BSTJ 50: 8. October 1971: Heavy Traffic Characteristics of a Circular Data Network. (Avi Itzhak, B. )

11 MB·English

Tleavy Teaflie Characteristics of
a Circular Data Network

iy ALITA

‘Teufir tehacion én the Plover bn for date toemsesien ie sui
under womens af bey inedtng. A Aearninatc matienatie! model
Ji diaertbing tee we is deetpst ond mentor. Thy matbemariad
poten sof © tone somplemantersty ype which Rs vot een seit
Ihe fhe Brace etic! pespramming, Ax oso provadae
te loatand wit ager, tor steaming afte Boxe Te prepsze.
‘the poomtore Nei ath fruits sutton fav eee Sows en reveals
‘putty of tation inte Leming case ee pete
‘ile ttn jam wong he yn Tie poopy Wi oe Ye lated
Lip nnmining appropriate sol, lo wep agen tie aorhgete tchasior
Wl the rape when Bessy Free taltone do nat pecait en thrfare
lever ears inseatipnonn. The pap ielede eo sewerica ean
wing wae of the Zonbaniesich wipssvas and nomriad rele
Frow aimnation akorivs ance nj tn efeta af deminatog ewes wn
oocytes cotton snot pet

im

“The macep o loop network for data tenemos hon pres
reef ly hy 1K, Pies" In uel oetwors she setians are connected
{ote on tnaa line om whic we wy bio allnged, reece
fo be deliver va ua station to aroehor is stem, a: t22 senting
Scien incvatariand packets sed exorsing we airs of che reeiring
non Those parsste age thon desvee een tle mais tne, one a2
Tne, nhere tsp Hom anwind in alld ei cieetion, Me saldeae
ff mich peekt shee 9. rh station nthe ay unl then
ie reese sition slur i i peravea from the rein tte. Tease
fu che tain ine eoneot he deayed: shorefors, a stulin en deliver
‘anak ant the zi vin alae lise on shen pezaitted be tie essence
‘agap acta or won weeviving» parkel fans che main ix, Principal
2902 a Au wvsunny rece nweaL onrmSat, ACTON HFT

Tuatoess of tae spam ry be expliond with the ald of the four
salinn seca show slemaells in Fig.

The four nese nauoture revel: tron th conte aie and stone
boty every lin he four passtearrying eomprinonts nthe ere
cf the grt gir aligted Ech tke four stoioas. Ducing exe «stop
trek stiion io ble tv taack the content of the aligned eomourtrsnt
Tithe romyacsanote He empty, fa meio ear Ise Li pe
Te fone few packet i the eampostmont, itil sl be reauernlnlere
iti ners so the aid station wrk xan Pe aio la pei
to Ton the eomparemont gui after url ihe A Pepa ln ean
Iie represented By the mechanic] sualag rliate abown ip Fig, 1;
however, the murchor of valving armen the stmuctuee ie nc nacos
favky eeual toe nuriber uf sal in te op. Teather, te ime
fof armas determined by shel nn f te aye (le Un ee
tow a bit to comp eo se round wo the loop. uiesnee af the loop
tim fe dscusos i mace din Been ¥.

“fs tas Pree loop ih i rirmulis aan teat zo ence contd
fs applied sad wad sation eves to ard ies mesonge Ht Us our
“Thasetan, 4 inn wl never mis the apportaily nd « mae
‘zis ales Thom ae 9 weamagee waiting for delivery a he sation

"This papi pieente a study of the Host earwsersics of wah
sostem hess teal, i, Wen te system Het able to deliver a
esatete, are ino yumins Yall up a= ser etetions. Iw Sees TL
the rosion n “stale cations” tor the tate ows is Finale

Fle 1 a shovat i
Loe so Ht 8

ucla wathercainlly ‘Tse cose of w olay saturated syste ie
analyzed iz Sooo TIT shee the basis is Ini tar the matermatical
theclepinent of she ood wal lt sorta weve in Section IV. A
Aecle’ decxiion of Piers Ieop for data tranmiecon ie aves i
Seetion V where the sllewalng priitien eect due to dominating
fate of stasis is laird by siulation The section sao diseases
‘te impor asovel a? the oder of stations iz the loop. Headers at
intersted in the mulvemabical elehorations tay akip Scetion TV.

