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.