2)
The x in 2.2 is the transmitted symbol, and vi and vD are Additive White Gaussian Noise (AWGN) at the relay i and the destination D, respectively. hSRi and hRi∗D are zero-mean circular symmetric complex Gaussian variables. i∗ denotes the selected relay for the second-stage transmission.
We realize that vi and vD should be independent random variables. However, for the simplicity sake, we will use σ2 as the power of the thermal noise for both vi and vD in the following text.
The received SNR at the selected relay i∗ and the destination is given by
γi∗ = Ps|hSRi∗|2
Then, we can easily find out the equivalent received SNR at the destination for the signals through the selected relay i∗ is [8]
Γi∗ = min{γi∗, γD} (2.5)
We will give a brief proof to illustrate that (2.5) is still exponentially distributed with coefficient λ = λ1+ λ2
As we can see from (2.6), the CDF for the minimum of two exponentially distributed random variables still takes the exponential random variable’s CDF form. That is, the minimum of two exponentially distributed random variable is still exponentially distributed, with the parameter λ = λ1+ λ2.
There is also a fixed rate constraint, C(bits/sec/Hz), caused by the service require-ment or the requirerequire-ment to make the transmission more power efficient. This rate constraint can also be represented by an SNR constraint, γcst on the equivalent receiv-ing SNR of the network. The relationship between the rate constraint, C and the SNR constraint, γcst is
C = 0.5 × log2(1 + γcst) (2.7)
2.2 Energy Limited Relays
The most innovative and challenging part of our work is to set energy limits on the relays so that each of them has only limited transmitting opportunities. Let Ei(t) be the remaining energy for the relay i at the beginning of the time slot t.
We set the initial energy of each relay to be the same. That is, Ei(t = 0) = E, for i = 1, 2, · · · , N . The source is, however, energy unlimited. We also assume that the energy consumed for decoding and processing the packet at the selected relay i∗ is very small compared to the one used to retransmit the packet to the destination, and thus it is ignored.
Combining the limitation on the energy and the fixed power transmission, we can easily see that the possible transmission attempts will be finite and the same for each relay. In the beginning of the first time slot, the possible transmission attempts for each relay is M = E/PR, where we normalize the transmission time in each time slot to 1.
Table 2.2: An example of how an idle time slot is dealt with.
Timeslot 1 2 3
Source (S) Broadcast X Broadcast
Selected Relay (Ri∗) X Send to Destination X
Timeslot 4 5 6
Source (S) X Idle X
Selected Relay (Ri∗) Send to Destination X X
2.3 Network Lifetime and Idle Time Slots
For each time slot, the source will choose the chosen relay according to different algo-rithms, and check if the instantaneous channel status of the chosen relay supports a rate that is greater than the rate constraint. If it comes back positive, the transmission will take place. However, if the chosen relay cannot support a rate that is greater than the rate constraint, that time slot will be abandoned.
A possible scenario is presented in Tab 2.2 where at time slot 5, the source finds out that the chosen relay does not possess a quality link to support a rate greater than the rate constraint, so the source decides not to transmit.
We omitted the getting CSI session in the beginning of the first stage to make Tab 2.2 more readable, and caution readers to deny any possible confusions.
We define the Network lifetime to be the total count of the time slots from the beginning to the last relay depletes its energy. This indicator, however, is not the longer the better as most other paper does.
This is due to the fact that the longer network lifetime represents that larger amount of idle time slots are generated. As a result, we will want this indicator to be as small as possible to reduce the total number of idle time slots. We will see more about how longer lifetime (more idle time slots) hurts our optimization goal in the next section.
2.4 Optimization Goal
In this research, we want to maximize the throughput of the network before all relays are running out of energies. The definition of the throughput η is
η = P∞
t=1
PN i=1
1
2ρi(t) log2(1 + Γi(t)) arg mint
PN
i=1Ei(t) = 0 (2.8)
ρi(t) =
(1, if i = i∗ 0, otherwise
The ρi(t) in (2.8) indicates whether the relay i is selected or not for the time slot t, and Ei(t) is the remaining energy for the relay i at the beginning of the time slot t. In addition, ρi(t) will be different due to different relay choosing algorithms.
The numerator of (2.8) is the total transmitted bits until all relays are out of energies, and the denominator is the network lifetime. The goal for each algorithm is to maximize the throughput η before all energy in the relays are consumed.
We can also acknowledge from the denominator of (2.8) that the idle slots should be avoided and the network lifetime should be as less as possible in order to maximize η.
That is, although we do not enforce a hard constraint on the total sum of the gen-erated idle time slots in this study, the penalty is already on the optimization goal η here.
Chapter 3
Conventional Relay Selection Algorithms
In this chapter, we introduce two relay selection algorithms, namely, mBRS (mod-ified Best Relay Selection) and Max-Life algorithms. Performance comparisons with the proposed algorithm are given in the ensuing chapters. BRS itself is a well-known algorithm and we develop an improved version for fair comparison purpose.
