BSTJ 40: 3. May 1961: Delays for Last-Come First-Served Service and the Busy Period. (Riordan, John)

3.3 MB·English

Delays for Last-Come First-Served Serviec
and the Busy Period

By JOHN HORDAN

Fo Psion Spat to nev caer, sa chon th the aan oy
Intron jatsort arose ie eal tn he i
tebe fre ey pra (Ue Sere of time daring whi the
server He eoutnoualy buay) any fr expnental dat fe
Sine Aimer apnea tha th vty got Jac 0
ccccnlte sera each with expnentit vis alinw wh asin foes—
ron wohih has bee a exriosity far some ie, Pin the dey tab
to for late Sonat ve aw ng err wth ened sree

iawibuon fet

lave fir Lat sume fisescved serve weae frst eoasidertd do

‘Suulo for syatom wilh Posen inp tw group of Fal seessble
seevers, cach wilh eaponratial service sve. Fur Ui onder of serve,
fe aeival which eat med inca oll the bibl edie,
fer Co tee head of the wig Line. Te eomitnnion Ins mst
thesetical incest, becase, v= le peo Fis-eon ostverved, it
seems Uo ea lal fe est a posse service seinen, a

Teast of hose tse see. Tiel Vt fs tt fe
sveloe delay Futian al sin assim,
Very bre iy, Go's fate elton, a x(t pros

avlity let wal ng-ccnna faz wevie, wheat a afven mete, he
Ji Tce — 1 ie, we ue ost #1 Ash complement of ©
‘etsitntie fasion nt} fe the complement of the conditions delay
fabutens tveticn, Then, We she neva rat, ac B the stevie

em iron of trem avons the eto! diferent zeustesee sete
oe the ffl)

1) Fei Frit ea,
FSO sam mt sort mHensHEAL FOUN HO oH

‘Woulo's sli of these eration wl he gon ler. Here i ut
Gent ta anti that tke wane "tions hag seedy spurl ie he
Termaultion nthe fey” penn, forthe sane vac system, Hy Pal
‘The eomreypmuent cf) $8 0, he pababiity that a busy pend
fal sere busy nbieh a6 gives momsen® tas a Ling stoner,
ontinmecom bine pera seus! J Since se huwtary cations al
agree ni = J(d)e ata in patvutay (8) — fll) Ut i ee cond
Steal delay livtation Fsnerion far List-oome fist sree servi i
fej he ditesnon feta of the bus’ pesad, for Une given eye
fem, "This te euros ad puetng rent messioned in Une abstet

"Tne few rest o this paper he pr thal, for Pon put vo
sing seven tho two dite ate aks wl for sg ial revi,

Por mote shar one sever, i psa thus the same ting fet,
Hn ao eney neuf rot ase For many’ eevers, each wit
texte servi ine, proof ieoy mie be gen it a Rey parallel

Tove singh verve ne, A sovomd seul ate determin nf The delay
lstbution far Ponape sigle scrver with consbanl sree
Time: thie eval legs the fw with that of Burke? on randons
sevice forthe ei ny aon

For let-come fist served servi, mse! finding the server busy ie
hole uni all sales evils Fog the corer bus or just eon
peting ervive have been sere. The waiting denn: mt he aerial
Fock hove so oft 6 thie dle brease the sew eval goes to the
tof the Tine

Tolling Tales ube dele Aseuten mage bo derive flows
‘Take tho arrival ite a, the service eate (esiprocal of he avenge
feice cts asi te srvie tine Ft bation tution ts 2, and the
Tllay tration Fmetion as G48) — sho stor oioting thet O* (0
{fers fram Gif, the slstribuon faneron for Ue ay peviod, ithe
Sarting epoch od sunrepandig satrvel

‘Nete Let that he stetiongay in fick fapeton Se:pten oa
nary time epoch i a senvine interval ant the epoch of next service
ompleion givens

ce) =o feo te a

{Tee sua ot of this oll now tw Paes 8 he has
redvediacatie
DSL Pom LASTeCOM TapSHeARIVED aves 787

otha

with

we) =

Note tha yo) ~ Ae ium only i ah = BO} V7 ara ee
sare thing, Beg) de

Nox atsices she inreval letaowr he amival epoch of a delay
demand ex the ep f Ln eat serve eamplesion, ‘he probability
Via tists Hs length yy ly! eC. The proba of m
sive in thie nervy, wh ie length ys te, Poison term,
lag)" /nt Th 0, the delay of the given aval for as-erne st
Seve series feng C2, Ly the delay rans othe ntoeval

lu tie few sevice eompletion pls He Ty inveval corned by fais
rival aad lf sebsuct avai vein neve aon pore, the
Mis ate Ceti oe ae fe e8) Hdenee Zar nT the iy othe

fenealation of @:01 a 28) Ky the same ayy for
the onto of CHE 8) and 1 al 9 9,

If Git) isthe estate fone hoe che saan of» washes, ech
oth dseibution ncton Gch, al Gh!) 1,080) — Ge, then the
onultionnl delay distribution Fanetion bs ive by

