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 Copyright © 1985 North-Holland=

 European Journal of Operational Res= earch 19 (1985) pp. 262-267

 Received December 1982:

 Revised = December 1983 and February 1984.

 Paper is posted with permission from Elsevier. http://ScienceDirect.com/science/journal/03772217

        =      Routing optimization in packet switching communication networks.

 

        =             &nb= sp;            =         Z.Flikop 725 Greenbrier Dr. Apt.14 Bohemia, NY 11756 USA

 

Abstract. For routi= ng assignments a special model and an optimization algorithm are proposed.  The efficiency of the routing asses= sments is evaluated by the average value of the total cost of delays for all packe= ts in the network. It is the objective function.  The main idea is the traffic, which= is transmitted from the source node to the destination node can be split betwe= en two or more logical paths.  The minimum of the objective function can be found by varying the traffic on ev= ery path and simultaneously from all the source nodes to the destination notes.= If this approach is applied, then the objective function is no separable and nonlinear. Because its shape is unknown in advance an adoptive nonlinear optimization algorithm is proposed. For evaluating its efficiency special s= et of test function has been used.

 

Keywords: Routing, communication, networks

 

  1. Introduc= tion

 

Nowadays complicated communication networks are developed and implemented. At the same time rout= ing optimization for this networks is studied very intensively and literature in this field is vast, for example [1-5]. This article discusses the same problem, and a special method to design he mathematical models of the networks is described. The first model has been developed and implemented in computer language in 1970 and published in 1971 [6]. For optimization in the early version of the model, a method similar to [7] has been used. Later a more efficient optimization algorithm was create= d [8]. This article is dedicated to it generalize the approach, which can be used = to simultaneously create and update the routing tables at all the nodes of the network. Before the approach is described, we analyze the network properties.

 

Usually a communication n= etwork consists of hosts and a transmission system [9].  The transmission system is a set of packet switches (notes) and channels which carry packets from the source no= des to the destination notes. The common situation is: The same note can serve different kinds of packets. As usual the same packet can be transmitted thr= ough the network by different paths. The total time of transmitting a packet from the resource note to the destination node dependents on the selected path. = Particularly this time depends on the traffic intensity ratios for all the nodes which constitute the path. In turn, the traffic intensity ratio depends on the fraction of the traffic that is transmitted through the path (i.e. assignme= nt all of the traffic flows within the network). Usually the delay in the netw= ork is the main factor used for evaluation the transmission system efficiency. =  Therefore the routing problem is a = common problem during network development. To control the traffic assignments a special model may be used. This model must reflect all principal properties= of the real network. For adaptation to the current situation of the network the model should be easily updated and have facilities needed for simulation and optimization the network. The main network property is the fact hat the com= munication network is a queuing network. These ideas are utilized in the approach described below.

 

  1. Mathemat= ical model

 

Assume that in transmissi= on system every node needs some time for pocket service. It is a service time  . Because  a queue of packets can build up in every node.  The time a packet to waits in the q= ueue depends on an average value and distribution of    and a= n average value and distribution of interarr= ivail  times.<= sub> Let  <= !--[if gte vml 1]>  and  . Assume th= at from every source note to the destination note more than one path can be selecte= d (i.e. traffic from the source to the destination node can be split). We will cons= ider only the essential paths between every source and destination node. (An ess= ential path does not contain all the nodes of any other path.) For instance, let us consider the network of Fig. 1.

 

 

This network has three so= urce nodes (1, 2 and 3) and two destination ones (8 and 9).  The nodes 2 and 3 generate only one= kind of messages each, but then node 1 generates two kind of messages for the sa= me destination node 8. (We will designate the messages as different if they ha= ve a different distribution of service time in any transmission node.)  For the model development, one can = use the idea of a logical path. By definition a logical path transmits only one kin= d of messages. Therefore, every physical path can contain one or more logical pa= ths. For example, the path 158 contains the logical path 1’5’8’= ; for the first kind of messages and the logical path 1’’5’R= 17;8’’ for the second kind of messages. The usual situation is when the same node belongs to different paths, i.e. it services different kind of messages (for example, the notes 4 and 6 of Fig.1)

