1、Chapter 3 Point-to-Point Multi-Stage Circuit Switching,3.1 Point-to-Point Circuit Switching 3.2 Cost Criteria for Switching 3.3 Multi-Stage Switching Network 3.4 Representing Connections by Paulls Matrix 3.5 Strict-Sense Non-blocking Clos Networks,3.6 Rearrangeable Networks3.7 Recursive Construction
2、 of Switching Networks3.8 The Cantor network3.9 Control Algorithms,3.1 Point-to-Point Switching,In this chapter, we shall deal with the problem of providing point-to-point interconnection between terminals with the same bit-rate such as telephony.,Because of equivalence of space division switching a
3、nd time division switching, we shall use only space division switching in our subsequent treatment of circuit switching(figure 1). A space division switching network can be represented by a graph G=(V, E), where V is the set of switching nodes and E the set of edges in the graph shown in figure2.,An
4、 edge is an ordered pair (u, v) for some u,v V. Let T in V be the set of transmitting terminals which have only outgoing edges, and R in V be the set of receiving terminals which have only incoming edges.,A connection requirement is specified for each t T by the subset of R to which t must be connec
5、ted.The subsets are disjoint for different t.,A path is defined as a sequence of connected edges (t,a),(a,b),(b,c),(f,g),(g,r) E where t T , r and a,b,c,f,g are distinct elements of V-T-R, the set of internal nodes. The set of paths from a t generally forms a tree in case of multi-casting.,Point-to-
6、point circuit switching is a special case of multi-cast switching in which contains at most 1 element for each t. The general problem of circuit switching is defined as the process of finding and setting up paths to satisfy a connection requirement.,3.2 Cost Criteria for Switching,Switching network
7、design involves the choice of the graph G for reducing the overall cost of implementing the switching system. This overall cost consists of a number of components which are highly technology dependent.,1 Crosspoint complexity Traditionally, most studies on switching networks focus on the number of c
8、rosspoints required to implement the switching network. With the economy provided by Very Large Scale Integration (VLSI) technology, a great deal of crosspoint complexity can be implemented on a single chip.,In this case, the designer may choose to minimize the chip count instead of crosspoint count
9、. A very important design criterion is the power consumption per chip.,2 Interconnect, fan-out, and logical depth complexity Given a graph G, we may have to partition the graph into subgraphs each with crosspoint complexity implementable on a VLSI chip. The length of the interconnection between chip
10、s is crucial.,If the length is too long, the power required to drive the interconnection would be high, or else we may have to lower the speed of the interconnection. Logical depth is defined as the number of nodes a signal traverses in the graph.,3 Network control complexity Given a graph G, we nee
11、d control algorithms to find and set up paths in G to satisfy the connection requirement between the terminals. Control complexity is defined by the hardware (computation and memory) requirement as well as the run time of algorithms.,There are graphs G with regular structure for which the control al
12、gorithms are simple. There are also classes of G for which the computation can be distributed over a large number of processors.,In general, the amount of computation depends also on the degree and kinds of blocking we may tolerate. 4 Degree and kinds of blocking tolerated Generally speaking, blocki
13、ng is defined as failure to satisfy a connection requirement.,Such a failure can be attributed to different causes. The earliest kinds of switching networks employed for telephony are blocking in the sense that not all connection requirement can be satisfied, simply because non-conflicting paths do
14、not exist for some connection requirements.,Thus a telephone call can be blocked even though the 2 terminals involved are free. The broadest class of non-blocking networks is that of the so called rearrangeable networks, for which a set of non-conflicting paths always exist for any connection requir
15、ement.,Rearranging paths for connected pairs involved extra control complexity, and may cause disruption of a connection.,A non-blocking network is termed strict-sense non-blocking if a path can be set up between an idle transmitter and an idle receiver (can be more than one receiver for the case of
16、 multi-casting) without disturbing the paths already setup.,In practice, we may tolerate a small probability of blocking. In return, we reduce the crosspoint complexity and control complexity.,In between these 2 classes of non-blocking networks, there are networks called wide sense non-blocking netw
