OPTIMUM NETWORK STRUCTURES WITH RANDOMLY-DISTRIBUTED NODES

Transcription

1 ITe 12 Trin, June 1988 OPTIMUM NETWORK STRUCTURES WITH RANDOMLY-DISTRIBUTED NODES Gyula SALLAI Research Institute f the Hungarian Psts and Telecmmunicatins Budapest, Hungary The intrductin f digital techniques int the telecmmunicatins netwrks tends t mdify the netwrk structure. Hierarchical and nnhierarchical netwrk mdels based n randmly-distributed ndes are develped and analysed. The paper presents the ptimum hierarchical netwrk structure with respect t parameters expressing the technlgical prgress, and the ptimum symmetrical nnhierarchical ruting scheme versus the density f tandem ndes and traffic lads. 1. INTRODUCTION Penetrating digital transmissin and switching facilities int the telecmmunicatins netwrk, in rder t utilize the benefits f the digital technlgy and stred-prgram cntrl, the netwrk structures, fundamental technical plans, netwrk planning methds are recnsidered and revised. Many studies have been published investigating the structure f natinwide and metrplitan netwrks [1-5J. The future natinwide netwrk structures are expected t be much simpler. The number f levels f the netwrk hierarchy will be reduced, cnsisting f a lcal and ne r tw transit levels having nnhierarchical traffic ruting within ' a hierarchical level; subscribers are cnnected t the lcal switches r their remte units. The netwrk evlutin studies are usually based n the use f extensive cmputer tls and rela~ed t a part f netwrk levels and a gegraphically limited prtin f the netwrk cncerned. The general cnclusins can be drawn frm a set f the case studies. The apprach used here is definitely different, lking directly fr a generic respnse f the netwrking prblems. The generic apprach prvides rules and guidelines fr netwrk planners, fcusing n certain planning aspects f the whle netwrk. T btain cnsistent evaluatin n a whle netwrk basis and use nly a descriptin f the netwrk as mderate as pssible, prbamdels are established. The cmmn bilistic netwrk as

2 sumptin. f the presented mdels is that the exchange 10- catins (ndes) are randmly distributed n the ttal cncerned area. Thus a prbabilistic descriptin f the varius distance characteristics can be derived, which gives sufficient accuracy fr ttal netwrk cst calculatins and an pprtunity fr the netwrk ptimizatin. In the bdy f the paper tw such generic netwrk mdels are shwn fr slving netwrk structure prblems. Fr hierarchical netwrk structures an ecnmetric mdel fr analytical evaluatin i~ presented t determine the ptimum number f hierarchical levels, the expected ptimum number f ndes n each level in the least-cst netwrk structure. The netwrk is suppsed t be star-cnnected, defined by structural parameters expressing the number f ndes fr each hieararchical level. The csts f the telecmmunicatin facilities including transmissin, switching, etc. are described by the Grsch's law f the ecnmics f scale {6}. Taking the average fr each level the expected ttal netwrk cst is derived as a functin f the structural parameters. The mdel is available t investigate the inherent relatinships f the hierarchical netwrk structure, and study the impact f the technlgical prgress thrugh the change f the parameters f Grsch's law n the ptimum netwrk structure. We nte that a prbabilistic hierarchical netwrk mdel t cmputer aided evaluatin was presented in /2J, which uses mre sriphisticated descriptins fr the demand distributins, cst functins etc. The results btained here are in cincidence.with the cnclusins f sphisticated generic and specific netwrk planning studies [1-5J. The nnhierarchical netwrk structures are fund t have great advantages in the upper levels f the lng-distance and the metrplitan netwrks having mre hmgeneus distributin f traffic lads [5, 7, 8l. Fr these applicatins the symmetrical, tw-link, sequential nnhierarchical ruting cncept with crankback is a natural slutin, where in a fully cnnected net-. wrk all circuit grups have the same cngestin (verflw) prbability and every relatin has the same number f tw link alternate tandem paths. Suppsing a symmetrical distributin f traffic lads [8-9} a symmetrical nnhierarchical ruted netanalysed t determine the ptimum number f the wrk mdel is path~, versus the density f the - ndes (the relative number f

