BSTJ 50: 10. December 1971: The Overflow Distribution for Constant Holding Time. (Burke, P.J.)

5.7 MB·English

The Overflow Distribution for Constant
Tlolding ‘time

By BJ BURKE

‘tenes cds ol 9,107

An injaie tek group apis into a Sait Boat ctee group au a
wine rou is ead. Tho equa diteibuton, aa trary
Stata of the wap of Busy tas fs the ao te eMatid forthe
ane of Piston sat leant hein Sine. Se ome coe
Derions of varies vu! iidittane for ecoinenti! ae eatant
outing iow ne gie, Te vrianon a he enn was foun fo beatae
(ha qa jared hal sine, aa the sate of ome tum 1m the
Jistcoice prop tha Soquatity Speen Bo fee wnall. sine
ane ater he varias der warkeaty—by 28 mich ax $0 pera
Tnpiiontnns of thee remaia jor He aie nypneving of befiow gre
‘eth noneswaraiadhlding Sine ave Sexes

‘We eons an itaito trans aroup whish is spit Sato a finite fet-
cheios yup and as overdee gro. Cede tha find all tras isn
the festenoice group av placed us Uw oven ye Tk waa
te tae ict i Poison end shat the system isin nguisunm, Ter
‘hase omzditinas the cistributne of Ue aurmber of tally Ue otal
group and of thas in the Zmieoige gen are ge aa for ve
“are sdependent of cha holding time dietrbation, Fa the oa of
fapanentcl Folding time the dithbation of the namber of calle in
fxsteuce at an arbi instant on the overRow group iv ala Kroxn,
having been four yy Kosten in 1957." "The later distribution,
particular it tron moment, i busie to the method of engineering
fovertow groups, often ealed the “equivalent random” metiod, pice
sneered by Wilkin aad Wretachnuide, (Ses IR 2 fr x deseiption of
thie eho) TF tis iseibation re also independent. of the beings
time dstrston, thon the equivalent randous aietlod could be pple
2106 sum nea everaas mucniest sound mo 18D)

vith uniform validity repurese of the underlying helding-ime <i.
tribution.

‘es inden tempting to speculate that the overflow diseibution hs
this indoponsem property, since the mumber of Duty trunks in the
foveriw goo 2 the diferengs betwvon lwo rundom variables: the
umber of asy trunks in the toto! group wnd Chal in the fist-meias
nou, sash of which is independant of the holding time ditebution,

‘Taforunstay the staple ample of constant holding tima studied
Ine pnves thatthe ovendow distribution se in feot depend en the
holding sie dstrioaton. ‘hie mul 3 rominiseont of that of Tings
and Wikel? wlio town Wh the nlolgag probably in a grading
depends om the form of the holngeimedisttion. Ln the eee of tho
tenuting the diferenens betwogn the blocking probsbiltis for eonstant
End expontitic! olling Lines ae too smal va be of practical sgnificr
lance, In contrat, Tim dWferenees between the seepective over
dlistibutions, an meanued by the alte differnt bees waranoes,
‘en fe quits Tange, mining elces to 50 porcen in wae ces,

published wack ny N. P Anchor indiatss, novela, chal when
‘the eqaivalsnt yandom method is bused on eprint hulding time, se
oslts of crgiwring a fist stage overfow group ace esta the
eons tbo es cml et expos) holding me. Tha i
for a aiven configaration of Hre-ctoige group wi Poison offered
Ty fis mpvnont eandora meted, eorssonily pled, esta
"ie same si avin group for w ion lowe probity heer tho
Dolding tira taken tbr vrolucl or expotrocal Thos the computing
ids—thae ie, algurcar, bles, au geal used in the application of
the equivalent random emethoi—wlish are bosed on. exponcntel-
Innlding-ane theory en be wl withcnt change, aF practical
for she pues of sogsering single overlow yeoup even when tho
ling inn i constane, A ord of tina, however, ie in order It
should be boa ta te nob iho of an exponent fora eoastant
holding-ine dscibuton must be shane Unnnughoat the prooodar, in
‘the ealulation both of thr uvedllow vasicaces and of te sine of 2
Ssquvaleat yrnup TY exponential theory is used fur te overflow vare
‘sheer wn epeslant theory for dimensazing the cguivalnt group, ere
Svil be u bie vowusd ow few trunk in tae owenfow group and the
serviee roy ne sigifioantty ones than that aimed for. 1F tho wpposie
terror ix made and the over variances ar based on constant old
ing-time seumption while th exynmeatal chara are used for estimating
the site of the equivalent geoup, Tie overfow greup vill bo over-
engineered, The Ittor erronmoun roll wil eeeur algo if constant
cron ooeramwnie s107

