Table Of Contentepra 20 Arton pan Tgh zeny
Transient Behavior of the Kendall Birth-Death
Process—Applications to Capacity Expansion
for Special Services
By 8. N. NUGHO
(neenuscrpt reid October 16, 1979)
In this paper we derive explicit expressions for the transient stale
probabilities of the Kendall birthdeath process, with and scithout
‘immigration, for any intial condition. We then propose this process
‘a8 a model for special serices point-to-point demand, ix which the
‘births represent ciruit “connects” and the deaths represent “discon
rect." This choice of mite is based on intuitive arguments and on
(he fact that the madel can represent the growth and twmnover
characteristics of special sercices demand. Thus, the mode! provides
‘a means by which special services demand, sith ite inherent wer.
tainty, may be approximately represented in various facility network
‘studies, 1 obtain, at the very least, useful qualitative reels, In
articular, we evaluate the probability of a held order (ie, the
‘Probability hat a service request is held for lack of spare facies)
teith Blocked Customers Held (Ron) as the quewe dieeipline. We also
‘op the model to capacity ezpaninn problems, introduce the con:
‘cept of margin, the extra eqpactty needed 19 meet the demand within
4 given helt-arder probabil, and examine its sensitivity with re-
"pect to growth, tumover (or churn), and system size, We find that
‘Geeregating small demands ino a single larger demand produces
‘significant reduction of the margin, because of improved statistical
properties,
1. iwrROBUCTION
In this article, the tansiant behavior of Une Kendall birth-deach
process" with immigration is examined, and aome applications of the
‘oasTig Kent eth presen inch nein ten opti
ss
process lo capacity expansion problems are discussed. The choice of
Ihc process was motivated by the search for a rode for special
fervices point-to-point cireult demand, 8 model which would be used
ts a tool for determining facility network cireut routing strategien
Special eervices demand ganeraly consis of demand for full-time
dedicated vrewits (og, foreign exchange lines, wars line, deta lines),
ts opposed to the measage-taffic offered load which consists of de-
fund for the use of common fucilities for a velatively short period of
time, Thusy the aystem examined is characterized by the stochastic
process 0) with realizations (stats) n= 0,1, -->9o% where might
fefer to the number of working eteuits or ome other facility, rather
‘han to the number of busy trunks, a in the message-tratfie case, By
‘efiniton, the bireh-death proce allows transions from some sale
hhton + Lvin a birth (cir connect), or to n 1 via a death (cireull
isconmnect). ‘The transition rates are‘, for the biths and uy for che
‘deaths, boll of hich are chosen proportional to 1 forthe fllowing
Tei clear, for special sorviw, chat the rate of daconneets, po is
state dependent. There sre n fact, indications that a i « monoton
‘cally increasing funetion of n.‘The simplest such function ism which
implies thatthe probability of disconnects is proportional to the size
fof thesystem. With this choice for Use deeth rato, a rember of possible
‘hoices exit fo the birch rae: Choosing it to be w constant eauses che
tenn namber of ciccuits to saturate in time, while choosing it to be
propotcional ton cares the mean to grow or decline exponentially
Since spevial services are presently characterized by significant net
frowth, it woud seem that m pliwible modal for special services
‘Semand sa bith and death proces in which bol the heh and death
‘ates are proportional to the sate
‘One consequence, however of assuming hy = 70a thet ifthe process
reaches the Hate n= Oat any time, by 4 teseaion of disconnect, it
rll stay there forever, sine the birth rate i zero. To eliminate this
Characteristic, the concept of immigration may be intrdiced by taking
Anamh + 8, where tthe Immigration factor. The cases wich and
‘without immigration wil be disused below.
