Genetic-algorithm-based reliability optimization for computer network expansion

IEEE TRANSACTIONS ON RELIABILITY, VOL. 44, NO. 1, 1995 MARCH 63

Genetic-Algorithm-Based Reliability Optimization for Computer Network Expansion

Anup Kumar, Member IEEE

Rakesh M. Pathak

Yash P. Gupta

University of Louisville, Louisville

Network Equipment Technologies, Redwood City

University of Colorado, Denver

Key Words - Network design, Computer network reliability, Genetic algorithm, Diameter

Reader Aids - General purpose: Describe a new network-expansion approach Special math needed for derivations: Probability Special math needed to use the results: Same Result useful to: System designers

Summary & Conclusions - This paper explains the development & implementation of new methodology for expanding existing computer networks. Expansion is achieved by adding new communication Links and computer nodes such that reliability measures of the network are optimized within specified constraints. A Genetic-Algorithm-Based Computer-Network Expansion Methodology (GANE) is developed to optimize a specified objective function (reliability measure) under a given set of network constraints. This technique is very powerful because the same approach can be extended to solve different types of problems; the only modification required is the objective function evaluation module. The versatility of the genetic algorithm is illustrated by applying it to solve various network expansion problems (optimize diameter, average distance and computer network reliability for network expansion). The results are compared with the optimal solutions computed using exhaustive search of complete solution space. The results demonstrate that GANE is very effective (in both accuracy and computation-time) and applies to a wide range of problems, but it does not guarantee the optimal results for every problem.

1. INTRODUCTION'

The advent of low cost computing devices has led to ex- plosive growth in computer networks. There are several benefits from this dispersal of computing across a network, eg, resource sharing and improved reliability. One of the major advantages of computer networks over the centralized systems is their potential for improved system reliability. The reliability of a system depends not only on the reliability of its nodes and communication links, but also on how nodes are connected by communication links. A completely connected network has the highest computer network reliability [ 11 while the simple loop network has the lowest computer network reliability.

'Acronyms, nomenclature, and notation are given at the end of the Introduction.

The topology of a network can be represented by a linear graph. These network topologies can be characterized by their network reliability, message-delay , or network-capacity . These performance characteristics depend on many properties of linear graphs which represent the network topology [2-51: diameter, average distance, number of ports at each node (degree of a node), and number of links. These properties are defined in section 2.3. Diameter directly relates to the delay in the network [5]: the higher the diameter, the larger the delay in the network. Diameter, average distance, and number of links directly im- pact the system reliability [3,5]. Reliability increases with the decrease in diameter and average distance, while reliability decreases with a decrease in the number of links.

Network expansion deals with an ever growing need for more computing across a network. In order to meet this demand, the size of the network is incrementally expanded according to the user requirements [6]. In such an environment, it is critical that the new nodes & links are added in a prudent manner to maximize the performance and reliability characteristics of the expanded network. A generalized framework is developed for expanding existing networks. The objective of this paper is to add new communication links and computer nodes to an existing network such that the cost factor(s) (representing the reliability measures of a network) are minimized/maximized and all the specified constraints are satisfied.

Performance-oriented cost functions have dominated the literature [3-81 for network design & expansion but there are few algorithms for optimizing reliability of the network. The algorithms in [9-121 provide a good starting point for designing reliable network topologies. In most network-design problems, the reliability & availability measures are used as constraints but not as the objective functions for optimization. Such algorithms deal with the problem of overall network design and do not apply directly to incremental network-expansion. In addition, the existing schemes do not optimize reliability [13] but instead try to satisfy the specified reliability bounds.

This paper addresses various types of problems for network expansion. These problems deal with the optimization of diameter, average distance, and computer network reliability during the expansion of an existing network. In each of these problems a set of given nodes is added to an existing network such that one of the these parameters is optimized. A Computer Network Expansion Methodology using Genetic Algorithm based optimization (GANE) has been developed that can be applied to solve all these problems. The GANE results are compared with the optimal solutions computed using exhaustive search. To demonstrate the generality of GANE, the network expansion problem is solved for optimizing diameter and average distance in addition to CNR optimization. First the results of diameter and average distance optimization are discussed and then the results of CNR optimization are provided.

0018-9529/95/$4.00 01995 IEEE

64 IEEE TRANSACTIONS ON RELIABILITY, VOL. 44, NO. 1, 1995 MARCH

Regarding future research, GANE can be extended for multiple criteria optimization, file allocation problems, and network topology design. The potential of GANE in distributed environment should be explored.

Section 2 defines the network expansion problem. Section 3 give the background of Genetic Algorithms. GANE is discussed in section 4. Section 5 provides the results of the problems from section 2.

