networks under node/link failure constraints. D Saha* and A Mukherjee+. The problem of selecting link capacities for different link failure probabilities in.
Research notes
Design of hierarchical communication networks under node/link failure constraints D Saha* and A Mukherjee+
The problem of selecting link capacities for different link failure probabilities in a hierarchical network is considered. Given a finite set of nodes, links and link failure probabilities, the goal is to obtain the set of link capacities in order to minimize the total design cost. A mathematical model for this problem has been formulated. The set of feasible solutions for this non-linear combinatorial optimization problem has been obtained using Lagrangean relaxation and sub-gradient optimization techniques. The applicability of the proposed formulation has been demonstrated by designing a two-level hierarchical computer communication network, and the corresponding computational results have also been reported. It has been found that the hierarchical design improves the cost efftciency upon the nonhierarchical design by an appreciable factor. Keywords: networks, hierarchical design
node/link
failure,
paper describes a method for designing hierarchical computer networks consisting of links with nonzero failure probabilities to meet the
This
*Department of Computer Science and Engineering, Jadavpur University, Calcutta 700 032, India ‘Department of Electronics and Telecommunications Engineering, Jadavpur University, Calcutta 700 032, India Paper received: 6 July 1993; revised paper received: 15 September I993
0140-3664/95/$09.50 378
computer
0
communications
requirements of link capacities at minimum cost. The topological design of geographically distributed autonomous computers connected by communication links in a computer networklm3, in general, assigns the links as well as their capacities for connecting the network nodes, and has several performance and economic implications. The concept of link failure probability introduced in this paper is also an important design parameter. During its operational phase, every network passes through a number of states when the links are forced to operate in sub-capacity conditions for several reasons, including node and/or link failures (i.e. changes in topology), network congestion, buffer overflow, packet loss or duplication, an inefficient protocol, a bad routing strategy, etc. It would be appropriate and mathematically tractable to represent this kind of degradation in network performance by specifying the individual link reliability yU,” between nodes u and v, which gives us the extent to which a link can be utilized to its specified maximum capacity over a period. Obviously y,, y will have a value in the range [0, 11. The inverse of yU,“, i.e. (1 - yU ,), is termed as link failure probability p,,, “, which also lies in the range [0, 11.
1995Elsevier volume
Science 16 number
B.V. All rights 5 may 1995
reserved
MOTIVATION
FOR
HIERARCHICAL NETWORKS Hierarchical computer communication networks (HCCN) are intuitively appealing when a large number of nodes are to be connected over a large, geographically distributed area. The principal motivations for HCCNs are: For very large systems the number of links needed with a single level network structure may beprohibitively large. come HCCNs exploit the locality that exists in the communication pattern in a multilevel network design to bring about a reduction in the required number of links. HCCNs reduce the degree (i.e. the number of links connected to a node) of the majority of nodes, keeping the degree of the remaining nodes the same as that of the corresponding non-hierarchical network. The hierarchical network reduces the link cost substantially at the expense of only moderately increasing the average internode distance and the average message delay4*5. Thus, HCCNs allow more nodes to be included in the network when constrained by the link cost.
Design of hierarchical
4. The hierarchical clustering structure5-7 is used in the design phase to minimize the computational cost of the design. Such a cost is assumed to have a polynomial growth with the number of clusters in the network. The optimal number of clusters and the optimal number of nodes in each cluster are determined when the same design strategy is considered at all levels. The optimal link capacities are also obtained for each cluster at every level of the hierarchy. The optimization problem is used to achieve these goals.
PREVIOUS
suitable for small to moderate sized networks, but become computationally very hard when directly applied to large networks’, lo. One modilication is to partition the problem into a hierarchy of subproblems that are solved one at a time, known as hierarchical design4, 8316. The formulations developed in the above-referenced papers on hierarchical design have used size and gate constraints to obtain the minimum design cost. However, there is no network design procedure that combines both hierarchy and link failure probability. This paper describes one way in which the link failure probability can be used to design HCCNs.
