Hindawi Publishing Co po a ion
Jou nal o Applied Ma hema ics
Volume 2012, A icle ID 186761, 18 pages
doi:10.1155/2012/186761
Resea ch A icle
E icien Me hod o App oxima ely Sol e Re ial
Sys ems wi h Impa ience
Jose Manuel Gimenez-Guzman,1M. Jose Domenech-Benlloch,2
Vicen Pla,2Jo ge Ma inez-Bause ,2and Vicen e Casa es-Gine 2
1Depa amen o Au oma ica, Uni e sidad de Alcal´
a, Alcal´
a de Hena es, 28871 Mad id, Spain
2Depa amen o Comunicaciones, Uni e si a Poli `
ecnica de Val`
encia, 46022 Valencia, Spain
Co espondence should be add essed o Jose Manuel Gimenez-Guzman, [email p o ec ed]
Recei ed 30 June 2011; Accep ed 18 Oc obe 2011
Academic Edi o : Nicola Guglielmi
Copy igh q2012 Jose Manuel Gimenez-Guzman e al. This is an open access a icle dis ibu ed
unde he C ea i e Commons A ibu ion License, which pe mi s un es ic ed use, dis ibu ion,
and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed.
We p esen a no el echnique o sol e mul ise e e ial sys ems wi h impa ience. Un o una ely
hese sys ems do no p esen an exac analy ic solu ion, so i is manda o y o eso o app oxima e
echniques. This no el echnique does no ely on he nume ical solu ion o he s eady-s a e
Kolmogo o equa ions o he Con inuous Time Ma ko Chain as i is common o his kind o
sys ems bu i conside s he sys em in i s Ma ko Decision P ocess se ing. This echnique, known
as alue ex apola ion, unca es he in ini e s a e space using a polynomial ex apola ion me hod
o app oach he s a es ou side he unca ed s a e space. A nume ical e alua ion is ca ied ou o
e alua e his echnique and o compa e i s pe o mance wi h p e ious echniques. The ob ained
esul s show ha alue ex apola ion g ea ly ou pe o ms he p e ious app oaches appea ed in
he li e a u e no only in e ms o accu acy bu also in e ms o compu a ional cos .
1. In oduc ion
A common assump ion when e alua ing he pe o mance o communica ion sys ems is ha
use s ha do no ob ain an immedia e se ice lea e he sys em wi hou e ying. Howe e ,
due o he inc easing numbe o cus ome s and ne wo k complexi y, he cus ome beha io
in gene al, and he e ial phenomenon in pa icula , may ha e a nonnegligible impac on he
sys em pe o mance. Fo example, in mobile cellula ne wo ks he impo ance o he e ial
phenomenon has been s essed in 1–3. An ex ensi e bibliog aphy on e ial queues can be
ound in 4. The modeling o epea ed a emp s has been a subjec o nume ous in es iga-
ions, because hese sys ems ha e a nonhomogeneous and in ini e s a e space. Howe e ,
i is known ha he classical heo y 5is de eloped o andom walks on he semis ip
{0,...,C}×Zbeing C he numbe o se e swi h in ini esimal ansi ions subjec o
condi ions o space homogenei y.
2 Jou nal o Applied Ma hema ics
When he space-homogenei y condi ion does no hold, o example, in he case o e-
ial queues, he p oblem o calcula ing he equilib ium dis ibu ion has no been sol ed
beyond app oxima e echniques when he numbe o se e s is highe han wo 6.Inpa -
icula , Ma san e al. 7p opose a well-known app oxima e echnique o i s analysis. In 8,
a gene aliza ion o he app oxima e echnique in 7was p oposed, showing a subs an ial
imp o emen in he accu acy a he expense o a ma ginal inc ease o he compu a ional cos .
Those app oxima ions a e based on he educ ion o an in ini e s a e space o a ini e one by
agg ega ing s a es. O he solu ions main ain he in ini e s a e space bu homogenize i beyond
a gi en le el in o de o sol e he sys em. These la e models a e known as gene alized
unca ed models 6and usually p esen he ad an age o p o iding a much be e accu acy
han he ini e me hodologies 9. In his ca ego y we ind he models p oposed by Falin
10, by Neu s and Rao 11, and by A alejo and Pozo 6. All hese app oaches ely on he
nume ical solu ion o he s eady-s a e Kolmogo o equa ions o he Con inuous Time Ma k-
o Chain CTMC ha desc ibes he sys em unde conside a ion.
Ve y ecen ly, howe e , an al e na i e app oach o e alua ing in ini e s a e space
Ma ko p ocesses has been in oduced by Leino e al. 12–14. The new echnique, named
alue ex apola ion, does no ely on sol ing he global balance equa ions. This echnique
conside s he sys em in i s MDP Ma ko Decision P ocessse ing and sol es he expec ed
alue om he Howa d equa ions w i en o a unca ed s a e space. Ins ead o a simple un-
ca ion, he ela i e alues o s a es jus ou side he unca ed s a e space a e es ima ed using a
polynomial ex apola ion based on he s a es inside, ob aining a closed sys em. The e o e, we
can compu e any pe o mance pa ame e as a as we a e capable o exp ess i as he expec ed
alue o a andom a iable ha is unc ion o he sys em s a e.