2 he ela ie

= [Een as = si acee o

rin — [erage

sual) ne ina sno, then she Goan of (8) fat

Pie) = aba orl a
wits yf defi ey 2). Nete Ful FS iy setocmine by Take
Phat = ae be atte o
TAR ue une sven ternsteat OMRNAL, HAY 186

ence Psi = Pia} when wad. aly when y{s) — pa). Ae already
soba heaps (20) — 3 =o tae exponen ial sete distbsion

he ramon tiki Sonetin for exporetial arsice vine his
Iv goven hy Twsses li ating 93 follow. Fas, si a) =
Mie es che solien (3) ie

rey 28

Nest, the iver af thi

au [Semen pa am 0)
oth 121 the Boose tut fth st nd al gina angten
neuter expression du We bic tebe add

‘sume work,

1-6 8)

we modcatons of ue angie: above fo aly aeseasibe ere,

ach with exponeitiel dstrfntion of svi tinue (exponential sven,
Tow Levy ave rolaively igen, Visti the deviation of ce busy
prvi distribution, a sviee pore i the singl-snreer ease is replied
Uy he intrval becroom Cesc at whie se Lis ile server bene
Tiny onal thr epoeh at sich Ree et a ¢ has reese becomes ile
lhe azeibation ome sof thi tere sat af Ue east of error
seaahce ene wilh ie aris exporeieia Calabcion, te see ne
Tis ibationy Ts tune 140) an ste ere ie sso
¥ ken

1a We) me @

ese the distribution Seti G20) fs obi From its ving wove
(expmecntial serve tine} voresponcest GU) simply by replacing b

"The leone fies enoved dey daibnton proved dete simpy
hy the reowrk thal 712i alsa she sebatinn of ta mntrved ev sen
fv aitcay poi in tap interval for whic #0) hw dstibution

faevem ned He tenaaation, bo
[es sb

pF} ie of expr for
Deis Woe Lastest anaes eax TS

‘ig. Lshowed onmparisonoteaenal delay earvastor varios onder:
of soree at an cuesuney 9 0-46 of 08. The otens see ace of a
sal sudo, cd inverse ade of eel Casa fi

belts ace valucsof an asia tse waeabse 1 ~ 6
fare values of Pad, the probaly chat she delay of a dela wee
lent, The ree mark “envelope” fea appr bound forall por
‘se endere of serves wesaly H is plot of 1 — ith

am Dev

sana uff i the froin
1 etait study of lays fr astro Fn-sved service al the
Ios ner, wth std without defetin frum waiting planed or 3

Tater popes,

ie Coston of del carve for sun erie avira was,
FD mr average muvee seniest, AY me

AV thowgl hie iain, enor (41, bat delay and sy peso di
sven neti cae aged anve eal, flowing, Toke f= the
“ter, the clagsiatic uf the Tey pri yume: Ferre

‘Sole fos tat toe generating Favetion fee cumbor 2f weiss
itor with citibution fre en C2) fs en bY

wy Eee
= Zee [mite ace ol

“The earopang gonorating fuetin Foruaabor of eras in a seevioe
interval egiven by
pix) ~ ata any im)
saul, by 2),
fa)
Nove the
pet bo ema
Now site fo fr the pratiios of w servis iv delay pes for

Iastcome Bincescaved serve, std

ri) - Do

Faris generating fiction; snd) ae tho eoresponaing eats for
=n pt loving De

sand
tO prin + Dbl he Die bon

ey ead = osha = ate oy
DELLS son Lasmen™ ninemsn eeevieR FAL

Note shat the eosreepering res fer fl easel by Takes ie

nee ay

Note that

2) _ cut Caysta ~ afl

Note aa thie (14 may he rien se

whl) ee fear
cu
Lb te Tigte
wots
34
Hones

Fu cansan service inn with service rte 6

way — 0,

wo salto ven ye
he

sothat

op
“Thea, using 160, the eset af the goers tunetion get, ae
ren by

nati oe
Beh | A AG Ut Do

“Thee resulls agate iting

fe nto) as)
From (15) it fullows Uae
sa)

von tsa (07) and (18 i fod hat

en (10% fn seston a gy = Jo

[in m2), aL af lr ar sonra i the sng frre Ge =
Cee itis icone abate S101 doe hed the long
fs been Heat

tation a the boon! fran ee spin ene of Pquation (1b) of
Sling which ivi tn Jensen? “of whore prapee to remember here
that be yas chie! wginee ofthe Capeshagen Telephane Compny a8
vl new rachersaiei of woe

"Tis Sally?

war Dina dae oo

hich, wt (1), deceries f.5.,
‘Note abs th H palehoanal gy = 6.

9) woman abo in the doa
dom of Hee Sy pv 9 phe sve wis expat verviee
tine a regtar arsvale enero Uy Chal.
"The st evi a he dst Suction

Su =
cea
an the
Des,
=, wee
8
orn = Darsen cy

isthe Couaitional) delay distribution Eanetion te Ista irs rye
Stine with Posi i tase severe wit ema seven tn
Tine isto nen fe he ne rt Eales

vith (he tego pare of

a nie Td Rate tm Sar nc ee Coat

Perec, Avia Math Aeoa. de Thang, & 80-6181 32)
8: nin Scene singe Redes Eom Ty
1. Wt SE Qo stein gh
2 empl Cd Pei Sy Rates Fett fr Pred da
s ster ity Ole Als" Veriigeniorsge tre Rgomigrbon Boemel, Ber

‘en ita Win Eat at Sala hp,
benny Fi sua ag Mle ol var Beaten
vent he tat Felina the Quon