Nonblocking Multicast-capable Optical Cross Connects based on the 4-stage Multicast Network Fangfang Yan, *Weisheng Hu, Weiqiang Sun, Wei Guo, and Yaohui Jin The State Key Laboratory of Advanced Optical Communication Systems and Networks, Shanghai Jiao Tong University, Shanghai 200240, China *Email:
[email protected] Abstract—In this paper, we investigated designing multicastcapable optical cross-connects (MC-OXCs) on a basis of the 4stage multicast network. Firstly, we derive the sufficient widesense nonblocking (WSNB) and rearrangeable nonblocking (RNB) conditions for the 4-stage multicast network with only two (the second and output) stages being multicast-capable. Both WSNB and RNB 4-stage multicast networks need O(N3/2) crosspoints. Then MC-OXCs empoying the WSNB and RNB 4-stage multicast network are proposed and proven to be power efficient in reducing the total power loss caused by light splitting. Keywords-multistage network, multicast, optical cross connects, wide-sense nonblocking, rearrangeable nonblocking
I.
INTRODUCTION
As bandwidth-intensive applications such as IPTV, triple play, video conferencing and 3G cell phones get popular; there rises a demand of supporting multicast communication directly at optical layer on next-generation optical networks. The multicast-capable optical cross connects (MC-OXC) are the necessary devices to support optical layer multicast. Light splitter is the fundamental element in the MC-OXC to construct light trees. In [1] a splitter and delivery (SaD) switch was proposed to support both unicast and multicast in the strict nonblocking (SNB) sense. However, it is difficult to fabric a splitter with many branches. Moreover, the number of small switches (including optical gates and 2×1 switches) required is extremely large, typically on the scale of N2 for the N×N SaD switch. Multistage network is a promising solution to construct a large dimension OXCs by cascading small unicast and multicast switches [5]-[7]. For the optical circuit (usually asynchronous) switching, SNB or at least wide-sense nonblocking (WSNB) is required, since an existing connection holds and can not be broken down during a relative long period. An SNB MC-OXC becomes more complex than an OXC supporting only unicast connections, as the cost of a SNB multicast switching network is lower bounded by O(N2)[5]. Using an appropriate routing algorithm, a WSNB MC-OXC can save the hardware cost. For synchronous optical packet switching, rearrangeable nonblocking (RNB) is sufficient as the connection states in the OXCs are updated at each timeslot. Many researches have been conducted to derive the nonblocking multicast conditions of the multistage network
[5]-[8], most of which focus on the Clos network [4]. The three-stage Clos network needs O(N5/3) crosspoints to achieve WSNB and RNB with the first and third stages being multicast-capable (MC) [5]. The Clos network with three MC stages needs O(N3/2log2N/(log2log2N)) crosspoints to be WSNB, if each multicast connection is restricted to use at most p (a predefined value) middle-stage switches [7]. A 4stage multicast network was proposed with the last three stages having fan-out capabilities in [8]. It has been proven that the 4-stage multicast network needs only O(N3/2) crosspoints to achieve WSNB. In general, optical switches (OSWs) are unicast ones and realized with several technologies such as microelectromechanical system (MEMS) devices [12]. Optical multicast switches (OMSs) have to use additional optical copy components, for example, light splitter, multi-wavelength converter [10] and active vertical coupler [11], to support multicast connections besides the unicast connections. Obviously, an OMS has higher cost compared with an OSW at the same dimension. Another difference between OSW and OMS is the physical performance. To copy a signal to multiple output ports using splitter causes power loss. Optical amplifier is alternatively used to compensate the loss but introduces noise simultaneously. Overall, the signal performance becomes worse undergoing multicasting. Therefore, it is expected to have as fewer MC stages in multi-stage network as possible. In this work, we investigate the 4-stage multicast network with only two (the second and output) stages having fan-out capabilities, instead of three stages in [8]. We derive the sufficient WSNB condition for a 4-stage multicast network under which about 16.1N3/2 crosspoints are needed at least. Then we propose the sufficient RNB condition under which about 7.4N3/2 crosspoints are needed. After introducing the nonblocking condition, we are able to construct MC-OXCs using the 4-stage network. We present a 4-stage MC-OXC with two MC stages. It can be WSNB or RNB with the numbers of second and third stage switches subject to the WSNB or RNB condition. The proposed MCOXC is power efficient compared with the MC-OXC based on the 4-stage network with three multicast stages [8]. The organization of the paper is as follows. Section II and III derive the sufficient WSNB and RNB condition of the 4stage multicast network respectively. The complexity in terms
This work was supported by the NSFC general and key (60632010) project, SRFDP, and 863 program. The authors would also like to thank the help of key laboratory of avionics system integration technology (Shanghai).
