3.4 Example
3.4.1 Steady-state probability
For the considered scenario, Fig. 3.2(b) illustrates the corresponding state transition diagram of the Markov chain. The relay buffer status of the Markov chain must conform to the constraints in (3.10) and (3.11), i.e.,
X1+ X2 = Ninit× K = 4, (3.15) 0≤ X1, X2 ≤ Lb− 1 = 5. (3.16)
Hence, this example has Ns = 5 relay buffer status of the Markov chain, (X1, X2) ∈ {(4, 0), (3, 1), (2, 2), (1, 3), (0, 4)} , {S1, S2, S3, S4, S5}, where each state, Si, i = 1, · · · , 5, is defined as Si = (X1, X2) with X1 and X2 being the buffer length of relays R1 and R2, respectively.
Depending on the buffer length and instantaneous SNR of the branches associated with each relay the transition probability matrix P can be computed as follows.
Case 1:
Case 1 considers the case when the best receiving and transmitting relays are different, including the transitions, S1 → S2, S2 → S3, S5 → S4, and S4 → S3. This occurs only when CRS operates SILO for relay selection (OR selects the same relay for transmission and reception). According to (3.3) and (3.4), RS and TS must be nonempty, and RS∈ {{R1}, {R2}, {R1, R2}} and TS ∈ {{R1}, {R2}, {R1, R2}} when SILO is adopted by CRS. We start from the transition S1 = (4, 0)→ S2 = (3, 1), corresponding to the case that R2 is the best receiving relay and R1 is the best transmitting relay. Thus, the transition probability from S1 to S2 can be written as
PS1S2 = P (R∗r = R2 ∩ R∗t = R1, SILO| S1)
= P (R∗r = R2, SI| S1)× P (R∗t = R1, LO | S1). (3.17)
The first term in (3.17) refers to the case that SI is performed to select R2 as the best receiving relay given state S1. In this case, the condition (R∗r = R2, SI| S1) only occurs when RS∈ {{R1, R2}, {R2}}. In RS = {R1, R2}, SI selects R2 with the shortest buffer length for reception from (3.1). In RS ={R2}, SI selects R2 for reception because only R2 can be selected by SI in RS. Thus, P (R∗r = R2, SI| S1) can be written as
P (R∗r = R2, SI| S1)
= P (RS ={R1, R2}) + P (RS = {R2})
= q1+ q2, (3.18)
where q1 and q2 are derived in appendix. The second term in (3.17) refers to the case that LO is performed to select R1 as the best transmitting relay given state S1. Similarly, the condition (R∗t = R1, LO | S1) only occurs when TS∈ {{R1, R2}, {R2}}.
In RS ={R1, R2}, LO selects R1 with the longest buffer length for transmission from (3.2). In TS ={R2}, LO selects R2 for transmission because only R2 can be selected by LO in TS. Thus, P (R∗t = R1, LO| S1) can be written as
P (R∗t = R1, LO| S1)
= P (TS = {R1, R2}) + P (TS = {R1})
= q1+ q2. (3.19)
Finally, (3.17) can be rewritten as
PS1S2 = P (R∗r = R2, SI| S1)× P (R∗t = R1, LO| S1)
= (q1+ q2)× (q1+ q2)
= (q1+ q2)2. (3.20)
Similarly, PS5S4 can be computed by the above procedure in case 1. The difference between PS5S4 and PS1S2 is that R1 is the best receiving relay and R2 is the best transmitting relay for transition form S5 to S4. In S5 = (0, 4), the buffer status of relays are opposite to S1 = (4, 0). Hence, PS5S4 is same as PS1S2, and can be written as
PS5S4 = P (R∗r = R1, SI| S5)× P (R∗t = R2, LO| S5)
= (q1+ q2)× (q1+ q2)
= (q1+ q2)2. (3.21)
Furthermore, the computation of the transition probabilities, PS2S3 and PS4S3, are same as that of PS1S2 and PS5S4, respectively. Besides, PS2S3 = PS4S3 due to the symmetry of the Markov chain. Thus, we have
PS1S2 = PS2S3 = PS5S4 = PS4S3 , C1. (3.22)
Case 2:
Case 2 is similar to case 1 in that different relays are selected for reception and trans-mission by using SILO, and includes the transitions, S2 → S1 and S4 → S5. The difference between case 1 and case 2 is that RS̸= {R1, R2} and TS ̸= {R1, R2} in this case. We start from the transition S2 = (3, 1)→ S1 = (4, 0), corresponding to the case that R1 is the best receiving relay and R2 is the best transmitting relay, and PS2S1 can be written as
PS2S1 = P (R∗r = R1 ∩ R∗t = R2, SILO| S2)
= P (R∗r = R1, SI| S2)× P (R∗t = R2, LO | S2). (3.23)
The first term in (3.23) refers to the case that SI is performed to select R1 as the best receiving relay given state S2. In this case, the condition (R∗r = R1, SI | S2) only occurs when RS = {R2} but not RS = {R1, R2}, which is different from case 1. If RS ={R1, R2}, R1 can not be the best receiving relay because the selected protocols of SI depends on the buffer length from (3.1). For example, if RS = {R1, R2}, SI would select R2 with the shortest buffer length rather than R1 in this case. Thus, P (R∗r = R1, SI| S2) can be written as
P (R∗r = R1, SI| S2)
= P (RS ={R1})
= q2. (3.24)
The second term in (3.23) refers to the case that LO is performed to select R2 as the best transmitting relay given state S2. Similarly, the condition R∗t = R2, LO | S2) only occurs when TS ={R2} but not TS = {R1, R2}. If TS = {R1, R2}, LO would select R1
with the longest buffer length rather than R2 in this case. Thus, P (R∗t = R2, LO| S2),
can be written as
P (R∗t = R2, LO| S2)
= P (TS = {R2})
= q2. (3.25)
Finally, (3.23) can be rewritten as
PS2S1 = P (R∗r = R1, SI| S2)× P (R∗t = R2, LO| S2)
= P (RS ={R1}) × P (TS = {R2})
= q2× q2
= q22. (3.26)
Similarly, PS4S5 can be computed by the above procedure in case 2. The difference between PS4S5 and PS2S1 is that R2 is the best receiving relay and R1 is the best transmitting relay for transition form S4 to S5. In S4 = (1, 3), the buffer status of relays are opposite to S2 = (3, 1). Hence, PS4S5 is same as PS2S1 due to the symmetry of the Markov chain, and can be written as
PS4S5 = P (R∗r = R2∩ R∗t = R1, SILO| S4)
= P (R∗r = R2, SI| S4)× P (R∗t = R1, LO| S4)
= P (RS ={R2}) × P (TS = {R1})
= q22
= PS2S1. (3.27)
Thus, we have
PS2S1 = PS4S5 , C2. (3.28)
Case 3:
Case 3 is similar to case 1 in that different relays are selected for reception and trans-mission by using SILO, and includes the transitions S3 → S2 and S3 → S4. The difference of case 3 from case 1 is that the buffer status of relays are the same in case 3, which let SILO select the relays randomly for reception and transmission respectively.
We start from the transition S3 = (2, 2)→ S2 = (3, 1), corresponding to the case that R1 is the best receiving relay and R2 is the best transmitting relay. As case 1, only SILO but OR, which selects the same relay to receive and transmit, can be adopted by CRS in this case. Thus, the transition probability from S3 to S2 can be written as
PS3S2 = P (R∗r = R1 ∩ R∗t = R2, SILO| S3)
= P (R∗r = R1, SI| S3)× P (R∗t = R2, LO | S3). (3.29)
As case 1, the condition (R∗r = R1, SI | S3) only occurs when RS ∈ {{R1, R2}, {R2}}, and the condition (R∗t = R2, LO | S3) only occurs when TS ∈ {{R1, R2}, {R2}}.
However, the buffer status of relays are the same in S3 = (2, 2), which is different from case 1, in this case. Hence, SI would random select the relays for reception if RS ={R1, R2} in S3, and so does LO. Thus, (3.29) can be rewritten as
2. The reason is that SI and LO would random select the relays for reception and transmission respectively in this case because the buffer status of relays are the same.
