3.5 Numerical Results and Discussion
3.5.3 Packet Loss due to Buffer Overflow
Now we compare the performance of MDP I policies with those of some other less adaptive schemes. All transmission is subject to a BER constraint of
0 1 2 3 4 5 6 7 0
1 2 3 4 5 6
Channel states
Packets transmitted/slot
Worst state Best state
1 packet in buffer 5 packets in buffer 10 packets in buffer 14 packets in buffer
Figure 3.3: Structure of optimal policies, i.e., transmission rates (packets/slot) for different channel states when the buffer occupancy is fixed at 1, 5, 10 and 14 packets. System parameters are: buffer length B = 15 packets, arrival rate λ = 1 packet/slot, average power constraint P = 16dB (the rest is given in Section 3.5.1). The fading process is i.i.d. over time. As can be seen, when the buffer occupancy is fixed, the transmission rate is non-increasing when the channel gain decreases toward outage (state 0).
Pb = 10−3 and we only care about packet loss due to buffer overflow. We consider two other classes of policies: channel inversion, i.e.,C Inv, and channel adaptive, i.e., C Adpt. For each C Inv policy, a fixed transmission rate is first selected. Based on this selected rate, the required SNR to meet the target BER is determined. Then, for each channel state with non-zeror gain, the transmit power is calculated based on inverting the channel gain to meet the required SNR. For channel state 0, i.e., when the channel gain is zero, the transmit- ter is turned off. In a C Adpt policy, we use the optimal link-adaptive policy that maximizes the transmission rate for our channel model under some aver- age power constraint and with the assumption that there are always packets to
64
12 14 16 18 20 22 24 26
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Power (dB)
Normalized Packet Loss Rate
MDP_I (BER = 10−3) C_Inv
C_Adpt Asymptotic Limit
Figure 3.4: Performance, in terms of normalized packet loss rate (due to buffer overflow only) versus average transmission power, for MDP I, C Inv, and C Adpt policies. System parameters are: buffer lengthB = 15 packets, arrival rateλ= 3 packets/slot, BER constraintPb = 10−3 (the rest is in Section 3.5.1).
Channel model is given by Table 3.1.
transmit. This scheme is equivalent to the variable-rate variable-power adaptive MQAM proposed in [GC97] (see Chapter 2, Section 2.1 for more detail). The performance of the three classes of policies, in terms of normalized packet loss rate (due to buffer overflow) versus the average power consumption are plotted in Fig. 3.4.
As it is expected, MDP I outperforms the other two classes of adaptive policies. For low values of average transmit power, the performance of MDP I policies and C Adpt policies are very close while that of the C Inv policies is much worse. This is expected, since at low power, the structure of an MDP I policy is similar to that of the C Adpt, and by focusing on conserving power, the system performance is improved. At high power, the performance of MDP I
and C Inv policies are close and it is interesting to see that, for the same average transmit power, a C Inv policy can result in less packet loss rate relative to a C Adpt policy. This means that at this high range of average transmission power, if we only adapt to the channel, the performance can be worse than not doing any rate adaptation at all. We have looked at the performance of MDP I, C Adpt, and C Inv policies for different values of Doppler frequency and observed that the performance of all schemes get worse when the Doppler frequency decreases. However, the relative difference between performance of different classes of adaptive policies does not seem to depend much on this parameter. We have also obtained results for longer buffer capacity and lower data arrival rate and observed that the performance trends of all schemes remain unchanged.
It can be noted in Fig. 3.4 that the packet loss rates of all policies, MDP I, C Adpt, C Inv, reach a floor when the transmit power is high enough. This floor is represented by the asymptotic limit in Fig. 3.4. When the power constraint is high enough, the transmitter will always empty the buffer except in state 0. In that case, the floor for the normalized packet loss rate can be calculated exactly by:
Lf loor = 1 λ
1 K
P∞
n=1PG(0,0)n−1(1−PG(0,0))La(n+ 1) P∞
n=1(n+ 1)PG(0,0)n−1(1−PG(0,0)) + K−1 K La(1)
, (3.44) whereLa(n) is the expected number of packets lost due to buffer overflow during an interval of n time slosts, given that the buffer is empty at the beginning of the interval and no packet is further transmitted. Similar to (3.10), La(n) can
66 be calculated by:
La(n) = (λnTs) 1−
B−b+u−1X
k=0
exp(−λnTs)(λnTs)k k!
!
−(B−b+u) 1−
B−b+uX
k=0
exp(−λnTs)(λnTs)k k!
! .
(3.45)
Note that for the channel models in which there is no outage state, i.e., all channel states have positive gain, the packet loss floor will be La(1). This discussion also holds for the problem in Chapter 4.