3 ' hdes n an area), the traffic lads, the permitted pint-ta-pint cngestin. The k-th path is suppsed ' t be the k-th shrtest tw-link tandem path. We find the expected rati f the lengths f the k-th tandem path t the direct path, and evaluate the wide range f practical cases. 2. HIERARCHICAL NETWORK STRUCTURES Let N be the number f ndes (eg. switching excanges, cncentratrs, crss-cnnectin pints) ri an area f size W. The ndes are assigned t ne f s hierarchical levels and star-cnnected. Let a nd~ n the i-th level prvide 1,2... i-th rder functins, and link with fj-l ndes n the j-th level (j=1,2... i-1). Then the number f the i-th rder ndes and areas, resp. are: s-l n. = ( f -1) II r 1 1 k=i+1 k ' (i=1,2... s-l) and n =m =1. We can write, that: s s s-l N= lr r. = f- n. = m 1.,i=l 1 ~, l where m i are called structural., parameters, r i are structura~factrs. Let fo=n $-1 dente the unifrm structural factr. The ttal netwrk cst is cmpsed as s-l C = L n. C(i) 1 i=l where C(i) is average cst fr an i-th rder nde and its link intercnnecting with its wn superir nde. The cst f any nde and its upward link is estimated accrding t the Grsch's law f the ecnmics f scale, using tw parameters in the frm f: ( 1 ) with arbitrary cnstant h. One parameter, p is related t the transmissin distance A,,ther parameter q is t the traffic capaci ty )}. In general O<p<l, O<q<l; Grsch' s hypthetical values are p=q=o.5. Practical values f the parameters p and q can be derived frm the analysis f actual csts [6]. The investigatins shw, that p and q are decreasing with the technlgical prgress [10} (Figure 1). Taking the average fr each level, suppsing that.a and V are independent we btain: 2.1B.4.3

4 T determine the dependence n the length, we suppse that the ndes, even the ndes f same rder are randmly dispersed n the examined area W. Thus, the distributin f ndes superir t i-th rder ndes has a Pissn distributin with mean value At l=m. 1/W. Then, designating the nearest nde f rder at '''''1+ 1+ least i+1 as a superir t an i-th rder nde, and denting the Euclidean distance between them by J. we btain:, 1 2, - b 11rX /- 1+. Prb C )i <x) = l-e i. e. J i has Rayleigh A =0. 5' i' where = P"i/02, we btain: distributin. We cnsider the actual length 0> 1 is cnstnant. Then, wi th /\)i = EiC,A) == Li E. CAP) = l"_4) 1 1T s -1 p.lf k=i+1 where IC.) dentes Gamma-functin, P cllects cnstants independent f i. T determine the dependence n the capacity, we suppse that and incming traffic lads f an i-th rder area the utging are prprtin t ' the number f the ndes f the area cncerned [ID}. Then we can write as: i-i E. C )) ):: 5. = 51 If r k 1 1 k=l We suppse that Ei C )} q) is in prprtin t 5 q, because the number f ndes in the same rder areas are apprximately i equal. Thus, intrducing a cnstant Q we btain: i-i E. ( 'Y q) = Q. If r k q 1 k=l Finally, the ttal netwrk cst is given as: s-l -q s-l t C = H fu. (ri-i) L 1 IT Lk k=l+l where t=1-q-p/2, H=h.P.Q.Nq. Nw -we can minimize C with res

