Part II: Real-Time Communication 63
6.3 Performance Evaluation
Table 6.1: Cluster Centrality Metrics.
Metric Definition
Degree DC(vq) =degree(vN q)
ℓ−1
Betweenness BC(vq) =∑q̸=r spr,s(vq)
spr,s
Closeness CC(vq) = 1
∑p̸=qdistance(vp,vq)
Eigenvector EC(vq) =λ 1
max(Ak)·∑Nj=1k aj,q·xj
densityΛ, whereN=80 is the total number of nodes including the gateway. For each centrality metric, i.e. DC, CC, BC and EC, we choose the node with the highest score (i.e. centrality) as gateway. The remainingN−1 nodes act as field nodes, i.e. as sensors and/or relays.
2) Real-Time flows. A random subset of n field devices are assumed as sensors periodically transmitting sensing data toward one gateway. Each sensor node produces a single flow of real-time packets. We assume n varies within [1,10] for all the cases. The result is a random set ofn real-time flows for each network topology. For each flow,Ci is obtained directly from the multiplication of the route length of φi (in hops) and the length of the time slot (configured to 10ms). We assume implicit deadlines, thus Di=Ti, whereTi is the flow period. Moreover, Ti
is harmonically generated in the range [24,27]slots. This leads to a direct computation of the super-frame length, a.k.a. hyper-period (H), here considered its maximum value of 1280ms.
3) Real-time assessment. For the evaluation of schedulability we assume a worst-case factor∆i,j
(as in [147]) and a time interval of evaluationℓ=H.
6.3.1.2 Simulation Results
Figure6.2ashows the schedulability ratio achieved with a gateway designation based on degree centrality and on a random baseline. These results suggest that, under varying topologies and workload conditions, the degree centrality metric is always better than (or equal to) the baseline.
Notably, the degree centrality achieves up to about 30% better schedulability under particular settings. These plots also suggest that higher gains are obtained under moderate workload, par-ticularly at the low and high network connectivity levels. Note that connectivity is varied here through node density, where a value of 1 targets a fully linked network, i.e. with each node linked (in average) to every other node in the network.
Figure6.2bshows the absolute deviation of the schedulability ratio achieved with the other types of centrality with respect to the degree centrality metric, which we consider as a reference for these plots. In all cases, our outcomes show that these other centrality criteria are always better than or equal to the degree centrality, achieving up to∼18% of additional improvement. However, none of these centrality metrics dominated the others in all the cases evaluated, thus exploring their synergy holds promise.
6.3 Performance Evaluation 71
1 5 10
Number of flows 0
0.5 1
Schedulability ratio
density = 0.1, m = 16, N = 80
Degree Random
1 5 10
Number of flows 0
0.5 1
Schedulability ratio
density = 0.5, m = 16, N = 80
Degree Random
1 5 10
Number of flows 0
0.5 1
Schedulability ratio
density = 0.5, m = 16, N = 80
Degree Random
(a)
1 5 10
Number of flows 0
0.1 0.2
Deviation
density = 0.1, m = 16, N = 80
Betweenness Closeness Eigenvector
1 5 10
Number of flows 0
0.1 0.2
Deviation
density = 0.5, m = 16, N = 80
Betweenness Closeness Eigenvector
1 5 10
Number of flows 0
0.1 0.2
Deviation
density = 1, m = 16, N = 80
Betweenness Closeness Eigenvector
(b)
Figure 6.2: The schedulability ratio under varying number of network flows n∈[1,10]; a) the comparison between the degree centrality and a random baseline when varying network density in {0.1,0.5,1.0}; b) the absolute deviation of the other centrality measures w.r.t. the degree centrality.
In particular, the betweenness centrality metric was able to achieve the largest gains for all densities, while eigenvector and closeness were better only under particular configurations. The eigenvector centrality was almost always equal or slightly better than the betweenness centrality for the lowest density case, but generally worse for the high and medium densities. Closeness cen-trality shows, in general, the smallest gains and a more unsteady performance, but still remarkably if compared to the random baseline. Degree centrality, though not dominant, remains the simplest, thus preferable from a complexity viewpoint.
