BSTJ 40: 1. January 1961: The Covariance Function of a Simple Trunk Group, with Applications to Traffic Measurement. (Benes, V.E.)

10.8 MB·English

The Covariance Function of a Simple
Trunk Group, with Applications
to ‘Traffic Measurement*

By YE, BENDS

(etna sre 8 1

Frdenss closioa! mate for telephone tae a lose system i eane
san rodN ante, calls anvci in 0 Polau pres ad eget exp
Menlad ating times ele tle cana! Se acted ef ones are demtsid
lett retrial, Let (1) tthe nner of tra Sent at lw exp
fowua or tha eosariance RK) of NU) ti forme of the hava
ates of the tawton mie f the ABortwo prooss No} Soto
“inn 2) ie enone paly im rv uf eonataate and he "reser
Sanction, ie the tention aroiabnty PAN(t) — NNO) = Nhs RO
fe accurately mppearimated by RIO Ie, wih rhe Large nate cr
federico wl apprasimad (underetimetad) by ~ HN

IU Rane wel epprerimete Joroas for sampiag errors ae mans-
feemend ve ee ro feat rout.

A theeretcal sniy of pling dtations in telephone ne meus
suontene i use both desing poeeedires for easing eric
Toads and in iotepreting Feld servations. Hywant and Palm’ have
fiven an nponoxitate ferauhs for the eempling oe incured_ when
tservations uf the ounsbers of eal, iu exitere are made ee ied i
erate of in Thee fran Yas the dst that 3 derived
for probe mel fof the tote) sv shih tere ie a infinite
‘umber of awsabietrusks, Thue there fn Timi to ebe number of
fei ean ho jogos te Kim, and no eongestio, Two i
Juttant poctanelery te uurbor 9 af raksin the grip, and te prob
oii pul ows ane Ice of arent. Porch wson the practical

snl myn camped in lilo he ahr we sing ene a
118 rm ante arene sxc TOMA, dats 101

pplication of sis modsl's wsully reset to large groups of frank
sich ae lightly fonds

1m this paper we cere and stu the covariance nstion of the si
ple stochastic model fw finite group of trunks. The sampling ear
In trae monaremants cat ho eeulted exsety fom the eatin
‘We fing formals to" the magia octane of ened “ath
for borh peiedie aud estintnus cnervtin, ‘The eet formas le
to staple appeosimatiens sul tTlapsees hick tke aout of
the munber nf Iruks, Out tesllsne susvarived el teu in
Saucon

‘We sal use A, Eshrg’s oh
of tn, described ae fol

ie Holding tines of unks are motually epee, vate with w
agtive exponent cistibation, ‘zw i mnwwaned in nts of mest
holding tie

i. lipoens a hich slle arrive Zorm a Pesan proses of dates
fe > 0, inepenslnily of the Today tina. ‘The foe tl Hehe
lene

ik Thore are V2 - taurie; calle whieh fal all 4 ofthese trues
Ino or", "ane are clare fy ye,

‘These aeumapions decernine a Markov slachtstic prooees Ni
na ey cay the rtmher of tanks in use at tine 1 anid,
“tep nstion fastating in unt steps et wees: 0 aid A swell now,
No) hea sletionesy prebabiitis (poy m= Waly 8 given by che
(rst) slang dissin

ca peobabilstic mel for «yew

wo

= equi prbuiy k (als ae as

With Mis stro of alate prcabilite, NC) ea eietly stationary
rovees, eos mean ssl ware ze sespetivay

m= elk — psi
Fm im apy = wh

‘The robebilty prof los eshowa in Dg, the fvctional ocspanry
2 i Fig. 2, sd the varias is Pig

120° em wn evsnmae eunteaL JOURS, aANCAMY L9BL

seaman (10)

Sai

‘The covariniee R(t) Vebveen samples NCDAV( of tho slorhaatie
preotee HC) he the average af fhe pendust of D7(8) and Ns}, mint
{he produc of the averages

Rita) = EN@N(A) — BME
Since N{-) isa etaconary ral proces, we have Riéal — RE|¢— 1
‘he function (+) i led ts ecarinsefaretion a the process)
Ccvomrasen HeNOHON aF SBME MEK arom NT

