Chapter 6 ATM Switching with Minimum-Depth Blocking Networks Architectures and performance of interconnection networks for ATM switching based on the adoption of banyan networks are described in this chapter. The interconnection networks pre- sented now have the common feature of a minimum depth routing network, that is the path(s) from each inlet to every outlet crosses the minimum number of routing stages required to guarantee full accessibility in the interconnection network and to exploit the self-routing property. According to our usual notations this number n is given by for a net- work built out of switching elements. Note that a packet can cross more than n stages where switching takes place, when distribution stages are adopted between the switch inlets and the n routing stages. Nevertheless, in all these structures the switching result per- formed in any of these additional stages does not affect in any way the self-routing operation taking place in the last n stages of the interconnection network. These structures are inherently blocking as each interstage link is shared by several I/O paths. Thus packet loss takes place if more than one packet requires the same outlet of the switching element (SE), unless a proper storage capability is provided in the SE itself. Unbuffered banyan networks are the simplest self-routing structure we can imagine. Nev- ertheless, they offer a poor traffic performance. Several approaches can be considered to improve the performance of banyan-based interconnection networks: 1. Replicating a banyan network into a set of parallel networks in order to divide the offered load among the networks; 2. Providing a certain multiplicity of interstage links, so as to allow several packets to share the interstage connection; 3. Providing each SE with internal buffers, which can be associated either with the SE inlets or to the SE outlets or can be shared by all the SE inlets and outlets; 4. Defining handshake protocols between adjacent SEs in order to avoid packet loss in a buff- ered SE; nN b log= NN× bb× This document was created with FrameMaker 4.0.4 ban_mindep Page 167 Monday, November 10, 1997 8:22 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) 168 ATM Switching with Minimum-Depth Blocking Networks 5. Providing external queueing when replicating unbuffered banyan networks, so that multi- ple packets addressing the same destination can be concurrently switched with success. Section 6.1 describes the performance of the unbuffered banyan networks and describes networks designed according to criteria 1 and 2; therefore networks built of a single banyan plane or parallel banyan planes are studied. Criteria 3 and 4 are exploited in Section 6.2, which provides a thorough discussion of banyan architectures suitable to ATM switching in which each switching element is provided with an internal queueing capability. Section 6.3 discusses how a set of internally unbuffered networks can be used for ATM switching if queueing is available at switch outlets with an optional queueing capacity associated with network inlets according to criterion 5. Some final remarks concerning the switch performance under offered traffic patterns other than random and other architectures of ATM switches based on minimum-depth routing networks are finally given in Section 6.4. 6.1. Unbuffered Networks The class of unbuffered networks is described now so as to provide the background necessary for a satisfactory understanding of the ATM switching architectures to be investigated in the next sections. The structure of the basic banyan network and its traffic performance are first discussed in relation to the behavior of the crossbar network. Then improved structures using the banyan network as the basic building block are examined: multiple banyan planes and mul- tiple interstage links are considered. 6.1.1. Crossbar and basic banyan networks The terminology and basic concepts of crossbar and banyan networks are here recalled and the corresponding traffic performance parameters are evaluated. 6.1.1.1. Basic structures In principle, we would like any interconnection network (IN) to provide an optimum perfor- mance, that is maximum throughput and minimum packet loss probability . Packets are lost in general for two different reasons in unbuffered networks: conflicts for an internal IN resource, or internal conflicts , and conflicts for the same IN outlet, or external conflicts . The loss due to external conflicts is independent of the particular network structure and is unavoidable in an unbuffered network. Thus, the “ideal” unbuffered structure is the crossbar network (see Section 2.1) that is free from internal conflicts since each of the crosspoints is dedicated to each specific I/O couple. An banyan network built out of SEs includes n stages of SEs in which . An example of a banyan network with Baseline topology and size is given in Figure 6.1a for and in Figure 6.1b for . As already explained in Section 2.3.1, internal conflicts can occur in banyan networks due to the link commonality of different I/O paths. Therefore the crossbar network can provide an upper bound on through- ρπ N 2 NN× bb× Nb⁄ nN b log= N 16= b 2= b 4= ban_mindep Page 168 Monday, November 10, 1997 8:22 pm Unbuffered Networks 169 put and loss performance of unbuffered networks and in particular of unbuffered banyan networks. 