4.4 Guaranteeing Quality of Service (QoS)
4.4.1 Formulations of Throughput Distribution and PDR
4.4.2.2 Multi-user Broadcast Case
Here, we start by obtaining the maximum achievable throughput threshold,ηth∗, for the min- imum and mean throughput of the system. To this end, we find the solutions to the problem in (4.49) for variousPthvalues and three different number of users (N =3,30and300). The
§4.4 Guaranteeing Quality of Service (QoS) 85 1−Pth )s p b M( h t η el ba ve i hc A m u mi xa M mean(·), N= 3 mean(·), N= 30 mean(·), N= 300 min(·), N= 3 min(·), N= 30 min(·), N= 300 10−6 10−5 10−4 10−3 10−2 10−1 100 1 1.5 2 2.5
Figure 4.11: Maximum achievableηthvs.1−PthforN =3,30and300users. The threshold constraint is considered for two cases: minimum of users’ throughput (min(⋅)) and mean of
users’ throughput (mean(⋅)).
results are depicted in Fig. 4.11. Each data point here corresponds to one set of{M, Ns}. In Fig. 4.11, it is observed that by increasing the value ofPth(i.e. tighter throughput con- straint), the maximum achievableηthdecreases for bothmin(⋅)andmean(⋅)cases. However, the effect of increasing the number of users N is different for these two cases. It results in smallerηth∗ for themin(⋅)case, as guaranteeing this QoS for larger number of users requires either sending more SRLNC packets (i.e. larger Ns) or choosing lower number of packets per block (i.e. smallerM), which in turn reduces the throughput values. In contrast, for the mean(⋅)case, increasingN interestingly results in largerηth∗. The reason behind this is the use of convolutions in (4.48). In fact, increasing the number of users in (4.48) results in a narrower PDF around the mean value of the throughput and as a result, a sharper rise in its CDF. Hence, when finding the maximum achievable ηth, choosing higher throughput with a fixedPthis possible for largerN.
Fig. 4.12 shows the corresponding PDRs of users with the worst PER (Pe =0.5), for the results depicted in Fig. 4.11. We note that these results provide upper-bounds on the PDR values, as they show the PDR of users with the worst channel condition and correspond to the maximumηth. In other words, better values of PDR can be achieved if we choose smallerηth and/or consider users with lower PER.
86 RLNC for Delay Sensitive Applications over TDD Satellite Channels 1−Pth P D R o f th e U se r( s) w it h th e W o rs t P E R mean(·), N= 3 mean(·), N= 30 mean(·), N= 300 min(·), N= 3 min(·), N= 30 min(·), N= 300 10−6 10−5 10−4 10−3 10−2 10−1 100 10−10 10−8 10−6 10−4 10−2 100
Figure 4.12: PDR of the user(s) with the worst PER vs. 1−Pth, for the results shown in Fig. 4.11.
From Fig. 4.12, it can be seen that for themin(⋅)case, lower (i.e. better) PDR bounds are achieved as the number of usersN increases. This is due to the fact that guaranteeing this QoS for largerN requires either largerNsor smallerM, as mentioned previously. Moreover, it is interesting to note that the value ofPthstrictly controls the maximum PDR. However, for the mean(⋅)case, regardless of the value ofN, the upper-bound of PDR is the maximum PER of the users in the system (i.e. 0.5 here). In fact, the optimum solution to (4.49) for this case is Ns=M, which results inPd=Pein (4.42) for every user.
Now, for any ηth smaller than thoseηth∗ depicted in Fig. 4.11, the optimumNs andM for the problem in (4.43) can be obtained. The results for four different cases are presented in Table 4.4. Considering these results, it can be concluded that designing based on guaranteeing the mean throughput provides a better performance for users with PER of 0.01 and 0.2, asηM is larger and PDR is still very small. In contrast, designing based on the minimum of users’ throughputs gives better performance for users withPe=0.5, as the system has actually been designed for these users. However, this better performance of users withPe=0.5comes at the cost of sacrificing the performance of users with lower PERs, sinceηM and consequently their throughputs become smaller.
In addition to the trade-off between the performances of users with different erasure con- ditions, the trade-off between the throughput and PDR can also be observed in Table 4.4. In
§4.5 Summary 87
Table 4.4: Examples of system design parameters forN =30users andPth=0.9. Pdshows the PDR of users withPe=[0.01,0.2,0.5]. PDRs of lower than10−16are shown as Pd =0.
Throughput values are in Mega bps (Mbps). g(⋅) ηth {M, Ns} Pd ηM min(⋅) 1.3 {59,156} [0, 0,5.5×10 −4] 1.33 1.1 {50,156} [0, 0,1.4×10−6] 1.12 mean(⋅) 1.8 {96,153} [0,9.9×10 −8, 0.49] 2.19 1.6 {85,153} [0,1.1×10−12, 0.46] 1.94
each of the min(⋅)ormean(⋅)cases, setting a higher throughput threshold (i.e. increasing ηth) results in higher values for the PDR, which is not desirable.