So a he alue ex apola ion echnique has been applied o mul iclass single se e
queues showing e y p omising esul s. I mus be no ed ha a key aspec on he applica ion
o alue ex apola ion lies on he elec ion o he ex apola ing unc ion o he ela i e s a e
alues. Indeed, in 14 he au ho s show ha by selec ing an app op ia e polynomial unc ion
he echnique yields exac esul s o he momen s o he queue leng h in a mul iclass Dis-
c imina o y P ocesso -Sha ing DPSsys em. Un o una ely, he app op ia eness o he unc-
ional o m o he ex apola ion depends on he sys em and also on he e enue unc ion, ha
is, he pe o mance pa ame e we a e in e es ed in. Hence, he e is no uni e sal good choice
o he ex apola ing unc ion. In his pape we add ess he applica ion o he alue ex a-
pola ion echnique o an impo an class o queuing sys ems, o example, e ial queues,
which a e essen ially diffe en o he ype o queues o which his echnique has been applied.
A po en ial d awback o alue ex apola ion compa ed o con en ional s a e space unca ion
me hods is ha , since he s a iona y s a e p obabili ies a e no ob ained, i one wan o
compu e se e al pe o mance pa ame e s, he echnique has o be applied once pe each o
hem. We apply well-known linea algeb a algo i hms o compu e se e al pe o mance pa-
ame e s simul aneously, and h ough some se ies o nume ical examples we show ha , a
leas o he ype o sys em ha we a e s udying, he ela i e impac in e ms o compu a ional
cos is ma ginal.
The applica ion o he alue ex apola ion echnique has only add essed p oblems in
which ela i e s a e alues a e expec ed o ollow a polynomial endency. In his pape we de-
elop he alue ex apola ion echnique o sol e a mul ise e e ial sys em, add essing also
he d awback o compu ing only a single pe o mance pa ame e e e y ime he echnique is
used.
In a i s pa o he pape , we de elop he analy ical pa o he echnique, de ining
he associa ed Howa d equa ions o he model and he e enue unc ions. In a second pa ,
Jou nal o Applied Ma hema ics 3
1
2
3
C
Re ial
o bi
μ
1Pi
kuse s
Pi
muse s
λ
···
μ
–
Figu e 1: Re ial model unde s udy.
we compa e ou echnique wi h o he p e iously p oposed echniques in e ms o accu acy
and compu a ional cos . Resul s show ha he p oposed echnique clea ly ou pe o ms he
es o he s udied echniques in e ms o compu a ional cos , and his imp o emen is e en
much highe in e ms o accu acy.
The es o he pape is s uc u ed as ollows. Sec ion 2 desc ibes he sys em unde
s udy, while Sec ion 3 in oduces he sol ing echnique used. In Sec ion 4, he nume ical
analysis is ca ied ou , e alua ing he alue ex apola ion echnique and compa ing i wi h
o he p e ious sol ing echniques p oposed in he li e a u e. Final ema ks and a summa y
o esul s a e p o ided in Sec ion 5.
2. Sys em Model
The sys em unde s udy is a gene ic e ial sys em including use impa ience, ha is, use s
lea e he sys em wi h ce ain p obabili y a e a nonsuccess ul e ial. A diag am o he sys-
emisshowninFigu e 1. Use s a i e ollowing a Poisson p ocess wi h a e λ o a sys em
wi h Cse e s and eques an exponen ially dis ibu ed se ice ime wi h a e μ. Wi hou loss
o gene ali y, we conside ha each use occupies one se e . When a new eques inds all
se e s occupied, i joins he e ial o bi wi h p obabili y 1. A e an exponen ially dis ibu ed
ime o a e μ , his session e ies. The ea emp is success ul i i inds a ee se e . O he -
wise, he use lea es he sys em wi h p obabili y Pio e u ns o he e ial o bi wi h p oba-
bili y 1−Pi, s a ing he e ial p ocedu e again. No e ha we conside an in ini e capaci y o
he e ial o bi .
The model conside ed can be ep esen ed as a bidimensional CTMC, S {K ,
M }, whe e K is he numbe o sessions being se ed and M  he numbe o use s in
he e ial o bi a ime . The s a e space o he p ocess is de ined by
S:{sk,m:k≤C;m∈Z}.2.1
Figu e 2 shows he ansi ion diag am o such sys em, showing wo impo an p op-
e ies in he dimension co esponding o he numbe o use s in he e ial o bi : on he one
hand i s in ini e ca dinali y and on he o he hand i s space-he e ogenei y p oduced by he
ac ha e ial a e depends on he numbe o cus ome s in he e ial o bi .
