--- title: One-Rate Bidirectional Clos Networks url: https://www.emergentmind.com/topics/one-rate-bidirectional-clos-networks type: topic --- # One-Rate Bidirectional Clos Networks One-rate bidirectional Clos networks are multistage switching fabrics in which all external and internal links operate at the same rate, while connectivity is supported in both directions under symmetric line capacities, full-duplex interpretations, or explicitly shared bidirectional capacities. In the classical setting they are three-stage Clos networks used for circuit switching; in later work they appear as packet-switch fabrics without speedup, optical ROADMs and all-optical datacenter networks, and centrally scheduled bidirectional Clos fabrics for optical DCNs. Across these settings, the central technical questions are nonblocking dimensioning, rate-1 throughput, path diversity, rearrangement cost, and the control mechanisms needed to exploit the fabric without introducing excessive scheduling complexity [1204.3180] [1812.11650] [2302.06238] [2507.12265]. ## 1. Formal definition and modeling conventions In its general form, a three-stage Clos network is denoted \(C(n_1,r_1,m,n_2,r_2)\), with \(r_1\) input crossbars of size \(n_1\times m\), \(m\) middle crossbars of size \(r_1\times r_2\), and \(r_2\) output crossbars of size \(m\times n_2\). The symmetric case \(C(n,m,r)\) sets \(n_1=n_2=n\) and \(r_1=r_2=r\) [1204.3180]. In one-rate circuit models, all calls use one unit of capacity, and all internal links have the same capacity as the external links. In packet-switch realizations, the same condition appears as rate-1 operation with no internal speedup and no central-stage expansion [1812.11650]. Bidirectionality is represented in more than one way. In the standard circuit interpretation, a physical full-duplex line can be modeled as two directed links of capacity \(1\), one in each direction; under symmetric traffic, one-direction results apply identically to the other direction [1204.3180]. In optical ROADM models, a fiber degree is explicitly a pair of bidirectional fibers on a directional degree, and the line side is described as a pair of ingress and egress modules [2302.06238]. In recent two-level datacenter formulations, bidirectionality is native: low-level switches \(L_j\) connect to top-level switches \(T_i\) through bidirectional links \((T_i,L_j)\) with shared integer capacity \(C_{i,j}\), and a unit connection between \(L_j\) and \(L_k\) via \(T_i\) is represented as \((T_i,L_j,L_k)\) [2507.12265]. The one-rate condition also varies by domain while preserving the same structural idea. In packet switching it means that all links \(IP\!-\!IM\), \(IM\!-\!CIM\), \(CIM\!-\!COM\), \(COM\!-\!OM\), and \(OM\!-\!OP\) operate at the same line rate and that neither memory speedup nor central-stage expansion is used [1812.11650]. In optical Clos-ROADMs it means homogeneous per-fiber capacity, equal numbers of wavelengths on line-side ports, and uniform traffic assumptions such as “the offered traffic load between any input-output port pair is the same” [2302.06238]. In the bidirectional DCN model, it means that a connection is an atomic transfer unit with rate \(1\), and each link carries as many unit connections as its capacity allows [2507.12265]. A key special case in the bidirectional DCN literature is the proportional one-rate bidirectional Clos network, defined by the existence of nonnegative integers \(W_{L_j}\) and \(W_{T_i}\) such that \[ C_{i,j}=2\cdot W_{T_i}\cdot W_{L_j}. \] The uniform model, in which all \(C_{i,j}\) are equal and even, is a special case of this proportional model [2507.12265]. This proportionality is what enables a reduction to a classical three-stage symmetric Clos network and thereby links modern bidirectional scheduling results back to classical nonblocking theory. ## 2. Classical nonblocking theory and middle-stage dimensioning The classical theory distinguishes strict-sense nonblocking (SNB), wide-sense nonblocking (WSNB), and rearrangeably nonblocking (RNB) behavior. SNB means that a new request between any free input-output pair can always be routed regardless of how existing connections are arranged. WSNB means that a new request can always be routed provided that past and new requests are routed according to a specified algorithm. RNB means that any permutation can be realized, possibly after rearranging existing connections [1204.3180]. For a symmetric three-stage Clos \(C(n,m,r)\) under one-rate circuit switching, the classical S-Clos condition is \[ m\ge 2n-1. \] A standard LP-duality derivation bounds the number of unavailable middle crossbars for a new request by \(2n-2\), so one additional middle module guarantees an available route [1204.3180]. Under the bidirectional full-duplex interpretation, the same bound