3.6 Numerical Results and Discussions
3.6.2 Average Gap Processing Time of the Window-Based Scheme . 60
Figure 3.9 shows GP Twindow against Eb/N0 for various window sizes. The accuracy of the derived average gap processing time (i.e. (3.13)) of the window-based stall avoidance scheme is validated by simulations. As shown in the figure, the analytical results match the simulations well. Most importantly, Fig. 3.9 also provides impor-tant insights into designing an admission control policy subject to the gap processing time constraint GP Tc. Consider a fair scheduling policy. The number of acceptable fully loaded users (Nu) in the system can be approximated by
Nu = bGP Tc/GP Twindowc , (3.46) where bxc is the largest integer less than or equal to x. For Eb/N0 = 14 dB in Fig.
3.9, one can observe that GP Twindow = 15.3, 19.2, 23.1 TTIs for W = 12, 15, 18,
2 3 4 5 6 1
2 3 4 5 6 7 8 9 10
Number of allowable retransmissions
Number of acceptable fully loaded users
V=4
V=6
V=8
Fig. 3.8: The number of acceptable fully loaded users of the timer-based stall avoidance scheme versus the numbers of allowable retransmissions (hr) for various number processes (V ) in the parallel SAW HARQ mechanism subject to a constraint of gap processing time 100 TTIs.
10 12 14 16 18 20 12
14 16 18 20 22 24 26 28 30 32
Eb/N
0 (dB)
Gap processing time (TTIs)
: Simulation : Analysis
Window: 18
Window: 15
Window: 12
Fig. 3.9: The average gap processing time of the window-based stall avoidance scheme with different window sizes for the 4-process SAW HARQ mechanism in the Rayleigh fading channel with Doppler frequency of 100 Hz.
respectively. For the maximal gap processing time constraint GP Tc = 100 TTIs, the allowable users are therefore equal to 6, 5, and 4 for the window of size of 12, 15, and 18, respectively.
As a matter of fact, the window size is a function of the allowable minimum retransmissions (hm) and the number of HARQ processes (V ). In the single user case, for an V-process SAW HARQ mechanism with a window of size of W , the allowable retransmissions (hm) of a missing packet is at least
hm = W
V − 1 − 1 . (3.47)
For V = 4 and W = 12, the missing packet can be retransmitted by V −1W −1 = 3 times.
Assume one process keeps transmitting a missing packet and the other 3 processes transmit new packets successfully. After 4 cycles, a window with a size of 12 is fully occupied. Because the window has no more space for the missing packet, this packet will not be transmitted.
Hence, the admission control policy subject to the gap processing time require-ment can also be designed for different combinations of parameters hm and V . Take V = 4 as an example. Figure 3.10 shows the number of acceptable fully loaded users versus Eb/N0 with various minimum allowable retransmissions (hm) subject to a gap processing time constraint of 100 TTIs. These curves are obtained by mapping Fig.
3.9 according to (3.46). From (3.47), hm = 3, 4, and 5 correspond to W = 12, 15, and 18, respectively. As shown in the figure, the acceptable fully loaded users is reduced from 7 to 4 as hm increases from 3 to 5 for 16 dB ≤ Eb/N0 ≤ 18 dB.
Figure 3.11 shows the number of acceptable fully loaded users of the window-based stall avoidance scheme against packet error rate (Pe) with various numbers of parallel processes (V ), where the maximal allowable gap processing time GP Tc= 100 TTIs and the minimum allowable retransmission hm = 3. These curves are obtained by substituting the parameters V , W , Ps, and PN →Ainto (3.13). Note that Pe = 1−Ps
and the corresponding window size W = 12, 20, and 28 is obtained from W = (n + 1)(V −1) according to (3.47). This figure can be associated with an admission control policy subject to gap processing time by observing Pe. According to the CRC results and Pe, suitable number of allowable users to maintain the QoS can be determined from the standpoint of meeting the gap processing time requirement. In the figure, we find that more parallel SAW HARQ processes results in fewer allowable users in the system subject to the total gap processing time requirement. For Pe= 0.15, the number of acceptable fully loaded users decreases from 7 to 3 as V increases from 4 to 8. Recall hm = V −1W − 1 in (3.47). For a fixed hm, a larger value of V also leads to a lager value of W and longer gap processing time. Hence, the number of acceptable fully loaded users is reduced for a larger value of V to satisfy the gap processing time requirement.
3.6.3 Average Gap Processing Time of the Indicator-Based Scheme
Figure 3.12 shows the analytical average gap processing time for the indicator-based stall avoidance scheme obtained from (3.21) and simulations. As shown in the figure, the differences between simulation and the analytical approximation are quite small.