Acronyms’

CNR computer-network reliability GA genetic algorithm GANE Genetic Algorithm for Network Expansion ST spanning tree.

Notation

N g G ( i ) generation i

total number of nodes in the computer network total number of strings in a generation

Node degree = number of links incident on the node. ST (of a network represented by graph G) = a connected subgraph of G that contains all the nodes of G. CNR 3 Pr {each node communicates with all the other nodes in the network} [14]. In order to compute CNR, the ST of a network are computed and then these ST are made disjoint using an algorithm in [14]. Any ST connects all the nodes of G with ( N - 1) links; hence it represents the minimum interconnection required for providing communication between the nodes. 4

This paper defines & solves the following network expansion problems for optimizing the computer network reliability, diameter, and average distance.

2.3 Problem 1 : Minimize Diameter, Under Degree Constraint

Given: Number of nodes, bounds on degree of each node (deg;), connectivity of the existing network

Objective function: Minimize diameter

Dij deg, degree of node i

minimum number of hops between nodes i & j max(D+. i = 1 to N; j = 1 to N )

9(T) AL, 4(a link exists between nodes & Design Variable: Communication-link addition for connecting

indicator function: 9(True)= 1, S(Fa1se) =O

AND 1 or 2 of the nodes are new) new nodes in an existing network

OLi

Cij TC ANi ON, AD average distance I

Other, standard notation is given in “Information for Readers & Authors” at the rear of each issue.

9(a communication link exists between nodes i & j AND 0 nodes are new) cost of communication between nodes i & j total cost of adding new nodes to an existing network new node i is added to the system old node i is already in the system

implies “or” when used in the text.

2. PROBLEM DEVELOPMENT

2.1 Assumptions

i. A network topology is modeled by an undirected graph G, where nodes represent the processing elements, and edges represent the communication links.

ii. A graph G does not have any self loops. iii. Failure of any link(s) are mutually s-independent. iv. A link has 2 states: operational or faulty. v. The network is s-coherent.

2.2 Definitions

Constraints:

2.4 Problem 2: Minimize Average Distance, Under Degree Constraint

Given: Number of nodes, bounds on degree of each node (deg,) , connectivity of the existing network Objective function: Minimize AD. Design Variable: Communication-link addition for connecting new nodes in an existing network

Constraints:

Cyel Lj,j 5 degi; 1 5 i 5 N

2.5 Problem 3: Maximize CNR, Under Diameter & Degree Constraints

Given: Number of nodes, link reliability (failure probability) values, bounds on the network diameter (D) , connectivity of the existing network Objective Function: Maximize computer network reliability Design Variable: Communication-link addition for connecting new nodes in an existing network

Constraints. diameter = max{DiJ: i = 1,2 ,..., N; j = 1,2 ,..., N } . AD E [E;”;’ CiN,i+1Dj,j]I[%N* ( N - l)].

Lij I deg;; 1 I i I N

’The singular & plural of an acronym are always spelled the same. max(Di,j: = to N; j to N , I

KUMAR ET AL: GENETIC-ALGORITHM-BASED RELIABILITY OPTIMIZATION 65

2.6 Problem 4: Maximize CNR, Under Cost Constraint

Given: Number of nodes, total allowable cost of adding the new nodes, cost of adding new links from a new node to every other node Objective Function: Maximize the CNR Design Variable: Communication link addition for connecting new nodes in an existing network Constraints: Total cost of adding various links to attach a given set of nodes I Total Allowable Cost

3. GENETIC-ALGORITHM BASED OPTIMIZATION

The concept of GA [15] was developed by John Holland. GA are search techniques for global optimization in a complex search space. As the name suggests, GA employ the concepts of “natural selection” and genetics. Using past information GA directs the search with anticipated improved performance. The traditional methods of optimization & search do not fare well over a broad spectrum of problem domains 1161. Some are limited in their scope because they use local search techniques (eg calculus based methods). Others, such as enumerative schemes, are not efficient when the practical search space is too large. GA can be applied to search a large, multimodal. complex problem spaces [15,22]. Thus, there is a good potential to obtain optimal and near optimal results using GA for network expansion problem. However, GA like other search algorithms [ 181 does not guarantee the optimal solution for the problem. Due to the random nature of GA, some problem-types (called GA deceptive [16]) might not be solved very effectively,

3.1 Concept of Genetic Algorithm

In GA, the search space is composed of all possible solutions to the problem. In one of the most commonly used representations, a solution in the search space is represented by a sequence of 0’s and 1’s. This solution string is the chromosome. Each chromosome has an associated objective function value, jitness value. A good chromosome is one that has a highllow fitness value depending upon the type of problem (maximizationhinimization). Fitness value indicates which chromosomes have a better potential of being carried to the next generation. A set of chromosomes and associated fitness values is the population. This population at a given stage of GA is a generation.