6.1.1.2. Performance In an crossbar network with random load, a specific output is idle in a slot when no packets are addressed to that port, which occurs with probability , so that the network throughput is immediately given by (6.1) Once the switch throughput is known, the packet loss probability is simply obtained as Thus, for an asymptotically large switch , the throughput is with a switch capacity given by . Owing to the random traffic assumption and to their single I/O path feature, banyan net- works with different topologies are all characterized by the same performance. The traffic performance of unbuffered banyan networks was initially studied by Patel [Pat81], who expressed the throughput as a quadratic recurrence relation. An asymptotic solution was then provided for this relation by Kruskal and Snir. [Kru83]. A closer bound of the banyan network throughput was found by Kumar and Jump. [Kum86], who also give the analysis of replicated Figure 6.1. Example of banyan networks with Baseline topology 1234 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 12 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 (a) (b) NN× 1 pN⁄–() N ρ 11 p N – N –= π 1 ρ p – 1 11 p N – N – p –== N ∞→() 1 e p– – p 1.0=() ρ max 0.632= ban_mindep Page 169 Monday, November 10, 1997 8:22 pm 170 ATM Switching with Minimum-Depth Blocking Networks and dilated banyan networks to be described next. Further extensions of these results are reported by Szymanski and Hamacker. [Szy87]. The analysis given here, which summarizes the main results provided in these papers, relies on a simplifying assumption, that is the statistical independence of the events of packet arrivals at SEs of different stages. Such a hypothesis means overestimating the offered load stage by stage, especially for high loads [Yoo90]. The throughput and loss performance of the basic unbuffered banyan network, which thus includes n stages of SEs, can be evaluated by recursive analysis of the load on adjacent stages of the network. Let indicate the probability that a generic outlet of an SE in stage i is “busy”, that is transmits a packet ( denotes the external load offered to the network). Since the probability that a packet is addressed to a given SE outlet is , we can easily write (6.2) Thus, throughput and loss are given by Figure 6.2. Switch capacity of a banyan network b n b n × bb× p i i 1 … n,,=() p 0 1 b⁄ p 0 p= p i 11 p i 1– b – b –= i 1 … n,,=() ρ p n = π 1 p n p 0 –= 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1 10 100 1000 10000 p=1.0 Maximum throughput, ρ max Switch size, N Crossbar b=8 b=4 b=2 ban_mindep Page 170 Monday, November 10, 1997 8:22 pm Unbuffered Networks 171 The switch capacity, , of a banyan network (Equation 6.2) with different sizes b of the basic switching element is compared in Figure 6.2 with that provided by a crossbar network (Equation 6.1) of the same size. The maximum throughput of the banyan network decreases as the switch size grows, since there are more packet conflicts due to the larger number of net- work stages. For a given switch size a better performance is given by a banyan network with a larger SE: apparently as the basic SE grows, less stages are needed to build a banyan net- work with a given size N . An asymptotic estimate of the banyan network throughput is computed in [Kru83] which provides an upper bound of the real network throughput and whose accuracy is larger for moderate loads and large networks. Figure 6.3 shows the accuracy of this simple bound for a banyan network loaded by three different traffic levels. The bound overestimates the real net- work throughput and the accuracy increases as the offered load p is lowered roughly independently of the switch size. It is also interesting to express π as a function of the loss probability occurring in the single stages. Since packets can be lost in general at any stage due to conflicts for the same SE outlet, it follows that Figure 6.3. Switch capacity of a banyan network ρ max bb× ρ 2b b 1–()n 2b p + ≅ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 100 1000 10000 b=2 Crossbar Analysis