‘We esmmes thas the dnvtiow en the min eirwlar nae oruesee
okie, Thorr are sation, ete Use ena Ts, mare
from: to ein euntuctkoriee inoneceing onder, The sugmnrt ume
Tine betwoes the th eh the {| Tiealalane = 1 doonn ct Ty
is alld the Sta breh, Shirly, es oh brea i the segment bee
(Gove etation ward scion 1, Tot, be Ha oruportion of Ua (oaeaas
feng Eta sn Seta fe stria §

prea a

Tee Xm square maceis P= |p| possess all Ue properties of
sotianiy moiris, Note, however, ligt she cements of Pare nt
neesrily probe,

"The enna a steinn 8 given he whieh i Ie meer mon
st Tw fyuckela pa tne srl governed wt the station. The enpaaity
‘i ono Ronn ele thats, eqs mie. ie capable of carrying
te maxis fe of sae eit, Tn fie shai dewription yer
Fig. I ssun that ote full volition tex: fur tines unite Bn geal
iu il take The number of eve urs wal ty ee ale rs
im ceity ef szeh brezen wil thow he ore yelp Fee wl.

"The avetaye Howe smering from elation #1 be ds

sd he vonage hve bra by, - Clete
Wsa Sa. FOL aw e
we
Nash FL erm ®
Tots Te t's soon of How emer from station J ond Mowing

aw then

a> Eps
Deed yun rs, sversse mremsiuaL suuusés, wowoHiM BF

_feenteaena HE og

lerisna, Layee i
eat an sy Oral hn edit es
Cnet) Fetes tended

reazonable applicetins p= 9.
i average Low bine anny ne i xtnee aHe fano-

Mion af maya 2e

nn Sew

THeocy which isa Senile slusen for Lae avrg Ma mse ay
retin 2) an @). he et ll Rie ad ue tren coe
iv in enor polybedeal et, eam anntze, ond selaet a
prtenln galuciee fon shi sok vs ai gies ableetive

Tr tle irene netorkesugsestal by Pier, never, [in i

entra eontin. Hatter each station ie srving to smaxiniae ty ase
Mine onta the maiz ius Hur this ome we dire w lable lution we
sv selmien som tie tbe mystien ila scar ethene

inaerceatioa, Suppose abon tive 7 feo etable selution. Cary =

aur iia rola oe (2 0 (0) andthe adavioral coacieoe. that if
fem Stu guet 1 llna ey — 8, Uo em teas tea ae a neczeey
‘condition sevume Hts gv & Lond rf <8, - However, station $ arives
fo maine ie fe i on fone We long a8 272, and p=

rte, ibe aut pssble hal p< Carat < 3. Thaadditonal
faction se not goncrul’y igi for ceraring chat 2 ie a ota

ssluten,
Trsuticoney ie hoa devon raed ya simpln xomacee. exemple.
Supper LS. he = 280
o 1 a a
pet ooo
oo ot
|
lo o a a

Tiss tn voi shat 24 — 10,0, 1,1) anes X7 = G11, 05 aa
sity ations (25 {0}, and tag addtional necevary eon ve
fee 8 aX" (at toe Sa SL Lor saplcty we assure
sriseause 2505

that the ows kenerored et the stations ete determininie fn tne
tnd coh soon generates exnetly 10 pete ons fn ani

upore now thal the system {ote Fig. 1) has one stating sera only
Ti such a eoee thes ane only tro ble lutions mama 8% Ge =)
sou 3","(@ ~ 0}. IF the system hes two rotating arms, an ditional
ube soliton, a = 4, ie added. Ifthe system aus F arms, there exit
1 seablo rautins, myie,on ~ OL, «=~ RT ayotem bell
elle forte sabe slain ce ~ ai Hat Gime tno enna
feniain packets From alain: 8 ad 4 wid & — Ww eorumrtinens conte
passes foun stations 1 dl 2, The reeessers condition for stilts
in aso a autisient vanditin if the Moss goremlea iy the stains

fee eontiuous fas in the tare nf annie Mier sal tis
uch ac vehiee wd w Jonp eonesting of pipe oe road). dn each a
foe th su! feb slitione 5 ntl the senvex pohedea
Se gv oy eatin 28 and 6

‘We veh to find ll lutions whieh aatety waetione (2, @), ena
the neces slay eunditinr (er ll he 2 the stale altinns
in tho eave of a syotemn veh eumtivanns Ms}. Que yrublem ae
redefined a= ono cf ting ai Fable vastions tothe fllewing st of

we Bh HB BO

hic Yaw mscibles a nee enmpleneataei paoblem,* where nd
spans shack va abs, easinte salienn 1 the set of equaviene (7)
ie anion sulucion se se ace of ations (9) ane 0 sree
Toast of the rials mint yi 2-6

Vorsuficonty-langevaluneu Careamnsle sy = Af Loo sa,
all the branehey are stormed wal gu = pS = 1 2 ms The
st enuctione (7 tales te forc

Sow a1.

