W.B.van den Hou t 3
J.P.C. Bla nc
Tilb urg Univ ersit y, Facul ty of Economic s
P.O.Box 90153, 5000 LE Tilb urg, The N ethe rl an ds
Abstract : A ne wversionofthePower-S eriesAl gori thmisdevel op edtocomputeth e
steady-stated istrib utionofari chcl assof Markovi anque uei ngnetworks. The arri valpro cess isa Mul
ti-queu eMarkovi anArri valPro cess,whi chi samulti -queu egen eral i zati onoftheBMAP.I ti ncl udes
Poi sson, fork and roun d-robin arrival s. At each queue the servic e pro ce ss i s a Markovian
Ser-vi cePro ce ss,wh ichin clu desse quence sof phase-typ e di stri buti on s,set-upti mesand mu lti -serve r
queu es. The rou ti ng is Markovi an . The resul ti ng qu euei ng network mo del is extremely
gen-eral, whi ch makes the Power-Se ri es Al gori th m a use fu l to ol to stud y load-bal an ci ng, capaci
ty-assignment and se quenc ingp robl ems.
1 Intr oduction
Networks of qu eues wi th ou t p ro d uct-form sol ution are usual ly d icu lt to anal yze, both an
a-l yti caa-la-l y and numeri call y. For Markovi an networks, the steady-state d istribu ti on is determi ned
by th e set of bal an ce equati ons, but because of the si ze of the mul ti- dimen sional state space
anynume ri cal metho d to sol ve the seequ ati on si s i nevi tab ly memory an d ti me c on su ming. The
Power-Seri es Al gori th m (PSA) ai ms to b e an eci ent wayto sol ve the b al ance equati ons. The
advantage of the PSA over other metho d s i s that tech niqu es l ike Pade -ap proxi mati on can b e
used toextrapol ate the power se ri es, and that th e behaviour of the p owe r- se ri es can be studi ed
toassess thecredi bi li tyof theresu lts.
Networksofqu eueswil lbeconsi deredwi thu nb ound edqueue si zes. Customersarri ve
accord-i ngtoaMul ti -qu eue Markovi anArrival Pro cess(MMAP),wh ich i sa mul ti -qu eue general ization
ofth eBatchMarkovi anArrivalPro cess(BMAP)i ntro d ucedbyLuc antoni[?]. O ntopofth eabi
l-i ty of the BMAP to mo de l de p end enci es between i nterarri valti mesan d batch si ze s, theMMAP
can al so mo del al l kin dsof de p en denci es b etwee n arrival s at the di erent qu eues, li ke fork and
round-robi n arri val s. At each queue the service pro cess is a Markovi an Servi ce Pro cess(MSP).
3
Thi s i ncl ude s for exampl e set-up times, sequ ence s of p hase-type di stri butions and mu lti -serve r
queu es. The rou ti ng of cu stomersi s Markovi an, whi ch i ncl ud esal arge varietyofne twork
struc-tures(li kethecl ass ofJ ackson n etworks, a smal l su b cl assofthe networksconsid eredhe re ). The
extremege nerali tyofthen etworkscontai nedi nthi sgen eral frameworkmake stheanal ysi sb el ow
a usefulto ol tostudyl oad- balanci ng, capaci ty-assignment and sequenc ingp robl ems.
Th e b asi c id ea of the PSA is li ke a homotopy: the tran sition rates of the ori gin al n etwork
aretransforme d witha parameter ,su ch that for =1 th etransforme dnetwork istheorigi nal
network an d the asymptoti cnetwork for i n ane ighb ourho o dof =0 iseasytoanal yze. Then
theinformati onfromtheprob lemnear =0canb eusedtosol vetheprobl emat =1. Thebasic
i deaof thePSAstemsfrom Ke an e(see [?]). I thasb e enapp li edtoque uei ngmo del swithqueues
i np aral lel[?,?],theshortest-que uemo d el[?],variousp ol li ngmo de ls[?,?],andtheBMAP/PH/1
queu e [?]. For an overview, see [?]. For al l these mo del sonl y the arrival pro cess need ed to b e
transformedand thetran sformati on parameter coul db ei nterpretedasthel oad ofthesystem.
Unfortu nate ly, thi s p ro c edure i s onl y p ossib le for feed forward ne tworks. For non-feed forward
networks, sets of equationswou ldh ave to be sol ved wi th a si zethat rapid ly i ncreases witheach
step of th eal gori thm. K o ol e[?] suggests to prevent thi sbytreating th equ euesasymmetric al l y.
Theap proach th at wi l lb eusedinthepresent pap e r,is totransformtherouti ngp ro c ess also. In
both app roaches, the parameter no l on ger hasa cl ear interpretati on. Thi s coul d be ove rc ome
byu sing moreth an one transformation parameter. Fore xampl e, aparametercoul db eusedto
transformthearri val pro cess, anda parameter for th erouti ngp ro c ess. Howe ve r,usi ngseveral
parame ters l eads to p owe r- series exp an si ons in more than one variabl e. Thi s impl i es th at more
co eci ents nee d to be cal cul ated an d that mul ti-d imen si onal Pade-app roxi mants are re qui re d.
For thi sreason, on ly asi ngle parameter wi ll beuse d here.
