ExerciseComparison of the series-parallel and parallel-series configurations:Assume N = M, that is, we have the same number of components in eachcolumn and row. Also assume that each component has the same reliabilityp with 0 < p < 1. Which of the two configurations tends to be more reliableif N grows very large (N → ∞)?SolutionWith N = M we obtain that the series-parallel configuration has reliabilityRs−p= [1 − (1 − p)N]N,and the parallel-series has reliabilityRp−s= 1 − (1 − pN)NWhich of these similarly looking expressions is larger if N tends to infinity?We can answer the question using the following known formula from calculus:limx→0(1 − x)1/x= e−1where e = 2.71... is the base of the natural logarithm. We use it for approx-imation: when x is very small, then(1 − x)1/x≈ e−1holds.For Rs−plet us take x = (1 − p)N. Then we have x → 0 if N → ∞. We canthen writeRs−p= (1 − x)N= (1 − x)(1/x)Nx= [(1 − x)1/x]Nx≈ e−Nx,using (1 − x)1/x≈ e−1in the last step. Since Nx = N(1 − p)N, therefore, wehave N x → 0, as the second factor tends to 0 exponentially, while the firstgrows only linearly. Thus, we obtain thatRs−p→ 1 (1)as N grows.For Rp−swe take x = pN. Then, similarly to the above, we have x → 0 ifN → ∞, so we can writeRp−s= 1 − (1 − x)N= 1 − (1 − x)(1/x)Nx= 1 − [(1 − x)1/x]Nx≈ 1 − e−Nx.Since now Nx = N pN, therefore, we have again N x → 0, as the secondfactor tends to 0 exponentially, while the first grows only linearly. Thus, weobtain that e−Nx→ 0, implyingRp−s→ 0 (2)Comparing (1) and (2), we can conclude that with N → ∞ the reliability ofthe series-parallel configuration tends to 1, while the reliability of the parallel-series configuration tends to 0. Thus, the series-parallel configuration will bemore reliable when N grows very large.InterpretationIt is interesting that the two apparently similar configurations behave sodifferently. The series-parallel tends to be more reliable as its size grows,while the parallel-series loses reliability with growing size. How could weinformally explain this difference?The series-parallel configuration has much more possible paths between theendpoints. We can pick any component from each parallel subnetwork tobuild a path, yielding NNpossible paths. On the other hand, the parallel-series configuration has only N possible paths between the endpoints, as eachseries subnetwork can serve as such a path, but there is no more possibility.Similarly, if we consider minimum cuts, that is, minimal subsets of compo-nents whose failure disconnects the network, then we find that the situation isreversed. The series-parallel configuration has only N possible minimum cuts(taking an entire parallel subnetwork), while the parallel-series configurationhas NNminimum cuts (taking one component from each series subnetwork).The above considerations show that the series-parallel configuration is muchmore connected, i.e., it has much more paths and much less cuts than theother. Thus, it is not surprising that it grows more
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