 

 Assume that note 1 generates the pa= cket traffic for the note 8’ and   is the traf= fic rate for this notes.  If the t= raffic can be split between two logical path 1’5’8’ and 1’= 6’8’ than

        =             &nb= sp;         =             &nb= sp;            =             &nb= sp;            =    (1)

 

Let  =3D {n | n = is an essential logical path between the source node A and destination node B}, A= ,B  S, where S = is the set of hosts in the network. If some source and destination notes are conne= cted by only one essential path, than one or more nonessential logical path can = be added. For the source node A and destination node B,

,        =             &nb= sp;            =             &nb= sp;          =             &nb= sp;            =       (2)

where  is the frac= tion of the total traffic from A to B that has been assigned to the path n;

 

        =             &nb= sp;            =             &nb= sp;            =             &nb= sp;         =          (3)

The entire network is

        =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp; (4)

A static model with stationery traffic inputs and unch= anging network configuration is considered. Under these conditions the average del= ay in path n is

 

        =             &nb= sp;            =             &nb= sp;            =             (5)

Where

 {z | z is a node, which belongs to the = path n};

 is the average value of the service time for the packe= ts at the node z (for the packets are transmitted from A to B ); the value of can be calculated in accordance with queuing theory [10];=

is the average value of the se= rvice time for the AB packets at the = note z: and

 is there an average value of the total propagation del= ay for the path n.

 

The waiting time depends on the total traffic intensity  for the nod= e z. For example the total traffic intensity for the note 6 = of  Fig. 1 is

 

Assume that

 

        =             &nb= sp;            =             &nb= sp;            =             &nb= sp;          =       (6)

 

i.e.          =             &nb= sp;            =             &nb= sp;            =             &nb= sp;         (7)

Let the average value of the coast of the n-path utilization be.

 

        =             &nb= sp;            =           =             &nb= sp;            =     (8)

 

where  represents = the weight (cost) coefficient and  the average number of packets in paths n. B= ecause [9]

 

 

it follows that

 

        =             &nb= sp;            =             &nb= sp;            =            (9)

 

Then, the average value of the cost for transmitting t= he packets from A to B is

 

        =             &nb= sp;            =             &nb= sp;            =             &nb= sp;         =             &nb= sp; (10)

 

The total average value of the cost of the network utilization is

 

        =             &nb= sp;            =             &nb= sp;            =     (11)

 

The traffic flow assignment can be described by a vect= or:

 

        =             &nb= sp;            =              (12)

 

Define the traffic flow assignment  to be optim= al if

 

        =             &nb= sp;            =             &nb= sp;            =                    =             &nb= sp;  (13)

 

Because the function  is nonlinea= r, the function (V) is nonlin= ear too; because the trffic intensity for every node depends on the traffic assignment for the whole network, the function = (V) is a no separable function. We assume that (V) is a well-behaved function admissible to nonlinear programming techniques. During optimization vector V may be ch= anged as follows:

 

        =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp; (14)

 

Where  is the step size and m is the step number. Because of the communication network is a queuing network, the optimization procedure must be implemented within the feasible area, where = the vector  provides a = steady conditions for every note in the network.[10]:

 

        =             &nb= sp;            =             &nb= sp;            =             &nb= sp;           =        (15)

 

with<= /span>  {w | w is a path, which contains the f= -node}. Usually it is very difficult to find the initial feasible point for the optimization that provides the above conditions. To resolve the problem, one dummy path which contains only one node with t=3D0 and a very big value of   may be added to a every  in the netw= ork. This technique is similar to the penalty approach [10]. In the very beginning of= the optimization procedure the whole traffic flow from a every source node can be assigned to the correspondent dummy path. The prop= osed method provides the condition (15) for the initial trial point. During opti= mization must satisfy (2) and (15) for every step m. For this kind of model the shape of the objective function depends on the choices of  and the cur= rent value of . Because the shape is nonlinear and unknown in in advance an advanced optimization technique must be= used. For the actual model an algorithm similar to that described below has been implemented.