applies per direction, so the structural requirement on \(m\) is unchanged. For \(C(n,m,2)\), Benes’s “reuse busy middle crossbar whenever possible” algorithm yields the WSNB condition \[ m\ge \left\lfloor \frac{3n}{2}\right\rfloor, \] and the result is stated as necessary and sufficient for that routing discipline [1204.3180]. This bound is substantially smaller than the SNB requirement and illustrates the traditional trade-off between stronger online guarantees and algorithm-dependent routing. The same LP framework extends to multirate settings. The paper on linear-programming duality introduces dynamic weighted edge coloring (DWEC) and obtains the multirate WSNB sufficient condition \[ m \ge 5.6355\,n + 4. \] For one-rate calls this remains sufficient but is not tight relative to the classical circuit-switching bounds [1204.3180]. The significance is methodological: a single LP/duality template covers strict-sense, wide-sense, multirate, and photonic variants of Clos-type networks. These results establish the canonical dimensioning question for one-rate bidirectional Clos networks: how many middle-stage resources are needed to ensure that uniform-capacity connections can be admitted without blocking, either unconditionally, algorithmically, or after rearrangement. Later packet-switch and optical results can be read as alternative ways of achieving or approximating the same objective under different control models. ## 3. Rate-1 packet-switch realizations without speedup A concrete packet-switch realization of the one-rate Clos idea is the split-central-buffered Load-Balancing Clos (LBC) switch, an \(N\times N\) cell-based switch with four stages: input, central-input, central-output, and output. It uses \(k\) input modules (IMs), \(k\) output modules (OMs), and a central stage split into \(k\) central-input modules (CIMs) and \(k\) central-output modules (COMs), with the symmetric choice \(n=k=m\) and hence \(N=k^2\) [1812.11650]. The split central stage separates load balancing from routing to outputs and inserts buffers between CIM and COM. The LBC uses pre-determined, periodic, disjoint permutations. At time slot \(t\), the IM configuration is \[ r=(s+t)\bmod m, \] the CIM configuration is \[ p=(i+t)\bmod k, \] and the COM configuration mirrors the CIM schedule in reverse: \[ j=(p-t)\bmod k. \] The combined IM-CIM connectivity is represented by an \(N\times N\) permutation matrix \(\mathbf{\Pi}(t)\), and the compound permutation over one period is \[ \mathbf{P_1}=\sum_{t=0}^{k-1}\mathbf{\Pi}(t). \] An analogous compound COM permutation is \[ \mathbf{P_2}=\sum_{t=0}^{k-1}\mathbf{\Phi}(t). \] These cyclic permutations evenly spread each input’s offered load across the \(k\) VOMQs associated with its destination OM [1812.11650]. The traffic model assumes admissible i.i.d. arrivals with rate matrix \(\mathbf{R_1}=[\lambda_{u,v}]\) satisfying \[ \sum_{u=0}^{N-1}\lambda_{u,v}\le 1,\qquad \sum_{v=0}^{N-1}\lambda_{u,v}\le 1. \] After IM-CIM load balancing, \[ \mathbf{R_2}=\frac{1}{k}\big((\mathbf{R_1}*\mathbb{1})\circ \mathbf{P_1}\big), \] so each row of \(\mathbf{R_2}\) has exactly \(k\) nonzero entries, each equal to \(1/k\) of the row sum of \(\mathbf{R_1}\). The analysis further shows \[ \mathbf{R_4}(v)=\mathbf{R_1}(v),\quad \forall v, \] which means that the traffic queued at the crosspoint buffers for output \(v\) reproduces the original external load [1812.11650]. The main throughput result is explicit: LBC achieves \(100\%\) throughput under admissible i.i.d. traffic without memory speedup or central-stage expansion. Stability is proved separately for VOQs, VOMQs, and CBs, with queue occupancies \(\mathbf{N_1}(t)\), \(\mathbf{N_2}(t)\), and \(\mathbf{N_3}(t)\) shown to have bounded drift under admissible load [1812.11650]. Simulations for \(N=64\) and \(N=256\) show \(100\%\) throughput under Bernoulli uniform, bursty ON/OFF, unbalanced, hot-spot, and two CB-stressing traffic patterns; delay is reported as close to ideal output-queued behavior in several cases [1812.11650]. A distinct contribution is in-order forwarding. Because load balancing sprays cells across multiple VOMQs and COMs, the switch applies a hold-back mechanism at the inputs. If a cell \(c_{y,\tau}\) is sent to a VOMQ with occupancy \(\delta\), then no later cell of flow \(y\) leaves the VOQ for the next \(\delta k\) time slots. Practically, each input keeps an input port counter per reachable VOMQ and a hold-down timer per VOQ; when the current VOMQ occupancy is \(\sigma\), the timer is set to \((\sigma-1)k\) [1812.11650]. The resulting theorem states that for any two cells \(c_{y,\tau}\) and \(c_{y,\tau'}\) of the same flow with \(\tau<\tau'\), the earlier cell always departs the destination output port before the later one. Although this switch is analyzed for