In se cti on 2, th e network mo de l i s i ntro du ced. I n sec ti on 3, the algori thm to calc ulate the
steady-state di stribu ti on and moments is d escribed. In section 4, two exampl es are given. The
rst consi ders th e op ti mal orde r of qu eues in seri es. The second shows that for cyc li c open
networks with symmetri c arrival s and equal loads the expected total nu mber of c ustomers is
mainl yd ete rmi ned bythesum of these con d moments of theservice -ti med istribu ti ons.
2 The Netwo rk M odel
The numb er of queue s i s S. U nle ss i ndi cate d otherwise , thefol lowin g notati on i s used . Vectors
arecol umnvectorsan dwri tteni nboldface. Thevectoreisavectorofones,0an de
0
arevectors
of zeros,e
s
areth eun it vectorsof siz eS for1sS and e
S+1 =e
1
. Foranyvectorn, de ne
jnj = e T
n. Matri ces are written i n capi tal s. The matrix O i s a matrix of ze ros and I
`
a uni t
matri xofsi ze`. Theop erator d enotestheKroneckerp ro d uctof twomatri ce s;th eoperator
tionthe rowve ctor and T =0Te (conformNeu ts [?], bu twi th outprobabi li tymassatze ro).
The cl ass of ph ase -type d istrib uti ons in clu des th eErl an g and hyp e re xp onenti al d istrib utions as
wellas ni te mi xtu re softhe se .
2.1 Multi-queue Ma rkovian Arriva l Process
Thearrivalp ro cessi sa Mu lti- queue Markovi anArrival Pro cess. Ithasanu nderl yin girreduc ib le
Markov pro cess withJ
0
states. I n th is und erlyi ng p ro c ess, atran siti on j !h i smade with rate
jh
(1 j;h J
0
<1). The set of possi bl e b atch arri vals is fb
m j0 m Mg, wi th b 0 =0 and b m 2IN S
nf0gfor1m M 1. Atransi tionj !h i n th eun derlyi ngp ro c esscausesan
arri valof batch b
m
withp robabi l ityq
mjh . A = f jh g; A = di ag(Ae); Q m = fq m jh g; P M m=0 Q m = ee T ; A m = A Q m ; P M m=0 A m = A:
The pure MMAP f(N MMAP t ;J t ); t 0g on state space IN S 2f1;:::;J 0
g i s i denti cal to the
BMAPif S=1and b
m
=m ( 0m1. I tthe n hasgenerator
Q MMAP = 0 B B B B B @ A 0 0 A A 1 A 2 111 O A 0 0 A A 1 111 O O A 0 0 A 111 . . . . . . . . . . . . 1 C C C C C A :
Lu cantoni [?] l ists a numb e r of speci al cases of th e BMAP, l ike the Poi sson p ro c ess,
Markov-mo dul ate dPoissonpro cesses,PH-ren ewalp ro c essesandpro ce sse swithcorre latedbatcharri val s.
Ifeachque ue hasan ind ep e ndent BMAP,th iscan bemo del le d as aMMAP. Othersp ec ialc ases
of MMAPsare:
1) Poi ssonarri val s: ind ep e ndent Poissonarri val swith rate
s atqueue s: M =S; A 0 0 A=0 P M m= 1 m ; b m =e m ; A m = m ; for 1m M:
2) Roun d-robi n arrival s: anarri valatqueue sisfol l owed byan arrival atque ues+1wi ththe
interarri val timee xp onenti al l y di stri butedwi th rate
s : M =S; A 0 0 A=0 diag(); b m =e m ; A m = m e m e T m+1 ; for1mM:
3) Fork arri vals: si mu ltaneous arri val sate achqueue withp hase-type i nte rarrivalti mes:
M =1; A 0 0 A=T; b =e; A =T 0 :
2.2 Markovian Service Pro cess
Theservi cepro cesse satal lqu euesarein depende ntMarkovi anServi ce Pro cesses. AMSPhasan
un derlyi ng Markovpro cess wi th J states, an d th e transi ti on rates are al lowe dto depend on the
numb e r of customers n at th at qu eue. A transi ti on j !h i s made with rate
jh
(n) and such a
transi tioncausesaservic ecomple ti onof`cu stomerswi thprobabil i tyr
`jh (n )( 1j;hJ <1 ;0`n1 ). B(n) = f jh (n)g; B(n) = d iag(B(n)e); R ` (n) = fr ` jh (n)g; P n ` =0 R ` (n) = e e T ; B ` (n) = B(n) R ` (n); P n ` =0 B ` (n) = B(n):
A pure Markovi an Se rvi ce Pro cess f(N MSP t ;J t ); t 0g on state space IN 2f1;:::;Jg h as generator Q MSP = 0 B B B B B @ B 0 (0)0 B(0) O O 111 B 1 (1) B 0 (1)0 B(1) O 111 B 2 (2) B 1 (2) B 0 (2)0 B(2) 111 . . . . . . . . . . . . 1 C C C C C A :
In thi spap e r,a numb er of assumpti onsi s made abou tth is se rvi ce pro cess. Fi rst, it i s assumed
thatall non- emptystatesare transi ent,sofromanyini tial state thee mptystateswi ll e ventual ly
bereached . Fu rthe rmore,i t i sassumed th at wh en th eMSP reachesthe emptystate s,it returns
tostate j wi th probabi li ty
j
,where i tre mai ns. Forthi s i tissuc ient that
B ` (`)=B ` (`)e T ; B ` (0)=O ; for`0:
For a queue thi s imp li es that at th e en d of each b usy perio d the MSP returns to state j with
probabi li ty
j
, where i tremain s unti l the ne xtarri val at th eque ue. Fi nal ly, it i s assumed that
customers arenotse rved in batch es:
B
`
(n)=O ; for`2:
Thi s assu mpti on i s not essential , bu t i s made because otherwi se a more compli cated routin g
pro cessne edstob ed e ne dan dnotationwou ldbemorei nvol ve d. Intheexampl esofMSPs l isted
below,theve ctors e
1 and e
2
aretheuni t ve ctorsof siz e2.
