ome 64 A Combinatorial Analysis of the Main Distributing Frame: Spare Requirements for Conversion to Preferential Assignment from Random Assignment Ry M, A. SCHUMER ons reseed Sartor 07) This paper preset 0 conbivaerial analycr of @ mathencatcel made Maw Distt Fran (ADR. Rel woe end eco ie cen of edna spree clad MDF toa gecervtnlcnsigeed eo ie aay es pevfrnat foram MDF doing io aventura aera {Phe sess ace wbusyantly rede teas MIDE wth ( eamvet a rihouty “The Blain Diss ing Fear (MTF) “thes aiont ia a eer= tos fies bxiling howe wi feta ee pert the exile Sater fonteeion of eae from ostel Ce Buing lo ential oor quip Sent side the billing. This primary. furetign aesouste for most fof fe terminal eanscily af iyi MDF. "The evmaiving sepaeisy Srrves «nite watiey uf oiler costcounestigg functions eanging Frat Lying tg e= Gv side plant esble pits to eress-ronnecting too tire pieces of eer ue einen ‘Phe eanvrntions) MDI ie « coublesided sto stature with pro ‘ecetors or terminal atripe rceunte on one side, and termina! strips on the other. These aro reHned ta ae che “verticals” and “horieastale? ‘ine to flee isonling orientation, The verial side i the pars of the [MDF where the outside cable is terminated. Ta hoszontal side isthe ure of the SIDE where exbles, whieh eonnee: tothe equioment of = Stra fre (nein fie and true epgpmest), are eave terme haved, To provide the masioum capacity of sntereouectione as well i coins Mesility in the emnestiay of any nite plant ep st to any entra oli conn, aiviedteaion of he MIDI cyst result in stall mimbers ef lov etoes-eonacetioss Thie een eal 2465 em nec, exarame TREMSLUAL JOURN, SHRM 1971 bb accomplished by having the auteice plsmt terminals opposite 02 nearly apposize the cent) uff eq:ipment terminsls. The MDP nt present dr handled on w cemplelely saul lass by Sememen end famerenen ko connect and dseanncot he Tange umbor at wires which seria on the MDE, ‘Cables terminated nn the vrtiea! side f the MDF are coneted to the orizonin! side by ‘wit refered co ae “enos-connertons™ oF “jumper” Wher ii tse plant terinale are not dirty epposte the enna off euuipment Lenina, thee wise 20 rim along hori= tontal sbelver in onder to enoaoet these vertical and horacnisl ter- tinals togither, Prony, wssygnment of eutede plane terminals to ental offoe equiptant vernal in oa random basi, eo stomp fi made to meke azsigmmeats wht jumper lengths are Kept to a otis! minimum. “erulore, when ersetonnections Yecome too Thumerais oon long tisy weeszaulaecxcerively, eau uumaagen ie wv exerts eapeeity Iumuper volume) of the Rerizontal heen ‘One proposed method of kesping erss-eonnceons hort fy Ube une of preftrn.in) ssixanrot, Iv thie tese of opersiion the MDP 3 tiviled Inco seve) mon, ond secon are op whit show in which ones ete seem ral alee nigarent are Yercincted. When oon- teetions age ta be mat, fo- exunle, beeen a cable pair end Tine {rel a etblo pvr Fretwell aval er nace pent ecomamica fad avatnsiity, Then, le eljment select whe ie Toeatad in the cate zoas us the wie ps" Fi Me eee! ae to at THs tnetod!reucce jumper pikap rl ine-rates the MDF eaneeity. Phere fries the querion oF law acy” aliens) spare Cemiva's ard, Uheretore, lst much centre. aie eqzipmnen: is requzed a voneert fro rauoualy coanected RIDE to « pr‘erersially aged moo of poration. Alnus some people reay fel teat x srobibiive ammount OF spare would oem He follwing gestions saow that weer the snceifod onions w elatively all amoust of epave ie =ezuized