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Unformatted text preview: THE OUTPUT OF A QUEUING SYSTEM PAUL J. BURKE
Bell Telephone Laboratorzes, New York, New York
(Received June 25, 1956) For a queulng system wrth Poxsson Input, 3. sxngle waltlng lme thhout
defections, and Identlcally dxstrxbuted Independent (negatlve) exponen
txal semce txmes, the equlhbnum dxstnbutlon of the number of semce
completions m an arbitrary tune mterval 1s shown to be the same as the
Input distrlbutlon, for any number of servers Thls result has applxca
tions In problems of tandem queulng The essence of the proof Is the
demonstration of the mdependence of an Interdeparture Interval and the
state of the system at the end of the Interval HE PROBLEM OF the output or ‘eﬂlux,’ as termed by MOBSE,m of a queumg system apparently has been prevmusly consrdered, but
not mvestlgated, 1n the hterature Thus Morse m the reference (31th states,
“A httle thought W111 convmce one that the eﬁlux from a smglechannel,
exponentlal semce channel, fed by Poxsson amvals, must be Poxsson w1th
the same rate as the amvals ” In s1mxlar vem, for the case of two gates 1n senes wrth amvals and serv
lce tlmes random at the ﬁrst gate, O’BRIEN has stated, “The amval of
customers at gate 2 w111 be random and the average arnval rate W111 be
exactly the same as that for gate 1 . . ” [2‘ Nelther statement 18 supported by analysus Smce the truth of these
statements 1s far from obv10us, a proof seems necessary It Is the mam
purpose of this paper to supply that proof and, although the quoted state
ments apply only to smgle—server queues, to generahze the result to multl—
server queues.* The motlvatlon for studying the output dlstnbutlon anses chleﬂy m
problems of tandem queumg, whlch occur m a vanety of apphcatlons
One example 1s that of customers 111 a store who must ﬁrst be wasted on by
sales clerks and then, after bemg served by these clerks, must then be
served by wrappers or cashlers Another, more comphcated example, 18
the settmg up of a telephone call through a smtchmg system. One type
of the latter problem has been studred by V E BENES m an unpubhshed
memorandum Th1s problem of Benes’ 1s concerned w1th a twostage queue,
the ﬁrst or reglster stage havmg random mput, exponentlal servme trmes,
and an arbltrary number of servers, wh11e the second or marker stage has
one server and general semce tlme The prmclpal result 1s that the equl
hbnum dlstnbutlon of queue length and elapsed semce tune 1n the second * A dlﬂerent method of proof which meludes the present result as a speclal case
was developed, after the completlon of this paper, by DR EDGAR REICH of the Rand
Corporation Dr Relch’s method wxll be published m the Annals of Mathematical Slammer
699 700 Paul J. Burke stage 15 the same as though the Input to the second stage were random In
Benes’ words, “If we had consrdered the marker.system by Itself w1th
P01sson mput at rate a calls per second, we would have obtamed preemer
the equatlon for the generatmg functlon of the number of people
present and the age of the marker ” Thus 1f 1t had been known that the
output from the ﬁrst stage was 1n fact random, the result of thls paper
would have followed from wellknown theorems w1th a cons1derable savmg 1n analys1s
Stlll another example of tandem queumg 1s analyzed 1n a paper by R R P JACKSON [31 In one case of Jackson’s problem, random mput to the
ﬁrst stage 1s assumed, and each stage has a smgle server w1th exponentlal
serv1ce t1me Agam, certaln of Jackson’s results follow nnmed1ately, 1f 1t
