Chuyển đổi lý thuyết P4 pot

Three-stage network The general scheme of a three-stage network is given in Figure 4.2, in which, as usual, n and m denote the number of inlets and outlets of the first- A and third- C s

Chapter 4 Non-blocking Networks

The class of strict-sense non-blocking networks is here investigated, that is those networks inwhich it is always possible to set up a new connection between an idle inlet and an idle outletindependently of the network permutation at the set-up time As with rearrangeable networksdescribed in Chapter 3, the class of non-blocking networks will be described starting from thebasic properties discovered more than thirty years ago (consider the Clos network) and goingthrough all the most recent findings on network non-blocking mainly referred to banyan-based interconnection networks

Section 4.1 describes three-stage non-blocking networks with interstage full connection(FC) and the recursive application of this principle to building non-blocking networks with anodd number of stages Networks with partial connection (PC) having the property of non-blocking are investigated in Section 4.2, whereas Section 4.3 provides a comparison of the dif-ferent structures of partially connected non-blocking networks Bounds on the network costfunction are finally discussed in Section 4.4

4.1 Full-connection Multistage Networks

We investigate here how the basic FC network including two or three stages of small crossbarmatrices can be made non-blocking The study is then extended to networks built by recursiveconstruction and thus including more than three stages

4.1.1 Two-stage network

The model of two-stage FC network, represented in Figure 2.11, includes matrices

at the first stage and matrices at the second stage.This network clearly has full sibility, but is blocking at the same time In fact, if we select a couple of arbitrary matrices at

This document was created with FrameMaker 4.0.4

net_th_snb Page 127 Tuesday, November 18, 1997 4:32 pm

Switching Theory: Architecture and Performance in Broadband ATM Networks

Achille Pattavina Copyright © 1998 John Wiley & Sons Ltd ISBNs: 0-471-96338-0 (Hardback); 0-470-84191-5 (Electronic)

128 Non-blocking Networks

the first and second stage, say and , no more than one connection between the n inlets

of and the m outlets of can be set up at a given time Since this limit is due to the singlelink between matrices, a non-blocking two-stage full-connection network is then easilyobtained by properly “dilating” the interstage connection pattern, that is by providing d linksbetween any couple of matrices in the two stages (Figure 4.1) Also such an FC network is a

factor required in a non-blocking network is simply given by

since no more than connections can be set up between and at the sametime The network cost for a two-stage non-blocking network is apparently d times the cost of

a non-dilated two-stage network In the case of a squared network

using the relation , we obtain a cost index

that is the two-stage non-blocking network doubles the crossbar network cost

Thus, the feature of smaller matrices in a two-stage non-blocking FC network compared

to a single crossbar network is paid by doubling the cost index, independent of the valueselected for the parameter n

4.1.2 Three-stage network

The general scheme of a three-stage network is given in Figure 4.2, in which, as usual, n and

m denote the number of inlets and outlets of the first- (A) and third- (C) stage matrices,respectively Adopting three stages in a multistage network, compared to a two-stage arrange-ment, introduces a very important feature: different I/O paths are available between anycouple of matrices and each engaging a different matrix in the second stage (B) Two I/

Figure 4.1 FC two-stage dilated network

Full-connection Multistage Networks 129

O paths can share interstage links, i.e when the two inlets (outlets) belong to the same A (C)matrix So, a suitable control algorithm for the network is required in order to set up the I/Opath for the new connections, so as not to affect the I/O connections already established

Full accessibility is implicitly guaranteed by the full connection of the two interstage terns Thus, formally describes the full accessibility condition

pat-The most general result about three-stage non-blocking FC networks with arbitrary values

Clos theorem A three-stage network (Figure 4.2) is strict-sense non-blocking if and only if

(4.1)