Finally, we compare the SRLNC scheme with RR, LT and ISRLNC schemes. For this purpose, we focus on the guaranteed minimum throughput and solve (4.43) for SRLNC, RR, LT and ISRLNC schemes for N = 30 users and different values ofPth andηth. The results showingηth versus their corresponding PDR for the user(s) with the worst PER are depicted in Fig. 4.13. These results show the superiority of SRLNC over RR and LT for the considered scenarios. For example, the maximum PDR of SRLNC forPth = 0.99andηth =1 Mbps is less than10−7, while it is larger than10−1 and10−7 for RR and LT, respectively. Moreover, it can be observed that feedback-free SRLNC performs nearly as well as the idealistic one (ISRLNC), especially for practical PDR values higher than10−4.
4.5
Summary
In this chapter, the problem of joint optimization of throughput and PDR in NC systems for applications with strict delivery deadline requirements was studied. We employed RLNC and targeted satellite systems with TDD erasure channels and non-negligible RTTs. Here, we proposed a general framework to analytically study the mean throughout and PDR of users, as well as their interactions under various system parameters and settings. Below we summarize the important messages and findings.
Firstly, using the proposed framework, the impact of feedback on the performance of the NC systems was studied in two stages. First, we compared the one-round (feedback-free) and two-round (with one feedback) schemes under different RTTs and delivery deadline re- quirements and observed that for single-user systems with large RTTs, the feedback-free NC
88 RLNC for Delay Sensitive Applications over TDD Satellite Channels
PDR of the User(s) with the Worst PER
)s p b M( h t η Pth= 0.8, SRLNC Pth= 0.99, SRLNC Pth= 0.8, RR Pth= 0.99, RR Pth= 0.8, LT Pth= 0.99, LT Pth= 0.8, ISRLNC Pth= 0.99, ISRLNC 10−12 10−10 10−8 10−6 10−4 10−2 100 0 0.5 1 1.5
Figure 4.13: Comparingηthvs. PDR of the user(s) with the worst PER for SRLNC, RR, LT and ISRLNC schemes withN =30users.
schemes provide better performances, in terms of the mean throughput and PDR. Second, for broadcasting to multiple users, we compared our feedback-free one-round SRLNC scheme with an idealistic SRLNC scheme, where immediate and complete knowledge about the users’ rDOF is available at the sender, and showed that by optimizing the NC design parameters (i.e. the number of packets per block, the number of transmissions for each block and the field size), performances very close to those of the idealistic scheme can be achieved, even without using feedback.
Secondly, for multi-user broadcast case, we investigated multiple scenarios, corresponding to different QoS requirements, and obtained the optimum SRLNC-based transmission strate- gies. The results shed light on trade-offs between performances of different users, affected by the required QoS types and NC design parameters. One important observation was that to de- sign a broadcast system, performances of some of the users should be sacrificed. For example, it was possible to provide an average performance for all users at the cost of sacrificing the per- formance of users with good channel conditions. As an alternative, a very good performance could be provided to users with good channel conditions, but at the cost of bringing down the performances of other users considerably below the average. Another important observation was that in contrast to conventional wisdom, choosing higher field sizeqwill not necessarily lead to better performance, hence it is beneficial to include its effect on overhead and decoding
§4.5 Summary 89
probabilities when optimizing system parameters. It was also observed that SRLNC scheme outperforms LT and RR schemes.
Lastly, we studied the problem of guaranteeing throughput. We focused on a feedback- free systematic RLNC scheme as the transmission model and considered two different classes of QoS requirements in the system, which required either the minimum or mean of users’ throughput to be higher than a thresholdηthwith probabilityPth. To satisfy these throughput requirements, we first formulated the PDF and CDF of the throughput for a single-user case and then extended the formulations to the multi-user case. We showed that by using the calculated PDFs and CDFs, the system can be designed such that the throughput and deadline require- ments of the system are guaranteed. Furthermore, we highlighted the trade-offs between the throughputs and PDRs of different users, which are affected by the system operating point and the channel erasure conditions. Comparing SRLNC with the RR, LT and ISRLNC schemes, it was shown that SRLNC is superior to the RR and LT schemes within the proposed framework, and it can work close to the ISRLNC scheme, where immediate feedbacks are considered to be available.
It is noteworthy that although the focus of this chapter was mostly on TDD channels with large RTTs, the proposed general framework can be used to quantify the performance of any similar system with arbitrary parameter sets, investigating which of the one- or two-round schemes works better. This is one of our contributions that no a priori assumption on the use of feedback is enforced for NC system design. Moreover, although we considered only up to one round of feedback, extending the proposed approaches to analytically obtain the exact expres- sions for schemes with more than one round of feedback is also possible for the single-user case, but can be challenging for multi-user broadcasting. In the literature, broadcast solutions are given by either heuristics [28], exhaustive search methods like Brute-force analysis [18], or simulations [15,16], but here we were able to obtain closed-form expressions for feedback-free schemes and evaluate them in the broadcast scenario.
By considering delay as a performance metric, as in Chapter 3, It is possible to extend the formulations in this chapter to scenarios where average delay constraint is also of interest.
Chapter 5