多頻及多向傳輸模型下的三級式不阻塞克勞斯網路

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(1)Å >¦×ç @àbçÍ î=d Öä£Ö²f_-íú.®-sgæ˜ On Nonblocking Three Stage Clos Networks under the Multirate-Multicast Model û ˝ Þ: "=• Nû`¤: ômp `¤. 2M¬Å. ûý~.

(2) Öä£Ö²f_-íú.®-sgæ˜ On Nonblocking Three Stage Clos Networks under the Multirate-Multicast Model û ˝ Þ: "=• Student: Huilan Chang Nû`¤: ômp `¤ Advisor: Frank K. Hwang Å >¦×ç @àbçÍ î=d. A Thesis Submitted to Department of Applied Mathematics College of Science National Chiao Tung University in Partial Fulfillment of the Requirements for the Degree of Master in Applied Mathematics June 2005 Hsinchu, Taiwan, Republic of China 2M¬Å. ûý~.

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(4) On Nonblocking Three Stage Clos Networks under the Multirate-Multicast Model. Student: Huilan Chang. Advisor: Frank K. Hwang. Department of Applied Mathematics National Chiao Tung University June, 2005. Abstract The purpose of this thesis is to survey the scattered piece-meal results on the multirate-multicast model for nonblocking three-stage Clos networks, to fill some gaps and to extend some results. We also do some numerical comparisons among existing results. It is hoped that our survey will facilitate future researchers to identify open problems and to make further inroads into this very important model with applications to communication and computer networks.. iv.

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(6) Contents 2d¿b. iii. Abstract. iv. Ðá. v. Contents. vi. List of Figures. vii. 1 Preliminaries 1.1 Model description . . . . . . . . 1.2 Multicast environment . . . . . 1.3 Multirate environment . . . . . 1.4 Multirate-multicast connections. . . . .. . . . .. . . . .. 2 Strictly and Wide-Sense Nonblocking 2.1 Strictly nonblocking . . . . . . . . . . 2.2 The no-split rule . . . . . . . . . . . 2.3 k-limited algorithm . . . . . . . . . . 2.4 R(l) algorithm . . . . . . . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. Network . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . .. 1 1 2 2 3. . . . .. 5 5 5 9 13. 3 Rearrangeable Nonblocking Networks. 15. 4 Numerical Comparison and Conclusion. 16. Appendix. 17. vi.

(7) List of Figures 1 2 3. C(2, 4, 3, 3, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . C(2, 4, 3) symmetric Clos Network with connection requests under multirate-multicast model. . . . . . . . . . . . . . . . . C(1, 2, 3, 2, 4) Clos Network with connection requests under open-end traffic. . . . . . . . . . . . . . . . . . . . . . . . . .. vii. 1 3 4.

(8) 1. Preliminaries. In this section, we introduce some general concepts, terminology and definitions that are used in this article.. 1.1. Model description. Multistage switching networks are composed of crosspoint switching elements or, more specifically, crosspoints that are usually grouped together into building-block subnetworks called switch modules. A three-stage Clos network, denoted by C(n1 , r1 , m, n2 , r2 ), has r1 switch modules of size n1 × m in stage 1, m switch modules of size r1 × r2 in stage 2, r2 switch modules of size m×n2 in stage 3, and exactly one link between every two switch modules in its consecutive stages. A link between two stages is called an internal link. Figure 1 shows a C(2, 4, 3, 3, 2) network. In a three-stage network, stage 1 is also referred to as the input stage, stage 2 as the middle stage, and stage 3 as the output stage. For the symmetrical case where n1 = n2 = n and r1 = r2 = r, the three-stage Clos network is denoted as a C(n, m, r) network. If each internal link in a C(n1 , r1 , m, n2 , r2 ) is replaced by d links, then the network is denoted by C(n1 , r1 , m, n2 , r2 , d). In general, the set of input ports is denoted as I = {1, 2, · · · , r1 n1 }, the set of output ports is denoted as {o1 , o2 , · · · , or2 n2 } and the set of switch modules in the output stage is denoted as O = {O1 , O2 , · · · , Or2 }. We also refer to links incident to input and outputs as external links.. Figure 1: C(2, 4, 3, 3, 2). 1.

(9) 1.2. Multicast environment. With the demand of multicast transmissions such as video-conference, internet games and distance learning, we need a multicast network to accomplish those demands. In the multicast network, a switch is said to have the fanout capability if the switch itself can route multicast traffic without blocking, i.e., any inlet can be connected to any number of idle outlets regardless of other connections. Usually, a switch module is assumed to have the fan-out capability. A multicast request is denoted as (x, {o1 , o2 , · · · , os }) where x is asked to be connected with output port oj , j = 1, 2, · · · , s. We often simplify its notation as C = (x, o(x)). If the cardinality of o(x), |o(x)|, is restricted to at most f , the traffic is called an f -cast traffic. If o(x) is the set of output ports for all x, the traffic is called broadcast. And we call a request point-to-point if |o(x)| = 1. f -cast traffics can be divided into two types according to whether additional receivers can be added after a multicast request is already connected. We will use open-end traffic (which allows additions) and closed-end traffic (which does not allow) to differentiate these two types. Suppose the input port x has generated r requests (xi , o(xi ), ωi ), 1 ≤ i ≤ r. If a new request from x, carrying the same message of xi , to some output ports unconnected yet is allowed, then we call it the open-end traffic; if not allowed, then the traffic is closed-end. Note that for either type of traffic, x is always allowed to generate a new request (xr+1 , o(xr+1 ), ωr+1 ) with a new message as long r+1 P ωi is under the capacity. as i=1. 1.3. Multirate environment. The need of a multirate network comes from the desire to integrate multimedia transmissions such as audio, data, image and video into one switching network. As different media require a broad range of bandwidths, each request is associated with its required amount of bandwidth, called its rate. A link in the switching network has a capacity and can carry as many requests as desired as long as the sum of their rates does not exceed the capacity of the link. There are two basic multirate models: discrete and continuous. The discrete model assumes that there is a finite number of distinct rates and the continuous model assumes that all rates are within a given interval. For 2.

