Vol.13 No.1 J. of Comput. Sci. & Technol. Jan. 1998

A Topology Des igning System for a Computer Network

Hou Zhengfeng (~iK~) Computer and Information Department, Hefei University of Technology

Hefei 230009, P.R. China Received August 5, 1995; revised June 20, 1997.

A b s t r a c t

In this paper, some problems on the topology design of network are discussed. An exact formula to calculate the delay of a line will be provided. In the design, the key problem is how to find some efficient heuristic algorithms. To solve this problem, a nonliner-discrete-capacity assignment heuristic and a hybrid perturbation heuristic are suggested. Then, a practical CAD system which helps design the topology of network will be introduced.

Keywords : Topology, network, delay.

1 I n t r o d u c t i o n

Most of Wide Area Networks (WAN) have an irregular topology. In the case of fewer nodes, the network designers may achieve a good design by means of their experience. Unfortunately, a WAN is often very large. There are many nodes in the subnet. This means that almost countless topologies can be chosen for the network. I t is impossible to achieve satisfied design by examining every potential topology. In order to produce a design with low-cost, high-throughput and high-reliability, it is necessary to develop a systematic method for topology design of a network. In this field, some people have made great contribution to the graph theory and queueing model. They have also proposed some algorithms, such as Kle i tman 's and Even's connectivity algorithms, the Max-Flow algorithm, the Saturated Cut heuristic algorithm, for designing the topology of network. But so far, these algorithms are not completely satisfactory.

Since 1990, we have studied some problems on network design including topology, de- lay, connectivity, routing and flow assignment, and proposed some efficient heuristic al- gorithms, such as a nonlinear-discrete-assignment algorithm, a reliability algorithm and a hybrid heuristic algorithm. We have also developed a Topology Designing System (TDS) for a computer network which has been implemented on a VAX/780 computer. Our research and simulation designing results will be described as follows.

2 De lay Analys is

In the past , the delay analysis was based on the M / M / 1 model. The following formula was used to calculate the line delay:

1 T - - - (i)

#c - A

92 J. of Comput. Sci. & Technol. Vol.13

(1) is very simple but not accurate enough. In fact, in addition to data packet, acknowl- edge packet and control packet also flow through the practical network. Furthermore, these packets in a network are not isolated. The other two kinds of packets certainly have an effect on the delay of the data packet. For this reason, we propose a more reasonable queueing model as follows:

first priority control packet

second priority acknowledge packet ~ servicing time line ,

data packet last priority

Fig. 1

In Fig. 1, there are three kinds of packets (control, acknowledge and data) waiting for transmitt ing in the queue of line (channel). The queue model is M/D/1 for data or acknowl- edge packet; D /D/1 for control packet. Any line of a node must provide service for these three kinds of packets. Each kind of packet has a servicing priority (see Fig.l). We also assume th&t this is 'not to try to be first'. In this case, the packet with higher priority can be inserted before the packet with lower priority. But if the lower is transmitting, it is not allowed that the packet with higher priority cuts off the current transmitt ing in order to obtain the channel.

Based on Fig. 1, we deduce (2) by using Little's Result.

T = A c / ( # c C ) 2 + A a / ( # a C ) 2 + 2 A d / ( # d C ) 2 + 1 (2) 2(1 - A c / # c C - A a / # a C ) ( 1 - A c / p c C - A a / # a C - A d / # d C ) # d C

where

Aa : arrival rate of acknowledge packet, 1/,ua :

Ac : arrival rate of control packet, 1/I.~c :

Ad : arrival rate of data packet, 1/#d :

C : l ine capacity.

mean length of acknowledge packet,

mean length of control packet,

mean length of data packet,

3 Nonlinear-Discrete-Capacity Assignment Heuristic Algorithm

After we have a proposed topology and a proposed set of flow assignments, it is not difficult to calculate the traffic of each line of the network. The question remains: how to optimize line capacity assignment which should meet the requirement of delay at minimal cost of the lines. This can be expressed as:

T~ _< Tmax (3)

1 m T = r ~ AiTi _< Tmax (4)

i=1

where, 7"/is the mean delay of line i; T is the average delay of the network.

No.1 A Topology Designing System for a Computer Network 93

Generally, it is assumed that the relation between cost and capacity of a line is linear, so as to reduce the complexity of the capacity assignment algorithm. This can be expressed as:

F~ : D~ x C~ (5)

where, Fi is the cost of line i; Di is the length of line i; Ci is the capacity of line i. Using Lagrangian multiplication, we get (6) tha t may be used to obtain the optimal line

capacity.