17、orks for which new connections can be add without disturbing existing connections, provided we follow some rules for choosing a path for the new connection. Very little is known so far about wide sense non-blocking networks.,Most recent switching systems are either rearrangeably non-blocking or stri
18、ct sense non-blocking. However, blocking still arises in practice because the control algorithms very often do not perform an exhaustive search for free paths or rearrangements of paths.,5 Network management complexity Network management involves the adaptation and maintenance of the switching netwo
19、rk after the switching system is put in place, to cope with failure events and growth in connectivity demand in the network.,Also, network management deals with the changes of traffic patterns from day to day, as well as momentary overloads. Diagnostic and failure maintenance measures constitute a s
20、ignificant part of the software programs of a switching system.,In order for switching cost to grow linearly with respect to total traffic, switching functions such as control, maintenance, call processing and the interconnection network should be as modular as possible. New modules could be plugged
21、 into the system as demand increases.,3.3 Multi-stage Switching Networks,In general, deciding whether a network G is non-blocking and finding non-conflicting paths in G is difficult.,However, efficient algorithms are known for certain multi-stage switching networks with very regular structures. We m
22、ay use a single crossbar interconnecting N transmitting terminals to N receiving terminals.,However, such an array would require crosspoints. More serious than this large crosspoint count is that each input and output is of linear (in N) length and may be affected by N crosspoints. A multi-stage net
23、work consists of K stages of switching nodes.,Stage k ( ) has switches.Each switch in stage k is named by a number j, . We represent this switch by S(j,k).Also, such a switch has input edges and output edges. We represent the input i of S(j,k) by e(i,j,k), and the output i of S(j,k) by o(i,j,k).,In
24、this chapter, we shall look at a very special class of interconnections: each switch in each stage is connected to each switch in the next stage by 1 edge. Thus, the number of outgoing edges , the number of switches in the next stage.,Likewise, the number of incoming edges , the number of switches i
25、n the previous stage. The goal of this chapter is to demonstrate efficient assignment algorithms for this special class of multi-stage networks.,3.4 Representing Connections by Paulls Matrix,In this section, we shall only be dealing with three stage networks, with each switch in a stage connected to
26、 each switch in the next stage by one edge.,Recall that three stage network is defined by five parameters(figure 3).We shall use a matrix notation devised by Paull for representing paths in the network.,Consider any switch a in stage 1 and any switch b in stage 3. Any input to a may be connected to
27、an output of b via a middle switch f(figure 4). Paulls connection matrix (figure 5) represents these paths by entering the middle switches f, g, h etc. into the (a,b) entry of an matrix.,The conditions for a legitimate connection matrix are given by : 1. Each row can have at most symbols. 2. Each co
28、lumn can have at most symbols. 3. The symbols in each row must be distinct. Therefore, there can be at most symbols.,4. The symbols in each column must be distinct. Therefore, there can be at most symbols. When we allow multi-casting, condition 1 and 3 may not be valid because a path from a first st
29、age switch may fan-out into a multi-cast tree at the second stage switch. However, conditions 2 and 4 remains valid.,3.5 Strict Sense Non-blocking Clos Networks,Let T be any subset of set T of transmitting terminals and R be any subset of the set R of receiving terminals. Each elements of T is conne
30、cted by a legitimate multi-cast tree to a non-empty and disjoint subset of R.,Each element of R is connected to one element in T. A network is strict-sense non-blocking if any t T-T can establish a legitimate multi-cast tree to any subset of R-R without changing the previously established paths, for
31、 all T and R as well as for all connection patterns between T and R.,Clos Theorem: A Clos network is strict-sense non-blocking if and only if the number of second stage switches . In particular, a symmetric network with is strict-sense non-blocking if and only if .,Proof: Suppose we want to establis
32、h a connection from a vacant input of a first stage switch a and a vacant output of a third stage switch b. We do so by putting a symbol in the (a,b) entry in Paulls connection matrix.,Now there can be at most distinct symbols in row a, because there are only inputs to switch a, less one input which
33、 wants to make a new connection. For a similar reason, there can be at most distinct symbols in column b.,Consider the worst case when these symbols are all distinct. If we have 1 more second stage switch, in other words a total of second stage switches, then we can enter a symbol into (a,b) while k