WORK
In recent years, computer communication networks (CCN) containing hundreds of terminals and many computers, and supporting largescale, on-line applications, have become a vital part of many industrial, governmental, financial and educational institutions. Today’s backbone networks may contain from just a few to thousands of nodes, with many hosts supporting innumerable users. The design of such networks is a complex procedure involving decisions on capacity assignments, computer protocols, buffer allocation, congestion and flow control, terminal selection, message routing, concentrator locations and topology. Previous research on the topological design of centralized CCNs has concentrated on issues regarding the location of switching nodes, the selection of channel capacities and the geographical layout of links and nodes, with a view to satisfy the given traffic and performance requirements under minimum cost ‘*3*8m’1.Much work has been devoted to developing efficient design techniques and operational procedures’,3. Recent methods’2p’5 are more realistic, in the sense that they strategically include the link failure constraints in topological design to enable the provision of maximum link utilization for computer networks during actual operation. Existing procedures are
PROPOSED STRUCTURE
HCCN
The structure of the HCCNs considered in this paper can be described as follows. Let N be the total number of nodes in the network. The location of these nodes are known precisely. They are divided into 6, clusters having nk nodes in cluster k (k = 1 . 8,). It is assumed, for convenience of design, that 61 evenly divides N, and that each cluster is of an identical size, i.e. ni =n2=..=&=..=?$. The nodes within each cluster are linked together by a level- 1 interconnection network. One (or more) node(s) from each cluster is (are) selected to act as the gateway node(s), and these gateway nodes from different clusters are then interconnected to form the level-2 hierarchy. However, for a multilevel hierarchical network, the gateway nodes are again divided into 62 clusters, where the jth cluster will contain nj nodes. The level-2 interconnection network is used to link each of these 82 clusters of level-l gateway nodes. Then again, one (or more) node(s) from each level-2 cluster is (are) selected as level-2 gateway nodes to be linked together by a level-3 network. In this way, the concept of a hierarchy can be extended to an arbitrary number of levels beyond the three-level hierarchy. Interconnection networks used at different levels may have different topologies
computer
communications
networks:
D Saha and A Mukherjee
and connectivities. Furthermore, networks used at the same level may also vary from cluster to cluster. We have, however, considered the optimization of cluster sizes in every level of the hierarchy. It is assumed to be always possible to group nodes together into clusters, with each cluster having a few gateways designated to handle intercluster demands. The gateways can aggregate demands from several nodes together to form a large demand to be routed to another gateway. The clustering is based on such factors as locality of interest and geographic area14. When the complete design of the network has been accomplished, the routing of the messages between inter-cluster nodes will be as follows. The messages are first forwarded to the gateway node which, after identifying the intercluster nature of the messages, routes them to the corresponding peer gateway node. From this node, the messages are distributed to the destination nodes. In the design, we have restricted the HCCNs to two levels. It is further assumed that the hierarchical network is required to have a connectivity of at least two, since connectivity is not a parameter of our design algorithm.
DESIGN
PROCEDURES
The procedure assumes the following input data. The number and locations of nodes (i.e. their distance matrix) are precisely known. The connectivity among the nodes and the reliability of each node are also known. The traffic density requirement for each node is given. Under these circumstances, it is necessary to design a two-level HCCN that satisfies the given traffic characteristics. The nodes are to be partitioned into clusters. In each cluster they are interconnected to form a small network. From each of these clusters, a subset of nodes responsible for inter-cluster communications is selected. These gate nodes, taken from all clusters, form the set of input nodes for the second
volume
18 number
5 may 1995
379
Design of hierarchical
networks:
D Saha and A Mukherjee
level design problem (i.e. the problem set in the next level of hierarchy). The same procedure can be continued further if the hierarchical network structure is to be extended to more than two levels. The input data set for the design problem includes: Number of nodes, N Distance matrix, [L&I,i,j E N Number of levels, H Number of gates per cluster in each level, g 5. Connectivity among nodes 6. Node reliability vector, [y,],
1. 2. 3. 4.
iEN
7. Traffic demand at each node, {c;}, i E N, and 8. Coefficient vector of the design cost, ri, i E N. Considering a two-level hierarchy, the complete design procedure consists of the following steps: Step I Determination of cluster size and number of clusters per level, Step II Cluster formation and intra-cluster link failure probability assignment, Step III Gate node identification, Step IV Inter-cluster link establishment and link reliability calculations, Step V Cluster capacity determination, Step VI Effective cluster capacity determination, Step VII 1st level link capacity assignment, and Step VIII 2nd level link capacity assignment. Step I is basically a minimization problem which has been solved using a subgradient optimization technique. We have developed a novel clustering algorithm that performs steps II and III with the help of input data (l), (2) and (4). Also in step II, the failure probabilities of the links within each cluster are determined using the following formula: failure probability of the link between intra-cluster nodes u and v = max[p,,p,]. In step IV, intercluster links are established by interconnecting the gate nodes identified in step III. Here, connectivity constraint has to be taken into account,
380
computer
communications
and any standard topological design algorithmlm3 like the Kleitman method ‘&I7 can be followed. The link reliability calculations based on input data (6) are done by simple mathematics explained below. In step V, the capacity of each cluster is determined by simply summing up the capacities of all the nodes of that cluster. In step VI, the effective capacity of each cluster is determined by adding the effective capacity of each node, where the effective node capacity is given as the product of node capacity and node reliability. Steps VII and VIII are again optimization problems solved using a subgradient optimization technique. Steps I-IV are discussed in detail below, whereas the mathematical analysis is given in the next section.