( C~ = "~--~ 1 + (6) # r T ~ /

This method of assigning capacity is simple and easy to implement, but two questions remain:

(1) In any practical network, the linear relation is not true. The relation between cost and capacity can be approximately expressed as Fi = Di x C~ (c~ varies from 0 to 1).

(2) The capacities achieved by (6) is successively distributed. However, only a discrete set of speeds is available, for example, 2,400, 9,600, 56,000 and 230,000 bps. As a result, if the op t imum capacity for a line turns out to be, say, 20,000bps, the line will not be available. Either a 9,600 or a 56,000 bps line will have to be used, the former providing too little performance and the latter too high a cost. Therefore, the capacity assignment using (6) may produce a configuration that is far from optimal.

To avoid these two defects, we propose a nonlinear-discrete-capacity assignment heuristic algorithm. The heuristic is described as follows:

(1) Setting up the matrix CTF which is shown in Fig.2. Here, assume that any line has k capacities available; m is the number of all lines in

the network; Cij is an available capacity of line i; tit and fij represent the delay and price corresponding to Cij.

The next step: pick up an element

the requirement of the network delay at

T = I k A - iTs <_ 7"

i=1

where, f{ is the cost of the line i.

cij ) tij in each row of the CTF, which must meet

k k s minimal cost, i.e.

m Tmax, ~ f/ is minimal, (7)

i=1

A simple method to solve this problem is exhausting the possibilities. But any way you look at it, the problem is huge. If there are rn lines and k kinds of capacities for any line, there are m k average delay calculations for this problem. For example, i f m = 100 and k -- 5, there are 1005 possibilities to be considered. Clearly, this exhausting search is practically impossible.

Our approach to solving this problem is using a heuristic search strategy which is called 's traight forword search'.

(2) Setting up a heuristic function Hij.

Ati _ t~j-ti(j+l) i = l , . . . , m , j = l , . . . , ( k - 1 ) . (8) Hij = -Ai A~i A~ fij fio+l~'

(3) The heuristic starts from the first rank of matr ix CTF. Clearly, the first rank contains a group of capacities with the lowest cost. Move the first rank into a variable BEST; subst i tute til of CTF into (7) (i = 1 , . . . ,rn) to calculate the average delay T. If T < Tmax, then the first rank is the best solution; otherwise goto Step (4).

94 J. of Comput. Sci. & Technol. Vo1.13

(4) Examine the existing elements in BEST and calculate each Hij for each element in BEST by (8). Assuming Hij is the maximum, the capacity of line i should be changed.

Tha t is, tij in BEST is replaced by ~ ti(j+l) . Then we calculate the average delay

\ f i j \ Yi(jq-1) T according to the BEST again. If 7" _< Tmax, the group of capacities in BEST is the best solution. Otherwise, repeat Step (4) until T _< Tm~x- The reiteration process is shown in Fig.2.

Cll

t l l

Ill[

c21

C T F = t21

f2t

Cml

trn l

existing elements in BEST.

~ ~ C 1 2 C13 Clk

t12 t13 tlk

I12 fla Irk

C22

t22

f22

Cm:

tm:

fr."

C23 c2k

t23 . . . . . . t2k

Y23 I2k

Crn3 Crnk

tin3 trnk

fro3 f mk

cij < ci(j+l)

(c 2) (c23) * if H22 is the maximum, t2~ in BEST is replaced by t23 .

f',~ .f~3

Fig.2

4 P e r t u r b a t i o n H e u r i s t i c

A per turbat ion requires the application of flow algorithm, capacity algori thm and reli- ability algorithm. There exist some per turbat ion heuristics. The most useful one of them is Saturated Cut heuristic. This heuristic is more sophisticated than the others. However, it is sometimes in a ' s top ' conditon or even failing. This 's top phase' is shown in Fig. 3. For a 30-node network, this 's top phase' may last more than 100 perturbat ions, i.e. during these perturbat ions, there is no improvement of the total cost, but a lot of computer time is wasted.

To improve the execution efficiency, another per turbat ion heuristic called 'Exchange' is suggested here. It goes as follows:

(1) Pick up an unconnected pair of nodes randomly. (2) If the distance between the two nodes is too far or the node degree of either is too great,

then goto Step (1); otherwise add the combination of the two nodes. (3) Check to see if the total cost becomes lower; if so, it is adopted as the new base topology.

Otherwise goto Step (4).

No.1 A Topology Designing System for a Computer Network 95

(4) Remove a line that wastes the most money; if the total cost is lower, it is adopted as the the new base topology. Otherwise, goto Step (1).