1) In depende ntp hase-type servi ce-ti medi stri buti on s:
B 0 (n)0 B(n)=T; forn1; B 1 (n)=T 0 ; forn1; T
2) As1), but wi th se t-u p: after each i dl ep eri o d th e rst servic e ti meh as i niti al di stri bution
1
, al l othe rserviceti mes have i ni ti al d istrib uti on
2 : B 0 (n)0 B(n)=T; forn1; B 1 (1)=T 0 1 ; B 1 (n)=T 0 2 ; forn2; = T 1 :
Any pai r of phase-typ e di stri butions ( ~ T 1 ; ~ 1 ) and ( ~ T 2 ; ~ 2
) can b e mo del ed by a si ngle
ge neratorT withtwod i erent i ni ti al di stri buti ons
1 and 2 ,bytakin gT bl o ckdi agonal : T =e 1 e T 1 ~ T 1 +e 2 e T 2 ~ T 2 ; 1 =e 1 ~ 1 and 2 =e 2 ~ 2 .
Example s 1 and 2 are speci al cases of se quence s of ph ase -type d istrib uti ons f(T
` ;
`
); ` 1g.
Bec au se the number of p hase s J of the MSP i s ni te, such se quence s must, after a numb e r of
set-up d istrib utions, start repeating i tse lf, eithe r i n a determi nisti c or i n a probabi li sti c sen se.
Bec au se the MSP starts anew atthe b e gi nn ing of e ach b usy p eri o d, also mixtures of sequ ences
arep ossib le. Thi sc ou ldbe usedtomo delfor exampl easituationwhereatthe b egi nni ngofeach
bu sy p e ri o d, ei th er a fast or a slow serveri s chosen . O the r exampl es of MSPs are mu lti -serve r
queu es:
3) c id entical expon enti al serverswi th rate :
B 0 (n)0 B(n)=0 mi nfc;ng; for n1; B 1 (n)=mi nfc;ng; for n1; =1:
4) c id entical phase-typ e serve rs:
B 0 (n )0 B(n )= P n s= 1 I s0 1 T I c 0s ; for1nc; B 0 (n )0 B(n )= P c s= 1 I s0 1 T I c 0s ; forc<n; B 1 (n )=[( P n s =1 I s 01 T 0 I n 0s )(I n0 1 )]I c0n ; for1nc; B 1 (n )= P c s= 1 I s0 1 T 0 I c0s ; forc<n; =( ::: ) T : Here, I s i s a un it-matri x wi th si ze ` s
for 0 s c, wh ere ` i s the nu mber of phases of
th e ph ase- type di stri buti on. The transi ti ons are d e ne d such that i f there are no wai tin g
custome rsi nth equeu e(nc),the nthe rstnserve rsareac ti veandtheothe rc0nserve rs
areid le;whenserve rscompl etesservi ce,the nthec ustomersatserverss+1;:::;nmoveto
serve rs s;:::;n01, conti nui ngservi ce in the same ph ase . Servern becomes i dl e, wi th the
serviceph asedi stri buted accordi ng to . Thi sway, no vari ab le s ne edto b e added tokeep
2.3 Markovian Rou ting Process
The routin g i s Markovi an: after servi ce compl etion at queu e sthe c ustomer join s queue t with
probabi li ty
st
and leavesthene twork wi th probabil i ty
s0 ( 1s;tS ). X =f st g; 0 =f s 0 g; Xe+ 0 =e:
2.4 Markovian Network P rocess
Theabovede scribedarrival ,serviceandroutingp ro c essesdetermin ethen etworkp ro c essf(N
t ;J t ); t 0gon state space = n (n;j) n2IN S ;1j s J s for 0sS o :
Thestate(n ;j)2d enotesthattherearen
s
customersatqueu es ,thearri valpro cess isinstate
j
0
and the servi ce pro ce ss at queu es i s in statej
s
( 1sS ). To i ntro d uce matri xnotati on,
i tisconve nie nttomapth e(2S+1)-d imensi onalstatesp aceontothe(S+1)-di me nsionalstate
space = n (n ;i) n2IN S ;1iI o ; whereI =J 0 2:::2J S
. Thi scan be done 'l exi cographi cal ly'with the map pi ng
i(j)=1+ S X s= 0 (j s 01) J s +1 ; where J s =J s 2:::2J S for0sS an d J S
+1=1. The reversemapp ing i s
j s (i)=1+ 2 (i01)mo d J s 3 di v J s+ 1 ; for0sS;
Thi smapp in gd ete rmi nesthen etworkp ro c ess f(N
t ;I t ); t0gon state sp ac e . Ifthen etwork
i s stabl e,thesteady-statep rob abi l iti esof th isp ro c ess
P i (n)= l i m t!1 Prf(N t ;I t )=(n;i)g
exi stforall(n;i)2
. The yarei nd ep e ndentofth ein iti alstate(N
0 ;I
0
)an duni quel ydetermi ned
bythe b al ance an d normal i zati on equ ati on s. For any matri x A, l et doubl e b rackets denote the
Kroneckerpro du ct [[A]] s =I J02 :::2Js01 AI Js+12 :::2 JS ; for 0sS:
Then th ebal an ceequ ati on sare
( hh A0A T 0 ii 0 + S P s =1 hh B s (n s )0B T s0 (n s )0 ss B T s1 (n s ) ii s ) P(n) = M P m= 1 hh A T m ii 0 P(n0b m ) + S P s =1 S P t=0 st hh B T s1 (n s +1) ii s P(n+e s 0e t );
for n 2 IN , with P(n) = 0 for n 62 IN . The matri ces A 0 an d B s 0 (n s ) i n the l eft-hand si de
corresp ond tochangesi n thearri val and servicepro ce sse s wi th out arri valor servi cecomp leti on,
and ss B s1 (n s