Algorithm for generating clusters and selection of gate nodes The clustering algorithm6 partitions a set of geographically distributed nodes over a region. The traffic pattern considered follows an inverse relationship with the distances among nodes. A pull function between two nodes u and v is defined as p(u, v) = l/d(u, v)~, where d(u, v) is the distance between nodes i and j, and m is an integer. A number of 6, root nodes is first selected such that The c;;; 1Ph 4 is minimum. nodes, except the root nodes, are calledfiee nodes. A set of 61 permanent clusters C_perm,and a set of 61 temporary clusters C_temp, are delined, where i E [l,S,]. We include one root node in each permanent cluster C_perm,, i E [l, S,]. Now each cluster C_perm, attracts a free node by a pu116.If the pull corresponds to C_ ternpi (io E [l, S,]) is maximum, then the free node is included in temporary cluster C_temp_ corresponding to that permane; cluster. From each temporary cluster C_temp,O,one node is selected to be included in permanent cluster C- Pe”.io for which the pull correspondmg to that cluster is maximum. All other remaining nodes are again declared free nodes. The procedure is continued until all the
volume
18 number
5 may 1995
free nodes are captured by the permanent cluster8. Once the clusters have been formed, it ‘is necessary to select the gate nodes for each cluster. This is obtained by using an algorithm called the gate selection algorithm5. This algorithm first finds the centre of each cluster. By adding the centres of all clusters, the centre of the network is obtained. Then the distance between the network centre and each node of every cluster is determined. Each node distance of every cluster from the network’s centre is multiplied by the node link failure probability. This is called the weighted distance between each node of the cluster and the network centre. A node in a cluster is selected to be the gate of the cluster if that node is at a minimum weighted distance from the centre of the network. In this way, the required number of gates for all clusters at each level of the hierarchy of the network is selected.
Assignment of failure probabilities to links The effective capacity of a network is defined as the sum of the weighted capacity of each link in the network, where the weighting function is the link reliability. We have already assumed that associated with an edge (u, v) between nodes u and v is a probability p,,” of failure. The inverse of this probability, i.e. (1 - pu,“), is the reliability y,, v of the link. The probability constraint requires that the effective capacity of the network is at least c, which is given as input. The link failure probability is assigned to each link in each cluster in following the manner: P“,” =
max(pM,p[vl), where PM is
the probability of node U. Then the reliability of each cluster is determined by averaging the link failure probabilities of the links of the cluster. This reliability is assigned to each gate node for inter-cluster communication in level 2 when connecting two gate nodes. When two gate nodes are linked together for inter-cluster communication, the
Design of hierarchical
inter-gateway reliability is similarly taken to be the minimum of the connected gate node reliabilities.
MATHEMATICAL FORMULATION The complete design has been organized into three sequential phases discussed below.
Phase II Clustering algorithm An informal description of the algorithm is given in the previous section, and here it is briefly reproduced in pseudo-Pascal syntax: begin while set of free nodes # {a} do Ctemp, + {a}, i E [1,&l for all free node k do for all root node i do
networks:
D Saha and A Mukherjee
tively, where and i E [1,6;] j E [l, H]. The link failure probability pik is assigned to the kth link of the ith cluster at the jth level of the hierarchy. The c{~ is to be determined for each cluster in every level of the hierarchy: Minimize
find pull p(k, i) for cluster Phase I
Subject to
end do
The set of N nodes is partitioned into 6, clusters. The size of each cluster is no, and the interconnection topology for each of these clusters is exactly the same. In the lowest level, a node is a single-node cluster. In the 1st level, N nodes are partitioned into 6, clusters. In the next level, 6, clusters are partitioned into a number of higher level clusters. The number of gate nodes may vary from cluster to cluster, and for each level the number of gate nodes per cluster is given for our problem. Thus, for the zeroth level (where elements are nodes), a gate vector g = (si, . . , a, 1, where gk rep= sents the number of gates in cluster k (gk d no,k = 1,611. The cluster size optimization problem is proposed7 as:
Subject to 61
and & d
nk
L(A) = G, + 2, subject to the constraint equation (2) where 2, is a Lagrangean multiplier. This problem is easily solved by a subgradient optimization procedure 9.1619
=
cc j
‘:
i
(3)
end do
assign the node from temporary cluster into a permanent cluster such that the average pull is maximum for this particular node. end do end do.