Proof Let us consider two tagged matrices in the first (A) and last (C) stage with maximumoccupancy but still allowing the set up of a new connection So, and connectionsare already set up in the matrices A and C, respectively, and one additional connection has to

be set up between the last idle input and output in the two tagged matrices (see Figure 4.3).The worst network loading condition corresponds to assuming an engagement pattern of the

matrices supporting the connections through matrix A are different from the ond-stage matrices supporting the connections through matrix C This also means thatthe no connection is set up between matrices A and C Since one additional second-stagematrix is needed to set up the required new connection,

sec-matrices in the second stage are necessary to make the three-stage network strictly

Figure 4.2 FC three-stage network

The network cost for a given N thus depends on the number of first-stage matrices, that

is on the number of inlets per first-stage matrix since By taking the first derivative

of C with respect to n and setting it to 0, we easily find the solution

(4.2)

which thus provides the minimum cost of the three-stage SNB network, i.e

(4.3)Unlike a two-stage network, a three-stage SNB network can become cheaper than a cross-bar (one-stage) network This event occurs for a minimum cost three-stage network when the

number of network inlets N satisfies the condition (as is easily obtained byequating the cost of the two networks) Interestingly enough, even only inlets areenough to have a three-stage network cheaper than the crossbar one By comparing Equations4.3 and 3.2, giving the cost of an SNB and RNB three-stage network respectively, it is notedthat the cost of a non-blocking network is about twice that of a rearrangeable network

4.1.3 Recursive network construction

Networks with more than three stages can be built by iterating the basic three stage tion Clos showed [Clo53] that a five-stage strict-sense non-blocking network can berecursively built starting from the basic three-stage non-blocking network by designing eachmatrix of the second-stage as a non-blocking three-stage network The recursion, which can

construc-Figure 4.3 Worst case occupancy in a three-stage network

1 2

n-1 1 2

m-1

n 1

A

m 1

≅

3 2

Full-connection Multistage Networks 131

be repeated an arbitrary number of times to generate networks with an odd number of stages s, enables the construction of networks that become less expensive when N grows beyond cer-

tain thresholds (see [Clo53]) Nevertheless, note that such new networks with an odd number

of stages are no longer connected multistage networks In general a squared network(that is specular across the central stage) with an odd number of stages requires

parameters to be specified that is

five-stage network the optimum choice of the two parameters can be determinedagain by computing the total network cost and by taking its first derivative with respect to and and setting it to 0 Thus the two conditions

(4.4)

are obtained from which and are computed for a given N

Since such a procedure is hardly expandable to larger values of s, Clos also suggested a

recursive general dimensioning procedure that starts from a three-stage structure and thenaccording to the Clos rule (Equation 4.1) expands each middle-stage matrix into a three-stagestructure and so on This structure does not minimize the network cost but requires just onecondition to be specified, that is the parameter , which is set to

(4.5)The cost index of the basic three-stage network built using Equation 4.5 is

(4.6)The cost index of a five-stage network (see Figure 4.4) is readily obtained considering that

, so that each of the three-stage central blocks has a size

and thus a cost given by Equation 4.6 with N replaced by So, considering the tional cost of the first and last stage the total network cost is

=

C3 (2 N–1) 3N 6N

3 2

2N

1 31–

14N

2 3+

n1 = N1 4⁄

Equation 4.7 with N replaced by So, considering the additional cost of the first and laststage the total network cost is

(4.8)

This procedure can be iterated to build an s-stage recursive Clos network (s odd) whose cost

index can be shown to be

2N

1 41–

2N

1 41–

46N

3 4

3N

1 2–+

s+ 3 2 - –k

N

2k

s+ 1 -

2N

2

s+ 1 -1–

s 1 2 -

N

4

s+ 1 -+

k= 2

s+ 1 2 -

∑

=

Full-connection Multistage Networks 133

which reduces to [Clo53]

with and An example of application of this procedure for some values

of network size N with a number of stages ranging from 1 to 9 gives the network costs of

Table 4.1 It is observed that it becomes more convenient to have more stages as the networksize increases