General GA Begin with initial set of chromosomes in generation G(0) Set i = O

repeat { Select good chromosomes from G(i) to be carried to G(i t 1) repeat {

From the pool of selected chromosomes { randomly select two chromosomes shuffle them to generate new offsprings compute the fitness value of new offsprings 1

add these offsprings to G(i t 1) } Until (enough offsprings are in generation G(i t 1)

i = i t 1 1

Until (Termination Condition is Satisfied)

The 3 main steps in the repeat loop for GA are:

1. Reproduction/selection: The process of selecting potentially good strings from the current generation to be carried to the next generation.

2. Crossover: The process of shuffling two randomly selected string to generate new offsprings. Mutation com- plements one or more bits of a chromosome to generate a new offspring.

3. Computation of fitness value using objective function. The population size is finite in each generation of GA,

which implies that only relatively fit chromosomes in generation i are carried to the next generation ( i+ l). The process of selection, crossover, and mutation is repeated till the termination condition is satisfied. Sections 3.2 - 3.4 explain these three basic genetic operators in detail [16].

3.2 Selection/Reproduction3

Since population size in each generation is fixed, only a finite number of relatively good chromosomes can be copied in the mating pool, depending on their fitness values. Chromosomes with higher fitness values tend to contribute more copies to the mating pool than those with lower fitness values. This is achieved by assigning a proportionately higher probability of getting copied to a chromosome with higher fitness value [16, 171. It uses a “biased” roulette wheel [14] for selection of chromosomes to be taken in the mating pool. Each chromosome i in the current generation is allotted a roulette- wheel slot - sized in proportion (pi,.) to its fitness value.

piJ = 0fiJ/Sumj

Notation

Ofij actual fitness value of a chromosome i in generation j (of g chromosomes)

Sumj Cf,, Of,,,: total of fitness values of all the chromosomes in generation j .

This proportionate allocation (biased according to fitness) of roulette-wheel slot size gives higher probability of selection to better strings in the mating pool. When the roulette wheel is spun, there is greater chance that a relatively good chromosome is copied into the mating pool because a good chromosome occupies a larger area on the roulette wheel. If the objective function is to be minimized, the winnowing operation [23] is performed on the objective function.

3This discussion presumes a muximization problem. I f it is a minimization problem, then interchange the words higher & lower.


3.3 Crossover

Crossover involves 2 steps:

1. From the mating pool, two chromosomes are selected at random for mating.

2. A crossover site c is selected uniformly at random in the interval [ 1, L - 11, where L is the length of the chromosome.

Two new chromosomes, oflsprings, are then obtained by swapping all characters between positions c & L. This can be shown using two chromosomes, say A and B, each of length 6 bits.

A:101 I101

MR, CR [mutation, crossover] rate PS population size NG number of generators.

4.1 Chromosomal-Coding Scheme

Since our problem involves the representation of connections between nodes, we use a coding scheme with binary numbers. The complete chromosome of network connectivity is divided into node fields which are equal to the number of nodes in the expanded network (after adding the new nodes to the existing network). One arrangement of connections for a 3-node network where one new node (AN3) is added to an existing 2-node network whose 2 nodes are (ON2, ON1), is:

B:010 101 1

AN3 ON2 ON1 Let c = 3. Then the two offsprings are obtained as follows: AL3.1 AL3,2 AL3,3 O b , l ob,2 Ab,3 oL1,l OL1,Z AL1,3

contribution from A contribution from B

contribution from B contribution from A C: 101 01 1

D: 010 101

This ensures that each new offspring gets a complete chromosome containing all the constituent information about the coded parameter (since each bit position represents some unique characteristic). The new crossover operators for GANE are developed in sections 4.3 & 4.4.

3.4 Mutation

The combined operation of reproduction & crossover sometimes loses some potentially good chromosomes. This can result in a non-optimal solution (local optimum). To overcome this problem, mutation is introduced. It is implemented by com- plementing a bit (0 to 1 and vice versa) at random. This ensures that good chromosomes are not permanently lost. The new mutation operator for GANE is developed in section 4.5.

4. GANE

This section discusses the detailed development of our GA for network expansion. The selection/reproduction scheme in GANE is adopted from section 3.2. The development of GANE requires (in the following order):

Chromosomal-coding scheme Initialization approach

This representation has 3 node fields: ON1, ON2, AN3. Each of these node fields has 3 bits because there are 3 nodes in the network. These bits OLi & ALij indicate whether there is a connection between nodes i & j . In a chromosome, each 0 L i j & ALjj, (1 I iJ 5 3), is represented by 0 or 1 (see Notation).