 

  1. Optimization algorithm

 

For the simplification goals it will be useful to chan= ge the description of V to

 

        =                      =             &nb= sp;            =             &nb= sp;            =         (16)

 

where  is the valu= e of the j-th coordinate at th step,  1, 2,…= ;

 

The optimization is divid= ed into sub-algorithms. These sub-algorithms are linked and used depending on their efficiency. The first and main algorithm is based on utilization of a trial vector which has a random number of changeable variables during every realization. According to this algorithm the value of   is ca= lculated by

 

 

  if       and         if          =             &nb= sp;       (17)

 

where  represents = the value of the objective function after the -th (last) successful step: t= he successful step is a step, that which provides the condition

;  is a vector, which corresponds to this step;

 

 if  =  ;    if        and     if           =             &nb= sp;            =             &nb= sp;   (18)

 

with  =3D [0,1] a= random number, P=3D[0,1] the probabili= ty that  will be cha= nged for this step, and  the step si= ze for the -th variable; and

 

 if  = ;     if         and      if            =             &nb= sp;            =         (19)

 

For the optimization a ra= ndom generator is used and one can see from (18) that the number of variables th= at will be changed during the -th step up is determined by P. During o= ptimization the value of P can be changed d= epending on the efficiency of search. The described approach helps to adapt the optimization procedure to the current shape of the objective function  If during the optimization two successful steps have been implemented,  than the next value  is calculat= ed by (17)  with

 

   if = ;

 if         =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             

 if         =             &nb= sp;            =             &nb= sp;            =   (20)

 

Then, the optimization pr= ocedure continues with .  This tech= nique helps to accelerate the search when the shape it is a parabola. Sometimes t= he shape changes very sharply or very slowly.=   In this situation it is expedient to cut of the acceleration in (20)= and continue the optimization procedure with a double step. For testing purposes the next inequality may be used:

 

        >         =             &nb= sp;            =           =             &nb= sp;            =             &nb= sp;            =             &nb= sp;    (21)

 

where  is the thre= shold to cut off acceleration if the shape changes very slowly and  if the shape changes very sharply.

 

If after few successive successful steps an unsuccessf= ul steps follows, then coordinates for the next step are calculated according = to

 

 if  {{   and   } or { and }};

 if  = ;        =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;      (22)

  if  {{  and  }  or  { and }}

 

This approach uses the id= ea that the current shape resembles a ravine. If the step is unsuccessful, than the procedure continues by rule (17). We will den= ote every trial value of P as . During optimization the two last values of ,  which per= mit us to find one or more successful points, are memorize  as  and  . When one or more successful steps follow a set of succe= ssive unsuccessful steps and their number exceeds a particular threshold the next value of  is calculat= ed by

 

  if  =    and     if          =             &nb= sp;            =             &nb= sp;            =            (2= 3)

 

 

This approach helps us to use the accumulated informat= ion about shaped behavior.  If aft= er  steps the v= alue of  does not pe= rmit any successful steps, than the next value of is calculated by

 

        =             &nb= sp;            =             &nb= sp;            =             &nb= sp;             =             &nb= sp;            =             &nb= sp;           (24)

 

where=  is incremen= ted by 1 every time if no successful step was done. As a result, the value of  is changed periodically by . This procedure continues until either one or more successful steps will be done (then we can assume =3D1 and  or whole se= t of P will be tested. Then the procedu= re is switched to the value of  which provi= des the last successful step; but for the condition

 

,                =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =          =             &nb= sp;(25)

 

where  is the numb= er of changes of .

