BSTJ 60: 1. January 1981: An Overflow System in Which Queuing Takes Precedence. (Morrison, J.A.)

4.9 MB·English

THE BELL SYSTEM
TECHNICAL JOURNAL

‘An Overflow System In Which
Queuing Takes Precedence
By JA, MORRISON
(aeanuscrit rain Jay 9.1980)

When calls offered to a primary group of trunka find all of them
‘bury, provisions are often made for these calls t0 overflow fo other
groups of trunks, Suoh traffic overflow systems have been of interest
for a tong time, bul recently overflow systems Unat allow for some
‘alls tbe queued have deen of importance. In this paper we analyze
€ traf overflow systam with queuing, which consists of a primary
luda secondary grog. The spnlem uaich we consider here differs
Irom the two eyetoms wre investigated earlier, in that no overflow fram
the primany tthe seconslory tx permitted if there is a watling pace
tvutlatte im the primary quave Ae cith the earlier invstigations we
adapt an analytical approach which considerably reduces the dimen-

‘ions ofthe problem, and simplifies the calzulation of various steady

tate quanutties of interest. Our rests include expressions jor the
lowe probabilities, the average teiting times inthe queues, al the
‘average number of demands ia service in each group.

1. iTRoDUCTION

In this paper a particlar overflow system with quouing is analyze.
“The eyolem consist of tro group primary and a secondary, with ne
servere und gs waiting spaces which receive demands from indepen-
ent Poison sources S, with arrival yates Ak > O, f= 1 and 2
reapectvely, an depicted in Fig. |. The aevie times of the demands
treindependent, nd exponentially distributed with mean service rate
ISO Mall servers inthe socnvlay are husy when a demand trom
‘Sp arrives, the demand is quoued if one of the qs waiting spaces is

Naa Is onic arene fanny 218 rime
svailable,othorwiso i is lost (blocked and cloared from the system).
Demands waiting in the secondary quous enter service (in some
prescribed order) as servers in the secondary bocome fre.

Teall my servers in Use primary are buy when a demand from Sy
amvves, the demand i queued in the primary, if one of the gy waiting
spaces is available. No overflow is permitted from the primary queue,
0 that demand in the primary quewe most wait for server in the
primary to become free, all n: wervers in he primary ary busy and
all g, waiting spaces ure occupied. when u demand from S) arrives. the
demand is served inthe secondary, if cere isn free server and there
fare na cemands walking in the secondary queue, otherwise ies lost.

‘The overflow system deocribed ahve differs fom he tw aystes
which we investigated earn," in that no avertiow from the primary
to the scoondary is permitted if there is a waiting space avallable in
the primary queue. Thi reition was one mvoked by Anderson.’ Th
the bvo systtas investigated earlier, arriving calls can overflow when.
the primary quoue isnot fll. The system considered in thie paper is
‘a particular cae ofthe one considered by Rah which wns curmposed
fof two queue, one of which i allowed Wo overflow vo Uie other under
spociied conditions involving the quove lengths. He obuained some
‘numerical aulutions using Gauss-Seidel erwton tchaigue, bul none
ofthese correspond to the paccularsprtam that we ate considering
He also developed en approximate procedure for analyzing hia system,
bd on the use of the Interrupted Poison Process.

Here wo analyze the overfnay system ting technique analogous
‘to the one introduced inthe earlier paper! Lec py denoce the steady
state probabilicy that there are { demands in the primary and j
demands in the secondary cither in service or waiting. These probs-
Iii sain a se of generale birtearvl-denth equations which
lake the form of patil difference eyualions connecting neighboring
states, We earry out an analysis that reduces the dimensions of the
problem, which may be eonsderahle in cases of incest. ‘The basie
Xechnigue isto aeparata variables in tha region away from a eesti
boundary of sate space. This leads tn an eigenvalue probln far the
separation constant. "The eigenvalues ate rnnis of polynornial uae
tion. The probabilities p, are then represented in ter of the corre
sponding eigenfonctions: The constant eoetlicints in these represen
lations arv determined from the boundary conditions and the normal:
{tation eanition that tho sam ofthe probnbilitia is unity,

Various steady-state quantities are of interest, which may be ex:
pressed in term ofthe probehiliie "The quantities include the lose
{er blocking) probebilities, the average waiting times in the queues,
and the average number of demands in serve in each group, These
‘quantities may be expressed directly in terms of the constant cneff-
‘ents which occur in the representaions for the probabilities, ‘Thus

{2 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1081
the sloudy slate quantities of interest may be calculated dirty
svithoat having to eslclate the probes py. Here again the reduc~
{ion i the dimensions ofthe proiem is valuable.

‘Only the theoretical results ure prowentod inthis paper. Numerical
results will be presented in a forcheoming paper by Kaufman, Seery,
and Morrion.° Result wll be give there forthe Lo overflow systems
‘onridered previously, based on the euier analysis as wel as forthe
System considered inthis paper.

Section If dlcusees the representation of the probabilities p, in
‘erms of the eigenfunctions, aa Uhe houndary and normalization
Cousens, Varin ateady-atat quantities of interest re calculated in
Section II, The appendix gives proweries of the eigenfunctions.