(10) the continuous model, it is customary to assume that each internal link has capacity 1 and each request has a normalized rate ω, b ≤ ω ≤ B, where b and B, 0 < b ≤ B ≤ 1, are bounds of ω. Each external link is assumed to have capacity β, B ≤ β, i.e., it can generate any number of requests in a frame as long as the sum of rates of these requests does not exceed β. We call this the β[b, B] model (which can also be used for discrete multirate model). We use β(b, B], β[b, B), β(b, B) to exclude b or B, and omit β if β = 1, respectively. Note that, [1,1] represents the classical model. In practice, β is usually less than 1, representing a 1-level speed-up.. 1.4. Multirate-multicast connections C1(0.4) C3(0.6). C2(0.2). 0.4 1. o1 o2. 0.4 + 0.6. o3 o4. 0.4 0.4+0.2. o5 o6. 0.6 0.6. 0.6. Figure 2: C(2, 4, 3) symmetric Clos Network with connection requests under multirate-multicast model. In a multirate network, we call an input port or an output port ω-idle, 0 ≤ ω ≤ 1, if the fraction of available bandwidth for that input port or output port is at least ω. A multicast connection with weight ω is a connection from an ω-idle input port to a set of ω-idle output ports using the same fraction of the bandwidth, ω, on every intermediate link of this multicast connection in the network. We will assume that every switch module in our multicast networks has fan-out capability. 3.

(11) 0.6 0.6. C1(0.6) C3(0.6). 0.6. I1. M1. 0.6. 0.6. 0.6. M2 0.4. C2(0.4). 1. 0.6+0.4. I2 M3. Figure 3: C(1, 2, 3, 2, 4) Clos Network with connection requests under openend traffic. In a C(n1 , r1 , m, n2 , r2 ) Clos network, a multirate-multicast request is denoted as C = (x, o(x), ω), where x ∈ {1, · · · , n1 r1 } is an input port, o(x) is a subset of {o1 , o2 , · · · , or2 n2 }, and ω is a required weight of the request. Figure 2 shows three multicast connection requests, where C 1 = (1, {o1 , o3 , o4 }, 0.4), C 2 = (4, {o4 }, 0.2) and C 3 = (1, {o1 , o5 , o6 }, 0.6). Again, if |o(x)| is restricted to at most f , the traffic is called a multirate f -cast traffic. Multirate f -cast networks are also divided into open-end traffic and closed-end traffic. Figure 3 shows three request connections on C(1, 2, 3, 2, 4) Clos network for (0,1] open-end 3-cast assignments: C1 = (1, {o1 , o6 }, 0.6) and C2 = (2, {o6 }, 0.4). Since |o(1)| = 2 < 3, we can add a new receiver o2 to request C1 , that is the request C3 . Notice that, the load of the link between I1 and M1 remains 0.6 after C3 is routed. We said a connection request is compatible to the existing configuration if adding this request does not cause capacity overflows for any external links. Nonblocking switching networks can be categorized into three types: 1) Strictly Nonblocking Switching Network (SNB): A connection request compatible to the existing configuration can always be routed. 2) Rearrangeable Nonblocking Switching Network (RNB): A connection 4.

(12) request compatible to the existing configuration can always be routed, although it may need to rearrange the existing connections. 3) Wide-Sense Nonblocking Switching Network (WSNB): An algorithm exists for setting connections such that a new connection request compatible to the existing configuration can always be routed. Since a closed-end traffic sequence is also an open-end traffic sequence, while a routing for open-end traffic is also one for closed-end traffic, we have Theorem 1.4.1. A network is multirate multicast nonblocking under the open-end traffic implies it is so under the closed-end traffic.. 2 2.1. Strictly and Wide-Sense Nonblocking Network Strictly nonblocking. There is no literature on SNB Clos networks under the multirate-multicast model. We now give a general result true for all networks. Theorem 2.1.1. A network is SNB under the open-end traffic for the multiratemulticast model if and only if it is so for the closed-end traffic. Proof. By Theorem 1.4.1, it suffices to prove the ”if” part. Further, it suffices to consider the new request consisting of a single output port since we can decompose a d-output-port request into d single output port requests. Consider a request C = (x, {o1 }, ω) which has the same message as the ith request of x, i.e., (xi , o(xi ), ω) already connected to some other outputs. Let T (x) denote the set of existing connections involving the ith request of x. Since this network is SNB for the closed-end traffic, C can be routed in a path p if T (X) is ignored. Further, when T (x) is put back and intersects p in a link, that link carries both paths with a combined load ω (not 2ω) since they can share the message. When the two paths need to split, then the fan-out capability of a switch is needed. So C is routed in the network in addition to existing connections.. 2.2. The no-split rule. The no-split rule specifies that output ports in the same output switch in a multicast request must be connected by using the fan-out of that output 5.