Bound Network throughput Switch size, N p=0.5 p=0.75 p=1.0 π i 1 p i p i 1– ⁄–= i 1 … n,,=() π 11π i –() i 1= n ∏ –= ban_mindep Page 171 Monday, November 10, 1997 8:22 pm 172 ATM Switching with Minimum-Depth Blocking Networks or equivalently by applying the theorem of total probability Therefore the loss probability can be expressed as a function of the link load stage by stage as (6.3) For the case of the stage load given by Equation 6.2 assumes an expression that is worth discussion, that is (6.4) Equation 6.4 says that the probability of a busy link in stage i is given by the probability of a busy link in the previous stage decreased by the probability that both the SE inlets are receiving a packet ( ) and both packets address the same SE outlet . So, the loss probability with SEs given by Equation 6.3 becomes (6.5) 6.1.2. Enhanced banyan networks Interconnection networks based on the use of banyan networks are now introduced and their traffic performance is evaluated. 6.1.2.1. Structures Improved structures of banyan interconnection networks were proposed [Kum86] whose basic idea is to have multiple internal paths per inlet/outlet pair. These structures either adopt multi- ple banyan networks in parallel or replace the interstage links by multiple parallel links. An interconnection network can be built using K parallel networks (planes) interconnected to a set of N splitters and a set of N combiners through suitable input and output interconnection patterns, respectively, as shown in Figure 6.4. These structures are referred to as replicated banyan networks (RBN), as the topology in each plane is banyan or derivable from a banyan structure. The splitters can distribute the incoming traffic in different modes to the banyan networks; the main techniques are: • random loading (RL), • multiple loading (ML), • selective loading (SL). ππ 1 π i 1 π h –() h 1= i 1– ∏ i 2= n ∑ += ππ 1 π i 1 π h –() h 1= i 1– ∏ i 2= n ∑ + 1 p 1 p 0 – 1 p i p i 1– – p h p h 1– h 1= i 1– ∏ i 2= n ∑ + p i 1– p i – p 0 i 1= n ∑ == = b 2= p i 11 p i 1– 2 – 2 – p i 1– 1 4 p i 1– 2 –== i 1 … n,,=() i 1– p i 1– 2 14⁄() 22× π p i 1– p i – p 0 i 1= n ∑ 1 4 p i 1– 2 p 0 i 1= n ∑ == NN× NN× 1 K× K 1× ban_mindep Page 172 Monday, November 10, 1997 8:22 pm Unbuffered Networks 173 RBNs with random and multiple loading are characterized by full banyan networks, the same input and output interconnection patterns, and different operations of the splitters, whereas selective loading uses “truncated” banyan networks and two different types of inter- connection pattern. In all these cases each combiner that receives more than one packet in a slot discards all but one of these packets. A replicated banyan network operating with RL or ML is represented in Figure 6.5: both interconnection patterns are of the EGS type (see Section 2.1). With random loading each splitter transmits the received packet to a randomly chosen plane out of the planes with even probability . The aim is to reduce the load per banyan network so as to increase the probability that conflicts between packets for interstage links do not occur. Each received packet is broadcast concurrently to all the planes with multiple loading. The purpose is to increase the probability that at least one copy of the packet successfully reaches its destination. Selective loading is based on dividing the outlets into disjoint subsets and dedicat- ing each banyan network suitably truncated to one of these sets. Therefore one EGS pattern of size connects the splitters to the banyan networks, whereas suitable patterns (one per banyan network) of size N must be used to guarantee full access to all the combiners from every banyan inlet. The splitters selectively load the planes with the traffic addressing their respective outlets. In order to guarantee full connectivity in the interconnection network, if each banyan network includes stages , the splitters transmit each packet to Figure 6.4. Replicated Banyan Network N-1 1 0 1xKNxNKx1 #0 #1 #(K-1) N-1 1 0 Banyan networks Interconnection pattern Interconnection pattern KK r = 1 K r ⁄ KK m = KK s = NK s K s K s nk– kK bs log=() ban_mindep Page 173 Monday, November 10, 1997 8:22 pm 174 ATM Switching with Minimum-Depth Blocking Networks the proper plane using the first k digits (in base b) of the routing tag. The example in Figure 6.6 refers to the case of , and in which the truncated banyan network has the reverse Baseline topology with the last stage removed. Note that the connec- tion between each banyan network and its combiners is a perfect shuffle (or EGS) pattern. The target of this technique is to reduce the number of packet conflicts by jointly reducing the offered load per plane and the number of conflict