Ivan be writen
INGE AND] — BIMUE + DIEING

‘where fpet ave she scationary Cor monn} proba give by
(yc

(0) = tin

Deane) = 9 [FCQ) =

denotes the tension pbb thal» tmake are busy wt ine (8
‘pete busy tre ®, he funetion 2-1} express the avg: depen
reo or eartlaton between ste nf A+) saan ab ee Fae

"The yrinrial pracien iw ie eavarianee fonetion 2{-) in tho
‘wong of telephone fri isin comping (hewretealetimstes of sten-
pling evr incusred in fe Load misaurements, "Two methods of meas
triage, the aedk et andthe fimeaverage, are considered in
thispuper, Pyle aitecount,» obeervtions (24y-">y 4.2% ~ Ur),
Jonesy! of tha elon peowesa are mode at inverse apert, the
‘over

LE a= WS

fs then sed as an estimate of the eared Jad m= @ apy. This
rthod s isporisnt ecmvomically Deeaus it sheaper to seu tra
irene paved clly chan la sere them eomnsovely. "Tho murbe >
ip the won isa and the tte Sy — zy = = — 2n i ele the
Cinna wader of pti im ere, fa cervations Table st etn
fez okdng tm, sean fers used nd routing values of + fo

yoni | —Monaase Tamms Sous Tormarats ax Vanune ov 7

1 Stas Cat sme |
Fiblatee Rese tdiag IBIS. | wer
22 eon, pram nnamwtest nTsaty FANN TOL

‘rious typas of eal, "The vaviance of a°'S, is expesible interme of
the eovariace fe) a8

08 E = bRGr, (ah

1 he nese the tou eal earl were
f

suse vo rine Un ie a, he vba of ee

aT) =

vet

yen [Oromia ay

‘Thos she man square crn of bth these metheds of traf miasne:
‘ant cane palate Lent wily i the evince (+) i korn,

1 forts (2) te eaveeinne fanotion iv exprestd in erm of the
stationary probabltie [p.| given by the Evang disbution, and hw
transition probabilities

Post) — Dafa} — we LEO) = mt

pitono tral, the particule tension probatility
Prt = NL} =X

Jang heen singled oa (in Reds, 9 and 4, fe
at ie chichetrie of the dynamic
fo pink of the alsin “all

Th the teary of te

soveey” or “eelasation? feo
Fehon of tne Marhow piv
tmaaks ney" eonition

Wie shrill hers tat ¢ mush air eogont reecon Ua thi a Be wl
hud ta support tho inporsiuee of tho resorery fmetion tn tele
Uo: the eovasianee function R¢-} ean be expressed atively i terse
lof the very fetion and the ered las Tn other wot gle
fone of the (N+ 1) tsmstion probabilities pearing in fortala 2)
races “ar determining the eovirianee funetion, aed tis one is he
eancery‘anetion pys()- Thief in thooneie asic of Une
ities ci ta the reootey fa Ln epee now te wae
vie oF 94S, atl af HFCT on expenile ing uly the eeoveey

‘We next give a summary of the enatente of the naomi setiont;
thie i followed by en aconat of epic resus ad socusions,

‘Ax ewes fons fer the covariance HI) i sacod and daessed in
ston TIL end deived in Seetinn VIE, ‘The foervla wally yiells 4
SOALANEH ENCHON OF eRERLE MESSE orouP 130

rigorous upper bound which apeats to give & lose approimesion to
We) ited, To Selina TV tle eecovery tion py) en, nit
ie shown lane Uhr nsaiance rag be express in eeu the neraceey
faetion ey salon antes Tho warioat of 97% i ied
Seotion Vea entt fruits, and an appresimating pneu [sed
fo the upyee bound for 12), are bth obtained. The varianee of che
tim average 17(7) is sidered in Sertion VI; nguin, an exaet formula
il an appresimstng upper bound are ound.
rhe eovarh nee fata Fe) ie bounded for hace acd awe
pproxzncid hy a sinyv exponential function