Gate selection algorithm The formal description of this algorithms is stated below. Given pi + ith node failure probability, i E [I,&]:
(2)
We obtain the Lagrangean relaxation of the problem as:
‘t
i
find average pull on a node
assign g: + n:, i E [l, S,], r E [l, IS,]], where gt is the rth gate node of the ith cluster.
i= I
.j
k of a temporary cluster by a node i of cluster Ci.
i E [l~.&l,j~ ;&,I.
n; = N
c c ck
end do do Vk do Vi
find mi, centre of ith cluster, i E [1,&l; find centre of the network M+Cimi,iE[1,6i]; find d(M, n’) +- d(M, ni) (1 - p,) find d(M, n:) = min d(M, nj),
Minimize
c
assign the free node k to a temporary cluster such that the average pull is maximum for this free node.
Phase III In this phase, H is considered as a hierarchical level and 6i is the number of clusters in level j, j E H, and Mai is the number of links in the 6, cluster in level H of the hierarchy”. The total capacity and effective capacity under link failure of the ith cluster at the jth level of the hierarchy are C{ and ??{, respec-
computer
communications
Here, the Lagrangean relaxations of these constraint equations are solved using a subgradient optimization technique’, ‘&19. Details can be found elsewhere17.
RESULTS The results obtained from the simulation are presented here to illustrate the usefulness of this method. As most of this discussion is oriented towards methods rather than results, this section is brief. The results presented are obtained using a model of a relatively small network of 22 nodes, shown in Figure I. The results for this network are provided in the Appendix. A two-connected topology is constructed, which is engineered using two methods: non-hierarchical and hierarchica12’. The total design cost (G = Gi + Gz) for each case is shown in Table 1. Table 1 Topology cost factor Design
cost
Two connected (non-hierarchical) Two connected (hierarchical)
volume
18 number
2.77071e09 15.67971e08
5 may 1995
381
Design of hierarchical
networks:
D Saha and A Mukherjee
bottleneck role played by the gateway nodes. However, the design cost reduction suggested above appears to economically alleviate the problem of congestion on intergateway links.
REFERENCES Figure1
Results
X-axis
I It can be seen that the cost of the non-hierarchical design is almost twice that of the hierarchical design when the total cost is considered.
that the performance of the H algorithm as compared with the NH algorithm monotonically improves with an increase in N (number of nodes)*‘.
2
3
4
HIERARCHICAL HIERARCHICAL
VS. NONCONCLUSION
To evaluate the performance of the hierarchical (H) algorithm versus the non-hierarchical (NH) algorithm, the percentage of cost improvement that can be achieved by the H algorithm normalized to the cost that can be achieved by the NH algorithm is calculated as follows: A = ((Co~t~u - COS~H) /COSqq”) 100% For the example of Figure found that A M 50%, i.e. algorithm performs much than the NH. Further data
1, it is
the H better shows
5
This paper has shown the use of hierarchical networks as a framework for developing network topologies under link failure constraints. Some example networks have been considered under the assumption of nonzero link failure probabilities, and evidence has been presented suggesting the suitability of the hierarchical networks. The disadvantages of HCCNs include the potentially high traffic rate on the inter-gateway links, resulting in a degradation in the network performance and diminished fault-tolerante of the network due to the
6
7
8
9
APPENDIX: CLUSTER-WISE LINK CAPACITIES HIERARCHICAL NETWORK For clusterI (level-l)
c[l] c[2] c[3] c[4] c[5] c[6]
= = = = = =
8854.074997 9107.349997 9445.049997 3535.299997 4210.699997 4886.099997
For cluster2 (level-l) c[l]=8546.124999
c[2) = c[3] = c[4] = c[5] = c[6] = c[7] =
8969.999999 9393.874999 9648.199999 3374.849999 4053.049999 473 I .249999
For cluster3 (level-l) c[I]= 9103.499973