Similar to the reason for the timer-based scheme, the receiver may possibly receive a packet with a smaller TSN than the previous missing packet. In this situation, the procedure of monitoring the status of the HARQ process can start only after another packet with a larger TSN arrives. Hence, Proposition III shows a lower bound on the average gap processing time for the indicator-based stall avoidance mechanism.
Comparing to Figs. 3.7, 3.9, and 3.12, the indicator-based scheme outperforms the timer-based and the window-based schemes in terms of the gap processing time.
For a 4-process SAW HARQ mechanism with Eb/N0 = 14 dB, the gap processing
10 12 14 16 18 20 2
3 4 5 6 7 8
Eb/N
0 (dB)
Number of acceptable fully loaded users
hm=3 (W=12 )
hm=4 (W=15)
hm=5 (W=18)
Fig. 3.10: The number of acceptable fully loaded users of the window-based stall avoidance scheme versus Eb/N0 with various number minimum allowable retransmissions (hm= 3, 4, 5) in the 4-process SAW HARQ mechanism subject to a gap processing time constraint of 100 TTIs.
0.1 0.15 0.2 0.25 0.3 0.35 0.4 1
2 3 4 5 6 7 8
Pe
Number of acceptable fully loaded users
V=4
V=6
V=8
Fig. 3.11: The number of acceptable fully loaded users of the window-based stall avoidance scheme versus Pe with various number processes (V ) in the parallel SAW HARQ mechanism subject to a gap processing time constraint of 100 TTIs. The minimum allowable re-transmission (hm) is three.
n=3 (W=12 )
n=4 (W=15)
n=5 (W=18)
10 12 14 16 18 20
4 6 8 10 12 14 16 18
Eb/N
0 (dB)
Gap processing time (TTIs)
: Simulation : Approximation
8−process
6−process
4−process
Fig. 3.12: Effect of the number of processes in the multi-process SAW HARQ mechanism on the gap processing time for the indicator-based avoidance scheme in the Rayleigh fading channel with Doppler frequency of 100 Hz .
time of the indicator-based scheme is 4.9 TTIs; the gap processing time is 21.7 TTIs for the timer-based scheme with a timer expiration of 20 TTIs; the gap processing time is 15.27 TTIs for the window-based scheme with a window of a size of 12. Also, it is found that the more the parallel HARQ processes, the longer the average gap processing time.
The developed gap processing time computation method for the indicator-based scheme can be applied to determine the acceptable fully loaded users through bGP Tc/GP Tindicatorc as the timer-based and the window-based schemes, where GP Tc is a given constrain on the gap processing time. Figure 3.13 shows the number of acceptable fully loaded users of the indicator-based stall avoidance scheme against Pe with various numbers of parallel HARQ processes (V ) under a constraint of the gap processing time 100 TTIs. One can find that with the aid of the indicator-based stall avoidance scheme, an HSDPA system can accommodate more users compared to the window-based scheme. For V = 4 at Pe = 0.2, the number of acceptable fully loaded users are 4, 6, and 24 for the timer-based, the window-based, and the indicator-based schemes, respectively. Furthermore, although more parallel HARQ processes can enhance throughput, the side effect of increasing gap processing time can not be ignored. As V increases from 4 to 12 at Pe = 0.2, it is necessary to reduce the acceptable fully loaded users from 24 to 6 if the gap processing time requirement is fulfilled.
3.6.4 Probability Mass Function of the Gap Processing Time
Figure 3.14 shows the probability mass functions of the gap processing time for the timer-based, the window-based, and the indicator-based stall avoidance schemes, where the timer’s expiration D = 24 and the detection window size W = 20 with
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0
5 10 15 20 25 30
Pe
Number of acceptable fully loaded users
V=4
V=6 V=8 V=10 V=12
Fig. 3.13: The number of acceptable fully loaded users of the indicator-based stall avoidance scheme versus Pe with various number processes (V ) in the parallel SAW HARQ mechanism subject to a gap processing time constraint of 100 TTIs.
V = 6 parallel HARQ processes at Eb/N0 = 14 dB. The analytical pmf for the timer-based scheme can be obtained by evaluating (3.6) and (3.9). For the window-timer-based scheme, we obtain the analytical pmf by evaluating (3.16), while the pmf for the indicator-based scheme can be obtained by calculating (3.35) and (3.43). From the figure, one can see that the analytical results can approximate the simulation results.
Most importantly, the gap processing time for the timer-based and the indicator-based schemes are centralized while that of the window-indicator-based scheme is widely spread.
For example, 78% and 71% of the gap processing time of the timer-based and the indicator-based schemes are lower than 26 and 8 TTIs, respectively, where the cor-responding average gap processing times are 26.8 and 8.4 TTIs. However, for the window-based scheme, only 53% of the gap processing time are lower than the aver-age value of 25.6 TTIs. Thus, we can conclude that indicator-based scheme is more capable of maintaining stable and better QoS for HSDPA.