978-1-4244-2324-8/08/$25.00 © 2008 IEEE. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings.
of crosspoint number is evaluated in Section IV for both WSNB and RNB networks. Section V illustrates the proposed MC-OXC employing the 4-stage network. Conclusions are given in section VI. II.
WIDE-SENSE NONBLOCKING CONDITION
Fig.1 depicts the 4-stage multicast network where the input stage consists of r1 n1×m1 switch modules, the second stage m1 r1×m2 switch modules, the third stage m2 m1×r2 switch modules, the output stage r2 m2×n2 switch modules. N = min (n1r1, n2r2). Only the second and fourth stages have fan-out capabilities. The multicast degree (d) of a multicast request is defined as the number of output switches the request heading for. A request with the multicast degree d is called a d-request. The switch module in the input/output stage is denoted as input module (IM) /output module (OM) respectively. The second stage switch module makes copies of an input request according to its multicast degree and is thus called copy module (CM). The third stage switch module distributes the request to their output switch modules and is thus called distribution module (DM).
Fig. 1. The 4-stage multicast network with the second and fourth stages being multicast-capable
A. Routing Algorithm The IM directs an input request to a CM that has maximum number of available output ports [8](e.g. m′2 ). At the chosen CM (e.g. the #i CM), the input request is replicated into d copies. As the DM is multicast-incapable, the d copies of the request have to be directed to d distinct DMs, each of which is responsible to route one copy of the request to one of the destination output switches. For each of the m′2 available DMs, if it has a free output port linking to one of the un-reached destination OMs (e.g. the #j OM), then connect the #i CM to the #j OM with this DM. At an OM, the request is further replicated and directed to the output ports. Therefore, no-split algorithm is implied in the routing algorithm. (In the no-split algorithm, only one path is needed to reach all the outputs of one OM [5]). An example is given in Fig.1 to illustrate the routing of a 2-request in the 4-stage network.
B. Sufficient Condition for WSNB Lemma 1: For a d-request, if there is at least one available CM with n2+d−1 out ports unoccupied, then it can be realized with the 4-stage network without blocking. Proof: Suppose that the current request is destined to d OMs, which are #i1, #i2…#id. It is directed to one of the available CMs with n2+d−1 out ports unoccupied. The routing algorithm attempts to route the request by d DMs out of the available n2+d−1 DMs. Firstly, consider routing the current request to the #i1 OM. At most n2−1 DMs have been occupied to route other existing requests to the #i1 OM and thus can not be used by the current request. Then choose a DM out of the other d DMs and route one copy of the request by the chosen DM to the #i1 OM. Then consider routing the request to the #i2 OM using (n2+d−1)−1 DMs. Similarly, At most n2−1 DMs have been occupied to route other existing requests to the #i2 OM. Then choose a DM out of the rest d−1 DMs to route another copy of the request to the #i2 