by Ua eonveesion ‘The approach wed in this anys i che following Initially, itis ‘asm that there 8 single elas of serview on the AIDE, Given the inival nomber of wise whieh eto Jrom one sone to snother, nd 8 ‘moe Tor Dre inteodurson of preferenta)assgament, Une probability Of cxhuasling xpute in a leat one none ea, be Sound exter for 8 ‘2-aone MDI ad bouded Tor the general case. Sinee running a few jumpers batreen sons isl onnsidorod a pelem, this probability {san upper bonnd on the paubbl.y of the NDP encouniering rouble ousisstonian axstiens oF nF 2467 hile converting to posternsal aseimment. For # given probity 7 MDP tnuitie, we ran, Levefory tnd at per bra on the Iinont of apie tqipmsat repi hy annvensnn. TE abe water af ove eres fs ot. known for a rshwnly emt MDE, the probity of exacting ence ot be averagad over Ue joi probe Ilkty of the mimes of exten of ene Kin ‘Wen dong wilt several slaw of serv? euch serien of be twoited net seancave MDF dite ene strutre do eis we take. simphfying sesomptier ala sn anhrer sn wien changes THsould notes thee the analysis veefermad here finds i a= sien] spare coouinad ‘lilly for votwusion te preleratal assign ‘hon, Howeser, one ty vonvsrsin ie sample all St a ated os eal arounk of spore ta efeively whist wnview changes, 1 SIATRMBUMIDS OF CRDESOU IN & BAKKE wi Ts this setion end in tas neve sotion, x sualel i ervelage tars ‘yo MDP, This se) ean he need tn determine how rnek epare fe reeuied for conversion to aeiersial asignrent by tte rs one Ine wane gevetal end moze soll ese ef slsone MDP is considered, ie heated io Hig 1, sou Tard sme 2 of Che MDW leripnce Xyund oy se plas eale pate ide 0) rspentivedy On the inside plaat side (side 1) tare age | Si and Sy FS teveion’s Me 1-Maled uf « ace BDF, i ta al i All Weer tata snes oP al epee ‘rable pe teat an the MDP. ™ 2483 au wean svarm opcwren sorawas, surreeaE respectively. S, and $ cre the nominal uoinber of suum equipment tevininalsevailable tor each anne of the MDE, inet, sevuras that she MDF is used fo eapsity (sy 0) 0s + Ne outed’ plant terminals are comnceted to eyipmient ut mide en terminals. inter, asmime thas the termioas re raadoun con nated, We wiah to Sa the probably Flv ne) 6 hsving ecuetly me Impers evnssing trom sie © of aexe | eo de Lof zone 2, end exncly ta lmperevvuering from se of anim 2 to re Ff none 1 hie tuntion i inured in Tig. 1. A‘ter 4inding a ital model to ooouat for proteeatial asiznnnl, the probaly u) ewhauning eguco ab fine sari Ue plea of me) ig enh wil be fd using gm. 4). Thie wall be done in Bectin TT Tn onder to eaoulte pine) we asa all sie way, TP 10 svne the MIDE ate equeliy kes fuzcom vernerione. ples» 2) then, ean of Wty, We, tbo torn, ramber of ees noone the MDP with n, and ng ecorovene ot the rapactive types, to We iow, el tnd Hy Ae a fab sep, wr choose as encom Ny ~ Ny ermal of cide T Uo be aml, We eas do aie in C4. 8)" ays he TaN, 4S =N,— 8, an ¥ —N) 48, In action, wo wil se Ie Hellowirg aration: S818, 95 = 8 Ny and Ty = By yu Now ll as ports Ue 2 bmsingte we exo ueing; there are ‘wigs. Shine the tn operations af shoeing sd permuting exe inde pete, he wlel muaber o” wy cones. Fie MDB isthe yoo, ofthe epuate exinralions ve We = LECT, 9) = T 8 ao News FFtoa 3) fom! ge fiona. Vrs, Jet us cow ng erin iow side To? sone 1, whieh wil azeorusadite jones fee ste O Df rue 2. Thove are CiT,, ta) dlstnoe trys tn ke the in ieee, hove aim C15, my) cstinet ways of nung The wrogous thicioe for yore 2. Now one