5 ITe 12 Trin, June 1988 pect t ri' taking the cnstraint (1) int accunt. The ptimized value f r i will be dented by Ri' Firstly, we suppse that r 1 =r 2 =... r s _ 1 =r. ' Then we btain R by slving the fllwing equatin: Rt 1-q + t 0 if t10 1 Rt-1 = R -1 In R if t=o. Accrdingly Ra depends nly n p and q, prvided that.1 R> N N-1 (Figure 2). If p=q=o, then the ptimum structure is indefinite. Under a given s,the ptimum structural factr Ri is given by a recursive frmula as: ] l-q. R i - l B 0.2 Switching TransmiS5in Figure 1. Typical values f parameters p and q. (Pints A and B describe link and line csts resp. Pint C represents ttal csts with high technlgy.) Figure 2. Optimum structural factr R versus parameters p and q. I f R 1 = R 0 r t = 0, the n R. = R. l' I f R 1~. Ran d t;> 0, the n R ~ R. 1; if t < 0, the n R.:;; R. l' 0 pt i m i z i n g s, the glbal ptimum is derived as Rl=R2="'~' and 2.1B.4.5 -

6 lgn Spt =min(n; 1 + l09r ) It has been fund, that the least-cst netwrk structure is unambiguusly defined by p and q, thrugh the ptimum unifrm structural factr R. Using this generic netwrk mdel, we can find ut the impact f the technlgical evlutin n the ptimum netwrk hierarchy. With the prgress the csts are less dependent n distance and capacity, p and q are decreasing (Figure 1). They have ppsite impact n R (Figure 2). Ultima telyj cmparing the figures, the prgress results in an increasing R, i.e. a decreasing tendency in the ptimum number f netwrk levels. Fr the sake f cmpleteness, it is nted, referring t [lo}, that when the higher-rder tandem functins are ' prvided by separate ndes, similar expressins and cnclusins are derived, under greater values f R' 3. NONHIERARCHIAL NETWORK STRUCTURES Nw let N be the number f switching ndes, fully intercnnecting by circuit grups with an verflw prbability b. Let n be the number f tw-link tandem paths (O$n~N-2).T find n' the ptimum f n with respect t the netwrk cst under a given pint-t-pint cngestin B, we suppse: a) the ffered traffic lad is A erlangs in any relatins; b) the sequence f chice f the tandem ndes is determined accrding t the length f tandem paths; c) the verflwed traffic cmpnents can be cnsidered a. randm. traffic. Thus the relatinship between band B is given as: where d dentes the cngestin f the tw-link tandem paths. In case f k tandem paths, the traffic A causes lad n 1+2k links, altgether as Mk=A.. ) k 170 di k=o, 1, 2... n. Due t the symmetry cnditins, netwrk cst is determined by summing the cst f carrying MO=A and traffic lad cmpnents Mk-M k _ 1, k=1,2... n, as fllws:

7 C=A(_l + a O n L k=l 'Pk -2- (2) where a k dentes marginal traffic capacity f. the links carrying cmpnents Mk-M k _ 1, ~k expresses the rati f the lengths f the k-th tandem path and direct path. The traffic lad ffered t a link in average is als Mk under k tandems, therefre a k is apprximated as' Mk-M k _ 1 a -=---=-- k Gk-G k _ 1 k=1,2... n ao=mo/g O ' where G k is the number f circuits, calculated by Erlang's lss frmula under lad M k, cngestin b. Fr calculating 'fk' let c =2"O be the distance f the surce and destinatin ndes, ck=2~ be the length f the k-th shrtest tandem path. Then the tandem ndes with ~ length c are lcated alng an ellipse with axes k ~, ~=Vcl2- ~, area 7: k= (l51f, ' andr--;=====::;:. f - c k - J.!. ~.!. C 7: k )2 k - c '1f.. 0 T. k=1,2... Fr a generic mdel, we find E( Y'k)' the expected value f 'f. Suppsing ' that the tandem ndes are randmly dispersed n the examined are W, then they have a Pissn k distribut in wit h m ea n )l = ( N - 2 ) I Wan d the d i s t rib uti 0 n 0 far e a lk' ha - ving k tandems is derived as: ~ (}l T)ie - ~T Prb('tk<T) = 1-L ~., = i.e. ~k has an Erlang-distributin f parameter k, with mean 2 T k=k/p. Thus, intrducing v=t] la 1r The E ('P k) ';; ~ k = J ~ + H + (k. v ) 2 estimatin f E( f k ) is asympttically undistrted in ca-. se f k. v --0. Applying ~k t (2), we minimize C with respect t n, under a given A, Band v. Eg., if v=4/1r, then ~1=1.42, ~2=1.82, ~3=2.l4, etc. Specially if v=2{3/k then ~k=2 fr all k, and C expresses the number f circuits f the netwrk. The ptimum numbers f tandem paths, n are shwn in Table 1, cllecting the values belnging t 8=0.5, 1 and 5 % int triples. Analyzing the table, it is fund under A> 5 erlangs, that =