1s known that the output of each stage has the same Porsson d1str1butlon as
the or1g1nal Input, and, furthermore, these results can be generallzed to any number of stages
It Is mtultlvely clear that, 1n tandem queumg processes of the type men tloned above, If the output dlstrrbutlon of each stage was of such character
that the queumg system formed by the second stage was amenable to analy
sxs, then the tandem queue could be analyzed stage—bystage msofar as the
separate delay and queuelength d1str1but1ons are concerned Such a
stage—by—stage analys1s can be expected to be c0ns1derably snnpler than
the srmultaneous analys1s heretofore necessary Fortunately, under the
condxtlons stated below, 1t 15 true that the output has the requ1red s1m
p11c1ty for treatmg each stage md1v1dually THE MODEL A SUMMARY statement of the theorem proved 1n th1s paper 1s that the out
put of a queumg system w1th P01sson Input and exponentlal holdmg tunes 1s
agam P01sson The detalls of the hypothes1s are as follows A smgle—stage
queue w1th random 1nput 1s assumed The average mterarrlval mterval has
length 1 / A That 1s, the probabrllty of the arrrval of a call (customer)
dunng an mterval of length dt 1s taken equal to >\ dt, w1thm mﬁmtesrmals of
h1gher order, 1ndependent of the state of the system, arrrval tlmes of pre v10us calls, or any other condltlons Whatever There are 8 servers (channels) each havmg an exponentlal holdmg tune
d1str1but1on w1th average 1/11 The holdmg tunes are completely mde
pendent of all condltlons Hence the probablhty that a call Wluch 1s re—
celvmg serv1ce at the begmmng of an mterval dt Wlll terrmnate durmg the
mterval rs [.1 dt W1thm lnﬁmtesrmals of hlgher order Under these assumpt1ons and the further cond1t10n su>)\, 1t 13 well
known that there 1s an ethbrlum d1str1but1on of the states (number of
calls 1n the system) Furthermore th1s d1str1but10n 1s the same as that of
the states encountered by calls enterlng the system Output of a Queuing System 701 All calls remam 1n the system unt11 they have rece1ved serv1ce Other
w1se, the queue d1sc1pl1ne, or order of serv1ce, 1s 1rre1evant, smce the output
and not the delay dlstrlbutlon 1s of 1nterest OUTLINE OF PROOF IN ORDER TO SHOW that the equlhbrlum d1str1but1on of the number of calls
completmg serv1ce durmg an arb1trary mterval of length T 1s P01sson \Vlth
parameter KT under the cond1t1ons of the model, an equu alent result, that
the t1me mtervals between success1ve call complet1ons are 1ndependently
dlstrlbuted WIth the same exponentlal d1str1but1on as the t1me mtervals
between arrlvals, 11 111 be obtamed meg to the randomness of the Input and to the exponent1al holdmg
t1me dlstnbutmn, the output process rs Markofﬁan w1th respect to the state
of the system, 1 e , glven the state of the system at any t1me t, no further
knowledge concernlng the output d1str1but10n subsequent to t 15 gamed
from the prev1ous h1story of the system It w111 be shown that an mter
departure mterval and the state of the system at the end of the mterval are
1ndependent at equ111br1um Together W1th the Markolﬁan property, thls
1ndependence lmphes that all mterdeparture mtervals are mdependent
It W111 be shown sunultaneously that the equ111br1um d1str1but1on of an
mterdeparture 1nterva1 1s exponent1a1, and hence 1t follows that the output
d1str1but10n, or d1str1butlon of call completlons, IS P01sson PROOF IT SHOULD BE noted ﬁrst that the probab111ty of the system bemg 111 some
state k 1mmed1ately after a call departs 1s the same as the probablhty that
