Part II: Real-Time Communication 63
7.3 Performance Evaluation
designation. However, when computing the centrality score of each node, we need first to adapt the respectivenetworkcentrality expression to the current interpretation of being acluster centrality metric, as presented earlier in Table6.1. Essentially, this adaptation is done by considering each clusterGℓas a sub-graph ofGcharacterized by its respective cluster adjacency matrixAℓand the number of nodes in the clusterNℓ, with∑kl=1Nℓ=N. In the case of Eq.7.1for the minimal-overlap centrality we reformule it as:
MOℓ(vq) = 1
∑ni,ℓj=1∧i̸=j∆qi,j+1 (7.2)
where nℓ <n is the number of sources which flows are directed to node vq ∈Gℓ, q∈[1,Nℓ], and thus MOℓ(vq)denotes the minimal-overlap (MO) cluster centrality for a given nodevq∈Gℓ. Note that whenk=1, the associated cluster corresponds to the whole network. Thus, this MO expression also generalizes Eq.7.1to multiple gateways.
7.3 Performance Evaluation
7.3.1 Simulation Setup
1) Wireless mesh networks. We consider 1000 network topologies built upon the synthetic gen-eration of random graphs. Each graph is generated using a sparse uniformly distributed random matrix with a target (average) node density ofd={0.1,0.5,1.0}. The dimension of the random matrix isN×N, whereN=75 is the total number of nodes in the network includingk={1,3,5} gateways. All graphs represent TSCH-based WSNs in multi-channel operation, here usingm=16 channels and time slots oftslot=10 ms.
2) Real-time network flows. We consider a subset ofn∈[1,30]nodes are source nodes while the rest acts as relays or gateways. Source nodes are selected randomly from the set of field nodes, i.e.
excluding thekgateways. All flows are consistently generated according to the real-time model 4-tuple parameter(Ci,Di,Ti,φi). Each flow is directed to only one of thek gateways through its shortest pathφi. Ci is computed by multiplying the time slot durationtslot with the length of the routing path (in number of hops).Tiis randomly generated in the form 2η, withη∈Nin the range [4,7], thus leading to harmonic periods. This implies a super-frame duration (or hyper-period) of H=27slots, i.e. H=1280 ms. All the respective deadlines (Di) are assumed to be equal to the respective flow periods, i.e.Di=Ti, thus assuming an implicit-deadline model.
3) Real-time performance assessment. We resort to the supply/demand-bound based schedu-lability test introduced earlier. We recall this real-time performance assessment tool assumes all transmissions are scheduled under a global EDF scheduling policy. We consider an interval equal to the hyper-period (ℓ=H) for evaluation, in all the cases. Concerning∆i,j, we use precise com-putation of node-path overlaps derived directly from network topologies.
1 15 30 Number of flows (n) 0
0.2 0.4 0.6 0.8 1
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d = 0.1, N = 75, m = 16
Min. Overlap Eigenvector Closeness Betweenness Degree Random
(a)
1 15 30
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d = 0.5, N = 75, m = 16
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(b)
1 15 30
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d = 1.0, N = 75, m = 16
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(c)
1 15 30
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0.5 1
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d = 0.1, N = 75, m = 16
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(d)
1 15 30
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d = 0.5, N = 75, m = 16
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(e)
1 15 30
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d = 1.0, N = 75, m = 16
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(f)
Figure 7.2: (Top) Schedulability ratio of 1000 random topologies for varying number of flows with target density 0.1 (a), 0.5 (b) and 1.0 (b) resorting to gateway designation methods based on i) classical network centrality metrics (e.g. eigenvector centrality) andii) the proposed minimal-overlap network centrality. (Bottom) The relative ratio in terms of schedulability from the best and worst possible gateway assignments. A random selection is included as reference.
7.3.2 Simulation Results
7.3.2.1 Traffic Schedulability,k=1
Fig.7.2(top-row) presents schedulability ratio results for the case of single-gateway designation, i.e.k=1, when varying the number of network flowsn∈[1,30]and when using different network densitiesd={0.1,0.5,1.0}(see left, middle and right plots of top row, respectively). These results correspond to different centrality metrics used as criteria to designate one gateway. Specifically, our minimal-overlap metric is compared against the classical eigenvector, closeness, betweenness and degree, as well as versus a random assignment for reference. As expected, it can be observed that the schedulability ratio decreases for larger numbers of flows in all configurations due to the larger channel contention and transmission conflicts. Conversely, higher network density increases the number of potential paths between any given pair of nodes, favoring schedulability.
The results also show that the minimal-overlap gateway designation method achieves higher schedulability for all numbers of flows and densities when compared with a non flow-informed methodbased on classical centrality metrics (e.g. betweenness or degree). We argue this is caused mostly by the minimal-overlap method’s ability to decrease, by design, the number of overlapping paths. This, in turn, allows for reducing transmission conflicts, thus improving the timely delivery of data messages. As expected, the proposedminimal-overlapmethod is also clearly superior to the random selection, further demonstrating the significance of judicious gateway designation.
