BSTJ 40: 3. May 1961: Stochastic Processes with Balking in the Theory of Telephone Traffic. (Takacs, Lajos)

7.6 MB·English

Stochastic Processes with Balking in. the
‘Theory of Telephone ‘Traffic*

By LaJos TaKses

1 wayne thot

elope ecg en ne arising acorn
i reuenent proce. ff en tacoma sh very jen be Phen
1 other coatice w eonsrtion eth paki p, ar alka with praaity
th het gy — 2). The hating Sonne ae muualy éndcponden raven
Stvunte wit common exponential dition 12 apap the socastc
‘ur of th oto f he untae 1) the cy ne tee

Mony eects ix tebplore seule vhowre (en ebaeer) may be
wwihec ey Ce inleetee of Sathuy, sal ee tk a ay
ous ithe vernon, thom] sande fo: baking
ost y ws etme GI ig dep dent only om he
Ste of the system tan sneoeing eal Std exe vee bg then
v eulinest eonuerlin il poobasiey pared balk wih prebabity
W(penp = TieTineif py buy O,1, =) hesyatem Fone with
Iie na of fet i re aft ml os ly, he eal for
lov any sevice, whike dp, = T= yyy m2} and py 0
Up img mck by ssl tho aslo ane aystean with ines

‘Ths ballanz' neds rvamnin lve to recurrent input aa pe
ential distin of Bolg ines. Mone soil, Th call eal
‘es re ko ss Mae stats my re ote 2225 whee hiner
fvval tes 80 = ein re Uy, soy np ~ 0) oe Mentally
titted, snatnilyeponciat, patie rand sasables with di

a
‘he bokng tenes are itil trite, rantually independent
tundom seaiabes we itil Sutin
may (26% HzO, &

{o He <0.

‘he holding times ore independent ofthe ra 8 wll.

Tt i denote (0) the uum of bua Hines ab the instant Do
fine", ~ ¢ 7 — 0); thats, £3 the numberof busy lines imsneiatele
before tho arrival of the ath eal "The sytem i sald ta be st
ie the stant 102) — Let an denots by a the sale nomnegstve
ineger sh Ha pe = 9. Io gy 3) — 3s) Hew me

Ti the pres: paper Re -hall gio» method to docemine the dis.
tuibution off far every n, the ditibucior of £0) for faite # wees,
and tho imiting aistrisuiers ef & und &(0) ese —> © and te
pwctvely. hunter, ou sll dacermine Ue sac ie Ine ofthe et
tous Be Be th

“The Laplsce-Sticltjostreasere: ot the dibution fonston of the
invesaival tine willbe eeeved

wee,

whic i converse if mah 0. Thy
ns wil be devote By

capeetation of the intecanival

nis - [sare
= 0 we t= B= Pa, Dae

whic i convaegen i (9) > 0.

Let Ply =

lim f= Py and lim Pole) = 2,
vid Hh Hae Ti ex

Define
rocrenome wmuusubs Wins MATING 7

where the empty pret menue 1; tit Cy — 1. We shall ae se
Te abveviat

1, w

Tenate by Affe the exported nusber of wl’ oeurrng iw Line
foueren (al which fun esti Fles base, The exported wer
Urns He > 2g oeeuring ‘nthe tine tore Og] elely

aC. Decne by Sif che espeted natn af Lions Bags —
‘conning in the tz intra 0

‘Let n)(F = 0, 152, --») be the diabution fupetion ofthe 1
ferences between screw lransitons Bia —» Bi tnd By > By
while Ute) = 0, 1, 2) --) the dauibutin fonction af the time

(erences between eomeestive tavtions By — He. If 0) — 0
then we sy that traumon Fh lakes pace at tine t= —0
Wie

[mean

aut
wie [Coram

he AK. Ralang

Tsang? ne prove that ff
pcthabis oe) = Le tne
y= Oven |, then

foams « Paison prone af tens
tse farther, ps = Lsiens <8

fe G1 3”

Ia thin eae P= Pat =
Ie ete,

om) la ld, Thief theelempest
TOR am way svsniag mucsicat aoa, MAY 108)

‘22 Comuy Pals
‘Paln fas genera the ale eat of Brg foe the ease when
raf tore reetsvent proees ad erie every asianpion reine
cishanged, Pa fae pe the

EC)

hay Pallieek? and Bene," and the transient behavior of the process
FRU0 by Benes! nd yt ae?

w

tn The Fait Sie Cn
‘Toe ego whan p= 1) = 0, 1, ©) hae tom investiga by eke
uthor!! ko fue prone Fat