It must be emphasized that ic is not the incent of this paper to
validate the model based on an examination of actual special wervices
“dna, Such statistical data anni is important for a fnal assessment
fof the accuracy of the moel and ie currently being undertaken, For
the purposes of this paper, ic shall be assumed that a study of the
‘proposed model is justified, based on the inmitive arguments given
Above and un the fact that the model captures the growth and turnover
‘characteristics of special services (se Section 5.1). The model provides
{means by which special cervices demand, with it inherent uncer
8 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1981
tainty, may be approximately represented in various special services
facility network studies, to obtain, atthe very least, useful qultaive
recut
‘Since the situation of intaast is that of net growth, ita clear that
statistical equilibrium does not exist and that iia the problem ofthe
‘transient aolutions ofthe Kolmogorov brth-death equations that is of
prime importance, Much liverature exets on the eubject of transient
lutions for bich-mnlsleath process and the ee in which the
Uranilion rates are state indepenilent ie completely elved.”™ The cane
in which A, and um are proparconal to the population is solved when
‘immigration i not included: The rule fora specific initial eondition,
namely staring from the state x ~ I, ae derived in Refs. 4 and 10 and
‘the expression for the general initial condition are quoted in Raf. 10.
For the nonsero immigration case, the form of the generaring function
For the state probabilities fs kos,” at ema that explicit expres
sons for the tate probabildes have not previously appeared in the
literature. In this paper, these expressions are derived for any non-
negative value of 8
Th special service, i an order for service is delayed because of lack
of spare feclices, the order is aid to be held, Thus, inorder to study
tapacity expansion problems, che probability of «held order is inte
AAvoed, as wel a the concep” of mang, the extra expacity needed to
rmeot the demand within x giver heldcorder probability. ‘This held-
order probability ix similar but 1 identical to the tran time
congestion ef the proves (see Appenilix B)- The «ew scipline
fotlowed herein Blocked Cuntomer Held (nen), in which an arriving
customer apends a tora time T (random variable) inthe sytem, aftr
‘which he departs rogurdless of whether he is waiting to be served (ie,
his service oder has boen delayed) or is actually being served (Le, be
thas hoen msignes a ere.
‘A furdamenal difference between this analysis and tletrafic muse
be emphasized. Tia diflerence arses because of the respective time
scale nthe 0 eases, Whereas the mean lifetime of 9 eall in afc
{+= 1/h i ofthe order of fen minutes, the mean lifetime of a circuit
in the process deserbed here is ofthe order of few yrs, I this
fact, couple with the relatively Go growth of special werviees demand,
‘hat tnales ic impossible o even approximately teat the process in a
Statistical equilum mode (pe growth) with a slomly varying enve-
lopereprecensng the row, Thus, the transontaspoct of the problem.
isto be concasted to the more conventional assumption, in eletrafc
Cheory, that statistical equilibrium prevails (it must be mentioned,
however, that some work has boon done concerning nonstationary
telephone traffic with time-varying Poisson offered load, eg, Refs. 12
to 18). Te tnust be further noted that, although the modal is being
APPLICATIONS OF SIRTH.OEATH PROCESS 59
proposed for special services demand, nevertheless, it may be applied,
trith an appropriate choice of parameters A, fan to Ay process
that behaves in similar manner.
"The held-order probability having been defined and the concopt of
margin introduced. questions concerning capacity expansion probledns
fre addremsod. Capacity xpancion i a problem that has been eeudied
by mang: In this paper, optimal eapacityexpansion policies are not
sought; enl¥ very specialized anpecis of the problem are considered,
For instance, the effects of aggregating demands into a larger single
demand are examinod, and the minimum capacity increment which
‘would mect the demand within a apecified interval of time and within
«given beld-order probability is determined, In addition, the relation
ship between spare capacity and lead time is diacusaed (are urnmary
ff result in Section TD. Some relevant work has been done by
Freidenfelde“* in which the author computes frst-passage tes 10
various levels of demand using general bsth-death proces, and
Aiseustes briefly ML-at-rliot problems. Work by Loss nd Whitt”
studies the impact of both deterministic and stochastic models on
Uutlization, ‘The authors use Browaian motion to model the stochastic
‘demand and follow a scheme siniar ta ours for determining che margin
needed ata future time.