OM. After d−1 iterations, the request is directed to d−1 OMs successfully. At last, consider routing the request to the #id OM using (n2+d−1)−(d−1) DMs. Similarly, At most n2−1 DMs have been occupied to route other existing requests to the #id OM. Then use the only DM available to route the last copy ■ of the request to the #id OM. Theorem 2: The 4-stage multicast network is WSNB if ⎧m1 ≥ n1 ⎪ (2.1) ⎧ N − r2 − n1 + 1 N − r2 − 1⎫ ⎨ ≥ + − + , 1 max m n r ⎨ ⎬ 2 2 ⎪ 2 m1 − 1 ⎭ ⎩ m1 − n1 + 1 ⎩ Proof: As the IM have no fan-out capability, any k0 (1≤k0≤n1) requests from the same IM have to be routed by k0 distinct CMs. To ensure any request can pass the input stage nonblockingly, m1≥n1 suffices. Consider the current request of multicast degree d (1≤d≤r2) from the #j IM. Suppose that k (1≤k≤n1−1) requests already exit from the same IM. Thus k out of m1 CMs can not be used by the current request. At most N−d−k out ports of the rest m1−k CMs have been occupied by existing requests. Let the function χ(m) denote the minimum number of busy out ports among m CMs. According to the routing algorithm, the current request will be directed to the CM with the minimum number of busy out ports χ(m1−k), which is given by(2.2). (2.2) χ (m1 − k ) ≤ ( N − d − k ) / (m1 − k ) According to lemma 1, the current request can be accepted, if there is a CM that has n2+d−1 out ports available. The current request can be realized nonblockingly, if the number of DMs m2 satisfies (2.3). Then WSNB condition for any request can be expressed in (2.4). m2 ≥ χ (m1 − k ) + n2 + d − 1 (2.3) ⇐ m2 ≥ G(k , d ) = ( N − d − k ) / (m1 − k ) + n2 + d − 1 (2.4) m2 ≥ max{G (k,d )} ∀1 ≤ k ≤ n1 − 1,1 ≤ d ≤ r2 As shown in (2.5), G′d ≥0, when m1≥n1. Given that d is
978-1-4244-2324-8/08/$25.00 © 2008 IEEE. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings.
invariable, (i.e. d=r2), the sign of Gk′ is invariable when k varies from 1 to n1−1.Hence, the maximum of G(k,d) can be found at k=n1−1 and d= r2 or k=n1−1 and d= r2. Therefore, the 4-stage multicast network is WSNB, if (2.6) satisfies. (2.5) Gk′ = ( N − d − m1 ) / (m1 − k ) Gd′ = 1 − 1 / (m1 − k ) m2 ≥ max{G (n1 − 1, r2 ), G (1, r2 )} (2.6)■ ⎧ N − r2 − n1 + 1 N − r2 − 1⎫ = n2 + r2 − 1 + max ⎨ , ⎬ m1 − 1 ⎭ ⎩ m1 − n1 + 1 III.
REARRANGEABLE NONBLOCKING CONDITION
Theorem 3: The 4-stage multicast network is RNB if ⎧m1 ≥ n1 ⎪ (3.1) ⎧ N − r2 ⎫ N − n1 N − n1 ⋅ r2 ⎨ m ≥ n + r + + r max , , 1 , ⎨ 2 2 2⎬ ⎪ 2 m1 m1 − n1 + 1 m1 − n1 + 1 ⎩ ⎭ ⎩ Proof: For the same reason with the WSNB case, m1≥n1 suffices to ensure that any request can pass the input stage. Consider routing k (1≤k≤n1) input requests from an IM through the 4-stage network. The multicast degrees of the k requests in the descending order are r2≥d1≥ d2≥… ≥dk≥1, with d1+d2+… +dk≤N. First consider routing the k requests in descending order of multicast degree through the first two stages. Suppose that no more than m′2 out ports of each CM can be occupied simultaneously. Take the di-request (1≤i≤k) for example. There are m1−(i−1) CMs left for it to be routed by, since the other i−1 CMs have been used to route the first i−1 requests. Then direct the direquest to the CM with the minimum number of busy out ports χ(m1−i+1), which is given by (3.2). To guarantee that the direquest can pass the first two stages successfully, the chosen CM must have more than di out ports unoccupied. Then we have (3.3). k χ (m1 − i + 1) ≤ N − ∑ d i / (m1 − i + 1) ≤ ( N − i ⋅ d i ) / (m1 − i + 1) (3.2)