As already mentioned, there is no known analytical solution to obtain the minimum cost

network for arbitrary values of N; moreover, even with small networks for which three or five

stages give the optimal configuration, some approximations must be introduced to have integervalues of By means of exhaustive searching techniques the minimum cost network can be

found, whose results for some values of N are given in Table 4.2 [Mar77] The minimum cost

network specified in this table has the same number of stages as the minimum-cost networkwith (almost) equal size built with the recursive Clos rule (see Table 4.1) However the formernetwork has a lower cost since it optimizes the choice of the parameters For example, the

Table 4.1 Cost of the recursive Clos s-stage network

N

100 10,000 5,700 6,092 7,386 9,121

200 40,000 16,370 16,017 18,898 23,219

500 250,000 65,582 56,685 64,165 78,058 1,000 1,000,000 186,737 146,300 159,904 192,571 2,000 4,000,000 530,656 375,651 395,340 470,292 5,000 25,000,000 2,106,320 1,298,858 1,295,294 1,511,331 10,000 100,000,000 5,970,000 3,308,487 3,159,700 3,625,165

Table 4.2 Minimum cost network by exhaustive search

N

100 3 5 5,400

500 5 10 5 53,200

1001 5 11 7 137,865 5,005 7 13 7 5 1,176,175 10,000 7 20 10 5 2,854,800

4.2 Partial-connection Multistage Networks

A partial-connection multistage network can be built starting from four basic techniques:

• vertical replication (VR) of banyan networks, in which several copies of a banyan network are

used;

• vertical replication coupled with horizontal extension (VR/HE), in which the single planes to be

replicated include more stages than in a basic banyan network;

• link dilation (LD) of a banyan network, in which the interstage links are replicated a certain

number of times;

• EGS network, in which the network is simply built as a cascade of EGS permutations.

In general, the control of strict-sense non-blocking networks requires a centralized controlbased on a storage structure that keeps a map of all the established I/O paths and makes it pos-sible to find a cascade of idle links through the network for each new connection requestbetween an idle inlet and an idle outlet

4.2.1 Vertical replication

Let us first consider the adoption of the pure VR technique that results in the overall replicatedbanyan network (RBN) already described in the preceding section (see also Figure 3.13) The

procedure to find out the number K of networks that makes the RBN strict-sense

non-block-ing must take into account the worst case of link occupation considernon-block-ing that now calls cannot

be rearranged once set up [Lea90]

Theorem A replicated banyan network of size with K planes is strict-sense

non-blocking if and only if

(4.10)

Proof A tagged I/O path, say the path 0-0, is selected, which includes interstage tagged links All the other conflicting I/O paths that share at least one interstage link with the tagged I/

O path are easily identified Such link sharing for the tagged path is shown by the double tree

(the stage numbering applies also to the links outgoing from each switching stage).Each node (branch) in the tree represents an SE (a link) of the original banyan network andthe branches terminated on one node only represent the network inlets and outlets Four sub-

trees can be identified in the double tree with n even (Figure 4.5a), two on the inlet side and

two on the outlet side, each including “open branches”: the subtree terminating the

inlets (outlets) 0-3 and 4-7 are referred to as upper subtree and lower subtree, respectively It is

quite simple to see that the worst case of link occupancy is given when the inlets (outlets) ofthe upper inlet (outlet) subtree are connected to outlets (inlets) other than those in the outlet

N×N

K

32 2

Partial-connection Multistage Networks 135

(inlet) subtrees by engaging at least one tagged link Moreover, since an even value of n implies

that we have a central branch not belonging to any subtree, the worst loading condition for thetagged link in the central stage (stage 3 in the figure) is given when the inlets of lower inletsubtree are connected to the outlets of the lower outlet subtree In the upper inlet subtree thetagged link of stage 1 is shared by one conflicting I/O path originating from the other SE inlet(the inlet 1), the tagged link of stage 2 is shared by two other conflicting paths originated frominlets not accounted for (the inlets 2 and 3), and the tagged link of stage (the lasttagged link of the upper inlet subtree) is shared by conflicting paths originatedfrom inlets which have not already been accounted for We have two of these upper subtrees(on inlet and outlet side); furthermore the “central” tagged link at stage is shared by

conflicting I/O paths (those terminated on the lower subtrees) Then the totalnumber of conflicting I/O paths is