Example

This example illustrates the organization & meaning of a chromosome used in solving the network expansion problem.

AN3 ON2 ON, 110 101 011

This chromosome shows that the new node AN3 is added to an existing network of 2 nodes (ON1, ON2). The chromosome implies that ON1 is connected to node ON2 (since OL1,2 = 1) in the existing network and also connected to new node AN3 in the expanded network (since AL1,3 = 1). Similar- ly, other fields express their connections. In general, if there are N nodes in the expanded network, and if the first ( i - 1 ) nodes represent the nodes in the old network, then the rest of the nodes are newly added to the network. The resultant chromosome is:

ANN ... AN; ON;-1 ... ON1 Field N ... Field i Field i- 1 ... Field 1

Genetic crossover operator Adjustment operator Node-modify operator ALN,l, .. ., AL,,j- 1, ALN,; . . ., A L , N Replacement-scheme and termination-rules. AL;,l, .. ., ALi,;-l, ALi,; .. . , AL;J

Acronyms Field 1 OL,,,, ..., 0Ll,;-,, ALl,; ..., ALi,p SNFSO, RNFSO [simple, random] node field swap operator NMO node modify operator NOM, NOT number of [mutations, tries] allowed.

Notation

Field N Field i Field i - 1 OL;-l,l, ..., OL;.l,i.l, ALi-l,i ..., A L ~ - ~ , N

AL,,; & OL;,; are always 0 because no-self loops are

KUMAR ET A L GENETIC-ALGORITHM-BASED RELIABILITY OPTIMIZATION

AN5 AN4

String 1 10110 01101 String 2 01110 10101

-

67

ON3 ON2 ON1 11011 10110 01101 11011 10101 01110

4.2 Initialization Approach

The initial population can be randomly created or well adapted [17]. For the traveling salesman problem, the use of adapted initial population provides very little advantage except that the convergence could be faster [18, 191. GANE does not use any criterion to generate biased or adapted population at initialization, but instead generation #1 is computed absolutely randomly.

4.3 Genetic Crossover Operators

ing GANE.

SNFSO: simple node field swap operator RNFSO: random node field swap operator.

Two new genetic crossover operators were used in develop-

4.3.1 SNFSO

The crossover is performed at the boundaries of the node fields. First, two chromosomes are randomly selected from the mating pool. Next, using a random number generator, an in- teger is generated in the range (0, N - 1 ). This number is used as the crossover site. The result gives two new chromosomes with information consisting of their parents.

Example

This example describes the crossover operation in detail.

If the crossover site is 2, the information exchange occurs as:

AN5 AN4 ON, ON2 ON, Child 1 10110 01101 I 11011 10101 01110

Child 01110 10101 11011 10110 01101 2 <- String 1 -> <-- String 2 -- >

<-String 2 -> <-- String 1 -- >

The crossover operator sometimes generates a chromosome which does not represent a valid network-connectivity. For example, node field ANS of child 1 shows that node 5 is connected to node 1, but ON1 shows that there is no connection between node 1 and node 5. This anomaly is corrected by an adjustment operator discussed in section 4.4.

4.3.2 RNFSO

The node fields are swapped randomly, one at a time, in the chromosome. A user can specify the total number of node fields to be swapped for the crossover. Random numbers, in [O, N - 11, are generated for swapping specified number of node fields. For example, if only 1 node field is to be swapped, then a random number is generated to decide which field is to be swapped.

AN, AN4 ON3 ON, ON, String 1 10110 01101 11011 10110 01101 String 2 01110 10101 11011 10101 01110

Let the field to be swapped be 2, then RNFSO will provide the following results:

ANS AN4 ON3 ON2 ON1 Child 1 10110 01101 11011 10101 01101 Child 2 01110 10101 11011 10110 01110

This operator can also create the anomaly in a chromosome as discussed for SNFSO. The anomaly created by RNFSO is corrected by the adjustment operator in section 4.4. For the network expansion problem, RNFSO is preferred because it provides an unbiased and equal opportunity for: a) each node to connect itself with other nodes, and b) maintaining their own connectivity using adjustment operator. On the other hand, using SNFSO causes lowered numbered nodes to become domi- nant in forcing the other nodes to change their connectivity through the adjustment operator.