FARO, neo,

Bi =

Hee
2 KO
‘iliiam,vatnnoe of N4-)
{lot cari} — load letra ae of dla trun

au the roriproal fine constant yim the exponent is tho dominant™
"haractedatie luo of he rals” or "ransiion” mati cf tho dite
nti equation sstsfisd Soy Ihe bation probabictie, ARercaely,
Fis the “oot of leas sogaitude us Paisson-Cbartiorpayomia. ‘The
ft ehonaae a faction of elena! Lae @ far Wms, 835 Hi
‘ad intel i ible TL,

A lower bound fore depeding ey om the mean an vrianee of NC),
ie riven im serie VAL! By nnn se of che ace that the met of
Ih difeoutial esanticns forse tra skin pechabiiee ie symmcctiza
Vie, Hale vale of eve ae pe tun, a7 <1, this banal
fan be use fo appronimate my. Tan’ ese, the bound is came

rting pice for the nee of Newton's met, ho found e Ue 2300
anv oie satis the iat

ith

1 = slam mews of N6-)
= taid earned — aft — pale
unin variance of 805)
haul eariad) (ld lose

rena srr of in eum)

anyeositnstion 2

ya rae in Wig.

Tae Phat est ait Ae nse arr aes)

eovamanex FONeTION tH SibuAe RENE GROM 12S

#4 ryt T

5g 6 Martino te position 2 at

iy the “five tr model se shall Rencforth me the sovhis
‘ue mel for tlophone tafe dstermazed iy all The sane asumaptione
tial we mae she Ielrduetion, except dhs: N= 3a unlined
hnuber of sunk fs ptated ir” aad Benes agro consid
this mel; Hayate tors pling error formule on

i sidly believed fh the "inknte trunk model is appeals to
Tooee grou af ight oad Leann. Sach a bebe i gratis unit
Fmparisots with a mace having Bnite mumaber of Crk ae rade,
Shudying the sovarianes ‘anetiew 0th azple ite trom gran om
les us to make ome ofthe aed umpadeons; eg, she varines of
fh i5, and ACT in the tu mls ee of pares inert. Knot ledge
Ur dhe gran fbn, zelovnat oo he ther Tv eas
tr ahich engineers avs Unldto py the Vinfite tran” model, vie

4. laree gue of arly Hee (an,
ii, small ruta lightly Ineo ean,
Si eal rns Beal ode Lea
TAS me saan svsvane rpmntEAL sowed, aanan LORE

‘The vance of 'S je mmc Htom above and appeal
she onal
rim 9) fot — LE et ah, ‘
Varin 83) tty — Ea 4)

where nis the mambor of obsswitions, and

{eoingn ehatacesatio vata)

“The eet oem forthe vorianes of 1¥S, fa the “infinite unl” model

we Cetuh tr co $+} oe

‘Tae upper Dour {4} for the finite eroup is comporod with the exaet
her (3) fr the “ini trunk rel in Fig 8, which showe euch

Thermal ax a aetna the sean inereal for vations, for a 20
ela fer Vo 20 (eke. The wares sages tits the wppes Bowne
fie Weve 18 for Nos smshionlly ess a Ue erresevale
seine inthe “ante trinket ga br eapoete mses
fhe sen inser» inmpeoves arcraey for Une cane aie of ebaer

tis, This Deets the envarienes faneton sive, and inant
in|
The

ve aul approsinata by

aoiat
Tes

vasl AT] = 2c wm
wee ie he lenge of She Limeinterval of wsations oberon,
fad of and re, as befor the varinnee of YC and ie demon
‘hameteistio value topeitively. The evaet formula for the verane
OLAMP) in te “fate trunk wad

eteaar
7
Since ry <2, sad oie alee Jess than @ if. < the “infinite
funk” mde! wureatimates the ravines of CT) i api to faite
coup. This condusix i ilustrted in Tig. 7, whieh shows the farmaa
{i} and (7) for ala of 0 cxhnge afer to 20-runks. Fos sn abervor
lon tia of 10 meup holding tes tho “infine trunk” formula (7)
applied hore wotld overestimite the variance by sboue 900 per ent
Tie is about as extemo a cus us would oceur in pructien Fig. 7 leo

a