opportunities. Providing multiple paths per I/O port, and hence reducing the packet loss due to conflicts for interstage links, can also be achieved by adopting a multiplicity of physical links for each “logical” interstage link of a banyan network (see Figure 4.10 for , and ). Now up to packets can be concurrently exchanged between two SEs in adjacent stages. These networks are referred to as dilated banyan networks (DBN). Such a solution makes the SE, whose physical size is now , much more complex than the basic SE. In order to drop all but one of the packets received by the last stage SEs and addressing a specific output, combiners can be used that concentrate the physical links of a logical outlet at stage n onto one interconnection network output. However, unlike replicated networks, this concentration function could be also performed directly by each SE in the last stage. Figure 6.5. RBN with random or multiple loading N-1 1 0 1xKNxNKx1 #0 #1 #(K-1) N-1 1 0 Banyan networks N 16= b 2= K s 2= KK d = K d 2≥() N 16= b 2= K d 2= K d 2K d 2K d × 22× K d 1× K d ban_mindep Page 174 Monday, November 10, 1997 8:22 pm Unbuffered Networks 175 6.1.2.2. Performance Analysis of replicated and dilated banyan networks follows directly from the analysis of a single banyan network. Operating a random loading of the K planes means evenly partitioning the offered load into K flows. The above recursive analysis can be applied again considering that the offered load per plane is now Throughput and loss in this case are (6.6) (6.7) For multiple loading it is difficult to provide simple expressions for throughput and delay. However, based on the results given in [Kum86], its performance is substantially the same as the random loading. This fact can be explained considering that replicating a packet on all Figure 6.6. Example of RBN with selective loading p 0 p K = ρ 11p n –() K –= π 1 ρ p – 1 11p n –() K – p –== ban_mindep Page 175 Monday, November 10, 1997 8:22 pm 176 ATM Switching with Minimum-Depth Blocking Networks planes increases the probability that at least one copy reaches the addressed output, as the choice for packet discarding is random in each plane. This advantage is compensated by the drawback of a higher load in each plane, which implies an increased number of collision (and loss) events. With selective loading, packet loss events occur only in stages of each plane and the offered load per plane is still . The packet loss probability is again given by with the switch throughput provided by since each combiner can receive up to K packets from the plane it is attached to. In dilated networks each SE has size , but not all physical links are active, that is enabled to receive packets. SEs have 1 active inlet and b active outlets per logical port at stage 1, b active inlets and active outlets at stage 2, K active inlets and K active outlets from stage k onwards . The same recursive load computation as described for the basic ban- yan network can be adopted here taking into account that each SE has bK physical inlets and b logical outlets, and that not all the physical SE inlets are active in stages 1 through . The event of m packets transmitted on a tagged link of an SE in stage i , whose proba- bility is , occurs when packets are received by the SE from its b upstream SEs and m of these packets address the tagged logical outlet. If denotes the probability that m packets are received on a tagged inlet an SE in stage 1, we can write The packet loss probability is given as usual by with the throughput provided by The switch capacity, , of different configurations of banyan networks is shown in Figure 6.7 in comparison with the crossbar network capacity. RBNs with random and selec- tive loading have been considered with and , respectively. A dilated banyan network with link dilation factors has also been studied. RBN with ran- dom and selective loading give a comparable throughput performance, the latter behaving a little better. A dilated banyan network with dilation factor behaves much better than an RBN network with replication factor . The dilated banyan network with nk– p 0 K⁄π1 ρ p⁄–= ρ 11p nk– –() K –= bK bK× b 2 kK b log=() k 1– 1 in≤≤() p i m() j m≥ p 0 m() p 0 m() 1 p – m 0= pm1= 0 m 1> = p i m() j m 1 b m 1 1 b – jm– p i 1– m 1 ()…p i 1– m b () m 1 … m b ++ j= ∑ jm= bK ∑ mK<() j h hm= j ∑ 1 b h 1 1 b – jh– p i 1– m 1 ()…p i 1– m b () m 1 … m b ++ j= ∑ jm= bK ∑ mK=() = π 1 ρ p⁄–= ρ 1 p n 0()–= ρ max K r 24,= K s 24,= K d 24,= K d KK d = K d 4= ban_mindep Page 176 Monday, November 10, 1997 8:22 pm . between adjacent SEs in order to avoid packet loss in a buff- ered SE; nN b log= NN× bb× This document was created with FrameMaker 4.0.4 ban_mindep Page 167 Monday, November 10, 1997 8:22