4.4 Adjustment Operator

Since crossover operators produce discrepancy in a connectivity chromosome, the following simple procedure is executed:

Adjustment Operator: for each swapped node field (AN/ON,) in a chromosome

f o r j = 1,N If bit Lf,j = 1 then {* L represents OL or AL * }

If Lj,f = 0 then Lj,r = 1; else

If bit Lj,f = 1 then Lj,, = 0; end for;

end for;

This is illustrated with the example from SNFSO, where child 1 and child 2 do not represent valid topologies. For child 1, node fields ON1 & ON2 are modified after the crossover, while ON, remains unchang- ed. Thus for swapped node fields ON, & ON,, each bit is to be checked for anomaly. For ON, field of child 1, bits b,4 = 0 and LQ, = 1, which implies that bit L4,2 in node field AN4 should be changed to 0 while bit L5,2 in node field AN, should be changed to 1. In the other modified field ON1 of child 1, bits L1,4 = 1 and L1,5 = 0, implying that bit L4,l of node field AN4 has to be modified to 1, and bit L5.1 of node field ANs should be changed to 0. The valid chromosome after adjustment is:

AN5 AN4 ON, ON, ON1 Child 1 01110 10101 I 11011 10101 01110

This adjustment operation preserves the crossover characteristics by adjusting the chromosomes according to newly acquired node fields.


4.5 Node Modify Operator (NMO) 4.7 Genetic Algorithm Operator

Four parameters affect the solution obtained from GANE: CR, NMO/MR, PS, NG. The detailed analysis & experiments for choosing the appropriate parameter values are considered in [ 18, 211. These parametric studies suggest that good GA per-

high CR very low MR moderate PS.

This operator is similar to the mutation operation in basic GA. This operation is performed to improve the connectivity of a node field, if it is appreciably reduced by the crossover operation. In order to use this operator, two user inputs are required: formance requires:

i. NOM (Number of Mutations): This specifies the number of bits to be complemented in various node fields of a chromosome represented as a fraction of total number of bits in the chromosome.

ii. NOT (Number of Tries): This specifies the number of times this operator should try to find a node field that meets

Based on these observations [16, 18, 211 indicate that the appropriate values are:

CR = 0.6

NMO = 0.03

PS = 30

NOT = N-1.

These values are valid for bit representation chromosomes. In this paper, these values are used for CR, MR/NMO, PS, NOT.

4.8 Algorithm for GANE

the criterion of degree constraint. It is fixed at (N-1). Four steps in the operator are:

1. Select at random a node field ANf from those nodes that are added to the existing network and have degree less than the upper bound.

2. Select at random any other node ANk/ONk which is not connected to ANf and has degree less than its upper bound.

3. If such a node cannot be found, go to step 1. If the number of nodes ANf tried for mutation exceeds NOT, then STOP.

4. If proper ANf and ANk/ONk are found, then create a new link by setting in node field ANf bit Lfk to 1, and bit Lk in node field ANk/ONk to 1. If number oimutations done s i

In this implementation NOM is specified in term of mutation rate. This is represented as a fraction of total number of bits to be modified in a chromosome.

The algorithm begins with an initial generation of valid

The initial generation contains a finite number of valid strings selected at random. The number of strings in any generation, population size, is kept an even number to ease the crossover.

far is than specified then go to 7 chromosomes which the constraints of the problem type.

4.6 Replacement Strategy and Termination Rules . .

The detailed steps for GANE are in figure 1.

The most common replacement strategy is to prob- abilistically-replace the poorest performing chromosome in the previous generation [ 191. On the other hand, the elitist strategy appends the best performing chromosome of the previous generation to the current population and, thereby, ensures that

5. RESULTS & DISCUSSION

This section provides the detailed results for the problems in section 2. The studies are conducted for each of the problems:

the chromosome with the best objective function value always survives to the next generation. Our GANE combines both these concepts. Each offspring generated after crossover is added to the new generation if it has a better objective function value than both of its parents. If the objective function value of an offspring is better than only one of the parents, then select a chromosome randomly from the better of the two parents and the offspring. If the offspring is worse than both parents, then any one of the parents is selected at random for the next generation. This ensures that the best chromosome is carried to the next generation, while the worst is not.

The execution of GANE can be terminated by using the rule:

The problem is solved using GANE to optimize the desired objective function for developing extended network configuration after adding a set of nodes. The objective function values for the topologies evaluated from GANE are compared with the optimal topologies computed from exhaustive search of the complete solution space. This study shows how good the GANE results are compared to optimal results. To demonstrate the effectiveness of GANE, we measured the time to solve various instances of each of the problems. These timing values are compared to the time required to get the same results using the exhaustive search in which all the valid network topologies were evaluated for obtaining globally op-

“When the average & maximum fitness values of strings in a generation become the same”.

timal solution. The results are also compared with the branch & bound approach.

In most cases, a very good objective function value for a problem can be obtained using GA when GANE is terminated using this rule [20].