"The organization ofthis paper is aa follows. Section II seta up the
problem and gives a summary of rsults. Tho explicic solutions forthe
fener cane ate derived in Secon TL, and this properties are exam
ined in Section IV. In Section V, gevwth, turnover, and churn are
defined, che concept of margin is introduced, and some of is applica:
tions to capacity expansion problems are discussed. Finally, Setion
‘VT contains the conclusions
|. BACKGROUND AND SUMMARY OF RESULTS.
121 General Birh-deeth equations
‘Consider a system deseribed hy «se of sates n=) 1,48, and
‘birth and desth process defined by a set of teanation rates (Mx pa)
‘Tho quantity Alia) + o(3) isthe probability of @ birch (death) in
‘the small inturval[,¢+ 6), given thatthe system isin state rat time
The probability of more than one birth or death in [+8] i 18).
‘The probabilities a(t of finding the system in tate matte # matt
satisfy the well-known infinite set of difference-differental equations
(p. 464 of Het. 1)
Phe Ha PAID + An Pa sl) wat Past
Vor 420, PQ=0, moO} a)
a
pao ~
ot Re Reach ae
60. THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1981
1f the intial number of cirouita ia my che intial condition may be
PaO By @
‘whore By if the Kronecker dla
"The particular birch-death process considered inthis paper are
the eates in which the rursition zaves are proportional to the pop
lation, n, wth oF without immigration **° The corresponding transi
tion raten, defined for all nonnwgativeintogers,m, are
er @
where 2, and faze nonnegative consinnts Tn the following sections,
Zeeuls fr the cane with no Immigration may be easily obtained by
setting,
12.2 Meen and vance
{thas been shown'* that the mean, m(e, and che variance, o(), of
‘processes such as thowe deseribed hy eqs (1) apd (2) may be ebtained
tvichout solving explicitly for Use 0). ‘The resulting expressions,
{atiafying nitil eonition 2), may’ be easly found vo be
(i) Case A
mit= (ws «
oe) = mt
Baie = ht wet a 6
emer ”
2 so
rope ee ir
APPLICATIONS OF SIRTHLOEATH PROCESS 61
and
(;)-teopeeomen ©
"This definition ofthe binomial coeficient is valid for any real number
‘rand any positive inter m (xp. 50 of Rf. 1) For m = 0, one defines
(Gy = 1, and for negative integers m, one define (5) = 0. The symbol
{f) ie not used ifm ie net an integer. Denoting » = A, where »fsany
‘ponnegacive rel number, the slations derived in this paper are
(Caso #u
cunmany "3" (") (" anton
(A Dhit metro, a9)
fay ify + =U a
PA =
Ly
FN,
Po = (money a on
fa mata a
(10) and (11) with no immigration (r= 0} are identical co
of Re. 10}
equa
the results quoted by Bally [gs (8
24 Applicstion to capacity expansion
In Section V, margin ia defined asthe capacity which n
in excees of the mean lo matt certain service roquirements, and the
proent marzin is delined ar che ratio ofthe margin to the mean in
peveent. The flloring ies summary of the main results
(i) By aggregating demands less percent margin is nooded than in
tha nonageregaved case, This effect is expecially significant for seall
demand,
(G2) Given w minimum desired time, 7; botween successive expan-
sions a procedure is establiahed for determining the minum eapacty
increment which would mest the given service requirement
Gi) Given a lead time,» bstoreen the moment facies are ordered
and the time they are available for us, a prowdure is established for
22 THE BELL SYSTEM TECHNICAL JOURNAL JANUARY 1081
Aelermining the threshold value of the remaining epare coreeponding
‘athe tine at which new facilities should be ordered.