f bas My ry unused Cena un sie O ‘sbich will weuire din eotneetion to tae renting taraical of ede Tof zone 1. Thene uae Py —nz wos te-minala an aie Lt sono 1 fe have heen taco connote 19 se © of tune 2), Therefore, there fate CUP. ming, thyme pools wy twas vert eet it fore 1 Simiasiy, tke disest warner fon irre 2 one wee fe wey S1CiPy ey, ety) ways. Tus ry X. termine Crom side T (0%, my rors goa Lona metros none 2) fw he lane ar ene CoMDINATORLAL O-9RIE OF seme 2460 heen toch 7 terminals an ide of zoze 1, Cpon chnosing the ferainals each permstation of the chosen terials eorceponi ta 6 Unique configuralion, There ane ay 1 auch permutations, Siler, there re Ny I pervitations forthe Ns cermizale of sie O o! rove 2 Since eacls of the enumeration stad cheve is independent of the otters, the totel sumer of confiqubions, (my, mf, the product [of eft of the indicia enneratione or Woy n) = EC WEEE E ~ M— dts = 8a = Wri nd C8 (moe mies, Sm EBL pinch ati Win er inn) = O08, nis aES Ss ECT IME MRoRapaLTY 07 RAIAvEMNG SANE I & DONE WOH ‘Nave auppoee that a Dsection MD das m, a8 my croeovere of th eco ypes neumed hava nth ieee to hea using prefer- Cruel resigns, Nove tha: simp tying sseuamptivas wre nade the MDP is Ae te capacity fia, aly 1 Me one plore terminals fae souretted to ceuiprent or ieee plot temo! ad thal all Tinea hee the sine ela werview™ With peeferenia assignee devine i op Keep al jumpers within the sone, Sa, 30820 have erosse ruin ore some Lo anater. Suppese ‘schenge pears fer jumper whit: Morn! re te dary. TF a9 yur is evailabo i tant aa, she change eapao: oacce funles the Tonrdany ie etceeed whch wo will suas is not slow}. However, ita spare ie avilable, chee will iy nk an pate avilebe ater the change, Clearly ther, a change fo> ich & He does aot ect the total mount of xevilahe spare. Thetoore, se op geal * to Snd te pobabiisy of ranging out of eonre while inpeaenting preter tential wesgarnen, wr aes only easider te my + ny linos ta eros the hoursenyhctwers seetons| ‘Se wo star ws m, erosnvers roca side Q of zone 120 aide I of 2470 uy niLs arm TRNTOAL SoORAL, SEPTEMBER 1971 sone 2 sad wth ne erasuvers Irom sie O of m2 to sd L of sone 3, the nnn ber of aetaining mare torasinele (ux side 1 is Sy + 2 ~ efor fone | and S; + me — my for roue 2, The proces: of Changing servi for the my + oy Hse cane desribed an © undera well starting wt (0, 0) will o1 + me eyuche, AU ears epoch (eneerponding to a change for one of fhe my + ms inns) wr rive tothe right and abe step op if cre ofthe me dines lus enge, la vep own i en othe es has a chenge. We denote the location af w point on the path by (ie r} ete he a the sight xe epoca This sitution dn dewribed in Pig. 2 The fins eine of the Fasdom wade will, in al ewes, bo (ey ee — Rel ce allstatel tg 2. Be step wp i the rantom. sralk suze tome 1 ano step corer “0 running ent of spore ard rane 2 fone atop farther Hom sunning out of spar, tae! conversely for steps Sen, The polity af exauusting spare, Brexboust/y } ia then Ue probaly sha & rally chosen pla ons (0,01 to Oy ~ te tin ng) (bere ave Clty — ny) ish paths) toutes om omesee (Se of balls of Uae Durer at fight Sy — and ele ‘More provsly staid, owe pebiem is vo dstezmine the pray Peo Sp bm meh SS bm eee) Prtextuntinc AAs sons is Append A, la pret in cyvalen to a form of Tho hall preblem® ann fee ro.utica to eur problem Seeomes Pr fost = EPO tem = 08} Ci be Sb RSLCfe a om ‘bers tho aonions are woor all “hlegane (postive, negative, aod rol