)withthee ve ntthatacu stomerjoinsthesameque ueagai n,whi chd o esnotchange
theque ue l engths. The rste xpression i nthe ri ght- handsi de corresp ond sto an arri val and the
second wi th a se rvi ce compl etion fol l owed by ei ther a dep arture from the ne twork (t = 0) or a
transi tiontoanoth er queue (t6=0;s ).
3 The Power -Ser ies Alg orithm
Thearri valandroutin gpro ce ss ofthene tworkdescri b e dabovewil lbetransformedwi tha
trans-formation parame ter in [0,1],in sucha waythatfor =1 th e transformed n etwork pro cess is
equaltotheori gi nalnetwork pro ce ss. Th is wi l lb edone issu chawaythattheste ad y- state
prob-abi li tiesasfun cti onof areanalyticat =0andtheco eci entsofth esteady-stateprobabi li ties
can becal cul ated recursi ve ly.
Let r
m = jb
m
j de note the numb e r of c ustomers in batch b
m
, for 1 m M. Re place the
probabi li ty matrice sQ m by Q m ( ) = rm Q m ; for1mM; Q 0 ( ) =Q 0 + M P m=1 (10 r m )Q m =ee T 0 M P m= 1 r m Q m :
Theprob ab il ityofanarrivalofrcu stomersi smulti pl iedby r
,andth eremaini ngp robabi l itymass
i s ad ded to theprob ab il ityof no arri val , so P M m=0 Q m ( ) =e e T for 2[0;1] and Q m (1) =Q m
for 1m M. For smal le r l essarrivals o ccuron average and for =0 no arri val s o cc urat
all .
LetX
d
de note th edi agonal matri xwiththesamedi agonal as the rou ti ngmatri x X,and X
o
the o -d iagon al part of X, so X
d +X
o
=X. In the transformed n etwork pro ce ss, the routin g
probabi li ties X and
0 are re pl ace dby X( )=X d + X o ; 0 ( )= 0 +(10 )(I0X d )e:
Theprobabi li tytogo fromque ue stoqueue t ,wi th t6=s,ismulti pl iedby ,and theremai nin g
probabi li ty mass i s ad ded to the p robabi l ity to leave th e network, so X( )e+
0 ( ) = e for 2[0;1] an dX(1)=X , 0 (1)= 0
. For small er ,th ecustomerson average vi si tl essqueu es,
becauseaftereach servi cecompl eti on theyl eave thenetwork withhi gherprobabil i ty. For =0,
customers on ly vi si t asi ngl e stati on ,possi bl y severalti mes.
Th earri val ratesat thequ euesfrom ou tsi de thenetwork are equal to
( )= M X rm b m T A m e;
where i sth estead y-statedi stri bu ti onofth eMarkovpro cess und erlyi ngtheMMAP,whi chcan
becal cul atedfrom(
A0A T
)=0;e T
=1. Thearri valratesb othfromoutsidethenetworkand
from theothe rqueuesaree qualto
( )= h I0X T ( ) i 0 1 ( )= ( 1 X r =0 r h (I0X d ) 0 1 X T o i r ) (I 0X d ) 0 1 ( ):
Thi s p ower seri esconverge s an d si ncei t has on ly non- negati ve co eci ents,( )i s i ncreasin g in
: for larger there are more arri vals an d customers l eave the network l ess often . The se rvi ce
pro cess at e ach que ue do es not depend on . From th is i t i s easi l y se en that i f the origi nal
network i s stabl e, the transforme d network i s al so stabl e for all i n[0,1], an d the steady-state
probabi li ties are,uptoa constant,uni quel y determi nedbythebalance equations. Thesec an b e
rearrangedinto ( hh A0A T ii 0 + S P s =1 hh B s (n s )0B T s0 (n s )0 ss B T s1 (n s ) ii s ) P (n) = M P m =1 rm hh A T m ii 0 fP (n0b m )0P (n)g + S P s =1 S P t=1 t6=s s t hh B T s 1 (n s +1) ii s fP (n+e s 0e t )0P (n+e s )g + S P s= 1 (10 ss ) hh B T s 1 (n s +1) ii s P (n+e s ); (1) forn2IN S
. Clearl y,thestead y-stateprob ab il i ti esarefuncti on sof . Becausearri val sofb atches
of siz erhave arate that isO ( r
), for #0,the ste ad y-stateprobabi li ties satisfy
P (n)=O jnj ; for #0;n2IN S : (2)
Noti cethat,for =0,on ly th eemptystateshave non-zerop robabi l ity,because the rearedep
ar-turesfromthe ne twork bu t n oarrival s. I n a futu re pap er several state me nts about convergence
and anal yti ci ty i nthe present pap e rwi l l be prove dfor amuch wi de rc lass of Markov p ro c esses,
amon gothersthat thesteady-state probabi li ti es are an al ytic functi on s of ,in anei ghb ourho o d
of =0,soth ey can bereprese nted bythei rp ower-se ri es exp an sions:
P (n )= 1 X r =jnj r U r (n); forn2IN s : (3)
The transformednetwork p ro c ess i ssuchthat theco eci entvectorsU
r
(n )of the se power-series
expansi onsc anbec al cul ate drec ursivel ybyth ePSA.Thiswil lbesh own rstforth ee mp tystates,
and the nforthenon-emptystates.