Similarly, PS3S4 can be computed by the above procedure in case 3. The difference between PS3S4 and PS3S2 is that R2 is the best receiving relay and R1 is the best
transmitting relay for transition form S3 to S4. In S4 = (1, 3), the buffer status of relays are opposite to S2 = (3, 1). Hence, PS3S4 is same as PS3S2 due to the symmetry of the Markov chain, and can be written as
PS3S4 = P (R∗r = R2, SI| S3)× P (R∗t = R1, LO| S3)
Case 4 considers the case when the best receiving and transmitting relays are the same for transiting to the same state, and includes the transitions S3 → S3. From (2.1), OR would select the same relay for reception and transmission. Besides, SILO may use the same relay for reception and transmission when both SI and LO select the same relay. Thus, OR or SILO can be adopted by CRS for transiting to the same state in this case, and the transition probability from S3 to S3 can be written as
PS3S3 = P (OR| S3) + P (R∗r = R1∩ R∗t = R1, SILO| S3)
+ P (Rr∗ = R2∩ R∗t = R2, SILO| S3). (3.33)
The first term in (3.33) refers to the case that CRS switches to OR given state S3. In this case, the condition (OR | S3) occurs when RS, TS, or both is empty from (3.3)
and (3.4). Thus, P (OR| S3) can be written as
P (OR | S3)
= P (RS =∅ ∪ TS = ∅)
= Q, (3.34)
where Q is derived in appendix. The second term in (3.33) refers to the case that adopts SILO and SILO selects R1 to receive and transmit given state S3. In this case, the condition (Rr∗ = R1∩R∗t = R1, SILO| S3) occurs when RS∈ {{R1, R2}, {R1}} and TS ∈ {{R1, R2}, {R1}} as case 1. The difference between the condition of case 4 and that of case 1 is that the buffer status of relays are the same in this case. In S3 = (2, 2), SI selects the relay randomly for reception in RS ={R1, R2} and so does LO because the buffer status of relays are the same. Thus, P (R∗r = R1∩ R∗t = R1, SILO| S3) can
Notice that the buffer status of relays are the same in S3 = (2, 2) resulting in that SI and LO would select the relay randomly when RS ={R1, R2} and TS = {R1, R2}, respectively. Hence, P (RS = {R1, R2}) and P (TS = {R1, R2}) have to be multiplied by 12 for letting R∗r = R1 and R∗t = R1 when RS = {R1, R2} and TS = {R1, R2} in S3. The third term in (3.33) refers to the case that adopts SILO and SILO selects R2 to receive and transmit given state S3. Similarly, the condition (R∗r = R2∩R∗t = R2, SILO| S3) occurs when RS ∈ {{R1, R2}, {R1}} and TS ∈ {{R1, R2}, {R1}}. In S3 = (2, 2), SI selects the relay randomly for reception in RS ={R1, R2} and so does LO because
the buffer status of relays are the same. Thus, P (R∗r = R2∩ R∗t = R2, SILO| S3) can
Case 5 is similar to case 4 in that the best receiving and transmitting relays are the same for transiting to the same state, and includes the transitions S2 → S2 and S4 → S4. The difference between case 4 and case 5 is that the buffer status of relays are different in case 5, resulting in the computations of P (Rr∗ = Ri∩ R∗t = Ri, SILO| S2), i = 1, 2, in case 5 are different from that in case 4. We start from the transition S2 = (3, 1)→ S2 = (3, 1), corresponding to the case that the best receiving and transmitting relays are the same one. Thus, the transition probability from S2 to S2 can be written as
PS2S2 = P (OR| S2) + P (R∗r = R1∩ R∗t = R1, SILO| S2)
+ P (Rr∗ = R2∩ R∗t = R2, SILO| S2). (3.38)
The first term in (3.38) refers to the case that CRS switches to OR given state S2. In this case, the condition (OR | S2) occurs when RS, TS, or both is empty from (3.3)
and (3.4). Thus, P (OR| S3) can be written as
P (OR | S3)
= P (RS =∅ ∪ TS = ∅)
= Q. (3.39)
The second term in (3.38) refers to the case that SILO selects R1to receive and transmit given state S2. In this case, the condition (Rr∗ = R1, SI| S2) occurs when RS ={R1} but not RS = {R1, R2}. In RS = {R1}, SI selects R1 for reception because only R1 can be selected by SI in RS. If RS = {R1, R2}, R1 can not be the best receiving relay because the selected protocols of SI depends on the buffer length from (3.1).