for which tas eumunence ext, I me krow my uni ny in edvance this is the desired revll. However, if we dat have thin kzowadg (egg, 3 we nel ah Lo oon fi the onsnvrar? cis aenult abou be waived ly” ae join: probe Aii'ty sl aml te. Now lt ne ase fiat we have 20 prior {nformation about #y and n-. To fine the grabnbili'y af exhnaming spam, equation (5 ie welabted by pies 2a). ie, Pefeshana) = 1 EE Pe enhanc, selmi md. 6) Schetioting cquotione (8) end (4) isto equation (6) and eitplity- ing yields congpueaTonn, amanvers Fa ar 2a wa nl Pe sua R12 DEL Tle tem 61 Cle Ss baths 4c CMasnidy gOS 3b. =a iy Ls wiih. TH) J “ Pr frst} be, = wee, TY SE ENO + Me Lb ASI, 8 HHO ERS, es FARRER, A mh o XO ERB, Ae, ei At hin pst Ube ocdet of ecrmscion i iniorehanged, seaming dst with seapect tomy tama vets pe ly ny, Tv rah age the ble ovinalnwah ileiey ot Appezeae B [cuation [235] is wl with Se fia! no 2472 ome mama aveTEN neue sua, seen 8 Pe [sham Tier, acer, aN. oF . tn, 1, a) D w ‘Ae ua exacuple at te ave of this eg, se Pig 8, Tho eres ane plots ct Prfeshuast) versa anainal oceon. wore, « (« = SéN), fr 2erann ‘egal size) MLU'e vita ls, I, and 10" nee” ad ene ces of serine Equaticn (9) givat che protubiiy of exbeasting eoure in one or doth of the tie zones. 0 Sd cho probability af cxheasting i w single ene, cay gone J, we Nia to i Lhe sobebility Pr ifextamue| = EE Pe tA ealmontera nab, ome ‘Toe pobbiityoortiened om my and sex be woven ad tad eS amen ese er) ded allo probe! with Pr i, exhaust Aa Slnureall Anpenty thie ie a om Fig, Bova of etaating sce ve wet yan fam Sige age eet one sed ere tren ble tina ny Kn on E rousssusomiaL asuesut OF uy 2ar3 Pr ett, 4 SH Pare tty pylons cit nl ao CA eM 8 Gite tks gure 4 shewa plus ef eyunion [1 for songs wie somorie wav Tet an MDP ith 10, 10% end J0* Ties nd uae ele ul servi Tu the ewe of a 2vore MIDE, a oxaet enatin hus aeen fou Jor tee probity of exhaneng foare 5 232 enti MDI aud fo> the Drobabeity sbal purlva’er wie coho spars. Ar wl be abowa fa tho nex Uso eottuns, (ve prasral elution fer the qeobablley ior ftansting pode ia au M7-scetion MDF ia Aicuto tad. Instead, be emk [eoustion (115) for a pactoular arotien aa bo uged to Bnd Uinper 6 lower Soule tle ese pen Pe FZ ean = an Tn this vosion the dettbuton of erosovers for nn Mame MDF swith «single linen sever ia fone mmc te ace wi i as foun for sone MD in Berton IL ie 4 aby tt pga seme ale pcs oe wal pt ATE rn am scorank uuUSiesL JUUAMAL, ARERR ‘The sanfigur:ion of the M-mv MDF fa Veested i ig. 8. Por sivpivty, the ol emingzotation wll be usd Roth, ron Tis alls ion, ye eine eI her of eaosenvers from ide O of arse $to cide Tf yore with y= M3 dation, Lo thie ence frimber «2 dite wags to ite the MDF ie gain given by Me ut feign (1), using Ihe rove goweral defini of eae § ahve "Now we east detewaine for given malrie fa} af eroearesa sho mimter, Wilms, oF cvtiret mamnentione of the MDE thst agve (i) exaeovere of ete ype, Let us euoebeaean The a ee. Bide of this rane must =p “he ober ones with Sy reruns eon snot Lo eige Of tzeen zones, ‘The member of msn do thie We pmcrme ame Te Ee sere the eorxstign is favs = 1 fo b= A. aa Zar dicot re Igrong ace converned, N.— Sp wy tezzinale wut: be seer from oho By S = 2a. cermivals rerwisng on ade Tale may be dene in GUT, = dete Me Se na! hes Te mlctian, afer ebowsing the porlvulas trina to Fe conaetted be X, tern iuale wn eile O 0” sone Fan be patmated fx Nylons, ath uf tle lain se oned above Hg, SMa af ox Me boston