“This. tm sovot ta tmeqnivabat tothe ae of tm | 1} nse ecantions
2506 maw man srs nacaimet sows, aes I

eae eens (9)

)

‘The fh eeation in (Qu) ie bloieuble by subtmoting Use sh euuntion
of et (zn the (6+ 19th ation af ses (8). Fguaion (0H) aay
be ovleted ae any nour combinitign uf oxstions (S, Nate text one
‘nation in {a} ia zedundentsineo Sy, py, — Tf all

Remark: The physical interpstation of eatin (0) i Coa how
in compen satzeation cetation ia able to deliver only when soseving
"The low sang from a given station murt equal tke Sew entering
1 elation rou tse mein ine. Taie equilibevan relation “2 exmorsed
Dy the m equetions of sat (0a). Te (v + 1jeh equacion, (tb), expresces
te eanaeiry lancation of the branches. If we ecleet = 3
welave

aden co,

satre és tho avenge number of bramesee Gistaree) trweced by
cae emerging fom lair j Peary (10) aes th the erie
ji toms parkole Fibs diane) demanded fn tho sytem

or tie ar md eid, sie all brow nr a4 ec
ravertes one pace per timo art, Ta cnt “orm ae hee
wee,
Ena-e ab

wwhove 27 je the tesnapons of X, (Once thee al vectors are dafined to
te ebm voces,

Pin a sceohasio matrix and, haters, the pnoblem repraserced
by equaciens (11) aaengly rezembler one oF astarciong tie steady
stace probablitien of a Tite atte spare Maroy chain. Me diferune
fathatn ce Markoy chain proleun dy ~ bora j hile inne lle
thie i not neeentariy ao, Im the flowing me aball rk ae of this
seanblae,

Daft:

(Sal be used to abbreviate Vatason 1”
(68) 8, id to be acoesbla fom S., 8, 4 8

it then enna
oH sorescane 2K

sequence of eeaneats of J auch that FyPaas “°° Paros Paes 0.
Tinh onset ie also customnury Woy Thal 5. leads to S;

(ii) TH. 8) and S, 1 8, both slats ae 2aid to comments
(8, o Sy). Olsniy # Spe 8, and Sy "2S, chen 8, oS,

(G9) Tat G4)" [8,28, Sih Cleary fe CC then CG) — C6.

10) A nomempty ease of stetions, C, is alles communicating

lace if for gorse station 8, # C, C — OG. Te follows that exo

‘communicating hows ain either identical or dij.

‘A oornmiaiesting cher is elowd if nn stetion outside the cass

{is anvil rom y sti te ela

(oi) A closed communivating sess shall be eid dumioating clase
dre elss of dominating stations, 4. sttsion not beng Uy
‘dominating das i dvaiaated statios

roi the theory of Marki eins we know thie thom exists at
Feast one ease of uninating setione ina given Pere loop. Hoover
itis possible thae thers wil be vm denivaled stations Is euch a ese
fll thesvetions ae dorsingting wn mn fons ino axe oF wore deminat=
jg ears, t-moy be sown Ill he somber of dominating classes
one if and only i there miss # station aeczetibi from ul wor
FHotione ia be loop, Tn the vace tat dousinated seri sie ch
ur oul tab Ls we dominating sation,

Tc domivalng ears of steions x repnesesal by a pineal sub-
macs of P fe domivating axomacrs), This sual is obtainable
by deoting all rows end enkurane of 2 eurresponding Ls stations noe
belonging to the rina eles, Simul, 1 dovainaied
stata may be roprecalel by une esbmatre of P.

hooray 1. Let B be a % & oun of Prepon a dmsivating
ase stations, den oft wet Tus the equation

vp-a ata)

Joon a iencar epace of one dinseion. Hattermre all he levsote af Y
at hace the ame ipa sy ile a poet, ara negate, le

Puof, Theory of finite Mackov chains (eye, Kemeny’ and Snel,

Carters, Asean Mot the Aemnating las sepregniel ty Riv =
Sey Sey co-sSq) then thce este a insque eee 7° ~ (ye Ve
5 basing epton (1) mult sei ogantion

a
2528 ym RRL TRL ZELULNICAL JOHN AL, GOBER ET

Clearly + > O and any aotution to eyution (110) may be obtained
by multiplying Y* hy some real number, We sine w veetor
Gite, urd wheres, ~ OE Sy¢C,and sy = ysite— i, The
Xeclor & igcalodw dominating snlution vo equations (1) enresponding
tothe matin wid Ve las C

