BSTJ 50: 3. March 1971: B.S.T.J. Briefs: A Telephone Traffic Model Based on Randomly Closing Crosspoints, and its Relationships to Other Models. (Benes, V.E.)

2.9 MB·English

1008 me seu srorme eimicen 2oumaL, ANCE 107.

A Telephone Trallie Model Bured on Randomly Closing
Ceosspoints, and Its Relationships to Other Models

By VE. BENES

rit wtsivd Beno 8 050

In the theory of traf in teleghone connecting networks it & on one
hand virtual nsesity, for practical purposes, to compromise the true
compleaty of the aystem wader stady and to intoduce drastic simplify-
ing wsurmptions Uist allow anu ealeulatiou to be done, and on the
father, i ie penfeotly feasible to pursue basic thooretial studios without
Such compromise and simpliication. For this rason, » spectrum of

sera sthesnstiea models for dereibing tafe im networe he eon
developed in rent yours

"These modele mange from “simple ono thal Fuenih wy ineomplete
dexeription bered on strong aachuste independence semumptione, 10
“complicated” ance that exaetly miszor networks srueture an routing,
ash yende of model has its usce: “simple” once Za easy computetion
tnd involved ove for general understanding

‘An exunple of = teful “Fimple™ mode! is the probsbility Hnear
graph suepestal by C. ¥, Laan 155, at otgnonth of elaewonk by
LE, Kittredge and H, C, Molina, At the other end of the vale, x0
‘example of e “complinted” model is tke Marko proves’ prapowed by
the author in 1968 ns an improvement ofthe “thermodynemio™ model."

‘We shall desribe hare nother "simple" mule), with a bese starting
point similar to that of Le, und then show how a certin natural rest
fion of this model yislés in many casos prostely the thermodynamic
‘model, Our presntation thus elfios the know relationships btvern
thaw motels, snd revels some unwell ney i seengthene out
‘underetnding of them by chowing how the epparently reais "oom
pllested” modes an vise through natnral and relatively minor mod-
Fcations of the "silos

‘Whereas Lee's model assumes that a lnk ¢ of the network is busy
tr idle with a probabil al inks being independent of cath othe,
‘re propose ead to aan thal ach (liv ewiteh ox) eratnpoin’
¢ is closed with some probability p. , ogsin independently. This
basis for enleleting probabilities hae the virtue of asiging prob-
ablity to every posible way of elsing swicehes physcaly measingf
fo not, We then modify the model by esleating all pobibitiot
eves cu 1098

condition! on tho syste’ being in a Hysielly mesningtal state,
“This procedure in elle ride our elec uf the irelowsnt tates by
sornalising them out, The resulting esr model oe ell the rosepoint
‘model. If, gi usally the case, every ell gow threugh the suse sur-
het of switches, then conditioning in this mex brings in the partition
Iuuetiou in natural way, snd the ecespeint model tarne out to be
formally equivaant tu the thermaslynanie one; when the iter i in
‘tm medied 0 a to take ralinie weaount of routing welling ees,
becomes the Markow proces move

‘We stecs that the suggestion mule Inne of a new model doce not
really improve our capacity to enleulste bovking, low-loss curves,
tr other prsctial item. Primal, it provides « now derivation of the
theemodynansie tadel tom simple {and strong) fst pincpls cimilor
lu thow used fr the probability lizear graph modal of Les.

‘The probity Tinear graph model has been extnsively discuss!
in the iterstury so we inule only a nesnme ofthe method: to ealenlste
the eangertion incurred by traf betcn an inlet and at cet,
etention i Secured ua the graph @ detuo by the peaittel the
rough tho network from w to @ eoasits ofall des und ranches
hrmgh ‘wbiee tome path from x tos pomee. Given aay complete
speiflation of which branches of @ ure busy nad which are idle (at 9
particular juncture of network operstion), itis posible to examine @
forage if tore fem th frm w ty 0 ny Brana which i busy. "The
tethod nox amigr rububility datatin tothe posible occupancies
bby postulating thet « Tink ¢ of G is boey with probabiley py inde-
one ofall othe Tks. The congestion fru and wie then eat
tthe probebifiy dish this distribution args wo the eveak “There
is no path from sto» compooad of ile ranches.”