‘We assume throughout thw analyse that gy 2 1, since the system
considered in this paper, and the two syatems analyzed earlier, are
dential if gy ~ 0) tey if there is no primary quoue. However, che
results ofthis paper rode to those obtained ean if gy =O. Tis
Tange, or even infinite, an alternate analysis, analogous Uo that pre
sented for the other two systems," ray be cared out forthe present
System, bur we do nol pursue tha here

1. REPRESENTATION AND BOUNDARY CONDITIONS

We let pu, denote the steady-state probebilty that there are i
demands in the primary ani j demands in the secondar, ether in
service or waiting. These probs ata a set of generalized birth
Snd-ceath equations” which may be derived in a straightforward
tanner, We define the trafic intensities

=
an(h ich ,

“Then the birdhand-death equations are
Jet.

pura) + al ~ Ba) + intima) + mint nlTP
eu = Snes ay 4 (1 Blan

‘OVERFLOW WHERE OUEUING TAKES PREGEDENCE 3
$A Ba dmin(é +1, mens + — Sadi + Amaia (6
for 0 si =h, 0 j= hs The normalization condition is
Dp cc)

For i x fy the variables in () may be separated, and there are
solutions ofthe form a, where
[ay + mingé,m) + pay = ai ~ dala + mint #2, mdaien (7)
forO-sis hy and
[eth = dg) # mina ~ 68:
a(t — Bolen + (1~ Balin + 1nd,
rom (0),

(a4 i+ pla, ~ an ~ Selans + (1+ Haus, o
for 0 =< my ~ 1, The solution of (0) may be expresoed in terms of
PPisson- Charlie” polynomials. We here denote the solution of (8) for
Which a: = 1 by (eq). The properties of sp, a) which we will nsed

sro given in the appendix
‘We assume that ¢, = 1. Then, from (7,

lat olar= aes + meen ao)

for 7 hy nnd pnw separation coment. He

for m =k, ~ 1. The solution of (10) may be expressed in tans of
‘Chebyshev pelyncmials of che second kind Ui} Ii convenient co

rine
(nomen) m

w= (2) ee o

“The appendix gives the properties of thew functions that we need We

role here, however, that Un(s) © 1 and U(x) = 0, From (9), (10),
(62), und (64, with a suitable normalization, follows chat

(2)'sman ontsn

- rey
(EY Toso. ainty so 030m
(Next, from (8}, a
(4) MB= alt Foes ++ Moy)

for 02 j= m~ 1 Te follows from (5 that fis proportional to
4-9, 0) for = j= Alo,

4 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1981
Los ~ dy) + me~ ADB) = aes # md Bien, 4)
for = 7s fb, Comeeponding to (11), we define

wo (2) u(t) ”

0) = Yo le) — War. 08)
“The appendix gives che proper of hese functions that we need It
follows om (0) and (2) thats proporinal to gf) for ne = 1=
dehe
‘Consequently, we take
_ [olcmasantod, Os Jsne
10 (eee weiiem an

We aio define

where

pai) = Saf ta AP 03)
“This equation may be written inthe form
salt ~ obo) ~ tyr ~ pp a, io] =O.

“The expresion in the square brackets in (19) polynomial in 9 of
dogree by = ne + glu was shown’ thi ile zeros ave positive and
fistinet,aid we denote them hy fm m= Ly oon, hy, Wy lv let po =.
follows that we may represent the probe min Ue fr
E eatdendsi-Pmadtalonls 0
peal 2)
E, cumbia -Pme tbhnd, meh

for 0.2 i= hy, where (a) is defined in (2), and the constants 6g are
10 be determined

remains to satisfy the boundary conditions corresponding to i =
‘yin (6) a8 well as the normalization condition (6). we ta f= hyn
(Gh, we obtain

Cast as tm FAR mPa (ait OT = BedPAG +

FU Upson OD

foros jem—1,

ohm + mh
ePhpin + eH adPHned + all = ByPHnre (2B)

‘OVERFLOW WHERE QUEUING TAKES PRECEDENCE 5
and, if ge 1,
[ealt = 5) + mst dns
0.06). t EPas vb RAL ~ BaP yn, (3)

form +1sjhe
TF we substitue (20) neo 21), afeer reduction with the help of the
recurrence relations inthe append, we ind that

clr +p a) ~ Sy + Py Ay foalf-Pe 89

+ efSaDm lon) ~ 5 ln, )By, ADaN 641 ~ Pm 3)
Anlen) = 0, (24)

or 0:= me — 1, Aloo ro (23), iis found that
E, Cobain (+ Ps 80nd

=, AL + om 01, lo AP) = 0,25)

formy+1= j= hy lemay be shown thatthe houndary condition (22)
in rodundans, as it en be exproted. ‘Thus the constants cn are deter-
mined by (24) an (25) only to within x ltipleative coneane, which

iedetermined from the normalization endition (8)
rom (20), withthe help of (16), (19), (62), and (58), i is found that
Spy — emi, a0) ~ tact ai%q AO), (26)

for0 = i= hy But, fom (12) and (65),
ozizm,

en
cs

Hence, from (26) and (27). with the help of (67), (58), and (65), the
normalization condition (s) implies that

oC, a8 0) ~ 8, a, a8, (0)
[5002 aN) ~ Gill eof = 1 28)