(4.11)

The number of planes sufficient for an RBN with n even to be strictly non-blocking is

then given by as stated in Equation 4.10, since in the worst case each conflicting I/Opath is routed onto a different plane, and the unity represents the additional plane needed bythe tagged path to satisfy the non-blocking condition An analogous proof applies to the case

of n odd (see Figure 4.5b for ), which is even simpler since the double tree does not

Figure 4.5 Double tree for the proof of non-blocking condition

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7

1 2

3 4

5 6

0 1 2 3 4 5 6 7

5 6

0 1 2 3 4 5 6 7

1 2 3

inlets

4 (a)

2 2

n

22–

n c+1

N = 128

have a central link reaching the same number of inlets and outlets Thus the double treeincludes only two subtrees, each including “open branches” Then the total num-ber of conflicting I/O paths is

(4.12)

and the number of planes sufficient for an RBN with n odd to be strictly non-blocking is

given by , thus proving Equation 4.10 The proof of necessity of the number of planesstated in the theorem immediately follows from the above reasoning on the worst case In fact

it is rather easy to identify a network state in which connections are set up, each sharingone link with the tagged path and each routed on a different plane ❏

So, the cost function of a strictly non-blocking network based on pure VR is

The comparison between the vertical replication factor required in a rearrangeable

that the strict-sense non blocking condition implies a network cost that is about 50% higherthan in a rearrangeable network of the same size

4.2.2 Vertical replication with horizontal extension

The HE technique can be used jointly with the VR technique to build a non-blocking work by thus allowing a smaller replication factor The first known result is due to Cantor[Can70, Can72] who assumed that each plane of the overall (Cantor) network is an Benes network Therefore the same vertical replication scheme of Figure 3.13 applieshere where the “depth” of each network is now stages

net-Table 4.3 Replication factor in rearrangeable and strictly

non-blocking VR banyan networks

Partial-connection Multistage Networks 137

Theorem A VR/HE banyan network of size built by replicating K times a Benes

net-work is strict-sense non-blocking if and only if

(4.13)(an example of a Cantor network is shown in Figure 4.6)

Proof Also in this case the paths conflicting with the tagged I/O path are counted so that therequired number of planes to provide the non-blocking condition is computed Unlike a ban-yan network, a Benes network makes available more than one path from each inlet to eachoutlet Since each added stage doubles the I-O paths, a Benes network provides

paths per I/O pair Figure 4.7 shows for these 8 tagged paths each including 6 tagged links for the tagged I/O pair 0-0, together with the corresponding channel graph (each node of the channel graph is representative of a tagged SE, i.e an SE along a tagged

path) The two tagged links outgoing from stage 1 are shared by the tagged inlet and by onlyone other inlet (inlet 1 in our example), which, upon becoming busy, makes unavailable paths for the I/O pair 0-0 (see also the channel graph of the example) The four taggedlinks of the tagged paths outgoing from stage 2 are shared by four network inlets in total,owing to the buddy property In fact the two SEs originating the links are reached by the samefirst-stage SEs Out of these four inlets, one is the tagged inlet and another has already beenaccounted for as engaging the first-stage link Therefore only two other inlets can engage one

of the four tagged links at stage 2, and each of these makes unavailable tagged paths Ingeneral, there are tagged links outgoing from SEs of stage i , which areaccessed by only network inlets owing to the constrained reachability property of a banyan