(
i =1
)
N − i ⋅ di (3.3) + di m1 − i + 1 Therefore, the nonblocking condition to route the k (1≤k≤n1) input requests through the first two stages can be expressed as in (3.4). The nonblocking condition for any requests to pass through the first two stages can be expressed as in (3.5). m2′ ≥ F (1, d1 ) , m2′ ≥ F (2, d 2 ), ..., m2′ ≥ F (k , d k ) (3.4) ⇐ m2′ ≥ max{F (i, d i )} ∀i, d i ∈ Z + subject to m2′ ≥ χ (m1 − i + 1) + d i ⇐ m2′ ≥ F (i, d i ) =
i ∈ [1, k ],1 ≤ d k ... ≤ d 2 ≤ d1 ≤ r2 , ∑i =1 d i ≤ N k
m2′ ≥ max{F (i, d i )}
∀i, d i ∈ Z + subject to
i ∈ [1, k ], k ∈ [1, n1 ],1 ≤ d k ... ≤ d 2 ≤ d1 ≤ r2 , ∑i =1 d i ≤ N k
(3.5)
⇐ m′2 ≥ max{F (i, d i )} ∀i, d i ∈ Z , i ∈ [1, n1 ], d i ∈ [1, r2 ] +
⇐ m′2 ≥ max{F (i, d )} ∀i, d ∈ Z + , i ∈ [1, n1 ], d ∈ [1, r2 ] We get a critical point i=i0=(m1+1)/2 and d=d0=N/(m1+1) by deriving Fd′ =0 and Fi′ =0. There is neither a relative maximum
or minimum point, since Δ 0 when a ≥ (1 + 5 ) 2 . g 4 (a ) = (3a + 1) (1 + 1 / a ) 2
p(n, r , a )min = min{p(n0 , r0 , a )} ∀a ≥
(
(4.19)
)
5 +1 / 2
(4.20) = p n0 , r0 , ( 5 + 1) / 2 ≈ 7.4 N The same minimum number of crosspoints (7.4N3/2) can be obtained in a similar way, when a ≤ (1 + 5 ) / 2 .
(
(4.8)
V.
)
3/ 2
MC-OXCS IMPLEMENTED BY THE 4-STAGE NETWORK
The WSNB 4-stage network in [8] has less crosspoints number (only 11.7N3/2) than 16.1N3/2. The WSNB 4-stage network with one less MC stage must have more crosspoints to be nonblocking. Therefore, there is a trade-off between the number of multicast switches and the number of total crosspoints when building a multicast switch fabric employing the WSNB 4-stage network. The 4-stage network with two MC stages is more preferable, when multicast switches lead to excessive cost or performance impairment compared to unicast switches. B. RNB According to theorem 3, the symmetric 4-stage multicast network is RNB under the condition in (4.9) and can be rewritten as in (4.10) with a>1. ⎧ ⎛ n − 1 ⎞ n(r − 1) ⎫ (4.9) + 1⎟⎟r , + 1⎬ m1 ≥ n m2 ≥ max ⎨n, ⎜⎜ − + 1 m m n 1 ⎠ ⎩ ⎝ 1 ⎭
(a) Demux (w/b)×m1
1
⎧ ⎛ n −1 ⎞ r −1 ⎫ (4.10) + 1⎬ + 1⎟ r , m 2 = max ⎨n, ⎜ ⎩ ⎝ an − 1 ⎠ a − 1 ⎭ To simply the analysis, let m1 and m2 satisfy: ⎧ ⎛1 ⎞ r ⎞⎫ (4.11) ⎛ r −1 + 1, m1 = an m2 = max⎨n, ⎜ + 1⎟r , max⎜ ⎟⎬ a a − − 1 ⎠⎭ a 1 ⎝ ⎝ ⎠ ⎩
λ1..λw
If a ≥ (1 + 5 ) / 2 , then g1(a)>0 and g2(a)>0 which make m2=max{n,(1+1/a)r}. If a ≤ (1 + 5 ) / 2 ,then g2(a)≤0 and g3(a) >n, in the wide-area WDM networks. α (SaD) = P ⋅ Ls + 10 log10 Q (5.1) Figure 2(a) shows the MC-OXC (I) based on the 4-stage multicast network. Every w/b input wavelength channels from the same input fiber are viewed as an input waveband and grouped into an IM of (w/b)×m1 OSW. The CM is built of (bn)×m2 SaD and responsible to split the input signal according to the number of destination output wavebands. The DM is m1×(bn) OSW and direct the input signal to different output wavebands. The OM is m2×(w/b) SaD switch to multicast the input signal towards different output