A complete example using GANE is solved in [ 131. In each of the tables, column 1 represents the number of nodes in an existing network and the number of nodes to be added to that

KUMAR ET AL: GENETIC-ALGORITHM-BASED RELIABILITY OPTIMIZATION

1 Iaitialite

NumkofGenetations-0 Cnate Initial Population

~

69

UsiOgRNSFo

compute fitness value of the new offsprings and add them to the new geoaation using U replacement policy

Copy new generation to old Increment the number of generations

1

Figure 1. Flow Chart for GANE

network. For example, 4+2 in table 1 represents an existing network of 4 nodes (N1 - N4) to which 2 new nodes (N5, N6) are added to yield an expanded network of 6 nodes.

5.1 Diameter Minimization Under Degree Constraint

The nodes are to be added so that: a) the diameter of the network is minimized, and b) degree of each node 5 specified upper bound. Four examples are solved for this problem.

TABLE 1 Topology for Node Addition with Diameter Minimization under

Degree Constraint

Node N1 N2 N3 N4 N5 N6 N7 NS N9

4 + 2 6,3,2 6,4,1 5,4,1 5,3,2 6,4,3 5,2,1 -

5 + 2 6,5,2 3, l 7,4,2 1,5,3 4,l 7 , l 6,4,3 -

6 + 2 7,6,2 7,3,1 8,4,2 8,5,3 6,4 5, l 2 , l 4,3 - 6 + 3 6,2 8,3,1 9,4,2 1,5,3 7,6,4 9,5,1 8,5,4 9,1,2 8,6,3

TABLE 2 Comparison of GANE with Exhaustive Search

[Both procedures give exactly the same optimal result]

Node Diameter

4+2 5 + 2 6+2 6 + 3

2.0 2.0 3.0 3.0

5.1.1 Example 1

An existing 4-node network (Nl, N2, N3, N4) is considered. Table 1, row 1 gives the connectivity. It is interpreted as :

node 1 is connected to nodes 2 & 3 node 2 is connected to nodes 4 & 1 node 3 is connected to nodes 4 & 1 node 4 is connected to nodes 3 & 2.

Two new nodes (N5, N6) are to be added to this network. The upper bound on the degree of each node is 3. The resulting network with 6 nodes is given in table 1, row 1. The new network connectivity is interpreted as:

node 1 is connected to nodes 2, 3, 6. Similarly connectivity for other nodes can be interpreted.

This network configuration obtained from GANE provides a diameter of 2.0. This is the same as the diameter of optimal topology computed using exhaustive search of complete solution space. The result is in table 2, row 1.

5.1.2 Example 2

Two nodes (N6, N7) are added to an existing 5-node network (Nl, N2, N3, N4, N5). Connectivity of the existing network and the expanded network is stated in table 1, row 2. The upper bound on the degree of each node is 3. The diameter of the topology obtained from GANE is exactly the diameter of the optimal topology evaluated using exhaustive search. The result is in table 2, row 2.

5.1.3 Example 3

A 6-node network (NI, N2, N3, N4, N5, N6) is expanded to include 2 more nodes (N7, N8). The degree specified for


all the nodes is 3. The original topology of the six node existing network and the topology of the 8-node expanded network are shown in table 1, row 3. The best topology for the new network from GANE with 8 nodes has a diameter of 3 - exactly the same as that from exhaustive search. The result is in table 2, row 3.

5.1.4 Example 4

An 6-node network (N1 - N6) is expanded to include 3 more nodes (N7 - N9). The connectivity of the original network; the 9-node expanded network is given in table 1, row 4. The diameters of the optimal topologies computed from GANE and exhaustive search are exactly the same. The result is in table 2, row 4.

5.2 Average-Distance Minimization Under Degree Constraint

TABLE 3 Topology for Node Addition for Average-Distance Optimization

Node NI N2 N3 N4 N5 N6 N7 N8 N9

4 + 2 5,4,2 5,3,1 6,4,2 6,3,1 6,2,1 5,4,3 - 5 + 2 7,5,2 6,3,1 6,4,2 5,3 7,4,1 7,3,2 6,5,1 -

6 + 2 7,6,2 7,3,1 8,4,2 8,5,3 7,6,4 8,5,1 5,2,1 6,4,3 - 6 + 3 7,6,2 8,3,1 9,4,2 7,5,3 8,6,4 9,5,1 4.1 9,5,2 8,6,3



is 3. The average distance for each of the expanded network obtained using GANE is the same as the average distance obtained using exhaustive search. The result is in table 4, and clear- ly shows that GANE can deliver optimal results.

5.3 CNR Maximization Under Diameter & Degree Constraints

The nodes are to be added in such a way that CNR of the network is maximized while the diameter & degree constraints of the network are satisfied. Four examples are solved. The existing topologies along with the expanded network topologies for each case are as shown in [13]. The constraints for each case are:

6-node network (ij = 1 to 6)


8-node network ( i J = 1 to 8)


degi I 4; max(Dij) I 3

degi I 3; max(Di,,) I 3

degi I 3; max(Dij) 5 3

degi I 3, i = 1 to 6; degi I 2 , i = 7 to 9 max(Dij) I 4.