 

The i= t is often the case that in an area close to the extremu= m point, the shape is almost flat and it is very difficult to find a successf= ul point. In this situation the area around the last successful point needs to= be tested carefully. How carefully depends on the value of. The large  the more ca= reful a search will be; on other hand the less often the value of P will be changed the more time wi= ll be needed for the optimization. To create the fair conditions for all optimiza= tion stages the value of  can be cont= rolled by

 

 =3D constan= t,        =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             (26)

where=  is the numb= er of successful steps realized with during the -th stage of optimization;  is the total number of steps at the same stage. Each stage of the search continues until= the inequality (26) is valid. As a criterion to terminate the optimization, the inequality  =3D constan= t ( for example ) may be used. The same algorithm can be utilized if the model is employed in real time mode. In this mode the optimization is a continuous procedure and information about network statue= s is updated currently

 

This =  optimization algorithm has b= een tested at this special set of test objective functions.  The functions included in the set represent a wide spectrum of possible shapes. In [11,12= ] one can find a detailed description of each function and its shape. The uni= versality of the search may be determined by the number of tests functions, for which= the extreme points or their close neighborhood can be found. The efficiency of = the optimization algorithm is evaluated by  the number of steps needed t= o accomplish the search. The test results (see the Appendix) have shown that the propose= d algorithm is universal and effective for all the test function. In addition, during t= he routing optimization the time needed to find  depends on = the distance between the initial point and the point that provide the optimal v= alue of Q, the choices of ,  the netwo= rk configuration, the service and interarrival rates.

 

The model developed here = will be useful in the study of the stability and sensitivity properties of the netw= ork as well as in optimization. The first property shows to what limits the val= ues of the network parameters can be changed upon the condition that the values dependable parameters will stay within described limits. The second propert= y shows how the speed of variation of some parameters depend on the speed of variation of the other parameters. This information can be = used together with information about the dynamic properties of the network for t= he calculation of different kind of thresholds, for the control and optimization procedures and for network improvement.

 

 

References.

 

[1] = Gerla, M.  Kleinrock, L., “Topol= ogical design of distributed computer networks”, IEEE Transactions on Communications 25 (1977) 48-60.

[2] = Gerla, M. Cantor, D.G., “Optimal routing in a packet switched computer netwo= rk” , IEEE Transactions on Computers 23(1974) 1062= -1069.

[3] Agnew, C., “On = the optimality of adaptive routing algorithms”, in :Conf. Rec. Nat. Telecommun., 1974, 1021= -1025.

[4] = Gallager, R.G., “A minimum delay routing algorithm using distributed computations”, IEEE Transactions on Communications (19= 77) 73-75.

[5] Segal, A. “The = modeling of adaptive routing in data communication networks” IEEE Transactions on Communications 25 (1977) 85-96.

[6] Flikop, Z “Opti= mal assignment of traffic flows into hierarchical communication network”,= Matematithceskie= metody issledovania sistem, =

        =             &nb= sp; Kiev, 1971, 249-255=

[7] Flikop, Z. “Random  search as a means for queui= ng network optimization”, Kibernetica 4 (1969) 75-79.

[8] Flikop, Z. Dolotov, A., “Method of the system optimization”, V= oprosy ekonomiki I upravlenia v stroitelnom proizvodctve&= #8221;, Kiev, 197= 8.

[9] Tanenbaum, A., “= Computer Networks” , Prentice-Hall, Englewood Cliffs, NJ. 1981.

[10] Wagner, H., Principles of Operation Research,. = Prentice-Hall, Englewood Cliffs, NJ. 1969.

[11] Zakcharov, V., “= ;Ten test functions for the optimization”, Avtomatica i vitchilitelnaia= technical 6 (1974).

[12] Lazarev, I= ., Valujsky, V., “Experimental comparison the random optimization algorithms”,  Avtomatica i vitchilitelnaia technic= al 2 (1974).

 

Appendix=

 

No.

Objective functions, constrains, initial points

Total # of steps

No.of success steps

Value obtined

1

358

80

2

47

20

0.0

3

750

149

4

1000

162

 

5

681

296

0.041

6

25

15

0.199787

7

683

348

0.057

8

683

364

0.037

9

681

296

0.041

10

2550

1167

11

174

48

12

 at 

55

34

0.0

13

 at <= !--[if gte vml 1]>

=3D(2.797,11,19,2.797,11,19,2.797,11,19,…,11,19)<= /p>

 

309

 

79

 

5007.2

14

8000

1478

0.0

 

 

 

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