(13) switch, i.e., using only one path to connect an input port and an output switch. Since a split routing can never help, we assume all WSNB and RNB algorithms use the no-split rule. Under the no-split rule, a request can be represented as (x, O(x), ω), where O(x) ⊆ O. Here, we call a traffic f -cast if |O(x)| ≤ f ≤ r2 . If the no-split rule is the only constraint in routing, we call the routing algorithm the no-split algorithm. Note that for point-to-point traffic, nonblocking under the no-split algorithm is equivalent to SNB. We first give a fundamental relation between the open-end and closed-end type of traffic under the no-split algorithm. Theorem 2.2.1. A three-stage Clos network is multirate f-cast WSNB under the no-split algorithm for the closed-end traffic, then it is so for the open-end traffic. The proof is analogous to the one in Theorem 2.1.1 by replacing o1 by O1 . All WSNB results in this Section are under the no-split algorithm. Svinnset [11] proved Theorem 2.2.2. A C(n1 , r1 , m, n2 , r2 ) network is WSNB for β[b, B] broadcast assignments if ¹ º ¹ º r2 (n1 β − B) n2 β − B m≥ + + 1. 1−B+² 1−B+² where ² is a positive number approaching zero. With d-fold internal links the corresponding result is ¹ º ¹ º n2 β − B r2 (n1 β − B) + + 1. m≥ (1 − B + ²)d (1 − B + ²)d Notice that, if 1−ω < b, 1−ω won’t be a request weight, so we can modify 1 β−ω) c+b nM2 β−ω c+1, Svinnset’s result in Theorem 2.2.2 to m ≥ maxb≤ω≤B b r2 (n M (ω) (ω) where M (ω) = max{1 − ω + ², b}. Following the concept used by Svinnset, we extend Theorem 2.2.2 (the modified version) to f -cast model.. 6.

(14) Theorem 2.2.3. A C(n1 , r1 , m, n2 , r2 ) network is WSNB for β[b, B] closedend f -cast assignments if ¹ º ¹ º f (n1 β − ω) n2 β − ω m ≥ max + +1 b≤ω≤B M (ω) M (ω) where M (ω) = max{1−ω +², b} and ² is a positive number approaching zero. Based on Theorem 2.2.3, we have the following two results: Corollary 2.2.4. A C(n1 , r1 , m, n2 , r2 ) network is WSNB for β[b, B] closedend f -cast assignments with b + B ≤ 1 if » ¼ » ¼ n2 β − B f (n1 β − B) m≥ + +1 1−B 1−B Corollary 2.2.5. A C(n1 , r1 , m, n2 , r2 ) network is WSNB for β[b, B] closedend f -cast assignments with b + B > 1 and b ≤ 1/2 if ¹ º ¹ º f (n1 β − 1 + b) n2 β − 1 + b m≥ + +1 b b We tighten the condition in Corollary 2.2.5. Theorem 2.2.6. A C(n1 , r1 , m, n2 , r2 ) network is WSNB for [b, B] closedend f -cast assignments with b + B > 1 if m > b1/bc(n1 − 1)f + b1/bc(n2 − 1). Proof. In the [b, B] with b + B > 1 model, once a link carries a request with weight B, it can’t carry a request anymore. This condition is the same with [b, 1] multirate condition. Let T (n, ω, t) denote the number of output links with weight greater than 1 − ω in the input switch associated with the new connection request, where n is the number of input ports in the input switch, ω is the weight of the new connection, and t is the maximum number of fanouts allowed for each connection at the input stage. Yang proved maxb≤ω≤1 T (n, ω, t) = b1/bc(n−1)t in [13] under [b, 1] condition. This implies maxb≤ω≤B T (n, ω, t) = b1/bc(n − 1)t for the [b, B] with b + B > 1 model. Since the sufficient condition is m > maxb≤ω≤B T (n1 , ω, f ) + T (n2 , ω, 1), this theorem follows. 7.

(15) Setting f = 1 in Theorem 2.2.3 implies the result in [6] for the symmetric Clos network. Corollary 2.2.7. A C(n, m, r) network is SNB for β[b, B] point-to-point assignments if ¹ º nβ − ω m ≥ 2 max + 1. b≤ω≤B M (ω) where M (ω) = max{1−ω +², b} and ² is a positive number approaching zero. The corresponding results of Corollary 2.2.4 and 2.2.5 for point-to-point (setting f = 1) symmetric model are the following two results in [6]. Corollary 2.2.8. A C(n, m, r) network is SNB for β[b, B] point-to-point assignments with b + B ≤ 1 if » ¼ nβ − B m≥2 +1 1−B Corollary 2.2.9. A C(n, m, r) network is SNB for β[b, B] point-to-point assignments with b + B > 1 and b ≤ 1/2 if ¹ º nβ − 1 + b m≥2 +1 b Setting f = 1 in Theorem 2.2.6 implies the sufficient side of the following result with β = 1 in [6]: Corollary 2.2.10. C(n, m, r) is SNB for the β[b, B] point-to-point assignments with b + B > 1 if and only if m ≥ 2bβ/bc(n − 1) + 1. Setting b = 1 in Theorem 2.2.6, then there is rate 1, i.e., it is the regular phone switching network model. Then Theorem 2.2.6 is comparable to Hwang [6] as follows except the boundary condition: Corollary 2.2.11. A C(n1 , r1 , m, n2 , r2 ) network is WSNB for closed-end f -cast assignments if m ≥ min{(n1 − 1)f + n2 , (N1 − 1)f + 1, N2 }.. 8.