(Ge) By introducing immigration, the sboorbing zero state is elim
‘ated and the percene margin needed to meet the eurvce requirements
is reduce for moderavely lnrge to large times (of the order of two years,
for more forthe particular values examined}
1, DERIVATION OF THE STATE PROBABILITIES
‘The approach followed (o wl the at of equations in (1) i the
generating function technique" Tn Ref. 10, sdiferential equation for
the generating function, Fs), defined below, is enablished and its
rolulion i derived. "The resulta are quoted in Section 8.1. Three well
Jmnown idenctien are given in Section 42 and are then used in Section
4.8 to derive explicie expressions for the state probabilicies. The pro
‘cedure flloweal in Berton 23s lo Wentiy Pl, €} a8 the generating
function for convolution of nwo Funetions,
‘8.1. The generating function
"The generating function, Fi, is vlated to the sate probabilities
‘through the fllowing exprerion
Fin = 3 s'Pate
oa)
“The differential equation for F(, 0, gan ine. (269) of Ref. 10 with
O88 ag) AFUE
3
TO ana) = 818 ~ F 0, as
woe
in) ~~e~ 90—
‘The wins igonncon ae
[_4_y (hams
(ea) les) Ati 08)
(a) Ge)
Fist
APPUCATIONS OF BINTH-DEATH PROCESS 62
where
a)
413)
‘The above solutions may be verified by direct substitution, Equation
(46) agrees with og, (671) of Raf, 10 lo, (17) with =O agro wich
99. (862) of Ret. 10.
In Section 1.8, use wil be made of the three following well-known
identities
3.2.1 Binomial iontty
or any a and and for any nonnegative intagor nthe following
ideotity holds (p.51, Re. 1:
(20)
3.2.2 Negative binomial ent
For any a and f auch that [8/2] < 1 and for any real number the
fullowing Wentity holds (seo pp. 51 and 269 of Tet.)
co-ar'= E car(Z)ra
‘fri strictly postive, identity (12.4) on p. 68 of Hef. 1 may be used to
write
Coes (ane cae ey
3.2. Generating function for a convolution
Let Fi(s) and Fis} be the generating functions for the sequences
(64 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1981
(PE nnes+ and (PR acon ropectivay,
Fin 3 sP2, Be)
“The fonction F(s) = F(s)Fe(s) i then the generating function for
(Pujeces ss the convolution of PS! and P2,and may be writen
cr
Be
Fo) - 3 e's 23)
where
Pr
Saveee $ one
‘The proof of this theorem is elementary (eg, see Chapter 11 of
Ret,
‘Note: Thie theorem applies to arbitesry sequences {Pi} and {P2")
(not necessarily probubility distributions) as lang as cheit respective
-enerating functions exis. Thus, th serits in es (22) must converge.
For the purpose of this theorer, however, is assumed that Fs) and
Fis) do exist.
3.3 Detvation of expo axpreasions
"The penerating function a eg (16! may be semaitzon as fllom:
Fig, t= 6, 1818.0, om
where
File) ad beso",
Fla (0H ast
fa) mete >o
Applying identity (20, it may be seen that #1, tis the generating
function for binomial type function,
Pose § (moreno
= Brey, (25)
APPLICATIONS OF BIRTH-DEATH PROCESS 65
where
paw (Se if men,
0 if mon
29
1a similar manner, may’ be seen frown dency (21) that Fs is
the generating funetion for x neyatve binomial type fanetion,
en
where
{may be shown tha | ca «<1 fr all values of ¢ = 0,05 9 1, and
Aya 0. Th, entity (21) applies in all che relevant casen
Te mow flows fone, (2) tae Fy) the generating function of
the convolution
pale = F PPMP LAD
ME [een]
[eras Near
‘where the upper liv on ths sum avisgs from the eapdition ma for
PEC establish hy e281
TRearranging one obtain
yay
I)
rm he ta band ef a a em. sh
{ore lr 10) vlan
mera
For this cass, Pls, 1) = 1. From the definition in eq (14), it ie then
(66 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1981