The results are in table 5. The optimal result is obtained using GANE in all 4 cases.

5.4 CNR Maximization Under Cost Constraint



Node Average Disrance Node CNR

4 + 2 5 + 2 6 + 2 6 f 3

1.4000 1.5238 1.6429 1 ,6944

4 + 2 5+2 6 + 2 6 + 3

0.9924 0.9913 0.9888 0.9580



Node CNR

4 + 2 0.9993 5 f 2 0.9814 6 + 2 0.9906 6 + 3 0.9579

The nodes are added so that: a) the average distance of the network is minimized, and b) degree of each node I specified upper bound. Four examples are solved for this problem. Table 3 gives the topologies of the existing networks and the corresponding expanded networks obtained using GANE. Each of these is interpreted as explained in section 5.1.

The constraints for each of the 4 network-expansion problems are the same, ie, upper bound on the degree of each node

Nodes are added in such a way that CNR of the network is maximized while the cost constraint is satisfied. Four examples are solved. The connectivity of the existing networks and that of the GANE-based expanded networks are shown in [13]. The cost of adding a communication link between two nodes is assumed to be directly proportional to the distance between those two nodes. No constraint is specified on the degree of any node, because the cost constraint itself limits the degree of any node. The cost constraint for each case is provided in [ 131. The CNR of the topologies obtained using GANE are compared to that of the best topologies obtained using exhaustive search; the results are the same and are in table 6.

5.5 Study Of Time Complexity Analysis

Tables 7-10 contain the results of the time comparison study for various instances the 4 problems in section 2.3. Each table compares the time required to solve that problem using GANE as well as using the deterministic optimal approach.

KUMAR ET A L GENETIC-ALGORITHM-BASED RELIABILITY OPTIMIZATION 71

Column 1 indicates the number of existing nodes and the number of nodes to be added, eg , 4+2 indicates an existing network of 4 nodes to which 2 nodes are to be added. Columns 2 & 3 represent the user time on a UNIX system to get the results using CANE and deterministic optimal case, respectively. For some problems (marked *), the exhaustive search was terminated after 6 hours of user time, because our purpose is just to highlight the processing time requirements for the two approaches. Column 4 lists the objective function values. The values were the same in optimal and CANE approaches except where optimal program was terminated at 6 hours. For those cases the value specified is for CANE.

TABLE 9 Time Comparison for CNR Maximization under Diameter &

Degree Constraints [Columns 2 & 3 give user-time in seconds; the * indicates

truncation before solution]

Nodes GANE Optimal CNR

6+2 20.4 48.6 0.9906 8+2 453 1835 0.9670 9+2 420 7219 0.9553 6 + 3 11.2 220 0.9579 8 + 3 3947 21.6k* 0.9765 9 + 3 2778 21.6k* 0.9643

A comparative study of branch & bound with CANE and exhaustive search is provided in [13].

Tables 7 - 10 show that an increase in the existing network size or in the number of new nodes to be added appreciably increases the time required to solve the problem using exhaustive search, unlike CANE where the time requirements are very low.

TABLE 10 Time Comparison for CNR Maximization under Degree & Cost

Constraints [Columns 2 & 3 give user-time in seconds; the * indicates


The time taken by CANE is smaller than by branch & bound [13]. Moreover, branch & bound requires large memory for the expanded tree and runs out of memory for larger problems.

The results for solving large problem using CANE are:

Problem 1 (25+2). The degree constraint for NI to N25 is 3 and N26-27 is 2. The diameter obtained using CANE was 8 and the execution time was 38.5 sec. Problem 2 (same initial topology as problem 1). The average distance obtained from CANE was 4.6553 and the execution time was 60.1 sec. Problem 3 (same initial topology as problem 1). The CNR obtained using CANE was 0.9953 and the execution time was 2925 sec.

TABLE 7 Time Comparison for Diameter Minimization under Degree

Constraint [Columns 2 & 3 give user-time in seconds; the * indicates


Nodes GANE Optimal Diameter

Nodes GANE Optimal CNR

6 + 2 19.4 48.2 0.9888 8 + 2 146 1788 0.9543 9 + 2 82 7217 0.9910 6 + 3 11.7 723 1 0.9580 8 + 3 49 1 21.6k’ 0.9941 9 + 3 369 21.6k’ 0.9923

6 + 3 0.1 19.4 3.0 13+2 2.3 3 42 5.0 15+2 1.1 1154 6.0 13+3 1.4 21.6k* 4.0 15+3 1.9 21.6k* 6.0

111

121

ACKNOWLEDGMENT

This research was partially supported by a grant from BellSouth Foundation. We are pleased to thank the Associate Editors and referees for their valuable suggestions on a draft of this paper.