(16) 2.3. k-limited algorithm. For the Multicast WSNB condition, Yang and Masson [12] first explicitly suggested an algorithm called k-limited algorithm, which is defined by the constraint that any request can use at most k middle switches. Notice that, the f -cast results in this subsection are all for the closed-end traffic. They gave the following result: Theorem 2.3.1. A C(n1 , r1 , m, n2 , r2 ) network is WSNB under a k-limited algorithm for f -cast assignments if m > (n1 − 1)k + (n2 − 1)f 1/k . Note that, the above theorem shows that the result depends on how we choose k. In [13], it was shown that k = O( lnlnlnff ) is an optimal choice for symmetric model. The k-limited algorithm can be extended to the multirate-multicast environment. Svinnset proved Lemma 2.3.2. For a request (x, O(x), in¦ C(n1 , r1 , m, n2 , r2 ) with β[b, B] ¥ ω) 2 β−B broadcast assignments, if there are n1−B+² + 1, where ² → 0+ , middle switches available for x to connect to in the existing configuration, then x can be connected to all output switches in O(x). It follows ¥ 2 β−B ¦ +1 < r2 , is WSNB Theorem 2.3.3. A C(n1 , r1 , m, n2 , r2 ) network, n1−B+² under this k-limited algorithm for β[b, B] broadcast assignments if ¹ º ¹ º k(n1 β − B) n2 β − B m≥ + + 1. 1−B+² 1−B+² ¥ 2 β−B ¦ where k = n1−B+² + 1 and ² is a positive number approaching zero. We describe the relation between Lemma 2.3.2 and Theorem 2.3.3: Since under a k-limited algorithm, fanouts ofj an inputk switch module is restricted 1 β−B) to at most k, maxb≤ωB T (n1 , ω, k) ≤ k(n . Hence, there are at least 1−B+² ¥ n2 β−B ¦ + 1, where ² → 0+ , middle switches available for this new request 1−B+² to connect to. Thus we can use Lemma 2.3.2. Yang [13] obtained a better result for B = 1. 9.

(17) Theorem 2.3.4. A C(n1 , r1 , m, n2 , r2 ) network is WSNB under a k-limited algorithm for [b, 1] multirate broadcast assignments if 1/k. m > b1/bc(n1 − 1)k + b1/bc(n2 − 1)r2 . For the f -cast model, Yang gave the following result. Theorem 2.3.5. A C(n1 , r1 , m, n2 , r2 ) network is WSNB under a k-limited algorithm for [b, 1] multirate f-cast assignments if m > b1/bc(n1 − 1)k + b1/bc(n2 − 1)f 1/k . Note that, setting b = 1 in this theorem yields Theorem 2.3.1. Kabaci´ nski-Danilewicz [7] gave a result for general B. Note that Yang’s result (Theorem 2.3.4) under symmetric model is a special case of this result: Define the following functions:   if i/j is not an integer or bi/jc = 0, bi/jc P (i, j) = bi/jc − 1 if i/j is an integer and i/j > 0,   0 if j = 0.   i − jP (i, j) for P (i, j) 6= 0 and i − jP (i, j) > b, R1 (i, j) = 0 for P (i, j) 6= 0 and i − jP (i, j) ≤ b,   β for P(i,j)=0. k (j j for R(i, j) ≥ b, R1 (i,j) R2 (i, j) = 0 for R1 (i, j) < b. ( i/j for j 6= 0, R3 (i, j) = 0 for j = 0. ( i for i ≥ b, R5 (i, j) = 0 for i < b.. Theorem 2.3.6. A C(n, m, r, d) network is WSNB under a k-limited algorithm for β[b, B] broadcast assignments if m > K(n, d)(k + r1/k ) 10.

(18) where 1 ≤ k ≤ min(K(n, d), r) and K(n, d) = Kβ,b,B (n, d) º ¹ (n − 1)bβ/bc   for B ∈ (1 − b, β],   d     ¹ (n − 1)bβ/bc + b(β − B)/bc º for B ∈ (1 − 2b, 21 ] = 2d    1 1   j k and 4 < b < 2 ,    (n−1)P (β,1−B)+bR3 (n−1,a)c+P (α(B),1−B) for other B. d. in which α(B) = [n − 1 − abR3 (n − 1, a)c]R1 (β, 1 − B) + R5 (β − B), a = R2 (β, 1 − B) + R3 (1, R4 (R1 (β, 1 − B), γ)), γ = max{ lim+ [1 − B + ² − R2 (β, 1 − B)R1 (β, 1 − B)], b}, ²→0. R4 (R1 (β, 1 − B), γ)   P (R1 (β, 1 − B), γ) if P (β, 1 − B) 6= 0, or P (β, 1 − B) = 0    and 1 − B − R2 (β, 1 − B)R1 (β, 1 − B) ≥ b, = ¹ º  R1 (β, 1 − B)   otherwise.  γ They also gave a result under discrete model: Theorem 2.3.7. A C(n, m, r, d) network is WSNB under a k-limited algorithm for discrete broadcast assignments with all rates in {ω1 , · · · , ωh }, where ω1 = b, ωh = B and b|ωi if ˆ m > K(n, d)(k + r1/k ). ˆ where 1 ≤ k ≤ min{K(n, d), r} and ˆ ˆ β,b,B (n, d) = (n − 1)bβ/bc + b(β − B)/bc . K(n, d) = K d(b(1 − B)/bc + 1) We now extend Theorem 2.3.6 to f -cast asymmetric model: Theorem 2.3.8. A C(n1 , r1 , m, n2 , r2 ) network is WSNB under a k-limited algorithm for β[b, B] broadcast assignments if m > K(n1 , 1)k + K(n2 , 1)f 1/k where 1 ≤ k ≤ min(K(n2 , 1), f ) and K(∗, ∗) is defined in Theorem 2.3.6. 11.