6.3.2 Centrality-Driven Multi-Gateway Designation
6.3.2.1 Simulation Setup
1) Network topologies. We consider 1000 random topologies built upon the synthetic generation of network graphs. Each topology was generated with a target node densityd=0.1 using a sparse uniformly distributed binary random matrix ofN×N, i.e. assuming lossless links, where N is the total number of network nodes, including thek gateways. Without loss of generality, we use k={1,3,5}andN=75 for all the simulation experiments.
0 320 640 960 1280 Time (ms)
0 100 200 300 400 500
W.C. Network demand (ms)
n=25, d = 0.1, m = 16, N = 75
Random
Degree Random
Degree
GW 1 Degree = 10
GW 2 Degree = 12
GW 3 Degree = 13
.
Figure 6.3: (Left) Worst-case network demand (ms) in one simulated case during a hyper-period.
(Right) An illustrative example of joint clustering and gateway designation;
2) Gateway designation. After clusters have been created, a number ofk nodes is selected as gateways using each of the cluster centrality metrics in Table6.1. In the absence of a better solution to compare against, a random designation ofk gateways is also considered, for benchmarking.
Note the random designation is done before clusters are created.
3) Network flows. A subset ofn∈[1,30]vertices is selected randomly as sensor nodes, i.e. to periodically transmit deadline-constrained data toward one of thekgateways. EachCiis computed directly by the product of the time slot, i.e., 10 ms, and the number of hops in the pathφi. Ti is harmonic and randomly generated in the range of[24,27]. This implies a super-frame length of H=1280ms. Finally,Diis set implicitly, i.e.Di=Ti.
4) Real-time assessment. We assess schedulability over a time interval equal to the super-frame, i.e., ℓ=H, and when all the m=16 channels are available. Concerning ∆i,j, we use precise computation derived from the network topology.
6.3.2.2 Simulation Results
Fig.6.3and Fig.6.4show simulation results based on the setup described above. Fig.6.4apresents the schedulability ratio overnflows for both the proposed multi-gateway designation framework (thicker lines) and a random baseline (thinner lines). As expected, results show that increasing the number of gateways improves schedulability in all cases. A schedulability ratio of 99% can be achieved with only 5, 5 and 6 flows using random designation withk =1,3 and 5, respec-tively. With our framework, these values increase to 11,17 and 21 flows, respectively, thus an improvement of 3.5 times in the number of schedulable flows with 5 gateways.
Fig. 6.3 (left) shows the network demand for one of the 1000 random topologies imposed byn=25 flows over an interval equal to the hyperperiodH, for both approaches. These results confirm that our framework significantly reduces the worst-case demand bounds.
Fig.6.3(right) illustrates the clustering and gateway designation in a concrete topology when using our framework. The clustering is coded in different colours and the corresponding desig-nated gateway is marked with a star.
6.3 Performance Evaluation 73
5 10 15 20 25 30
Number of flows (n) 0
0.5 1
Schedulability Ratio
d = 0.1, m = 16, N = 75
Random
Degree
(a)
0 5 10 15 20 25 30
Number of flows (n) -0.05
0 0.05
Deviation
k = 1, d = 0.1, m = 16, N = 75
BC CC EC
0 5 10 15 20 25 30
Number of flows (n) -0.05
0 0.05
Deviation
k = 3, d = 0.1, m = 16, N = 75
BC CC EC
0 5 10 15 20 25 30
Number of flows (n) -0.05
0 0.05
Deviation
k = 5, d = 0.1, m = 16, N = 75
BC CC EC
(b)
Figure 6.4: a) Schedulability ratio of 1000 random topologies with target density 0.1 and k= {1,3,5}, withdegreecentrality versus random designation, and b)schedulability ratio deviation of other centrality metrics w.r.t. degree centrality fork∈ {1,3,5}
Figures6.4bpresent the deviation (difference) in the schedulability ratio achieved by the other centrality metrics w.r.t. degree centrality, namely, betweenness centrality (BC), closeness central-ity (CC) and eigenvector centralcentral-ity (EC), fork∈ {1,3,5}. These results should be correlated with those in Fig.6.4a. All centrality metrics perform equally well for low loads, given the low mutual interference, with all flows meeting their deadlines (100% schedulability). With high loads, mutual interference grows and all metrics perform poorly, with few or no flows meeting their deadlines (0% schedulability). The differences, in any case small (<5%), appear when schedulability starts degrading, thus being of low interest from a system design point of view.