helding-time overflow variances ate ertimated by aotaal moaeusoments
Gf overflow caffe rather than, at i done prsenty, from theonoticel,
fonsienations any, and if wnsideraten i nol ven to theft othe
hellingtime distibution on the vero vain

Tis fortunate that fhe estan Tolding tie eae i tractable, since
constant teling-line rearsonts am exteme point inthe srt of heliing-
time ditibutons whim thm ate nen according to thie coocionts
of vanntion. Tew monahle fo aonjecture, for example, cbt an overex
Yariaice forny Tolding-lime dicribation whose eorfeeat of variation
‘less than unity will fer les Ino thal fv expneatsl balding ine
than dors lie oven vatinee Zor coment heli tire, Tha the
Cyivalont rai snethod i spaeabl the euse of wunatant tli
itn, is fort’ sn wea the eetieent of variction of tze holding
time dstibusou ia batiean zer0 ane ono, haeteermom, the fac that
1 conantHaldingsimne equivalent non owslare virk’s reals
‘hie ars aloe indstinsisuble Fon he fr exjmmeatiad Holding
tine leds support to toe eonjonture Chak in Cae apaliation of tae
oer che form ofthe holdin debe ny be inane een
‘Wher is vlc’ of vacation i gzeater est ut

‘Sthough the chief parpise of fale stuay was to giv information
shew: the extent uf fhe dependones of the parametecs of usorfow
flstebusiece om holding time rian, Ure orga vation wae
Storided otherwise, In fac the peeecut investignian waa spanked
fhe obsecvat on hal the fants ft desompetioa of the variance
tf over trie resul Inve uperpritinn of indepindent Liston
‘apt stron flere] to rhe game fstencie group, desied rigs
by A, owt for exponectial Loidisg time im uneublished wok, fs
ald or any holy te seeann, hi omnia

Var fy) = af Yee ah bath ~ POEL o

where

17.7 the numberof eas in ta overflow belonging to cha th stm,
22, = the proportion of th feed al iw erm,

end

= Ley,

‘The question immediately srses whether the obsorvation dal (1) ix
independent ofthe holding-time dstbton hes ew wppleation, vie
iWinvolves Var (yy whies was naw ferotore only forthe expeneatish

fase, To satle this seen, » ehurulevaation of the evesiow trafic
A198 uy wus oYaTEW THEA TONED, DREENER 1971

fn monaxyeneai exe i eagiend, AG it turas ont, im the eonstent=
Tldingi esse not only the vara bul ke Ue ital ill
‘may be found exactly

“Kosten formule for the pibabiity of y trunks busy inthe ovatow
for gm oer Toal of « evangs offered to ¢ trunks may be written,
ser some simplifies,

1
(titty,

e

whem fag = Vand (e).~ of = We fe 5 + D)

ong foveplas for the couse caso ure

Secby- st neat nyo,
a

ny (Ag al we

LEQ) Chiles}

{A trie wating of Uns erivation ot @) and (4) aves in Sostion I

Algsnie dais af the devas oF veo std wubeoqasnt foreoul
‘ag given in the Anettic)

TE mit be ronal Ut the stato probebiity formulas in the
constant rate ave the simple for enepputscinal purpoets,sinee they
lnwelve ony Gn suis of pitive comms.