Thenem # Lal Q bv wk XE mabmasis of P repreentng dominated
aon, th the cnt onbsion to YQ = YF aa Y ~ 0,

Proof, "Uhcory of ite Markow chains,
Theorem 8. Swen that We wiatrés P conve exartly s dauating

Inbwuies Bee Beye, Ha repeomnting daménating elses Ch Cy

(Cx loupe that X7 isthe urinating aluion corresponding

(eR o = Hoe sme ia lie to equations (8) sf and only 4)
x- Dex,

oa

1 as

‘Povof, ‘This 8 wn ioenesies result of Thro | and 2

We conclude taat in tae enae of eomplety wuration. fo) = ty =
1,2, cw tle alisagn eer a zeanezaive sutinn lo eur aroha
1 ehere ls rly um seninating eles oF etuion the Lp, taare
cxets a urique soution te equine (6). Tf howe i more Than one
{Homing less in the Jo thew thers exist iniaitly emany”nen-
‘oxutivenulims obeaingble ay vine enbinstion9 ofthe dominating
selutions

‘Retuining tothe more ners ener, we decribe in tie sostion the
Io ondtitt seceecuiry ur sulving Whe get of equations @) whon
pe bet 12.0 i The ago ie based on the results obsainesd
i the foregeing anulvs, ous that the eapaiey of the Branches,
fe moy be varied Wolwson D and 1, we stort with + = and inetease
St until ¢ — 1, While ws doing we simaltsazousy leod the rye sd
‘iain the fasble solutions for any given value o°0 + = 1

‘We alll tent by outlining the procelue for Finding just one scltion,
Looe stew 259

Sorts 0,2 = 08 = Ae

Stop 42 Inerease v by 1, Tr PO! fod a square subpar: BI sopre-
senting © dominating clu» of stations O°", 08" = 8,
Bac Sub

Step 2: Tn the waique positive vacter ¥"" —
sty the tof enters

creyn = oy

re
“Tusdetermsne the value of dh, woileeh Cau Ma he 2
nd td, =

TKolargs Pt che fon 2"
Oi 8, 20 and 2

shove the misimiastion ic yer al ach that See
Tat ged each A" = 1 gna 8 ~ 5, and gate hart
Stop.” Gthere’ae inewaze chy 3" gd canta.
Sing 4 (RS Ey nes Nm and go to “Last Stop." Otkervise
onatruct the n> — 0 x fe —¢ | 1h equarecaeis
bp adding hos anlumn ot PO vais Gh | TIE eve i
isthe st veh, ie aded tothe tone) and thon desing,
fue il evi and tha jth tow, Mos = ae aa
Fea aera ene of rk oration, Gur baa ta "Stop 1.

ast Blas Th slain i
we Davy ag

aroP.

rosadn> Acecibat above wil alone yt! a over

‘reece 4s The
Ge} ea mand met

Proofs 3 geass ates 2° ere saute, Theron shore aly
triacs at Just one duanmting eubmatris of 2" deneted by BP
Sine A" Bund a!" 2 Oren X~ De, aa = 0. um te

1 raed for dosing 9 aie Ape
2500p oun erst aMHstOne JOM, ano EH

agovithm we have x — Sok aN XO fA = 2". the algo
Flin als engnes that STR hg = 1,2 00+ Nand therefore X 33.
Sinos P is w Ue numberof ieruious ean exorel

Theorem § (Exisons}: X ebixaed by sr Laaond Shih yrenby
Se fous i i ain (

Proof oe enlange ¥"" by adding sao clamoats corrspondig to
foams of P° mnt saeludrd ia. Wan enlarged reer, denoted
bby Fie dominating eortion to

ve ay
zw

sehere 1 the sob of indies of cou indued in P", Theve equse
tons ar equivalent [ace equations {8} ee to

as

Lotiyat, tek ag
Thincusily rite hy he une of eeustion (4, shah
ap =e an

“The vector X©"inchtained by acding ta Y*"2em element comeapend-
ing to oolamag of P not inladed in B"."Thanefore

ae Ean = a!) forall cet, any
From che defvition of w, eguntions A) wn (61, we Rowe that
duis 8 eveforal jeseepe i ~ ¥, (wheres —1= wil = DIETER,
ten 22 5 0, 1 Gallows

se) Sah sa, ioral oat, a

Suuuming euuatione (18) tl
ing oustion (11) 6 obtain

ih ema vowed en oxbetial
ven

sek, fowl r= 128,

From dhe wzortin we ew On ie, fall =,
dos Otierviee 2, — 2, « Tha completes the proot Cor 3