(19) Proof. Yang [13] gave a result that m > maxb≤ω≤1 {T (n1 , ω, k)+T (n2 , ω, 1)f 1/k } is a sufficient condition for a C(n1 , r1 , m, n2 , r2 ) network to be WSNB under a k-limited algorithm for [b, 1] f -cast assignments, where 1 ≤ x ≤ min(T (n2 , ω, 1), f ) and T (n, ω, t) was defined in proof of Theorem 2.2.6. We can modify this result to: a C(n1 , r1 , m, n2 , r2 ) network is WSNB under a k-limited algorithm for β[b, B] f -cast assignments if m > maxb≤ω≤B {T (n1 , ω, k) + T (n2 , ω, 1)f 1/k } where 1 ≤ x ≤ min(T (n2 , ω, 1), f ). Kabaci´ nski and Danilewicz showed max1≤ω≤B T (n, ω, t) = K(n, 1)t in the β[b, B] model. The theorem follows. Note that, setting β = B = 1, we obtain Theorem 2.3.5. Similarly, we also extend Theorem 2.3.7 to asymmetric f -cast model: Theorem 2.3.9. A C(n1 , r1 , m, n2 , r2 ) network is WSNB under a k-limited algorithm for β[b, B] discrete f-cast assignments with all rates in {ω1 , · · · , ωh }, where ω1 = b, ωh = B and b|ωi if ˆ 1 , 1)k + K(n ˆ 2 , 1)f 1/k . m > K(n ˆ 2 , 1), f } and K(∗, ˆ ∗) is defined in Theorem 2.3.7. where 1 ≤ k ≤ min{K(n Chan-Chan-Yeung [2] proved Theorem 2.3.10. A C(n1 , r1 , m, n2 , r2 ) network is WSNB under a k-limited algorithm for β[b, B] broadcast assignments if ¹ º ¹ º k(n1 β − B) n2 β − B 1/k m> + r2 . 1−B+² 1−B+² 2 β−B where 1 ≤ k ≤ min(b n1−B+² c, r2 ) and ² is a positive number approaching zero.. With d-fold internal links the corresponding result is ¹ º ¹ º k(n1 β − B) n2 β − B 1/k m≥ + r . (1 − B + ²)d (1 − B + ²)d 2 j k n2 β−B where 1 ≤ k ≤ min(r2 , (1−B+²)d ) and ² is a positive number approaching zero. Kim and Du [8] proved. 12.

(20) Theorem 2.3.11. Under a k-limited algorithm, a broadcast request with 1 weight ω ≤ p+1 cannot be blocked in C(n, m, r) for the β[b, B] model if m>. (βn − ω)(p + 1) (k + r1/k ) p. We now extend Theorem 2.3.11 to the f -cast asymmetric model and the proof is given in Appendix. Theorem 2.3.12. Under a k-limited algorithm, a f -cast request with weight 1 ≤ p+1 cannot be blocked in C(n1 , r1 , m, n2 , r2 ) for the β[b, B] model if m>. 2.4. (βn1 − b)(p + 1) (βn2 − b)(p + 1) 1/k k+ f p p. R(l) algorithm. Kim and Du extended a routing algorithm R(l), first used by Gao and Hwang in multirate point-to-point model (see Corollary 2.4.4), to broadcast traffic. It is the algorithm of reserving l middle switches only for large requests (which can overflow to other middle switches) where a request is called large if its weight ω > 1/(q + 1), q = b1/Bc. Notice that, the f -cast results in this subsection are all for the closed-end traffic. Theorem 2.4.1. A C(n, m, r) network is WSNB under R(l) and a k-limited algorithm for β[b, B] multirate broadcast assignments if  βn(q + 1)(Bq + B + q − 1)   (k + r1/k ) for B < 23/32, 2 q m>  ( 15βn + n − 1)(k + r1/k ) for B ≥ 23/32. 8 and l = d(βn(Bq + B − 1)(q + 1)/q 2 )(k + r1/k )e. where q = b B1 c. We now extend the above theorem to f -cast asymmetric model and the proof is given in Appendix.. 13.

(21) Theorem 2.4.2. A C(n1 , r1 , m, n2 , r2 ) network is WSNB under R(l) and a k-limited algorithm for β[b, B] f -cast broadcast assignments if m ≥ s + l, where s> & l=. (βn1 − b)(q + 1) (βn2 − b)(q + 1) 1/k k+ f , q q. [(βn1 −. 1 )k q+1. + (βn2 −. 1 )f 1/k q+1. − s(1 − B)](q + 1). '. q. and q = b B1 c. Our result slightly improves over [8] which replace the two b terms in s and 1 two q+1 terms in l by 0. To get the point-to-point result, we set f = k = 1. For b → 0 and the symmetric case, we obtain Corollary 2.4.3. A C(n, m, r) network is WSNB under R(l) for β(0, B] point-to-point assignments if m≥ and. 2βn(q + 1)(Bq + B + q − 1) 2 − . q2 q. 2 l = d2βn(Bq + B − 1)(q + 1)/q 2 − e. q. For this model, Gao and Hwang’s result [5] proved: Corollary 2.4.4. A C(n, m, r) network is WSNB under R(l) for β(0, B] point-to-point assignments if m≥. 2βn(q + 1)(Bq + B + q − 1) . q2. and l = d2βn(Bq + B − 1)(q + 1)/q 2 e.. 14.