TABLE 8 Time Comparison for Average-Distance Minimization under

Degree Constraint [Columns 2 & 3 give user-time in seconds; the * indicates


Nodes GANE Optimal Average Distance

[31

141

151

161

6 + 3 1.7 19.0 1.694 13+2 16.8 342 2.867 15+2 10.6 1162 3.152 13+3 18.1 2 1.6k’ 2.750 15+3 24.7 21.6k8 3.014

171

181

191

REFERENCES

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[IO] D.K. Pradhan, S.M. Reddy, “A fault tolerant communication architecture for distributed systems”, IEEE Trans. Computers, vol C-31, 1982 Sep, pp 863-870.

[ I l l K.K. Aggarwal, Y.C. Chopra, J.S. Bajwa, “Topological layout of links for optimizing the overall reliability in a computer communication system”, Microelectronics & Reliability, vol 22, num 3, 1982, pp 347-351.

[12] R.S. Wilkov, “Design of computer networks based on a new reliability measure”, Proc. Symp. Computer Communication Networks & Teletrafic, 1972 Apr, pp 371-384. A. Kumar, R. Pathak, Y. Gupta, “Computer network expansion using genetic algorithm approach”, Technical Report, AK-EMCS-94-3, 1994; University of Louisville.

[I41 K.K. Agganval, S. Rai, “Reliability evaluation in computer communication networks”, IEEE Trans. Reliability, vol R-30, 1981 Jun, pp 32-35.

[I51 J.H. Holland, “Genetic algorithm and the optimal allocation of trials”, SIAM J. Computing, vol 2, num 2, 1973 Jun, pp 88-105.

[ 161 D.E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, 1989, p 412; Addison-Wesley.

[17] L. Davis, e ta l , “Genetic algorithms and simulated annealing: An over- view”, Genetic Algorithms and Simulated Annealing (L. Davis, Ed), 1987; Morgan Kaufman.

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[13]

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AUTHORS

Dr. Anup Kumar; EMACS Dept; Univ. of Louisville; Louisville, Kentucky 40292 USA.

Anup Kumar received a BE (1983) in Computer Science from Allahabad University and a PhD (1989) from North Carolina State University. He is an Assistant Professor at the University of Louisville. His research interests include fault-tolerance evaluation, distributed system modeling, high-speed networks, multimedia system, and genetic algorithm application. He is an Assoc. Editor of International Journal of Engineering Design and Automation. He has edited special issues on performance evaluation in Simulation Digest and on genetic algorithm application in Computers and Operations Research. He is the Vice Chair’n of IEEE Computer Society TCSIM. He served on the program committee of the 1994 IEEE Symposium on Parallel & Distributed Systems, the Seventh (1994) International Conference on Parallel Distributed Systems, and other computer conferences.

Rakesh M. Pathak; Network Equipment Technologies; 800 Saginaw Dr; Red- wood City, California 94063-4740 USA.

Rakesh M. Pathak received his BE in Electronics Engineering from Maharaja Sayajirao University. He is working at Network Equipment Technologies. His research interests include distributed processing, computer networks, and high-speed computing.

Dr. Yash P. Gupta, Dean; College of Business Administration; Univ. of Col- orado; Denver, Colorado 80217 USA.

Yash Gupta is Professor & Dean, College of Business Administration at the University of Colorado at Denver, and was the Frazier Family Professor at the School of Business, University of Louisville. He holds a BS (Engineer- ing), MTech, and PhD from the University of Bradford. His research interests are in planning & control systems for advanced manufacturing systems, information-system strategies, TQM, and applications of quantitative techniques in industry. He teaches courses in management information systems and operations management. He is a founding member of the Total Quality Transfor- mation Committee of the Louisville Chamber of Commerce and a member of the Manufacturing Technology Committee of the Louisville Advanced Technology Council. He has written over 115 articles published in international journals, serves on the editorial boards of four journals, and has been invited to be the guest editor for three journals. Yash serves on the faculties of the Speed Scientific School at the University of Louisville, the School of Engineering at the University of Manitoba, and the School of Business at Baptist College (Hong Kong). He was Senior Research Fellow in the Telecommunications Research Center at the University of Louisville. He received numerous awards of excellence from several institutions and professional organizations including the University of Louisville President’s Award for Outstanding Scholarship, Research , and Creative Activity for 1991.

Manuscript received 1994 July 5.

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Genetic-algorithm-based reliability optimization for computer network expansion