“With crepe ts conglent, the formule for the moments are
snocheratany. The marti, of enum, re ual vd ae ive iy

Although the sasiaace fe the seand-ordor moment of direct incest
the sncnd festorial momiect ie equivalent for ost purpose nnd il be
ven here ence iia implan, Hop exponential weve Gime this moment
tay be written

rw ofiee

-9 EG seta] ©
cemeriow omMBUTIEN 3109

(rom this exprotion itis immediate that 2!" — 6 | Lora =e I,
‘foot that wear found nec. i cher computer progeams,) Kot
Comm aking ne the secon fetes some fe
n

a

Io allt suerga eases sting, st saras ot that Py > FEE In the
face © ~ 1; the formulas ave simple enatgh to allow an eosy anclstio
‘too! tht this relationship true foray a, fact thet #> 3°
Innpie thot the sorelation between te number of bus trunk in the
ovorlow group and tht nthe frst-rhoies grenp i ler inthe constant
thn in the exponential ease, "This fact x peraupe Jae surprising to the
intuition than nthe reslt fat the esto ie mee vaviale fr eonscant
‘ha for exponential hog ie

Onin tte eusaney te Halling in taker. se al eli
lobe afunt ngth, He ell esentin she overflow eeu at an arb
tiers inant fg vanepreiselythow thaloverfirerd during Ue pegeding
tina ivsereal of length arity, he sner uf wale pron nthe Fate
choice group at the instant fy — 1 i knw to ave the treated
ison cistributon {ass kao we Belong’ fret distcbusien). We
‘condition frst on the number of ells present inthe fist-ehoie eroup
bt. — 1, Next, we canton on the mombor of sorrals dune ‘he
Interval | — 1 4), whic, of nse, ts te urinary Pun eelie
tion, We nove observe thatthe hing eprtare itante together
wth the oral instant a9 eonditoned, are sual independently
Unirnly distributed on he unit interval. This observation enables ws
tn eampete the ealeulaton by en spplieation of ballot theorem.

Let

(muse of nck i the fint-choies group

(0 = cfered Lond in ering

= probability of ca.2c tae frt-eion group at an anbitrery

‘pe = probe ily of Furivls desing w unit of times py = ea

sss Srobsbilty of 7 alle om the Festhee grou and y elle
tre thheaerflow. roy a a wary da
5200 THE BELL sun woeMIONE TOPRNSE, DECREE 172

sy = pobability of y alle on the overflow group at sn arbitrary
ksi) = probability of y ovedluwe dosing a unit intrval ot whore
Frisia indrt chore ate # eos 0a the frt-hoiee gxoup and

sting whieh j now coll aeive

hot we may write

ro
sn
Moe = Peay wl > 05 @
snd wt een utr satin ve a
ove De Laois, ow
Sec ft hp of te starve et a

12

P¢o

[a= meay aw fe)

vse Bf) i the seriee-time distribution funetion (wth unit meu).
‘The Test two results have een pubes by several antors; an ele-
riencary prof has bs i by T. Tian whose papor ineiudce &
Uibligeapiy on tho problem.

‘The implinscn of (1S) the present cam is that the remaining
seine times of the eall iaially = the Praetoioe tanks 2p inde-
vim opseawesies 201

enemy sniformly elise on the unis imterval. Purthermore, i
Sllknown that when he mamber of Poiscon arrivals curing a lise
time interval is give, eho individual strives ure depends
uniformly distbuted over tia interval. Honee any spec nayuene
ff thei denarcines ant jaivale sarin he inerval la 1, has the
fame probability, namely 1/01,

"The sanity 0; ,) ithe probaly Ubwe w sequane of astra
fad deparcines Tis the papers ht all teas the exeoss of the
fcoumualed pervs over the semumalated departures is tity less
‘han one more than the nial ribo of ile trunk, hy Tes thn
2 i FL Thus the probes of ealeating f(D; 3) ean be recoeied
fs a “ball.” problen. Suscessive arrival wed deparires exe elled
vents," and we donate ym, ail A ropoelvel, she wecumuluted
umber of arrival snd departures atthe rk event, Wit thie notation,

FO:8,j) = Pei ca bem EFL Pah tba UD

“The rcs palit ib ven oa aon Ye case Ckaptor 1
of Ref 5. We have, sith the neal eomventivaw gonoe-nirg Dito
‘oetieta,

i
ce)
(3 i

(etn)