Figure 4.6 Cantor network for N=8

network This property also implies that the network inlets reaching the tagged SEs at stage

are a subset of the inlets reaching the tagged SEs at stage i Therefore, the tagged links

outgoing from stage i can be engaged by inlets not accounted for in the previous stages.Each of these inlets, upon becoming busy, makes unavailable tagged paths Therefore,the total number of paths that can be blocked by non-tagged inlets is

where is the number of additional stages in each Benes network compared to

a banyan network Based on the network symmetry across the central stage of the Benes works, the same number of paths is made unavailable by non-tagged outlets Theworst case of tagged path occupancy is given when the tagged paths made unavailable by thenon-tagged inlets and by the non-tagged outlets are disjoint sets It is rather easy to show thatthis situation occurs in a Cantor network owing to the recursive construction of the Benesnetwork, which results in a series–parallel channel graph of the Cantor network (the channelgraph of the Cantor network of Figure 4.6 is shown in Figure 4.8) In order for the Cantornetwork to be non-blocking, the number of its tagged I/O paths must not be smaller than thetotal number of blocked paths plus one (the path needed to connect the tagged I/O pair) The

net-total number of tagged I/O paths is clearly given by K times the number of tagged I/O paths

Figure 4.7 Extended banyan network with m=3

Partial-connection Multistage Networks 139

per Benes plane, that is The total number of blocked I/O paths in the Cantor work is still in spite of the K planes, since the generic network inlet i is connected to the inlet i of the K Benes planes and can make only one of them busy Therefore

net-which, owing to the integer values assumed by K, gives the minimum number of planes

suffi-cient to guarantee the strictly non-blocking condition

thus completing the proof of sufficiency The proof of necessity of at least planes forthe network non-blocking stated in the theorem immediately follows from the above reason-ing on the worst case In fact it is rather easy to identify a network state in which all the tagged

The cost index for the Cantor network is

where the last term in the sum accounts for the crosspoint count in the expansion and tration stages

concen-So, the asymptotic growth of the cost index in a Cantor network is ,whereas in a rearrangeable Benes network it is Notice however that the highercost of strict-sense non-blocking networks over rearrangeable networks is accompanied by theextreme simplicity of control algorithms for the former networks In fact choosing an I/Opath for a new connection only requires the knowledge of the idle–busy condition of theinterstage links available to support that path

We further observe that the pure VR non-blocking network has an asymptotic cost

whereas the cost of the Cantor network is Therefore, the

Figure 4.8 Channel graph of the Cantor network with N=8

n b = n bi+n bo

K N

2 ≥n b+1 2(log2N–1) 2log 2N 2

1+ (log2N–1)N

2 +1

K≥log2N

N

2log

K–1

N×N

C 4log2N N

2 2( log2N–1) 2N+ log2N 4Nlog22N

Cantor network is asymptotically cheaper than a pure VR strictly non-blocking network.However, owing to the different coefficient of the term with the highest exponent

Theorem A VR/HE banyan network with K planes each configured as an EBN where the m

additional stages are added by means of the mirror imaging technique is strict-sense blocking if and only if

cor-in the EBN plane Therefore the non-tagged cor-inlets, upon becomcor-ing busy, engage a tagged lcor-ink

paths Unlike the Cantor network we have now other tagged links to be taken into accountthat originate from stage , , …, until the centre of the EBN is reached If is

unavailable only one tagged path Analogously to the pure VR network, we have to considerthat an even value of implies that the “central” tagged links (i.e., those of stage

) reach the same number of inlets and outlets, so that the paths made unavailable

by these links must not be counted twice An example is again represented by Figure 4.9,where the central links are those of stage 3 Therefore the number of blocked paths in an EBNplane with even is now

(4.15)

Note that it is legitimate to double the number of blocked paths from each side of theplane (the first term in Equation 4.15) In fact in the worst case the blocked paths from oneside of the plane are disjoint from the blocked paths originated from the other side of the

n–m

2 -

m–1

2

n–m+ 1 2 -

∑

n+m

2 - 1

n+m

2 - 1+

i=m+ 1

n+m

2 - 1

∑

+