Once the constants ive been devermined, the steady atace prob
abilities py, may be calculated Irom (20), We reruark chat the number

(6. THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1961
of eonstanca to be determined ix ouly £1 + 1, whereas the nomber of
probabilities py i Uh + Ike Uh

UU, SOME STEADY-STATE qUANTMES

‘We proceed now tw the calculation of various steady. stato quantities
of inert, ‘These quantities are depited in the diagram of Fig
‘which indicates the mean flowrtates. Te lass probabilities By andl La
sre givon by

LS rn bam S Pity em)
and the probabiiies chat demand from the primiry, or secondavy,

sourve i queued on arrival are

aE km &

aad E Ere om

The probability that a demand smiving from the primary source

ig Men ren neti

OVERFLOW WHFRE QUELING TAKES PRECEDENCE 7
overflows Gimmedively) is

Days
Since the mean service rae ia, the mean departure rate frm the

primary queue to the primary servers in
Renn XS pw (32)

while che mean deparcure vate ftom the secondary queue to the
secondary servers it

Res = mail ~ Bad SE Be (a3)
"The average number of demands inthe primary and secondary queues
EW-mipn VAY YG- man (ae

‘leo, the average number of dewanda in service in the two groupe ane

A= EE ming mia, = EE minty mada, 6)

‘Now, according o Lil's Ubeoremythewveruge numberof domands
in queving eyacem ix equal to the wveragu rate of arrival of demande
to that ater tines the average Lime apent in that aystem. Lf we apply
thie vesule to the primary and secondary queues, we find thae the
‘average waiting times ofthe demand mhich ate queued in the primary
(ria the cecondary axe given by

l= 0, 36)

‘heorem tothe primary and secondary groups

+ hita= 0 6m

“The stendy state quantices of interact may be expressed in term of
the constants cq with the help of the representations in (20), From
(23) ie is found, with the help of (12) and (18, that

r= J ealta(oa, #3800) ~ 8, lm 8M, lbe]

iaiepm defo 8

18 THE GEL SYSTEM TECHNICAL JOURNAL, JANUARY 1901
We define

fe), 00) el) ~ soe oe]
Ist 0 @)

from (25). Then, om (80) ito found, with the help of (26), 27), and
(65), that

= deta, a0 (0) ~ 1h )
“Moreover, fom (29) and (21) i flows that
Ltd de, 0, a

1 from (32) it follows that Ry, = XQ. as isto be expected, since in
the steady state the moan deperturo rato fom the queue i equal to
the mean arrival rat to

‘We detine

auer= fare (8s VE EVO. EFL Gay

Ya@ed, E21
“Then, from (4, with the help of (26), (27, and (8), ite found that

v= ds 0, 08, (2) co

Alun, from (36), with che help of (5), (58), (58), and (65, it fellows

cane

ernay be vii, with the help of (29), (67), and (65), that (41) and
(4) ure consistent with (87). "The explicitness of the expressions for
the quantities in (0), (41), (43), ana (44) is due to the fact that these
‘quantcies are nt affeced by the secondary. This, ofcourse, is nol the
‘ase forthe lees probability Z, which is given by (38).

(Next, from (28), since jal = 1, ik found, with che help of (201
and (63), lt

Tam Seatal-pm allel + 95 0 loa)

a +400 =}.

ay, AL + bm Oy, lanl. (8)

Alo, from (8, with che help of (6) and (68) i follows chat

ame — F catecsl2 = hm dele!

[lL F be Ohba) — See + Oy a fant (66)

OVERFLOW WHERE QUEUING TAKES PRECEDENCE 9
In view of (98), (1), (48), and (46), the second relationship in (37
provides a usef numerical cesk.
‘We now define

nape mis he an

fe sum on fin (3), we obtain

(l= Ba) nde erm = a
form +15 Ilo

ramaiten med
ons se =r Hom (28 ant

oa ow

“Then from (90) and (i), with he help of (63), we obtain
= [%a0) ~ Ta, 1)

and z= AQ is to be expected. Also, from (94) an (42), it follows
that

52)

‘Thie completes the calculation of expressions for the steady-state
quanton of interest.
1V. ACKNOWLEDGMENT

‘The author ia grateful to G.M. Anderson for bringing this problem
to his attention
APPENDIX

‘We defn a0, a) by the xecurrne relation

(as 14 NO, a) = alt dol Pa) + EF Da, a
sik a) = 3)

fori =0,1, +, Thus ss, a) isa polynomial of degres nin both A and
t,and it may be related to a Poeeon-Charir polynomial.” However,
se pve here the properties of (4 which we sll need. An explicit
Formula i

aa) = y Oe

Lt on

10. THEBELL SYSTEM TECHNICAL JOURNAL, JANUARY’ 1863