On the other hand, Max-Life algorithm serves as a reference algorithm we put forth to show what would happen when we are completely blind to the CSI information.
As a result, the Max Life algorithm’s performance should serve as a lower-bound for comparison with any reasonable CSI-aided relay selection algorithm.
3.1 Modified Best Relay Selection
Conventional BRS (Best Relay Selection) is a very intuitive algorithm. By selecting the relay to obtain a Source-Relay-Destination link with the best instantaneous equivalent SNR, we should be able to achieve the best possible performance for each time slot.
It is trivial and true that the BRS is indeed the best algorithm in maximizing the network throughput when the relays are not energy-limited and there is no delay bound constraint. However, when the relays are energy-limited, it has only finite transmission
fact, as we explain later in this section, it may yield unsatisfactory performance in the presence of energy limitation.
First, we would like to explain how we modify the conventional BRS algorithm and why. We can first see how the conventional BRS different from our modified BRS algorithm in Fig.3.1.
We can see that the conventional BRS algorithm will definitely transmit in every time slot. That is, after finding out the selected relay i∗, the source will start the transmission even if the channel status of the selected relay will cause an outage. This
Figure 3.1: mBRS vs. BRS algorithm.
is something we would like to change in the conventional BRS algorithm, since it isn’t allowed to transmit if the transmission does not fulfill the rate constraint C assuming having knowledge of the complete CSI. Therefore, we change the conventional BRS algorithm to make the source to stop the transmission if the selected relay i∗ can not support a link with equivalent rate larger than the rate constraint. We call this revised version of the conventional BRS algorithm the modified BRS algorithm (mBRS). An illustration of the modified BRS algorithm can be found in Fig.3.1
The reason why the conventional BRS works fine in tons of other researches is the fact that they do not consider the energy consumption. As a result, there is no benefits to stop the transmission.
3.1.1 Analysis of Modified BRS algorithm
To further understand exactly why modified BRS algorithm is not the best choice when it comes to energy-limited relays, we give some mathematical facts in this section.
First, we claim that the mBRS tends to exhaust all the energy of the relay with better Γi first, and then continue on accordingly. This fact is directly from the theorem below.
Lemma 3.1.1. For two relays m, n, if Γm Γn, then mBRS will deplete all the energy in relay m first before any energy being consumed from relay n.
Proof.
P [relay m to transmit | relay m, n still have energy]
=
P [mBRS choose relay n to transmit | relay m, n still have energy] = 0
From the above lemma, we can derive the following theorem:
Theorem 3.1.1. Modified BRS algorithm tends to empty the energy of the relay i with the best Γi first and then accordingly.
Proof.
Let’s suppose there are M relays with Γ1 Γ2 · · · ΓM,
then let’s assume that for any m < n, relay n transmits when relay m still has energy, that is : P [relay n transmits | relay m, n still have energy]6=0
But we know from the assumption that Γm Γn,
which is a direct violation to the lemma 3.1.1. As a result, P [relay n transmits | relay m, n still have energy]=0
3.1.2 Conclusion on Modified BRS
Therefore, we can see that mBRS tends to deplete the energy of the relays with better average SNR first, and save the worst ones for the last. This behavior immediately gives mBRS a huge disadvantage since it will certainly suffer a lot of idle time slots after the relays with good link qualities are all firstly consumed.
Let’s check again our optimization goal in (2.8). It is not hard to find out that the excess idle time slots will hurt the throughput and thus decrease the total performance.
As a result, it is both intuitive and mathematically proven that mBRS will not perform well when the relays are energy constrained.
Yet another disadvantage this property brings is the fact that even for those relays with good average link qualities (high average SNR), mBRS tends to use all their energies consecutively without any diversity.
In other words, the relay with high average SNR can have a bad instantaneous
gain still wins over the other remaining relays. On the contrary, the other relay may have a little worse instantaneous channel gain, but compared to its own distribution, it may be in its lucky days. However, mBRS will choose the former to transmit since it only takes instantaneous channel SNR into account.
To sum up, mBRS not only generates excess idle time slots by leaving those relay with disadvantage links to the last, but also uses the energy of the relays with better average SNRs unwisely.
3.2 Max Life algorithm
What we would like to do in this section is to introduce a CSI -free relay selection algo-rithm, and thus should serve as a lower bound toward all robust CSI related algorithm when relays are energy-constrained.
We will introduce a Max Life algorithm, which uses no CSI information : At time slot t :
i∗M L= arg max
i Ei(t)
We can see that the Max Life algorithm choose the relay with the most energy left, so that it can make the rate of the energy consumption of each relay to be very close. The most special property of this selection algorithm will be the fact that the number of alive relays will be almost the same from the beginning to the end of the network lifetime.
This property also allows us to reset the relay network easily, since you can replace all the batteries in relays in one trip.
Max Life algorithm should set a lower-bound on all reasonable CSI aware relay selection algorithms, since it does not use any form of CSIs to do the relay selection.