and states of the se rvi ce pro cesses must therefore ren der the ste ad y-state d istribu ti on of the
arri valpro cess:
h I J 0 e T i X n 2IN S P (n)= ;
wheree isavectorofone s wi th si zeJ
1
2:::2J
S
. Wh enth en etworki semp ty,thestatesofthe
servi cepro ce sse sat thequeues ared istrib ute daccordi ngtothe ini tial di stri buti on s
s : P (0 )= h I J 0 e T i P (0) ; where = 1 ::: S
. Comb in ingboth re nders
P (0)= 8 < : 0 h I J 0 e T i X n>0 P (n ) 9 = ; :
In serti ngthepower-serie sexp ansi ons(??) andequatin gtheco eci entsofc orrespond in gp owe rs
of one ither sid eof th ee quali tysignshows that theco eci entsofthe exp an sionsoftheempty
statesP (0) satisfy U 0 (0) = ; U r (0) =0 ( h I J0 e T i P 0<jnj r U r (n) ) ; for r1: (4) Noti ceth at e T U 0
(0)=1,sofor =0 all p robabi l itymass i satth eemptystates.
Insertin g the p ower-series expansi ons (??) i nto the balance e quati ons (??) and equatin g
the c oe cie nts of corresp ond ing powers of on ei ther si de of the e quali ty si gn, shows th at the
co eci entsofthep ower-seri ese xpan sionsofthenon-e mp tystate ssati sfythefol lowingrec urrence
rel ati on s: ( hh A0A T ii 0 + S P s=1 hh B s (n s )0B T s0 (n s )0 ss B T s1 (n s ) ii s ) U r (n) = M P m=1 hh A T m ii 0 fU r 0r m (n0b m )0U r0 r m (n)g + S P s=1 S P t=1 t6=s s t hh B T s1 (n s +1) ii s fU r 01 (n+e s 0e t )0U r 01 (n+e s )g + S P s= 1 (10 ss ) hh B T s1 (n s +1) ii s U r (n+e s ); (5) forn2IN S
;rjn j. The matrix i nthele ft-hand sid e,
hh A0A T ii 0 + S X s=1 hh B s (n s )0B T s0 (n s )0 ss B T s1 (n s ) ii s ;
i s i nverti bl efor all n2IN S
nf0g. The c o ecie nts U
~ r
(n)~ in theri ght-hand si de ei ther have ~r<r
prop e rty (??). Toge th er, th is i mpl ies that th e co eci ents of the expansi ons of the steady-state
probabi li ties up to theR-th power of can b e cal cu latedrecu rsi vel y, for in creasing value s of r
and,for each xed r, fordecreasi ngval ues ofjnj,starti ngwithjn j=r :
Power- Seri esAl gori thm
Cal cul ateU
0;0 from(??) forr=1;:::;Rdo for N =r;:::;1 d o foral l n2IN S withjn j=N d o Cal cu lateU r (n) from(??)
Cal cul ate U
r
(0 )from (??)
Usuall yoneisnotsomu chi nte re stedi ntheste ad y-statep robabi l itie s,b utmorei nmomentsof
thep ro c ess. Theexpansi ons of momentscan b eob tai nedfromtheexpansi on softh esteady-state
probabi li ties: li m t!1 E ff(N t ;I t )g= X (n;i)2 f(n ;i)P i (n )= 1 X r =0 r X (n;i)2 :e T n r f(n ;i)U r i (n); forfu nctions f : !IR. Example s are
f(n ;i)=jn j, thee xpected total nu mberofcustomersi nthenetwork,
f(n ;i)=n t
s
, thet -thmome ntof thequ euelen gth atqueu e s,
f(n ;i)=n
s n
t
, thec ross-pro du ct of thequeue le ngths atque ues san d t.