unidirectional traffic, its building blocks and its rate-1 scheduling are symmetric. This suggests that the same deterministic scheduling and buffering principles can be applied per direction in a bidirectional Clos fabric without changing link rates [1812.11650]. ## 4. Optical and all-optical interpretations In optical switching, one-rate bidirectional Clos networks appear in three-stage Clos-ROADMs and folded Clos datacenter networks. The relevant parameters are directional degree \(D\), fiber degree \(L\), and middle-stage size \(M\). A fiber degree is a pair of bidirectional fibers on a directional degree, and a Clos-ROADM is denoted \(v(M,L,D)\), in contrast to a Spanke-ROADM \(s(D,L)\) [2302.06238]. The ingress and egress stages use arrays of \(L\times M\) and \(M\times L\) WSSs, the middle stage uses \(M\) switches of size \(D\times D\), and neighboring stages are fully connected by short-reach fibers. The nonblocking condition for a WSS-based Clos-ROADM is the spatially strictly non-blocking requirement \[ M>2\cdot L-1. \] For comparison with a strictly non-blocking Spanke-ROADM, the paper chooses \(M=2\cdot L\) [2302.06238]. With wavelength continuity, a further wavelength-dimensional requirement is stated: \[ W_{\text{net}}\ge 2\,W_{\text{line}}, \] where \(W_{\text{net}}\) is the total number of wavelengths supported by the Clos network and \(W_{\text{line}}\) is the number of wavelengths per line-side port [2302.06238]. These conditions are optical analogues of classical one-rate Clos dimensioning, with wavelengths acting as unit-capacity circuits. The paper also derives a blocking benchmark using the Erlang-B formula \[ EB(p,w)=\frac{p^w/w!}{\sum_{k=0}^{w}p^k/k!}, \] together with \[ B_i=EB(p,w),\qquad B_o=EB\big(p(1-B_i),w\big),\qquad B=1-(1-B_i)(1-B_o). \] The underlying assumptions are Erlang traffic on each input-output pair, homogeneous ports, and the same number of wavelengths \(w\) per port [2302.06238]. This is explicitly a one-rate regime. The complexity comparison with Spanke is one of the main quantitative results. For Spanke-ROADM \(s(D,L)\), \[ \#WSS_{\text{Spanke}}=2\cdot L\cdot D,\qquad \#fibers_{\text{Spanke}}=(D^2-D)\,L^2. \] For Clos-ROADM \(v(M,L,D)\), \[ \#WSS_{\text{Clos}}=2\cdot D+M,\qquad \#fibers_{\text{Clos}}=2\cdot L\cdot M. \] If \(M=2L\), the Clos fiber complexity becomes \(4L^2=O(L^2)\), while the Spanke fiber complexity remains \(O(D^2L^2)\) [2302.06238]. In the case \(D=L=10\) with 5 wavelengths per fiber and 2 Erlang per fiber degree, the blocking probability of \(v(M,10,10)\) becomes “very close” to that of \(s(10,10)\) when \(M=6\); the internal fiber count is then 120 for Clos versus 9,000 for Spanke [2302.06238]. Different middle-stage technologies alter the blocking regime. For the base network \(v(5,5,5)\), WSS Clos-ROADM matches CDC-ROADM performance, TWC-WSS reaches the theory limit defined by the Erlang benchmark, AWG performs worse because of limited routing flexibility, TWC-AWG provides slight improvement, and TWC-AWG-TWC approaches the theory limit [2302.06238]. In all-optical DCNs, the same three-stage Clos can be folded into a Spine-Leaf architecture, and the middle stage can also be replaced by optical splitters for multicast or by deterministic round-robin direct connections to form a Torus-like topology [2302.06238]. A plausible implication is that one-rate bidirectional Clos theory extends naturally from electrical circuit interpretations to wavelength-level optical switching whenever capacities remain homogeneous. ## 5. Centralized scheduling and replacement chains in bidirectional DCNs A recent formulation treats the one-rate bidirectional Clos network as a two-level complete bipartite fabric with low-level switches \(L_0,\dots,L_{m-1}\), top-level switches \(T_0,\dots,T_{n-1}\), and bidirectional links \((T_i,L_j)\) of integer capacity \(C_{i,j}\) [2507.12265]. A connection between low-level switches \(L_j\) and \(L_k\) via \(T_i\) is \((T_i,L_j,L_k)\), and the demand matrix \(D\) is symmetric with \(D_{j,k}=D_{k,j}\) and \(D_{j,j}=0\). For a new routing scheme \(Y\), feasibility requires \[ \forall i,j:\quad \sum_{k=0}^{m-1}Y_{i,j,k}\le C_{i,j}, \] and \[ \forall j,k:\quad \sum_{i=0}^{n-1}Y_{i,j,k}\ge D_{j,k}. \] The demand itself is feasible iff \[ \forall j:\quad \sum_k D_{j,k}\le \sum_i C_{i,j}. \] The associated optimization objective is minimal rearrangement. Given an original routing \(X\), the number of rearrangements needed to obtain \(Y\) is \[ \mathrm{NumRearr}(X,Y)= \sum_{i=0}^{n-1}\sum_{j=0}^{m-1}\sum_{k=0}^{m-1}|X_{i,j,k}-Y_{i,j,k}|. \] FastReChain attacks this problem through replacement chains [2507.12265]. A connection is redundant when \[ E(X)_{j,k}>D_{j,k},\qquad E(X)_{j,k}=\sum_{i=0}^{n-1}X_{i,j,k}, \] and a link is explicitly available if \[ \sum_k X_{i,j,k}