If an algorithm uses CSI and still loses to the Max Life algorithm, it is definitely not a reasonable one. We will see both our proposed algorithm and mBRS outperform the Max Life algorithm by much in the simulation part.
Chapter 4
A New Relay Selection Algorithm
We have shown why mBRS and Max Life algorithms perform poorly when the relays have energy constrains in previous chapter. We will now propose a new relay selection algorithm in this chapter and try to analyze and predict its performance.
Unlike mBRS and the Max Life algorithm in the previous chapter, our proposed algorithm uses a more reasonable way to choose the selected relay i∗ by considering both the instantaneous channel status and the statistical SNR mean of each relay.
As the result, the proposed algorithm utilizes the energy of the relays more efficiently and reasonably than the other two algorithms, and thus we call it a greener technology.
4.1 Introduction to the proposed algorithm
In this section, we explain how the proposed algorithm works and give better perfor-mance when relays are energy-constrained.
We can see that mBRS tends to choose the relay based on the instantaneous chan-nel gains only. On the other hand, Max Life algorithm chooses the selected relay by how much energy is in each relay without any CSI information. Both of them have a disadvantage as we either stated or proved in previous chapter.
The proposed algorithm, on the contrary, chooses the selected relay i∗ by both the instantaneous SNR performance and the statistical SNR mean. By comparing the
instan-taneous SNR performance to its statistical mean for each relay, we can select a desirable relay so that its performance will be the best with the statistical mean concerned.
Actually, our proposed algorithm uses the CDF (Cumulative Density Function) to do the selection to ensure that a fair comparison is done among all relays. In each time slot, the source will calculate the CDF values Di(Γi(t)) of the current instantaneous channel gains for every relay i, and the greatest one will then be chosen as the selected relay i∗. The transmission will take place if the selected relay i∗ supports a rate greater than the rate constraint C.
The following is the selection algorithm of the proposed algorithm At time slot t :
i∗ = arg max
i Di(Γi(t)) Di(Γi(t)) = 1 − e
−Γi(t) Γi
Let’s make clear here that our proposed algorithm uses no more information than the mBRS algorithm mentioned in the previous chapter but statistical means, which are usually assumed known in related researches and easy to obtain.
By converting the instantaneous channel gains to the CDF values, we can now elimi-nate the chance of blindly choosing the relay with a better instantaneous SNR to transmit which is a major defect for mBRS stated and proved in 3.1.1.
More importantly, every relay has a better chance to be selected when its instan-taneous channel equivalent SNR is performing better with respect to its own statistic distribution. This gives a fair chance for the relays with weaker average SNR to be chosen as a transmitted relay if they are performing well according to their own statistics.
What’s more, for those relays with high average SNR, this selection mechanism prevents them from transmitting when their instantaneous CSI is the best among all relays but weak with respect to their own statistic distribution.
The above facts result in that the energy of the relays are used much more efficiently
4.2 Markovian Formulation of the Relay Energy Con-sumption Process
There are many relay-related researches, but few of them try to analyze the performance of their proposed algorithms. However, in this research, we will try to give some mathe-matical analysis about the algorithm we proposed. It is already very difficult to analyze the performance of a relay system and merely impossible when the system has a limited lifetime. In this study, the system lifetime comes to an end when all relays depleted their energies.
This section is composed of four subsections. In the first subsection, we analyze the total transmittable bits under no rate constraint for our algorithm during the network lifetime. For the second subsection, we analyze the network lifetime for our algorithm.
As for the third part, we analyze the total transmittable bits for our algorithm under rate constraints.
For the first and third parts, we use the Laplace transform to estimate the distribu-tion of total transmittable bits and then calculate their means. However, in the forth part, we use Gauss approximation by directly estimating the mean and variance of the distribution of total transmittable bits.
We would like to mention that in the first and third subsection, the numerical Laplace transform performed contains extreme high computational complexity, and it will be practically impossible to commence if the total number of the relays or the transmission opportunity of each relay becomes large.
This is in fact the motivation for the forth part, since the Gaussian approximation method can give us an approximation with much less complexity.
We will define here the state Am = (TN −1, TN −2, · · · , Ti, · · · , T0). i is the relay index, and the state index m is determined by m =PN −1
i=0 Ti×(M +1)i ,where Ti = 0, 1, · · · , M is the remaining transmission opportunity of relay i, and M is the initial transmitting opportunity for each relay as defined in Sec 2.2. In fact, the state index m is the decimal
value of (M + 1)-ary tuple (TN −1, TN −2, · · · , T0).
It is not always possible for a state Amto transit to another state Anin one transition, or in other words, in one transmission, since we only select one relay to assist the transmission, and we should only consume one transmission opportunity.
That is, the only possible transition for one transmission from the state Am to state An is logM +1(m − n) ∈ {0, 1, 2, · · · , N − 1}, which represents that exactly one
That is, the only possible transition for one transmission from the state Am to state An is logM +1(m − n) ∈ {0, 1, 2, · · · , N − 1}, which represents that exactly one