an arrlvmg call Wlll ﬁnd the system 1n state k A departmg call that leaves
the system 1n state k represents a trans1t1on from state k+1 to state k,
wh11e an arr1v1ng call’s ﬁndmg the system 1n state 19 1s a trans1t1on from
state I: to state k+1 The number of trans1t1ons from state [C to state
k+1 cannot dlﬁer by more than one from those from state k+l to state 10
1n any t1me 1nterval Hence the proport1on of calls leavmg the system 1n
state k approaches the same hm1t as the proportlon of calls ﬁndmg the
system 1n state 10 The latter hm1t 1s known to be equal to the equ1
11br1um probab111ty of the system bemg 1n state k at an arbltrary mstant
These equ111br1um probab111t1es comprlse the d1str1but1on,[4l Wh1ch 1n the
present notatlon may be wr1tten Pk=Po (VIM/’9', (0§k<s) Pk=Po (MM/(sI 8""), (has)
Where 170 IS determmed by the reqmrement
11:: Pk = 1 Let L denote the length of an arb1trary mterdeparture 1nterval and 702 Paul J Burke n(t) the state at a tune t after the last prevrous departure Let Fk(t) be
the probablhty that n(t)=k and 101ntly that L>t , It may be helpful to note that
2:: mt) =F<z> 1s the fallmg dlstnbutlon of the length of an mterdeparture mterval and Fk(0)=Pk
at equlhbrlum
For an 1nﬁn1tes1mal 1nterval of length dt, Fo(t+dt) =Fo(t) (1 —>\ dt), w1th1n mﬁmtesunals of hxgher order, smce L>t+dt If and only If L>t
and no calls arnve durlng dt Sumlarly, Fk(t+dt)=Fk(t) (1—x dt—Jp. dt)+Ft_1(t) A dt, where j=k for k<s and J=8 for has These equatlons reduce to the
dxfferentlal equations Fn'(t) = “A 170(1),
(1)
[PI/(t) =)\ Fk.1(t) '— Owl#1) Fla), subject to the 1mt1al condltlons (whlch Imply the emstence of equlhbmum)
F140) =17);
Equatxons (1) can be solved by Inductlon to yleld
F10) =1»: 6—“ (2) as the umque solutrons subject to the 1mt1a1 cond1t10ns The result 1m
plles that the margmal dlstnbutxon of the mterdeparture intervals 18 ex
ponentlal W1th parameter A, re, the same as mterarnval d1str1but10n
Also, the 1ndependence of L and n(L) can be readlly estabhshed from It The probablhty that
t+dt>L>t and n(L+O) = It IS Fk+1(t) (k+1) u dt for k+1§s
and Fk+1(t) s p dt for k+1 >s
But these expressrons reduce to (woo (WY pa e‘“ A dt
and [1/(31 8H» (x/p)‘ g»“ e‘” A dt respectlvely, whlch are factored mto the margmal probablllty functlons
of n(L) and L, thus provmg the 1ndependence of L and n(L). The 1nde— Output of a Queuing System 703 pendence of the length of an arbltrary mterval and all subsequent mter—
vals follows from the last result together wrth the Markoff property, as
1s shown by the formal argument followmg Let A represent the set of lengths of an arbrtrary number of 1nterde parture mtervals subsequent to the arbltrary mterval of length L, and let
P(——) represent the probablhty functron of the chance vanable(s) repre
sented mthm the brackets The Markolf property 1mphes P(Aln(L))=P(Aln(L), L) (3) where to av01d ambrgmty n(L) may be taken to mean n(L+0) The
1ndependence of ML) and L IS eqmvalent to P (n(L),L) = P(n(L)) P (L) (4) The Jomt probabrhty functlon of the m1t1al mterval—length, the state at
the end of the mterval, and the set of subsequent Intervallengths may be wrltten as
P(L,n(L),A) =P(AIL,n(L)) P(L,n(L)) (5)
Subst1tutmg (3) and (4) mto (5), one has P(L,n(L),A)=P(An(L)) P(n(L)) P(L)
(6)
=P(A,n(L)) P (L)
Whence from (6), P(L,A)= 233:5” P(A,n(L)) P(L) =P(L) PM) From thls result follows the mutual 1ndependence of all Intervals, wh1ch concludes the proof It may be remarked that for the ethbnum output to be P01sson Ulll