Pe Ee

ang, if #42) not lio stron and a <a chew thelial
Aish (2449 exis aa

®

‘he tratsieut behavior uf the press [2(0%) sal treated in Refs, 1
snd 1

ba The Case by m= PG AN aE

‘This coro, whore 9} 9 ~ 1, plays ta important role in
petticie cowatere and bis oe ineawiztel by the wathe
ow that

bear of
‘oho he
LosSe ore

If Pts) ie no 9 tee Stain
station {PQ exe aa

and ia < =, fhe the Hiting

me ves
win |
ve p any
Beat ie
sn tie ps ak 0A
ca ie Di, Pin 0
hn sayin ee isin eon
ai ea se id nein een
Fen ee tne
ee
WE Tet tele a a, Ga)
tar) = wats a ce we od
REALL Rt nn
po) iusto =U
sn) bel)
WATS SET fate = ad Fe
i)
‘pi nt ore 12
we wo

RCO YGE ts

vere the snpy prot sons [The ferme (143 proved We
Ms
Tn the yuriouae sae ge <1 pe = GL hm
800 tae sxe ne

ete) = Pu }, (15)
Ceca
cr aersenall|

1 Ss eu to oe th sue

1 of mando variables (| forms &
boringenecie Merkay chan wi tration probelilicies

earth, OT

the cantina Irasiton pri lve tha te inlernerivl Gime
8. = Cooma). Fong iff — Jud Py ~ 2 then Ey hs w Bernoul
Aliststution, ther with acme jb Yaad > when tho wth eal
realizes a enuneeion, wt sets gies fade wher Ue wth el
Ages uo The cyst i ado he sweats Bp a the tl atey fy =

‘Starting from the iitial catriowcen 2") she estributious [Ps]
san be doveemized suecesively by th following Formulas:

Pom Epa wR as
Howocer tien wit thas in samy ete ik fe mao wom
sieve the Dire cme a [POY dst, By definition,

ws (EO) ne on anacs oo
serena uczeses WAH TATRINE su.

fs the rh Vinod moment of 2°"). eo pane va 1 C4

whore C. i natant, shen issu He pre Us every! exists
fn 59 <0 Jel hone 46 consent. Thue te dtebution Wy"
Is uniquely doereined by 102°). Weecbtain from (20) cst

Pe

hie the inversion foreuln f Jord!
Tis coenival toe tn reload queen

(On -E0)o

ven ef

(OA Be

ene hy inversion

pee = we
Now we rll pave
hore i We hase (9° — Le = 1,2, 008) aad
ES 8 ob mL eek Uae
sere os = eine rt

mB bem, Po

shen

‘oa, Wit of alle rs HT he eh ial saameat the Hee
nocllitibution {4th parameters wad 9 ttt hme

= (Q)raan ab

Using 27), we get by (18) a

oo eo eee
S02 tate anak ween mecasiran sor, MAY Het

when

fey

Toe multiply Loe wd wf 28) ye PL anal a hea far ev
Teg). We lay 25) pa (21 a (22, The eos
tos the pa he Ene

Stovting doom Oy — 1 2 <=} on biel moar 2% =
2,3, +01) can ve obtained reeusivaly by (24) ard (25) spots,
EU) = Send ry — a chen fbee 4 Herma: dition with pavarcters
ante "and seg tur = 3

cere wears. en

Reman 5. Move alice she gens tone

sien a
Erve fe u() ‘0

vi ant Fo) — tse tur 0, 1,2
ak €{0) = § then by (32) we get

[Qn ese] ons

we auppone
rocsstie PROCESSES WHR WALA nn

equations ie given by

whee

fal (8) = 1 Binal, 1s
806 tom nua song TaemeNt mat, aA “ae

Sm tunes euTEAMOTON [Pay

dn the Mankor sbain (8) the stetee Ly, diy +55 ly form an ire
ieble ose set, while Uy, Hy, ato tani tte. Hither
am sccm =e, but we neem gurseves to che eaten Ey i,

By, teen tae Maskow skain fy) boedueible. "Tee Nlarkow chai [él
ji always aperiodic. According

im =P 12)
lmaye exits ad i indepeudent of the iat distsibution. There axe
tito posite: etter wvery Py (4 = 0,12, =) or [Pad ea
prohutity distribution, (Tn the evnnd cate Py'> OF ES ws and Pe —
DICE > ma.) Tn the soil ene [Py the unique statiosey distin
tio of he Markov: chain fy al roustely i here ex a statonuy
Aistefomion chen i urge ant agree with che Fring disebution
Pa

Tv te paetenlar cave ny — 143 — 0 2,1 the siting ei
‘lon aluoge exes, chad heen. pueved ia Ref, 10. Tu thi apeeal ese

ao Ei vee.

Tee sna an arbiteary eequence {pj then evidently

whence it follows chet {fy Relongy to the cecond eases that, {Py ea
probably dstibutiea,

‘the stationary deibution {74} 8 uniquely determined by the fol
lowing system uf ince

r= Z par, os)

ad

Re «59

Since inthis ao 26" — 15 fer avery oe gt (38) by {48}, Now let
var Invoccuce the Eiseunad moments

uB Oe 6

WA 2) a)