(22) 3. Rearrangeable Nonblocking Networks. For the multirate-multicast model, Kim and Du gave a rearrangeable algorithm. The algorithm orders the requests by their weights and routes each of them using at most k middle switches. The requests are routed in the order from heavy to light. To route the next heaviest request, the algorithm would not disturb the heaviest requests which were already routed. It continues to route the other requests until the lightest request is successfully routed. Let us introduce a multirate model, the recursive half channel model. In this model, there are h rates ω1 , · · · , ωh with ω1 > ω2 > · · · > ωi−1 > 1/2 ≥ ωi > ωi+1 > · · · > ωh and ωj divides ωj−1 for i + 1 ≤ j ≤ h. Kim-Du [8] (1998) proved Theorem 3.0.5. C(n, m, r) is rearrangeable for the (0, 1] recursive half channel model and broadcast assignments if m > (n − 1). min. (k + r1/k ).. 1≤k≤min(n−1,r). We extend Theorem 3.0.5 to (i) f -cast, (ii) asymmetric Clos network and give a proof in Appendix. Theorem 3.0.6. C(n1 , r1 , m, n2 , r2 ) is rearrangeable for the (0, 1] recursive half channel model and f-cast assignments if © ª m> min (n1 − 1)k + (n2 − 1)f 1/k . 1≤k≤min(n2 −1,f ). Setting f = k = 1 in the above theorem, we get the point-to-point result. And it is the same with the result given by Lin, Du, Hu and Xue [9] under the symmetric model. Corollary 3.0.7. C(n, 2n − 1, r) is rearrangeable for the (0, 1] recursive half channel model and point-to-point assignments. Kim and Du observe a case: Theorem 3.0.8. C(n, m, r) is rearrangeable for broadcast assignments with weights chosen from {ω1 , · · · , ωh }, where 1 ≥ ω1 > · · · > ωh > 0 and ωj divides ωj−1 for 2 ≤ j ≤ h if m > (n − 1). min. (k + r1/k ).. 1≤k≤min(n−1,r). 15.

(23) 4. Numerical Comparison and Conclusion. In this section, we compare some necessary conditions of C(n, m, r) Clos networks which are WSNB for [b, B] broadcast assignments. Let us denote Kim-Du [8] as KD, Yang [13] as Y, Kabaci´ nski-Danilewicz [7] as KaD and Chan-Chan-Yeung [2] as CY.. r 4 12 20 28. KD b 112 147 159 168. n = 10 Y and KaD 0.1 0.4 0.6 360 72 36 477 96 48 515 103 52 544 109 55. KD b 227 299 323 342. n = 20 Y and KaD 0.1 0.4 0.6 760 152 76 1005 201 101 1086 218 109 1147 230 115. KD b 457 603 652 689. n = 40 Y and KaD 0.1 0.4 0.6 1560 312 156 2063 413 207 2229 446 223 2355 471 236. Table 1: B = 1 Note that, for B = 1, Yang [13]’s result is the same with Kabaci´ nskiDanilewicz [7]’s. And Kim-Du [8]’s result is better than Yang [13] and Kabaci´ nski-Danilewicz [7]’s result only when b <= 1/4 or 1/4 < b ≤ 1/3 and n > 16. The following two tables show some comparisons between the results under B < 1 ( except Yang’s result which constrains B = 1). n r 4 12 20 28. 10 CY KD KaD for all b ≤ 0.25 146 112 149 193 147 196 208 159 212 220 168 224. 20 CY KD KaD for all b ≤ 0.25 306 227 309 404 299 408 437 323 441 461 342 465. 40 CY KD KaD for all b ≤ 0.25 626 457 629 828 603 831 894 652 898 944 689 948. Table 2: B = 0.75 Finally, we compare the results of Chen-Chen-Yeung and Kim-Du.. 16.

(24) n. 10 CY KD for all b 62 67 82 88 88 95 93 100. r 4 12 20 28. 30 KaD 0.01 0.3 65 57 85 75 92 81 97 85. CY KD for all b 197 199 260 262 281 283 296 299. 100 KaD 0.01 0.3 197 177 260 233 281 252 296 266. CY KD for all b 662 661 875 873 945 943 999 997. KaD 0.01 0.3 665 597 879 789 949 852 1003 900. Table 3: B = 0.4 Let n−B , 1−B  n(q + 1)(Bq + B + q − 1)   for B < 23/32, q2 hKD (n, B) = ,  ( 23n − 1) for B ≥ 23/32. 8 ¹ º 1 q2B h(B) = 2 , where q = . q − (q + 1)(Bq + B + q − 1)(1 − B) B hCY (n, B) =. For B ≥ 23 , hKD (n, B) < hCY (n, B) for n large. Hence Kim-Du’s result 32 23 is better than Chen-Chen-Yeung’s for B ≥ 32 . Table 2 shows this property. 23 For B < 32 , hCY (n, B) < hKD (n, B) for all n < h(B) and hKD (n, B) < hCY (n, B) for n large. Hence Kim-Du’s result is better than Chen-Chenand n large. Table 3 shows this property, where h(0.4) = Yeung’s for B < 23 32 40. But notice that, for some B, h(B) is very large.. Appendix Let Mj denote the vector (Mj (1), · · · , M j (r2 )), where M j (k) is the sum of weights loaded on link between middle switch j and output switch k. In Figure 2, for example, M 1 = (0.4, 0, 0), M 2 = (0.6, 0.4, 0), M 3 = (0, 0.2, 0.6) and M 4 = (0, 0.4, 0) . Proof of Theorem 2.3.12. Let C = (x, O(x), ω) be a new f -cast request. 1 . Under the k-limited algorithm let m0 be We have |O(x)| ≤ f and ω ≤ p+1 17.