formly for all values of the parameter p, the Input (assumed to have a ﬁxed
average) must be Pmsson T1118 follows from the fact that the output
dlstnbutron may be made to approxrmate the 1nput by allowmg p to be— come 1nﬁn1te
AN EXAMPLE As AN ILLUSTRATION of the apphcatlon of the result of 131115 paper, an
Ideahzatlon of the Sltuat1on 1nvolvmg sales clerks and cashlers mentloned
above Wlll be cons1dered It w111 be assumed that customers have access
to any of the clerks who may be free and that the clerks have equal access
to all the merchandrse After servwe by a clerk, a customer proceeds to
cashlers and has access to all of them regardless of the type of purchase
Also, there 1s a smgle queue In front of the clerks and another smgle queue
1n front of the cashiers SerVICe by the clerks and by the casluers 1s order of arr1va1 704 Paul J Burke It W111 be further assumed that ev1dence ex1sts that the serv1ce t1mes of
clerks may be satlsfactorlly apprommated by an exponent1al d1str1but1on
w1th average 1 5 mmutes, whlle the serv1ce t1mes of cashlers are almost
constant at 1 mmute The problem 1s to determlne the numbers of clerks
and cashlers necessary so that the probab111t1es of a customer’s belng de
layed more than three mmutes In front of the clerks or more than two
mlnutes 1n front of the cash1ers are each less than 0 05 for a per1od of sev
eral hours durmg whlch customer arrlvals are random (Po1sson) W1th an
average of two per m1nute Smce the serv1ce tunes of the clerks are exponent1al, 1t may be 1nferred
from the theorem of th1s paper that the 1nput to the cash1ers 1s P01sson
w1th an average of two per m1nute regardless of the number of clerks, pro—
Vlded only that these exceed three, and hence the number of cash1ers can
be determmed 1ndependently of the number of clerks The requ1red number of clerks can be found W1th the a1d of plotted delay
d1str1but1ons for exponent1al hold1ng tlmes [51 For ﬁve clerks, the occu
pancy 1s 0 60, s1nce arnvals average three per serV1ce t1me, and the prob—
ab111ty of delay greater than two serv1ce t1mes (three mlnutes) 1s found
graphwally to be 0 0047, wh1ch more than meets the cr1ter10n For
four clerks, however, the occupancy 1s 0 75 and the probablhty of delay
greater than two serv1ce t1mes 1s 0 070, whlch does not meet the crlterlon
Hence ﬁve clerks are necessary Smnlarly, to determme the number of cashlers, plotted delay (hstnbu—
tlons for constant hold1ng t1mes may be used 1‘” If there are three
cashlers, the occupancy w111 be 0 67, and the probablhty of delay greater
than two serv1ce tlmes W111 be less than 0 01 Hence ﬁve clerks and three
cash1ers W111 be requ1red to meet the serwce cr1ter1a ACKNOWLEDGMENTS THANKS are due to W S HAYWARD, JR for suggestmg the method of
d1fferent1al equatlons, to E REICH for cr1t1c1sm and correctlon of part of
the proof, and to R H DAVIS for helpful d1scuss1ons REFERENCES 1 PHILIP M MORSE, “Stochastlc Propert1es of Wa1t1ng Lmes,” Opns Res 3,
256 (1955) 2 GEORGE G O’BRIEN, “Some Queumg Problems,” J Soc Indust Appl Math
2, 134 (1954) 3 R R P JACKSON, “Queuemg Systems w1th Phase Type Serv1ce,” Opnl Res
Quart 5, 109120 (1954) 4 T C FRY, Probabzhty and Its Engmeenng Uses, 382, Van Nostrand, New York,
1928 5 E C MOLINA, “Apphcatlon of the Theory of Probab1hty to Telephone Trunk
mg Problems,” Bell System Tech J 6, 461—494 (1927) 6 C D CROMMELIN, “Delay Probab111ty Formulae,” Post Oﬁee Elec Eng J 26..
266—274 (1934) Copyright 1956, by INFORMS, all rights resenred. Copyright of CJIperations Research
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