wavelengths inside a waveband. Totally nw fixed wavelength converters (FWCs) are added before the multiplexers to converter the input signal to the output wavelengths and avoid wavelength contention. Let a=1, b=(w/n) 1/2 and w/b=bn =N1/2 for simplicity in section V. The MC-OXC (I) is WSNB if (5.2) satisfies according to theorem 2, and is RNB subject to (5.3) according to theorem 3. m1 = (a + 1)w / b = 2 N 1 / 2 m2 = w / b + (1 + 1 / a ) ⋅ bn = 3N 1 / 2 (5.2) ⎧⎪m1 = ( a + 1) w / b = 2 N 1 / 2 (5.3) ⎨ ⎪⎩m2 = max{w / b, [1 / (a + 1) + 1]bn, (bn − 1) / a + 1} = 1.5 N 1 / 2 The insertion loss α is an important issue for MC-OXC. Loss caused by power splitting of the SaD switches in the second and fourth stages contributes the main factor of insertion loss in MC-OXC (I). As the MEMS switch features at very low loss (a 64×64 3-dimensional MEMS switch has a loss of only 1.9dB [12]), the power loss of the MEMS switches is not counted. The insertion loss of MC-OXC (I) is written in (5.4) in the WSNB and RNB cases. α (I ) = 10 log10 (m2 ⋅ w / b ) + (m2 + w / b )Ls (5.4) ⎧⎪10 log10 (3 N ) + 4 N 1 / 2 Ls (WSNB) =⎨ ⎪⎩10 log10 (1.5 N ) + 2.5 N 1 / 2 Ls (RNB) MC-OXC (II), see Fig.2(b), based on 4-stage network with three multicast stages is similar to MC-OXC (I), except that the DM is built of SaD switch instead of OSW. MC-OXC (II) is WSNB if (5.5) satisfies according to [8]. The insertion loss in MC-OXC (II) is mainly caused by power splitting in the last three stages and is expressed in (5.6). m1 = (a + 1)w / b = 2 N 1 / 2 m2 = w / b + bn / a = 2 N 1 / 2 (5.5)
α (II) = 10 log10 (m2 ⋅ bn ⋅ w / b ) + (m2 + bn + w / b )Ls
(
)
= 10 log 10 2 N 3 / 2 + 4 N 1 / 2 Ls
VI.
(5.6)
CONCLUSION
In this paper, we consider designing the multistage MCOXCs using the 4-stage network. The WSNB and RNB conditions are presented and proven for the 4-stage multicast network. Under these conditions, the complexity of the 4-stage
network evaluated by the number of crosspoints is O(N3/2). Afterwards, we introduce the WSNB and RNB MC-OXCs based on the 4-stage network. A comparison of MC-OXCs employing the 4-stage network with two or three MC stages is given in table I. In general, WSNB MC-OXC (I) has more OSW switches, but less SaD switches than MC-OXC (II) based 4-stage network with three multicast stages. The RNB MC-OXC (I) provides almost 50% lower cost compared with its WSNB counterpart. The WSNB MC-OXC (I) with two MC stages saves 10log10N1/2 dB optical power than the MC-OXC with three MC stages approximately. Therefore, the proposed MC-OXCs are power efficient in reducing the power loss produced by light splitting. TABLE I.
Structure MC-OXC(I) WSNB MC-OXC(I) RNB MC-OXC(II) WSNB
A COMPARISON OF MC-OXCS WITH TWO OR THREE MC STAGES OSW
SaD
NO. [Dimension]
NO. [Dimension]
(dB)
N1/2[N1/2×2N1/2] 3N1/2[2N1/2×N1/2] N1/2[N1/2×2N1/2] 1.5N1/2[2N1/2×N1/2]
2N1/2[N1/2×3N1/2] N1/2[3N1/2×N1/2] 2N1/2[N1/2×1.5N1/2] N1/2[1.5N1/2×N1/2] 2N1/2[N1/2×2N1/2] 3N1/2[2N1/2×N1/2]
10log10(3N) +4N1/2Ls 10log10(1.5N) +2.5N1/2Ls 10log10 (2N3/2) +4N1/2Ls
N1/2[N1/2×2N1/2]
Insertion loss
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978-1-4244-2324-8/08/$25.00 © 2008 IEEE. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings.