Compared with Saturated Cut, Exchange heuristic execution efficiency is lower, but its execution is more smooth.

We combine Saturated Cut with Exchange into a Hybrid. At the beginning, the Satu- rated Cut is used. When the heuristic is in a 'stop' condition or failing, it is replaced by the Exchange. After several executions of Exchange, the Saturated Cut is used again.

total c o s t ~

~ .--stop phase---, - - - ' - -

0 50 I00 No. of perturbations

Fig.3

5 S imula t ion D e s i g n i n g M o d e l s and R e s u l t s

Our experiment models are characterized by the following hypotheses: (1) Store-and-forward method is used in the subnet. (2) There are three kinds of packets (control, acknowledge and data) in the subnet. (3) Only consider the queueing delay (neglect the propagation delay). (4) The total cost only consists of all lines in the network, neglecting the cost of the

nodes. (5) Shortest-path routing is used. The following parameters are used:

number of nodes: 30; the topology is shown in Fig.4;

(a) Initial topology. (b) Final topology.

l Fig.4

the distance matrix between nodes; traffic: multiplying a fixed traffic matrix by a scale factor (from 0.8 to 2.0); available capacities for any line: 2,400, 4,800, 9,600, 56,000bps; the cost matrix of lines; mean arrival rate of the control packet: 0.1/sec; mean length of the control packet: 400 bits; mean length of the data packet: 1008• bits; mean length of the acknowledge packet: 40 bits; line delay maximum (congestion constraint): 0.2;

96 J. of Comput. Sci. & Technol. Vol.13

�9 ne twork average de lay m a x i m u m : 0.3; �9 connec t iv i ty : 2; �9 n u m b e r of pe r tu rba t i ons : 200. W h e n the traffic is changed by mu l t i p ly ing scale factors , the t o t a l cost as a funct ion of

traffic is shown in Fig.5. The hybr id p e r t u r b a t i o n execut ion efficiency is shown in Fig.6. The s imula t ion takes abou t 1.5 hours of V A X / 7 8 0 c o m p u t e r t ime for 200 p e r t u r b a t i o n s .

6O 45 l total cost . ~ 50 40 ~ �9 40

35 30 30 20 25 i0

I 0.0 1.0 1.2 1.4 1.6 1.8 2.0 traffic scale factor

total cost

number of perturbations, 20 40 60 80 100 120 140 160 180 200

Fig.5 Fig.6

6 C o n c l u s i o n s

Our research is based on G r a p h T h e o r y and Queue ing Theory�9 We have p r o p o s e d some heur i s t ic a lgo r i thms for the des ign of ne tworks . These heur is t ics are feasible and efficient�9 Using TDS, we have made some s imula t ion designs for a 30-node ne twork . T h e resul t is sat isfying�9 Each p e r t u r b a t i o n only takes a b o u t 20 seconds of VAX/780 c o m p u t e r t ime. If there a re 1,000 p e r t u r b a t i o n s in a des igning process , a b o u t 8 hours of C P U t ime m a y be t aken up. Th is is not in tolerable .

R e f e r e n c e s

[1] Mischa Shwartz. Communication Network Design and Analysis. Englewood Cliffs (ed.), Prentice-Hall, N.J., 1977.

[2] Nagle John B. On packet switches with infinite storage. IEEE Trans. on Communications, 1987, COM-35(4).

[3] Bezalel Garish, Irina Neuman. A system for routing and capacity assignment in computer communica- tion networks. IEEE Trans. on Communications, 1989, COM-37(4).

[4] Charles Knessl. Two parallel M/G/1 queues where arrivals join the system with the smaller buffer content. IEEE Trans. on Communications, 1987, COM-35(11).

[5] Tropper C. Priorities and performance in packet switching networks. Computer Networks and [SDN Systems, 1986, 12(2).

[6] Bakry S H, BEI-Redaisy, MAL-Turaigi. Computer simulation of a packet switching computer network. Computer Communications, 1988, 11(3).

[7] Kershenbaum A e t al. MENTOR: An algorithm for mesh network topological optimization and routing. IEEE Trans. on Communications, 1991, COM-39(4).

[8] Thilakam R K, J Ashok, The design and flow control of a high speed integrated packet switched network�9 Computer Networks and Systems, 1992, 25(3).

H o u Z h e n g f e n g , born in Feb. 1958, received his B.S. and M.S. degrees from Hefei University of Technology in 1982 and 1988, respectirely. He was a visiting scholar at University of Geneva, Switzerland from Jan. 1992 to Dec. 1993. He is an Associate Professor at Hefei University of Technology now. His major research interests are computer network and database.