c[2] = 9543.124974 c[3] = 3827.999973 c[4] = 4883.099973
382
computer
p[l] p[2] p[3] p[4] p[5] p[6]
= = = = = =
0.890000 0.920000 0.960000 0.260000 0.340000 0.420000
p[l] p[2] p[3] p[4] p[5] p[6] p[7]
= = = = = = =
0.850000 0.900000 0.950000 0.980000 0.240000 0.320000 0.400000
p[l] p[2] p[3] p[4]
= = = =
0.900000 0.950000 0.300000 0.420000
For c[l] c[2] c[3] c[4] c[5]
IN THE 10
cluster4 (level-l)
= = = = =
9008.124982 9368.024982 9637.949982 9817.899982 3879.549981
p[l] p[2] p[3] p[4] p[5]
2nd levelclusters (level-2) p[l]= c[l]=6303.784995
c[2] c[3] c[4] c[5] c[6] c[7] c[8]
= = = = = = =
6213.942495 6213.942495 6213.942495 6213.942495 6426.297495 6426.297495 6426.297495
p[2] p[3] p[4] p[5] p[6] p[7] p[8]
= = = = =
0.850000 0.890000 0.920000 0.940000 0.280000
11
12 0.642000
= 0.63 1000 = 0.631000 = 0.631000 =0.631000 = 0.657000 = 0.657000 = 0.657000
13
14
15
communications
volume
18 number
5 may 1995
Bertseks, D and Gallagar, R Data Networks, Prentice Hall, Englewood Cliffs, NJ (1987) Tanenbaum, A S Computer Networks, Prentice Hall, Englewood Cliffs, NJ (1988) Chou, W (ed) Computer Communication, Prentice Hall, Englewood Cliffs, NJ (1985) Dandumudi, S P and Eager, D L ‘Hierarchical interconnection networks for multicomputer systems’, IEEE Trans. Cornput., Vol 39 No 6 (1990) pp 786797 Mukherjee, A, Bhaumik, B B and Bandyopadhyay, A K Gate selection algorithms in hierarchical networks, Technical Report JU/ETCE/AM/92/1, Jadavpur University, India (1992) Mukherjee, A, Bhaumik, B B and Bandyopadhyay, A K ‘An energy function and its application to a new clustering algorithm’, Proc. IEEE Conf. (TENCON) Compur. Commun. Automation, Beijing, China (1993) Mukherjee, A, Bhaumik, B B and Bandyopadhyay, A K ‘Hierarchical clustering technique for the minimum cost design of natural clustered computer communication network’, Proc. IEEE Conf (TENCON) Comput., Commun. Automation, Beijing, China (1993) Kleinrock, L and Kamoun, F ‘Optimal clustering structures for hierarchical topological design of large computer networks’, Nefworks, Vol 10 (1980) pp 221248 Gavish, B and Hantler, S L ‘An algorithm for optimal route selection in SNA networks’, IEEE Trans. Commun., Vol 3 1 No IO (1983) pp 1151-1160 Gerla, M and Kleinrock, L ‘On the topological design of distributed computer networks’, IEEE Trans. Commun., Vol 25 No 1 (January 1977) Gavish, B ‘Topological design of centralised computer Networks-Formulations and aleorithms’. Networks, Vol 2 (1982) pp 3557-377 Cardwell, R H, Monma, C L and Wu, T H ‘Computer aided design procedures for survival fibre optic networks’, IEEE J. Selected Areas in Commun., Vol 7 No 8 (October 1989) pp 1188-l 197 Rahaman, M and Latif, M ‘Topological design of a distributed computer communication network in a noisy environment’, Comput. Commun., Vol 12 No 16 (December 1989) pp 331-336 Gavish, B et al. ‘Fibre optic circuit network design under reliability constraints’, IEEE J. Selected Areas in Commun., Vol 7 No 8 (October 1989) pp 1181-1187 Saha, D, Mukherjee, A and Dutta, S K
Design of hierarchical ‘Design of computer communication networks under link reliability constraints’, Proc. IEEE Tencon, Beijing, China (October 1993) 16 Held, M, Wolfe, P and Crowder, H ‘Validation of subgradient optimization’, Math Program., Vol 6 (1974) pp 62-88 I7 Poljack, B T ‘A general method of
solving
extremum
problem’,
I8 Held, M and Karp, R M ‘The travelling salesman problem and minimum spanning trees: Part II’, Math. Program., Vol I (1971) pp 6-25 I9 Geoffrion, A M ‘Lagrangean relaxation and its uses in integer programming’,
computer
communications
D Saha and A Mukherjee
Math. Program Study, Vol 2 (1974) pp
Soviet
Math. Doklady, Vol 8 (1967) pp 5933597
networks:
8221 I4 20 Saha, D, Mukherjee, A and Dutta, S K Hierarchical design of large computer communication networks, Technical Re-
port JU/CSE/AM/93/DS-2, Department of CSE, Jadavpur University, India (April 1993)
volume
18 number
5 may 1995
383