We hive diserbed the probubility Tir grag male os assuming
vaomething only about certain events having to do with the graph
‘of athe for a parcieala cll from w tov, and not as providing a prob-
tbiletie description ofthe bury or idle condition of all the lik in the
network. oweves. itis entirely possible tp extend the probabilistic
description, wed mt Las moe. Zor Tinks of @ to all he Tks is Whe
ebwerk. This extension ie nacural because te desertion inboleva
fer one inle-atles pai, iv should be so forall och puis, aud Jor
‘links, Te wl of eours wll give only an icomplete stochasie made,
since it says rather Iie about whst erospoints are clos’, x9 that
in genorl itis not porible to tll wht inti connected to whut outlet.
1006 tr mz wvicaaeLuouCTow. SOCAL, uN LHL

Hoonever, the extension dos shed some light on the charsster and. ac-
uraey uf Lee's model, as me mata in the nexh parraph, and it alo
iste the now mode! tne peoposed

"The feshion in which thie extended verson of Loe’s mol works ix
clear the tater of the network, i, all the poasble ws of elsing
frospoints whether phyalealy menniogin or not, are partitioned ae
fording to the equvalenon seleGon of "basing the tone links Busy
tind probabliter are atsgoed (o dhose equivalence clases, Tn his
Situation, ib is unfortunetely true thw physically mesaingfal ond
pysiea ieevant (iereeoopic) states opeur inthe sume equivlence
bikes. Were this not vo, une emul try to remove the effet of the Im
eleva state by inviting tat all probabilities be bnken coditioasL
fon being inthe et of relevant states, ‘Thine, Inwever, does nat hove
probity signe Lo it

‘Whore jt, neverthles, «isle modifetion of Lav apprateh in
which the normalization devi for eliminating tho “ireelevat”states
an be ured." change to be mide i this: heres ess mo asst
2 probability p, of being huey lo each If, we proqmne to wssigh
robebilze p. of boing cowl to each eresspoint wll ilependently
Jn both eases, The poles ie thie it Se knosn wa: ercapeints aro
tloced, then i kann what Has are bey, but at alae sera. This
fpprosch fins the propesty of seigning probability te ewer stato of
‘the network, shyouly menaingl or not.

‘in particular, the set of muaningful state is assignod probability.
Once this ix trie we eam renrict atlention to these stats, We shall
tliminate the eet ofthe imlcennt states by cimaly normalising them
fut of sh plore, Le, 29 euleulting ll probablitiss of interest comix
anal ov Tag in fee act of roving ul tatas, The dieetbution of
probebilly (over the at 8 of phyaially mening satay) obtained
In thie way we shall eall Gir Poroaspoin” model, berstse the basie
tent to which probability iv wasiged are closings or openings of eros
points Tes simian vein, Tos tnadel might bo elle he “link” med,
Jnscase the basi probes 2 assignad tothe busy or ile rondition
of Tinks

Let $ be the set of physially messing ways of elsing crosspoint:
sand for ce Slee) be the amber of switcher ur eevapoints closed
asst a 17

in 2 Let ue suppose that all emespuinia ave the same eines po being
lowed. Ui ie lkely to bo true ifthe network tree ie uniform end
1 ther iv nethor eyoentratin row expansion ) The basic ancowsional
brobebilty wsigvel to then

"nes

o_o

wero Ca the tals number of croepains inthe network: The prob-
bist of» conditional en being in the aet Sof physically mening
tates then sero rg 8, and

at = pit

1 tw 208,
Err

or with y = pf = a,

‘his is of the familiar Maxwell-Reltamaon fora, with the fovetion
6) playing the poe of tho cnengy. The aor Familiar with the tharmno-
Udyzamie model” wil at wave reoogyse the resablenee of the above
trivesion tod bnse (quiron) sate probabilities in thet model,
‘high aacgna a mocauingful eta ze Sa probability

pag

vith |] = number of elle in progress ix stato p, and A a positive
Frntant, At we have gointad au, thi ditrbation ix obtsinot by
Insiniting the entropy? Manetional subiece ta a fixed moan saber
ff cals in. progres Exactly the wimo angumont® claractoize the
Aiscribution over Sin the eresspaint modal, as fellows

Theorem: ‘The distribution (g,, 2 «S| af probability or 8 whic wax
Sinata te entapy funcional

Hi) = Zan ne

jee he ear

er)
1008 Rapanui ayes rEeMSIOML UCR}, WARP 197

lithe Bord asnber >0, i gies

cee > Oi he clan detained snip By the enaton

tos Be

“Thus the crosspoint model dite ffom the thermedynnais node
only in St The average number of closed rosspoit, rater than
hat of ells in progeny, i ised while maximising the entropy. Ina
lrpportant clas of eos the two models formally evincide, even. This
cin he arn from the

Ccaretary: Tf ery cut goa though exactly » ovitvies, ten the ale
robes cased by te raped mada wih parameter pare ema
the come cs those assigned by the rma modal with parameter

- (Gy

For evidently in this cate (2) — a [2). The property that evary call
ots through the same momber of ewitehes iv posisand by virtually
Al the ooreeting networks ised fs paeticn.

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