For example, if RS = {R1, R2}, SI would select R2 with the shortest buffer length rather than R1 in this case. However, the condition (R∗t = R1, LO | S1) occurs when TS∈ {{R1, R2}, {R2}}. In TS = {R1, R2}, LO selects R1 with the longest buffer length for transmission from (3.2). In TS = {R2}, LO selects R2 for transmission because only R2 can be selected by LO in TS. Thus, P (R∗r = R1∩ R∗t = R1, SILO| S2) can be
The third term in (3.38) refers to the case that SILO selects R2 to receive and trans-mit given state S2. Similarly, the condition (R∗r = R2, SI | S2) occurs when RS ∈ {{R1, R2}, {R2}}. In RS = {R1, R2}, SI selects R2 with the shortest buffer length for reception from (3.1). In RS = {R1}, SI selects R1 for reception because only R1 can be selected by SI in RS. However, the condition (Rt∗ = R2, LO| S1) only occurs when TS = {R2} but not TS = {R1, R2}. In TS = {R2}, LO selects R2 for transmission because only R2 can be selected by LO in TS. If TS ={R1, R2}, R2 can not be the best
transmitting relay because the selected protocols of LO depends on the buffer length from (3.2). For example, if TS ={R1, R2}, LO would select R1 with the longest buffer length rather than R2 in this case. Thus, P (R∗r = R2∩ Rt∗ = R2, SILO | S2) can be written as
P (R∗r = R2∩ R∗t = R2, SILO| S2)
= P (R∗r = R2, SI| S2)× P (R∗t = R2, LO| S2)
= [P (RS ={R1, R2}) + P (RS = {R2})]
× P (TS = {R2})
= (q1+ q2)× q2. (3.41)
Finally, (3.38) can be rewritten as
PS2S2 = Q + 2× (q1+ q2)× q2. (3.42)
In S4 = (1, 3), the buffer status of relays are opposite to S2 = (3, 1). Hence, PS4S4 is same as PS2S2 due to the symmetry of the Markov chain, and we denote the probability as
PS4S4 = PS2S2 , C5. (3.43)
Case 6:
Case 6 is similar to case 5 in that the best receiving and transmitting relays are the same, and includes the transitions S1 → S1 and S5 → S5. The difference between case 5 and case 6 is that CRS may switch to OR when RS and TS are nonempty. In other words, although RS̸= ∅ ∩ TS ̸= ∅, CRS still switches to OR because the buffer of the best receiving relay selected by SI is full or the buffer of the best transmitting relay selected by LO is empty. In this case, from (3.4), CRS switches to OR when Nl,R∗
t = Nl,R2 = 0, which means R2 is the best transmitting relay selected by LO and the buffer of R2 is empty, in S1 = (4, 0). Thus, the transition probability from S1 to
S1 can be written as
PS1S1 = P (OR| S1) + P (R∗r = R1∩ R∗t = R1, SILO| S1)
= P (RS =∅ ∪ TS = ∅) + P (R∗t = R2, SILO| S1)
+ P (Rr∗ = R1∩ R∗t = R1, SILO| S1). (3.44)
The second term in (3.44) refers to the case that CRS reduces to OR because the buffer of the best transmitting relay, R2, selected by LO is empty. Hence, the system would select the same relay to receive and transmit by the condition that CRS switches to OR when R∗t = R2, which is selected by SILO. In this case, the condition (R∗t = R2, LO| S1) occurs when TS = {R2} but not RS = {R1, R2}. In RS = {R2}, LO selects R2 for reception because only R2 can be selected by LO in TS. If TS ={R1, R2}, R2 can not be the best receiving relay because the selected protocols of LO depends on the buffer length from (3.2). For example, if TS ={R1, R2}, LO would select R1 with the longest
where q3 is derived in appendix. The third term in (3.44) refers to the case that SILO selects R1 to receive and transmit given state S1. In this case, the condition (R∗r = R1, SI| S1) occurs when RS ={R1} but not RS = {R1, R2}. In RS = {R1}, SI selects R1 for reception because only R1 can be selected by SI in RS. If RS ={R1, R2}, R1 can not be the best receiving relay because the selected protocols of SI depends on the buffer length from (3.1). For example, if RS = {R1, R2}, SI would select R2
with the shortest buffer length rather than R1 in this case. However, the condition (R∗t = R1, LO | S1) occurs when TS ∈ {{R1, R2}, {R2}}. In TS = {R1, R2}, LO selects R1 with the longest buffer length for transmission from (3.2). In TS = {R2},
LO selects R2 for transmission because only R2 can be selected by LO in TS. Thus, same as PS1S1 due to the symmetry of the Markov chain, and we denote the probability as
PS5S5 = PS1S1 , C6. (3.48)
Then, the transition probability matrix P can be obtain by (3.22), (3.28), (3.32), (3.37), (3.43), and (3.48). Moreover, the steady-state probability can be obtained by (3.13), and can be written as
where the initial probability vector can be known, v(0) = [
0 0 1 0 0 ]
, by the initial buffer length, Ninit = 2, for each relay as shown Fig. 3.2(a).