We also analyze how the proposed method deviates from the system’s optimal gateway election
7.3 Performance Evaluation 81
1 5 10 15 20 25
Number of flows (n) 0
10 20 30
Avg. Overlaps
d = {0.1,1.0}, N = 75, m = 16 Min. Overlap
Degree
(a)
1 5 10 15 20 25
Number of flows (n) 0
0.5 1
Avg. Exec. Time (s)
d = {0.1,1.0}, N = 75, m = 16 Min. Overlap
Degree Best
(b)
Figure 7.3: (Left) Average number of overlaps of 100 random topologies when varying network flows for two gateway designation methods (minimal-overlap and degree) and two extreme den-sities, namely 0.1 (solid line) and 1.0 (dotted line). (Right) Execution time for different gateway designation methods, namely, minimal-overlap, degree and best, considering up to 25 network flows and two extreme target densities, namely 0.1 (solid line) and 1.0 (dotted line).
Fig.7.2(bottom-row). The metric relative ratiois defined as the ratio between the schedulabil-ity ratio of a given method to the schedulabilschedulabil-ity ratio of the best and worst performing nodes in the network, with a value of 1 denoting best and 0 the worst performance. The results show the performance of the proposed method is only slightly below the best method, having the maxi-mum degradation of∼24% for a density of 0.1 and 20 simultaneous flows. We highlight that this degradation is lower for larger densities (e.g.d=0.5), becoming negligible for highly connected networks (e.g. d=1.0), because the overall overlapping degree decreases for increasing density;
as shown also in [11]. Additionally, these results reveal that the performance improvements of the proposed method are, in general, able to increase for higher density and higher number of flows when compared with other centrality-based gateway designation methods or random gateway se-lection.
7.3.2.2 Computational Cost,k=1
Fig.7.3 depicts the average execution time for different gateway designation methods versus the optimal gateway designation. Regarding the classical centrality-based designation method, we solely present the result for degree centrality for visual clarity and because this has the lowest execution time among all metrics. We also present results for two extreme density values of 0.1 (solid line) and 1 (dashed lines). The setup for this experiment used MATLAB R2020b on Ubuntu 18.04 LTS on a laptop with an Intel Core i7-6500U CPU at 2.5GHz and 4GB of DD3 RAM.
These results confirm the low execution time of the degree-centrality gateway designation.
On the other hand, minimal overlaps designation considerably decreases the execution time when compared against optimal gateway designation, particularly for higher number of flows. Note that the optimal method uses extensive search with full schedulability analysis for each case, while the MO metric just requires computing the number of overlaps in the network given a set of flows. The results also show that the density has a minimal impact on the average execution time. Overall, the proposed method provides a good trade-off between achievable schedulability ratio (near optimal) and computational cost (about half the value of the optimal method).
1 15 30 Number of flows (n)
0 0.5 1
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d = 0.1, N = 75, m = 16
Worst
Best
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d = 0.1, N = 75, m = 16
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Best
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(a)k=2
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d = 0.5, N = 75, m = 16
Worst Best
Min. Overlap Eigenvector Closeness Betweenness Degree
(b)k=3
Figure 7.4: Average schedulability ratio of 1000 random topologies for varying number of flows when using (a)k=2 and (b)k=3 gateways. The solid blue line curves show the results for our minimal-overlap centrality used as a cluster centrality metric and when compared against classical centrality metrics. Best and worst performing nodes are also shown as benchmarks.
. 7.3.2.3 Schedulability Ratio, k>1
Fig.7.4presents different schedulability ratio results for theminimal-overlapmulti-gateway des-ignation framework when compared to classical centrality-driven methods. Also, both the best and worst possible gateway selections at each cluster are presented (in dotted lines) for compari-son. Overall, these results show our minimal-overlap framework is always dominant over all the classical metrics assessed while achieving optimal (best) or near-optimal real-time performance, especially when the node density is increased. As expected, it is also shown to be clearly superior that the worst performing gateway selection at each cluster.
Fig. 7.4a shows the results for k=2 gateways while Fig. 7.4b the case for k=3. These results confirm the expectation that an increase in the number of gateways (and clusters) also fa-vors schedulability. As shown in the previous chapter, a greater number of gateways significantly reduces network demand, positively impacting the overall network schedulability. This benefit can be observed here too, for example, in the case study withd =0.5 (bottom-row), which al-lows passing from 10 to 30 schedulable fal-lows with 99.9% of schedulability, which represents an improvement of 200% w.r.t. to the same framework, same configuration, but one more clus-ter/gateway. A similar effect can be observed when usingd=0.1 (top-row) but with lower gains, due to the reduced node density. Note that this density-related effect is consistent with the idea of having more diversity for paths favors schedulability, especially when using aminimal-overlap