34、eeping all symbols in row a distinct, and all symbols in column b distinct as well.,This condition on is also necessary by considering a special sequence of connections.The preceding proof also gives the procedure for making connections shown in figure 7.,3.6 Rearrangeable Networks,The Slepian-Dugui
35、d Theorem:A three stage Clos network is rearrangeable if and only if . In particular, a symmetric network with is rearrangeably non-blocking if and only if .,Proof: Suppose we want to establish a connection between switches a and b. We shall prove that if , we must either have 1. A symbol which is n
36、ot found in both row a and column b; or2. There exists a symbol c in row a which is not found in column b, and a symbol d in column b which is not in row a(figure 8).,For if case 1 is not true, the symbols must be found either in the row or the column or both. Using the assumed condition on , we hav
37、e . Now there are at most symbols in row a. Hence there must be a d in column b not found in a. Using a symmetrical argument, there must be a c in row a not found in b.,If case 1 is true, then we can place the unfound symbol in (a,b), thus completing the connection without any rearrangements. Otherw
38、ise, we look at the row where d appears in column b to see if the symbol c appears in that row as shown in figure 8. If such a c is found, we check that column to see if a d appears, as shown in figure 9.,We continue the process alternately until we cannot found a c or d. The rearrangement is facili
39、tated by putting a d in (a,b), and replacing all c with d as well as all d with c in the chain as shown in figure 10. Obviously, the four conditions for a legitimate connection matrix is satisfied with such rearrangements.,Since there are inputs to a first stage switch outputs to a third stage switc
40、h, it is also necessary to have at least second stage switches in order that the first or third stage switch may be fully utilized.Paull reduced the number of rearrangements, as stated in the following theorem.,Paulls theorem: The number of circuits that need to be rearranged is at most . Proof: Wit
41、hout loss of generality, let us assume that . First, we look at a c in the row for the d that appears in column b as shown in figure 8. Next, we start the chain from c in row a, instead of extending the chain from d in column b.,We now extend the two chains alternately as shown in figure 11, so that
42、 at each step, the chains have lengths differing by at most one. When either one of the chain cannot be grown further, we choose that chain for rearrangement.,In each step for which the length of both chains are increased by one, we visit one new column. Hence there can be at most steps in extending
43、 the two chains. Hence we need at most rearrangements.,3.7 Recursive Construction of Switching Networks,Suppose we want to construct an switching network.Let . Strict-sense and rearrangeably non-blocking networks can be constructed in the manner shown in figure 12.,The crosspoint count for the rearr
44、angeable construction is . The crosspoint count for the strictly non-blocking construction is .,A recursively constructed rearrangeably non-blocking Clos network is formed by implementing each or switch (figure 12) as a three-stage rearrangeably non-blocking network.,A recursively constructed strict
45、-sense non-blocking Clos network is formed by implementing each or switch (figure 12) as a three-stage strict-sense non-blocking network.,Suppose N can be factored into p and q in many ways, which can be further factored.Which factor p should be chosen first, and how should the subnetworks be furthe
46、r factored?,For the special case of , n being a positive integer, we can recursively construct a rearrangeably network by factoring N into p=2 and q=N/2. This resulting network shown in figure 13 is called Benes Network, which has stages. Each stage consists of N/2 switches of size . Therefore, the
47、number of crosspoint according to this recursive construction is roughly .,A baseline network is defined as the first half of the Benes network, namely the left part of the network from the inputs of the Benes network to the outputs of the binary switches at stage .The baseline networks belong to th
48、e more general class of banyan networks.,An inverse baseline network is defined as the second half of the Benes network, namely the right part of the network from the inputs of the binary switches at stage to the outputs of the Benes network. Unfortunately, the recursive construction of a strict-sen
49、se non-blocking network does not give a network with fewer than crosspoints, for all N and a fixed constant C.,Let , and factor N into 3 stages, each consisting of switches of size . Let be the crosspoint complexity of the switch.For convenience, we assume there are second stage switches.We have then the recursive relationship:,