Th estoragerequi rements oftheal gori th mcan b esubstanti all yreduc edi fth emaximalbatch
si zer=sup
m r
m
i s nite. From (??) i t can be see n that i nstep rof th e al gorithm,c o eci ents
U
~ r (
~
n) withr~<r0rarethen nolongerne eded to cal cul ated theremain ing c o ecie nts.
Th e MSP at que ue s depend s only on the qu eue l en gth at que ue s . The state depend ence
coul db emad emoregen eral ,noton lyforth eservi cepro cessesbutal soforth earrivalan droutin g
pro cesse s. Thestatedependen ceofth ese rvi ceandrouti ngpro ce sse smustbesu chthat,for =0,
all non-emp ty states of the transformed network p ro c ess are tran sient, so eventu al l y th eempty
states are reached. Then the se rvi ce p ro c esses mu st be stop p ed an d the di stri buti on ove r
the servi ce-ph ases must be known (b ut n eed not b e the K rone cker pro du ct
1
:::
S ).
The Markov pro cess underl yi ng th e MMAP must b e state i nde p en dent toc al cul ate , but the
probabi li ty matri ces Q
m
c an be state dep e ndent. Thi s way,the co e cie nts of the empty states
can sti l l b e cal cul ated from (??) and the co eci entsof th en on -empty states can b e calc ul ated
from (??) i fthe p arametersare re place dbythestate depende ntp arameters.
Suppose that by the algori thm d escribed ab ove, for ei ther p robabi l iti es or moments, the
co eci ents fv
r
; 0 r Rg are obtai ned . To c ompu te th ese rst R co eci ents the number of
co eci ents of th estate p rob abi l iti esthatn eedtobecalc ul ated is
# 8 (r;n ;i)2IN2 jjnjrR 9 =I2 R+S+1 ! :
Bec au se th is numbergrows fast in R,i t wi l lbe obviousthat it i s worthwhi l e toobtain as much
i nformationfrom th eco eci entsfv
r
; 0 rRg as p ossi bl e. Thati s whytechni ques to make
seri esconve rge or converge fasterform an essenti al part of the PSA. Theep sil on algorithmand
bi l ine ar mappi ng wil l sh ortly be di scussed . For a more thorou gh disc ussion, see [?]. De ne the
parti al an d i n ni tesum
V R ( )= R X r = 0 r v r ; V( )= l i m R!1 V R ( ):
Si nce for = 1 th e transforme d network is equal to the origi nal ne twork, one is i nte re sted in
V(1). The radiu s of convergenc eof V( ) i s al ways stri ctlyp osi ti ve but c anbe arbi trari ly small ,
so V(1) ne ed not converge. O ne way to obtai n convergence i s by th e epsi lon al gorithm, wh ich
i s an e cie nt and stabl e way to cal cul ate Pade-app roximants. Pade-approxi mants rep lace the
parti al sum V
R
( ) by a quoti ent of p artial sums V 1
S
( ) and V 2
R0 S
( ), i n such a way that they
coi nci de i nal l rstRco e cie nts:
V R ( )= P 0r S r v 1 r P 0r R0 S r v 2 r +O ( R+ 1 ); for #0:
Thi sway,sin gu lariti esofV( )canb einc lud edi nth edenomi nator. Th eepsi lonalgorithmusual ly
i mprovesthespeed of c onvergenceconsid erab ly,as can be see nfromthe example s in section 4.
Anoth er waytoreme dysin gu lari ti es i sbyusi ng thebi l in ear c on formal mapp ing
( )= (1+G) 1+G ; ()= 1+G(10) ; forG0:
Thi smappi ngmapstheuni ti nte rval[0,1]ontoitsel fand forG!1i tmaps th ed iskj 0 1
2 j
1
2
onto theun it di sk. IfV( ) hasnosi ngul ari tiesin j 0 1
2 j
1
2
,thi smappi ngcan b eusedto map
allsi ngul ari ti es outsi detheuni tdi sk,tomakethep owe r- se ri ese xpansions convergeat ==1.
IfV( ) isanal yti c at =0,thepower-seri esexpansi oni n is
V ( ( ))= X r 0 [ ( )] r v r = X r 0 r w r ; where w 0 =v 0 ; w r = G 1+G r r P s= 1 0 r 01 s0 1 1 vs G s ; for r1:
In steadofcal cul ati ngtheco e cie ntsoftheexp an sioni nfromtheexpansi onin ,theycanal so
becal cul ate dd irectl y. Anadvantageof di re ctcal cul ati oni s thattheseque nce fv
r
; r0ggrows
faster in r th an the sequen ce fw
r
; r0g, so cal cul ati on of fw
r
func ti onof also sati sfythe orde rproperty. More over, th eP
(n) are an al ytic i n,sothey can
berepresentedbyth eir power-seri esexpansi ons
P (n) = X r jnj r W r (n); forn2IN S :
As b e fore, the co eci ents of theempty state s W
r
(0);r0, satisfy (??). Be cause th e map pin g
i s a quotie nt of ni te p ol ynomial s, the co e c ients can stil l b e c al cul ate d by a l i near re cursive
algorithm. Assu methatr=sup
m r
m
i s nite. Rep laci ng by () inthebalanceequ ati on s(??),
multi pl yin g b oth side s by [1+G(10 )] r
, an d e quati ng c o ecie nts of corresp ond ing p owers of
rend ers the n ew recu rsi ve e quati ons for the non-empty states. The mapp in g was n ot u sed in
the examp le si n se cti on 4, because the power seri es were regular e nough to obtain convergence
bymeansofonl yth eep si l on algori thm.
