We then provide improved lower bounds in the directed and half-duplex cases for many well-known network topologies, such as Butterfly, de Bruijn, and Kautz graphs.
In particular, given any network of n processors and any systolic period s, in the directed and the undirected half-duplex cases every s-systolic gossip protocol takes at least log (n)/log (1/λ) − O (log log(n)) time steps, where λ is the unique solution between 0 and 1 of λ·p⌊s/2⌋·p⌈s/2⌉="1, with pi = 1 + λ2 + ⋯ + λ2i − 2 for any integer i > 0.
The half-duplex case was studied in [8, 9].
A similar study for both the full-duplex and the half-duplex case with TDD is presented in [13].
In the half-duplex case, for each distance, the optimal transmission probability lies inside the accessible region.
In that specific case, the optimum probability of transmission (20) in the accessible region (in a probabilistic sense) saturates, that is, it reaches in the half-duplex case and in the full-duplex case.
Therefore, the femto link throughput for the half and full duplex cases is written as follows begin{array}{c}hfill {T}^{(half)}=pleft 1-pright)left(1-{p}=pleft 1-pright left 1-{peta right)hfileft 1-{p{T}_{OF}rightplogt(1-{p}_{OF}right) leftleft(1+theta right)hfill end{array}. in whfillp = p RB p tx.
In Figures 3(a) and 3(b), the accessible and optimal terminal probabilities of transmission are presented as functions of, in the cases with (a) half duplex and (b) full duplex communications, respectively.
Therefore, there is no difference between this time division approach (half duplex) and the full duplex one, and the analysis and conclusions hold in this case too.
Therefore, the femto link throughput for the half and full-duplex cases is written as follows begin{array}{rcl}{T}^{left(mathrm{half}right)}& =& pleft 1-pright)left(1-{pleft 1-pright}right) left 1-{p1+theta right) {T}^{left 1-{prm{full}_{mathrm{OF}}right1-{p}_{mathrm{OF}}right) log left(1+theta right)end{array} in which p = p RB p T}^{left
Full duplex Half duplex (opt).