(25) the number of middle switches blocking the new request from the x’s input switch B. Then, m0 (1 − ω) ≤ (βn1 − ω)k implies m0 ≤. (βn1 − ω)(p + 1) k, p. (1). p since 1 − ω ≥ p+1 . Consider the m00 × |O(x)| destination matrix M by discarding at most m0 rows whose corresponding middle switches are blocking the new request from the input switch B where those |O(x)| columns are corresponding to output switches in O(x). Suppose that any k middle switches cannot satisfy this new multicast request. Let t1 (j) be the number of elements in the j-th row whose values are greater than 1 − ω and t1 = min1≤j≤m00 t1 (j). We obtain, m00. X p m00 t1 ≤ m00 t1 (1 − ω) ≤ t1 (j)(1 − ω) ≤ (βn2 − ω)|O(x)| ≤ (βn2 − ω)f p+1 j=1 implying m00 ≤. (βn2 − ω)(p + 1) f , p t1. (2). since t1 6= 0. Assume the s-th row has the minimum, i.e., t1 = t1 (s). We can route a part of the request to f − t1 output switches by using the middle switch corresponding to the s-th row and delete those f − t1 columns from M for finding the next middle switch to route the remaining destinations. Generally, assume there are only ti−1 output switches which are needed to be routed by using m00 × ti−1 destination matrix M (i−1) for i < k. Let ti (j) be the number of elements in the j-th row whose values are greater than 1 − ω and ti be the minimum of ti (j) for all j. Then, m00. X p ≤ m00 ti (1 − ω) ≤ ti (j)(1 − ω) ≤ (βn2 − ω)ti−1 m ti p+1 j=1 00. implies m00 ≤. (βn2 − ω)(p + 1) ti−1 . p ti 18. (3).

(26) where ti 6= 0 for i < k. Otherwise, it is a contradiction to the assumption that any k middle switches can not satisfy the new multicast request. When i = k, each row vector has at least one element whose value is greater than 1 − ω. Therefore, m00. X p m ≤ m00 (1 − ω) ≤ (1 − ω) ≤ (βn2 − ω)tk−1 , p+1 j=1 00. implies. (βn2 − ω)(p + 1) tk−1 . (4) p Since a geometric mean is not less than the minimum of a sequence, the minimum m00 can be obtained from (2), (3) and (4) as m00 ≤. m00 ≤. (βn2 − ω)(p + 1) 1/k f . p. (5). (βn1 − ω)(p + 1) (βn2 − ω)(p + 1) 1/k k+ f reaches its maximum at ω = b. p p (βn1 − b)(p + 1) (βn2 − b)(p + 1) 1/k Hence, if m > k+ f , then the new rep p quest can be routed through this network.. Proof of Theorem 2.4.2. Under the algorithm R(l), assume we partition middle switches M into MS and ML whose sizes are mS and mL , respectively. 1 The algorithm forces a small call (ω ≤ q+1 ) to use only MS but allows a large 1 call (ω < q+1 ) to use not only ML but also MS . Let C = (x, O(x), ω) be 1 a request compatible to the existing configuration. First assume ω ≤ q+1 . From Theorem 2.3.12, setting mS ≥ s > (βn1 −b)(q+1) k + (βn2 −b)(q+1) f 1/k , C q q 1 can be routed. Next assume ω > q+1 . Let MS0 be a subset of MS blocking 0 C from x’s input switch B and ML be a subset of ML blocking the request from B and their sizes are m0S and m0L . Since q = b B1 c, we have each link from B to ML0 carrying exactly q calls. Because of the compatibility, the maximum total weights going to the middle stage out of the input switch is at most (βn1 − ω)k. Therefore, m0L. q q + m0S (1 − B) ≤ m0L + m0S (1 − ω) ≤ (βn1 − ω)k. q+1 q+1 19. (6).

(27) Let MS00 ⊆ MS \ MS0 and ML00 ⊆ ML \ ML0 be the subset of MS and ML which are available for C, respectively. Their sizes are denoted as m00S and m00L . To find out the maximum number of blocking links to output switches in O(x), let us consider (m00S + m00L ) × |O(x)| destination matrix M. Suppose that any k middle switches from MS00 ∪ ML00 can not satisfy this new multicast request. We will use the same notation for ti (j) and ti as Theorem 2.3.12 but ti = minj∈MS00 ∪ML00 ti (j). X X p m00L t1 + m00S t1 (1 − B) ≤ t1 (j) + t1 (j)(1 − B) p+1 00 00 j∈ML. j∈MS. ≤ (βn2 − ω)|O(x)| ≤ (βn2 − ω)f implies m00L. q f + m00S (1B ) ≤ (βn2 − ω) , q+1 t1. (7). since t1 6= 0. We apply a method similar to Theorem 2.3.12 to contruct the destination matrix and obtain the minimum number of middle switches as q ti−1 m00L + m00S (1 − B) ≤ (βn2 − ω) for i < k, (8) q+1 ti q m00L + m00S (1 − B) ≤ (βn2 − ω)tk−1 for i = k. (9) q+1 From (7), (8) and (9), we get m00L. q + m00S (1 + B) ≤ (βn2 − ω)f 1/k q+1. (10). Set m∗L = m0L + m00L ≤ mL and m∗S = m0S + m00S ≤ mS , we obtain the following by summing up (6) and (10): m∗L ≤. [(βn1 − ω)k + (βn2 − ω)f 1/k − m∗S (1 − B)](q + 1) q. (βn2 − b)(q + 1) 1/k (βn1 − b)(q + 1) k+ f suffices to route q q all small requests. For this m∗S , if & ' 1 1 1/k [(βn − )k + (βn − )f − s(1 − B)](q + 1) 1 2 q+1 q+1 m∗L > l = , q Setting m∗S = s >. 20.