4 Ex amples
The examp le s in secti on s 4.1 and 4.2 c on side r th e opti mal order of que ues i n series an d the
depend enceonhi gh ermomentsofth eservic e-timed istrib utionsofthetotalnumberofcustome rs
i ncycl i cnetworks.
4.1 Optimal O rd er
- j- j
-An importantdesi gnp robl emi n queu ein gth eory i show,fora gi ven arri valpro cess an dservi
ce-time d istribu ti ons, the qu eues shoul d be ord ered i n serie s, such that the mean sojourn ti me of
customers i s min imi zed, or equi vale ntl y th e mean queue l ength. Exac t analysi s is i n general
very di c ul t, even for 2 qu eues. Whi tt [?] prop oses a heuri sti c based on the app roximation
of the dep arture pro cess of each qu eue by a renewal pro cess, charac te ri zed by th e rst two
mome nts of the re newal interval . Gree nb erg an d Wol [?] propose d a h euristi c based on li ght
trac b ehavi ou ran d gave some exampl es whe re b othhe uristi cs di dn ot give th esame soluti on.
They warn ed that extreme cauti on must b e usedi n appl yin gapp roximations tod evelop d esign
pro ced ures and stated th at a heuri sti c based onl y on mean and squared c o ecie nt of variation
cannot b e e xp ected to work we ll . However, th ey di d not i nd icate how l arge the d i erence in
performance ofboth su ggestedsol utionswoul db e .
Consi der th e fol lowi ngmo d el. Ac cord ing to a Poi sson pro cess with rate, c ustomers arrive
toobtai n se rvi cefromtwo serve rs. Bothservers h avean Erlang(2)se rvi ce-ti me d istribu ti on,one
with mean 1 and the other wi th mean 4. Shoul d the customers rst visi t the fast or the sl ow
serveranddo estheoptimalord erdepend onthearri valrate? Accordi ngtoWhi tt theoptimal
orderi s tovi sitthefast se rver rst;GreenbergandWol suggestthat,i nli ght trac,theslowe r
server shoul d b e visi ted rst. In the tabl e b el ow, th e expected total numb er of custome rs is
1 ; 2 =0:1 =0:3 =0:5 =0:7 =0:9 1, 4 0.1337 0.4745 1.009 2.118 7.231 4, 1 0.1335 0.4734 1.005 2.111 7.218
To visi tthe sl ower se rver rst i s b e tter in al l cases, but cl early thedi eren ce i s n egl i gi bl e.
Nu-meri cal e xp eri ments i ndi cate that visi ting the sl owerserver rst i ssti ll sl ightly b e tter whe n the
expon ential interarri val ti mes are repl aced by Erl angor hyperexpon enti al d istrib utions or when
themeansof theservic e-time distrib utionsare takenfurtherapart(b utwithequal co eci ent of
vari ati on ).
Convergence of the p owe r series was slowe stfor the mo d el wi th =0:9 and the fast serve r
rst. Inth etabl eb e lowtheori gi nalse ri es andtheseri esafter appl yin gtheep sil onal gori thmare
shown . R 1 5 10 20 40 60 V R (1) 0.2250 1.766 3.245 4.877 6.410 6.945 [V R (1)] 0.2250 1.766 7.362 7.232 7.231 7.231
The ori gi nal series seems to c onverge monotoni cal ly, bu t after appl yi ng th e epsi l on al gori thm,
converge nce is much faster. In general, conve rgen ce i s sl ower i fthe load of theori gin al n etwork
i s hi gher an d the paramete rs of the mo del are more extreme. For exampl e, hyperexpon enti al
di stri buti on sresul ti nsl ower convergenceth anErl angdi stri buti on s.
4.2 Insens itivity for Higher Moments
j j j ? ? ? - - - 8 9 ? ? ? A A A U A A A U A A A U Consid erthefol lowin gmo d el . Customersarriveac cord ingtoap ro c ess
that i s a mi xtu re of i ndepende nt i denti cal Poi sson arri val p ro c esses
and fork arri vals. The i ndepend ent Poisson p ro c esses have rate
1 ,
thesimul tane ou sforkarri val sh ave exp one nti al interarri valti meswi th
rate 2 : I =1; 11 =S 1 + 2 ; M =S+1; b m =e m (1mS); b S+1 =e; q 0;1;1 =0; q m ;1;1 = 1 S1+ 2 (1mS); q S+ 1;1;1 = 2 S1+2 :
Therouti ngi s suchthat,afterservi cecomp leti onata queue ,custome rs eitherleavethen etwork
withprob ab il ityp,or gotothenext queu e wi thprob ab il i ty10p:
st = 8 > > < > > : p t=0; 10p t=smo d S+1; 0 otherwise.
Dep e ndi ng on th e co eci ent of vari ation, the service -ti me di stri bu ti ons at the d i erent queues
queu es, 1 = 2 = 0:09;p = 0:2 and 1 = 2 = 3
= 1. Thi s way, al l queue s have i de nti cal
l oad 1 = 2 = 3
= 0:9. The d i erence b e tween the four mo del s i s in the vari ance 2
s
of the
servi ce-time di stri butionsatthequ eues.