8 ITe 12 Trin, June 1988 dubling the density f the tandem ndes results in ne additinal tandem path at mst and the case f unifrm ~k may be inserted between v= ff and 2 f3. I t is fund in practical cases (20S. A :S er 1 a n 9 s, 0. 5 ~ B=:::l %, 1"S v::;;4) t hat 1~ no 'S 2, and a p ply i n 9 nly n=l, the extra cst is, if any, less then 4.2 and 2.3 % under B=0.5 and 1 % resp. Additinally, the design and imp1e- Table 1. Triples f n (B=0.5%), n (1%), n (5%). A(erl) v 2{3" 3,3,1 2,2,1 2, 2, 1 2,1,0 1, 1,0 1,1,0 1,1,0 '2{3/k 5,4,2 3,3,1 2,2, 1 2,1,0 2,1,0 1,1,0 1, 1,0 {3 4,4,2 3,3,1 3,.2, 1 2,2,1 2,1,1 2,1,0 1,1,0 4/'[ 5,4,2 4,3,2 3,2,1 2,2,1 2,2,1 2,1,1 1,1,0 2/11' 7,6,3 5,4,2 4,3,2 3,3,1 3,2,1 2,2,1 2,2,1 mentatin f the nnhierarchical ruting scheme with n=l are simplified (eg. crankback cntrl is eliminated) and the mst favurable verlad features are prvided [8, 101. Therefre, n=l is generally prpsed in the symmetrical nnhierarchical ruting. 4. CONCLUSIONS The netwrk mdels based n randmly-distributed ndes are apprpriate t find theretical relatinships and ririentate the chice f the structure f bth the hierarchical and nnhierarchical netwrks. We pinted ut the dependence f ptimum hierarchical structure n the ecnmics f scale and that the technlgical prgress results in a decrease f the ptimum number, f hierarchical levels. It is als fund, that the intrductin f the symmetrical nnhie~archical ruting is favurable with ne tandem path, taking bth the netwrk ecnmy and prtectin as well as the simplicity f technical implementatin int accunt

9 ITe 12 Trin, June 1988 REFERENCES 1 Larssn, T.: Technical lng-range scheme f a cuntry advanced in telecmmunicatin.3rd Telecmm Frum, Geneva, Sallai G., Papp Z.: A statistical methd fr ptimizing hierarchical netwrk structure. NETWORKS' 80, pp Paris, September Rsma, A.H.: Optimizatin f digital netwrk structures. 11th ITC, Kyt, Olssn E., Rudberg A.: Impact f fiber ptic technlgy and high capacity switches n metrplitan netwrk structure. NETWORKS' 86, Innisbrk, Flrida, June Clse, W., Darling P., O'Nell, P.: Develpment f the Australian integrate digital netwrk and strategy twards ISDN. ISS'84, Flrence, May Ellis, L.W.: The law f the ecnmics f scale applied t telecmmunicatins system design. Electrical Cmm N.1, 7 Ash, G.R., Kafker A.H., Krishnan K.R.: Intercity dynamic ruting architecture and feasibility. loth ITC, Mntreal, Krner, U., Wallstrm B.: On symmetrical tw-link ruting in circuit switched netwrk. 11th ITe, Kyt, Yum, T.K., Schwartz M.: Cmparisn f ruting prcedures fr circuit-switched traffic in nnhierarchical net,wrks. IEEE Trans. n Cmm. Vl.35. N Sallai G.: Digitalizatin f telecmmunicatins netwrks Prc. f ' PKI, Vl.2.Budapest, in print. 2.1B.4.9