‘We are particulary intrested in the marginal overt state probabil
ties, maely the qantities dnote ard hase may be pt into
«form suitable for numerical ealeulaion by cubetitutng the values of
Ahoexpressions ry, and int {10} and {11}. Ths nl role,
8202 nee aka sure toUsTOAT son, DpeEMBER En

biaine after small arsount of manipulation, wre sown aa (8) and
(See Appaudix)

realy (8) and (1) ae eonsidersbly simpler than their analors
for exparcntal hoking time, shown a (Zl, whieh wus obtained from
equation (8) of Ie, 1s It does not fellow, however, Unt the moments
‘ofthe distsbatin dlr by (@) nd (4) axe silo form than those
for exponential holding Gi, Tn fet though in prteple each of the
moments ean be wea a2 nite smn, st sero to ew ewthorvaliout
tusk to win eloeed-form expression Tor them. Sines aly the
‘iro mesninte of woetlow distributions ure preset of pease ileret,
‘ye shall cosine or attention tothe tw Ins onder moments or the
(hse of sonata henge,

"The men af the dstaibution fwny yv ~ 0, 1, ~o-} i obviously the
ame ae thos for exponsnlil Wwhling tine, Nevertaelse, «direct
flelation of te mse Trow the expresion for tho staze probabilities
is efa. ar a okedk un the neuney of the slgobraie manipulation
Sndiotad poeviously, ‘Thalia n heck, che equst

Ee [E9]" ws

sould bo shana tobe an dati, indo ts. Bow Apps}
‘The eal ioe the seroma factorial momen: woe le iy ioe
swunnbion, The usta ane givea im the Anpordis
Tan rmend above that « siwple proof that Fy > #2" ean be
ive for @ =P do this, we observe thet iy subatitasing ¢ — 1
fity (we obtain

Poy L glee tee thm <9

Ly exbatitating ¢ = 1 inte @), we obrain

nea yt 0)

‘Thus we must prove

en
Arter cate nut tiplyng and simplising, chre ull, eouivaentiy,

Bho ean

a

‘si eee fora pits fo garth

A comoarios of the avert vatiser from find he tans group
ze ranging om 1 te TOD ie ae eV oble Late ani were
‘lute oh the basi 3 (@) aod (-"The starved varies, “yrtow ened
nals tol igh values of, were ealeuleted sepetnto's Ineeane uf
26 0 annarvey in AP) nh see values It shelby ed thee Bh
heving 1 aublewsians dees wut ear from re of atrniary fou te
‘exazo sad sen wus all Lo obnin the ves By lve rere
‘The dintiaiens of taller several trambogrosp sacs ex fees
Ila she Pi Tal abou be herd th over wees
the varacsea ifr tio, foe wera far © — a, hier bt
‘eas shown the varies diserones i gecter thu 3 oeeeut. the tala
dlitibuniors ore viv lis a ll ete bites of pense: inert
‘Sings she falmcetindscingsivhahiey wives fore = 10, @— 2
frumested a» bist proboblt level, Ht may le well ts ptt ot ta
ny ile noxtizbly oe ue Jw yanuca. Tha the prubability
festoeding. 95 ouey Leane “a she avselow 3¢ QUUMG ti exponential
sd GAIT sensu ang bane A ggg fore nee athe #2
robot distinriens,wisuen iv tl mess ote — 1. — 2 we
ree ives ees fur whiek-a = 2eandv > W Wore — 2Dazd eM
veneer couric evald aot be made wenn ef complet los
of acousacy ia (2, Sino ty lative irene inthe wares fs 2023
Tees Uae ene has for «sali ogame tn esr Ht
vides ar cxeellene appeasimation Se £2 im tis gea6 Ls septs
hat {51 ew bl ne apyrniation t0 @) fe w > 5 yoisey ee
vehote (2) fe malian ‘er one) eommamtetion
‘Vama: 1 Couraaisis oF Veruacean or Ovguciow ron ConstaNt
Hovoine sen ano Trower, Houint Tt