2 1 ; 2 2 ; 2 3 PrfN =0g EfN 1 g E fN 2 g E fN 3 g E fjNjg 1;1;1 0.0033 10.28 10.28 10.28 30.84 2; 1 2 ; 1 2 0.0038 12.95 9.928 7.760 30.63 1 1 2 ;1; 1 2 0.0035 11.39 10.97 8.426 30.78 1 2 ;1;1 1 2 0.0035 9.141 9.701 11.95 30.79
Thec onvergenceof th ep owe r seriesof E fjNjgin thesecond mo d elwaspo orest:
R 1 5 10 20 40 60 V R (1) 0.2700 2.007 4.326 8.387 14.79 19.37 [V R (1)] 0.2700 - 0.2523 0.1951 32.41 30.59 30.63
Again,bothseries se emtoc onverge,but moreco eci entsnee dtob ec al cul ate dtostabi li zethan
i nsecti on 4.1.
It can b eseen that th e meanqueu e l ength of e ach queue i s i ncreasin g i nthe variance of the
servi ce-timedi stri buti onof b oththequeu ei tse lfandthepreced ing qu eue. From th el astc ol umn
i tcan beseenth attheexpectedtotalnumberofcustomersi nthenetworki sapp roxi matel yequal
for al l fourmo del s. For p=1, thi sfoll ows i mmedi ate ly from the Pol laczek- Khi ntch in e formul a,
becausethen al l queu esare M/ G/1qu eueswi th i dentic al l oad an d meanservi ce ti me ,so:
EfjNjg=S 1 + 2 1 2(10 1 ) 1+ 1 2 1 S X s =1 2 s ! :
Thevari an cesoftheservi ce-timed istribu ti onsarechosen,su chthatthe irsumisequal to3foral l
fourmo de ls. N umericale xp eri ments i ndi catethatthepropertythattheme antotalque uelen gth
i smainl yd etermi ne dbyth esumofth evari an cesal sohold s formoregeneralmo del s, namel yfor
networkswi thasymme tri carrivalpro cess,cycl ic rou ti ngan de qual loadsatthedi erentqueu es.
Here, symmetri c means th at I and A can be arb itrary, but i fb
m
i s a possi bl e arri val, theneach
permutati on b
~ m
of b
m
is also a possi bl e arri val and Q
m = Q
~ m
(more i ntui ti vely, that at each
arri valofab atch,anarrival of anyp ermutati onof th isb atchwouldhave b eenequal lyl i ke ly). A
cycl i crouti ngmatri x X i sa matri xsuchthat
s;t =
smo d S+1;t mo d S+ 1
; for1s;tS:
If th earri val pro ce ssi s symme tri cand theroutin gcycl ic , th entheloads atthe queue sare i
Fo r networks of ./G/1 queues, with a symmetric arrival p rocess, cyclic routing and equ al
loads ata llqu eues, the expected totalnumber of cu st omersin the net work ismainly
deter-mined by the su m o f th e varia nces of the service- time d istribu tions, and not so mu ch by
their sh apes.
Of c ou rse such a hypoth esis c ou ld never be prove d bythe PSA, but i t can be used to eval uate
vari ou s'randoml y'ch osen mo del sand mo de lsthatare li kely to be counter-exampl es.
5 Conclusio ns
Ametho dwasp ropose dtoanalyz e awi dec lassofMarkovianque uei ng networks. Bec au seofthe
'curseof d imen sional ity',th esiz eof thene tworksmustn ecessaril y b emo de rate. Networksof up
to4or5qu euescanbeanal yzedi ftheal gori th mi sprogrammedcareful ly,metho d stoimp rovethe
converge nce ofp owe rserie sare emp loye dand theparametersof themo del are notto oextre me.
Withago o duser i nterfacetode termi ne th ep arametersforap artic ul ar mo d el ,thePower-S eries
Al gori th m provi des a means to easi l y eval uate many di erent mo del s. Th erefore, it can b e an
aid for stud yin g the i nteracti on b etwe en qu eues and for te sti ng and devel op ing app roxi mations
of p erforman cemeasures and heuri sti cs.
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wi thmore th an one wai tin gli ne,J. Compu t. Appl. Math.2 0(1987), 119-125.
[2]Blanc ,J.P.C.,R.D.vanderMe i,O ptimi zati onofp ol li ngsystemsbymeansofgradi entmetho ds
and thepower-serie sal gori thm, Tilbu rgUniversity, ReportFEW 575 (1992).
[3] Bl anc , J .P.C., Performanc e e valu ati on of p ol li ng systems byme an s of the power-seri esal
go-ri thm,Annals of Operations Research 35(1992), 155-186.
[4] Bl anc, J.P.C., The p ower-seri es algorithm appl i ed to the sh ortest-queu e mo del , Operations
Resea rch 40(1992),157- 167.
[5] B lanc, J .P.C., Performance analysi s and optimi zati on wi th the p owe r- series al gori thm, in
Perfo rmance Evaluat ion o f Com puter a nd Comm unicatio n Syst em s, ed s. L. Don ati ell o,
R .Nel son ,S pringer-VerlagBerl in ,1993,53-80.
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Management Sciencebf 34(1988), 500-508.
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