(28) the request can be routed. Therefore, m = mL + mS ≥ m∗S + m∗L > s + l middle switch modules suffice.. Proof of Theorem 3.0.6. Assume ω1 , · · · , ωh are those h rates with ω1 > ω2 > · · · > ωi−1 > 1/2 ≥ ωi > ωi+1 > · · · > ωh and ωj divides ωj−1 for i + 1 ≤ j ≤ h. We will prove this theorem by induction on h. For h = 1, each link can carry no more than one call due to ω1 > 1/2 so this three-stage Clos network is nonblocking and rearrangeable if m > min1≤k≤min(n2 −1,f ) (n1 − 1)k + (n2 − 1)f 1/k (see Corollary 2.3.1). Assume that this Clos network is rearrangeable for h = h0 − 1. Consider i integers u1 , u2 , · · · , ui−1 and v such that ωj +uj ωh0 ≤ 1 < ωj +(uj +1)ωh0 for j ≤ i−1 and vωh0 ≤ 1 < (v+1)ωh0 . If a link blocks a new connected request C = (x, O(x), ωh0 ), then the blocking link is carrying either one ωj -call for some j ≤ i − 1 and some ωk -calls, h ≥ k ≥ i, with total weight uj ωh0 (Uj -blocking), or no such ωj -call but all ωk -calls with total weight vωh0 (V -blocking). Let us assume that m0 middle switches are blocking this new ωh0 -call from input stage with m0 > (n1 − 1)k. Because all connected requests were able to duplicate their messages at most k times at the input switch, at least n1 input ports should have carry full weights that are ωj +uj ωh0 or vωh0 . This is a contradiction to our assumption for the compatible new ωh0 -call. Hence, m0 ≤ (n1 − 1)k. Consider m00 middle switches by discarding those m0 middle switches blocking this new connection request from input stage. Suppose that no k middle switches among m00 can route the new ωh0 -call. Because each output switch in O(x) has at most (n2 −1) output ports which are either Uj -blocking for some j ≤ i − 1 or V -blocking for the new call, the total number of blocking links between middle stage and output switches in O(x) is no more than (n2 −1)|O(x)| ≤ (n2 −1)f . By using the similar approach as Theorem 2.3.12, we can obtain, m00 ≤ (n2 − 1)f /t1 , (11) m00 ≤ (n2 − 1)ti−1 /ti for i < k,. (12). m00 ≤ (n2 − 1)tk−1 for i = k.. (13). A minimum of a sequence is not larger than its geometric mean so that, from. 21.

(29) (11), (12) and (13), we can obtain, · ¸1/k f t1 tk−2 m ≤ (n2 − 1) · (n2 − 1) · · · · · (n2 − 1) · (n2 − 1)tk−1 = (n2 −1)f 1/k . t1 t2 tk−1 00. We showed that the nonblocking multicast Clos network for the switching network is also rearrangeable for multirate-multicast communications for the recursive half channel model.. References [1] K.S. Chan, S. Chan, K.L. Yeung, Wide-sense non-blocking multicast ATM switches. Electronics Letters,Vol. 33, No. 6, (1997), 462-464. [2] K. S. Chan, S. Chan, K. L. Yeung, Design of wide-sense non-blocking multicast ATM switches. IEEE Communications Letters,Vol. 2, No. 5, (1998), 146-148. [3] S. P. Chung, K. W. Ross, On nonblocking interconnection networks. SIAM J. Comput., Vol. 20, (1991), 726-736. [4] P. Feldman, J. Friedman, N. Pippenger, Nonblocking networks. ACM Symp. Thy. Comput., Vol. 18, (1986), 247-254. [5] B. Gao, F. K. Hwang, Wide-sense nonblocking for multirate 3-stage Clos networks. Theor. Comput. Sci., Vol. 182, (1997), 171-182. [6] F. K. Hwang, The Mathematical Theory of Nonblocking Switching Netwoks. Word Scientific, Series on Applied Mathematics, Vol. 15, 2nd. Edition, (2004). [7] W. Kabaci´ nski, G. Danilewicz, Wide-sense non-blocking nulticast ATM switching networks. Performance Evalution, Vol. 41, (2000), 165-177. [8] Dongsoo S. Kim, Ding-Zhu Du, Multirate Multicast Switching Networks. Lecture Notes In Computer Science, Vol. 1449, (1998), 219-228. [9] G. H. Lin, D. Z. Du, X. D. Hu, G. Xue, On rearrangeablility of multirate Clos networks. SIAM J. Comput., Vol. 28, (1999), 1225-1231.. 22.

(30) [10] R. Melen, J. S. Turner, Nonblocking multirate networks. SIAM J. Comput., Vol. 18, (1989), 301-313. [11] Inge Svinnset, Nonblocking ATM Switching Networks. IEEE Transations on Communications, Vol. 42, No. 2/3/4, (1994), 1352-1358. [12] Y. Yang, G. M. Masson, Nonblocking broadcast switching networks. IEEE Trans. Commun., Vol. 9, (1991), 1005-1015. [13] Y. Yang, An analysis model on nonblocking multirate broadcast networks. In International Conference on Supercomputing, (1994), 256-263.. 23.

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