4 Jou nal o Applied Ma hema ics
···
···
···
···
···
μμλ λ
λλλλ
μλμλ
λ
λλλ
μ
(0, 0)(0, 1)(0, 2)(0, 3)
2μ 3μ
μ 2μ 3μ
(1, 0)(1, 1)(1, 2)(1, 3)
2μ2μ2μ2μ
(2, 0)(2, 1)(2, 2)(2, 1)
.
.
.
.
.
.
.
.
.
.
.
.
(C–1, 0)(C–1, 1)(C–1, 2)(C–1, 3)
μ 2μ 3μ
Cμ λ
Cμ λCμ λCμ
(C, 0)(C, 1)(C, 2)(C, 3)
μ Pi2μ Pi3μ Pi
Figu e 2: T ansi ion diag am.
3. Sol ing Technique
In his sec ion we de elop he alue ex apola ion echnique o he model p esen ed in
Sec ion 2. Addi ionally, we p esen some pa icula i ies ha should be aken in o accoun
when using his echnique.
3.1. MDP Se ings
As i has been a o emen ioned, he p oblem unde in e es has no a closed o m solu ion
when C>26, so app oxima ion echniques a e manda o y. To he bes o ou knowledge,
all he app oxima e echniques ha ha e appea ed in li e a u e compu e he s eady s a e
p obabili ies πs using he balance equa ions in o de o compu e he desi ed pe o mance
pa ame e s, ha is, sol ing he linea sys em o equa ions:
πs
s/
s
qss
s/
s
πsqss∀s∈S,3.1
along wi h he no maliza ion condi ion sπs1, whe e qss ep esen s he ansi ion a e
om s a e s o s.
No wi hs anding, alue ex apola ion is no based on he p obabili y o being in a ce -
ain s a e, bu on a new me ic called ela i e s a e alues. Rela i e s a e alues appea when
we conside he sys em in he se ing o an MDP. Fo mally, an MDP can be de ined as a uple
{S,A,P,R}, whe e Sis a se o s a es, Ais a se o ac ions, Pis a s a e ansi ion unc ion, and
Ris a e enue unc ion. The s a e o he sys em can be con olled by choosing ac ions a om
A, in luencing in his way he s a e ansi ions. The ansi ion unc ion P:S×S×A → R
speci ies he ansi ion a e be ween a pai o s a es when a ce ain ac ion is aken a he o ig-
inal s a e. The i s cha ac e is ic o he alue ex apola ion echnique is he necessi y o he
de ini ion o a e enue unc ion ha mus be a unc ion o he sys em s a e, ha is, s.
Following he de ini ion o he e enue unc ion o e e y s a e, in he s eady s a e, a mean
e enue a e o he en i e p ocess can be in oduced as s∈Sπs s. In he alue
Jou nal o Applied Ma hema ics 5
ex apola ion echnique, he e enue unc ion Rhas o be de ined so ha he esul ing a e -
age e enue coincides wi h he desi ed pe o mance me ic.
Once we ha e de ined he MDP amewo k as well as he e enue unc ion, we a e in
a posi ion o de ine he ela i e s a e alues. I is ob ious ha a e pe o ming an ac ion in
s a e s he sys em will collec a e enue o ha ac ion  s, bu as he numbe o ansi ions
inc eases, he a e age e enue collec ed con e ges o . The ela i e s a e alue  s is equal
o he diffe ence be ween he o al e enue incu ed when he sys em s a s a s a e sand he
o al e enue incu ed in a sys em o which he e enue a e a all s a es is :
sE∞
0
 S  − d |S0s.3.2
The Howa d equa ions ela e e enues, ela i e s a e alues, and ansi ion a es:
s− 
s
qss s− s0∀s∈S.3.3
The Howa d equa ions ep esen he policy e alua ion phase o he well-known policy
i e a ion algo i hm, he mos widesp ead dynamic p og amming echnique, p oposed in 15.
The Howa d equa ions ha co espond o he sys em unde s udy a e
k,m− λ k1,m
− k,m kμ k−1,m
− k,m
mμ  k1,m−1− k,m 0i k<C,
C, m− λ C, m 1− C, m Cμ C−1,m
− C, m
mμ Pi C, m −1− C, m 0i kC.
3.4
As we can obse e he numbe o s a es is in ini e because mcan ake any alue in Z,
hus we need o unca e he s a e space o 
S. In ou case, he unca ed s a e space is de ined
by

S:{sk,m:k≤C;m≤Q}.3.5
In gene al, Qis known as he unca ion le el. As we choose a highe alue o Q,we
can expec a highe accu acy as he sys em is mo e simila o he o iginal one, bu we will ha e
a highe compu a ion cos oo. The e o e, he objec i e will be o achie e a ce ain accu acy
wi h he minimum alue o Q.
The e will be as many Howa d equa ions as numbe o s a es, |
S|. The numbe o un-
knowns will be he |
S| ela i e s a e alues plus he expec ed e enue , ha is,|
S| 1un-
knowns. Howe e , as only he diffe ences in he ela i e alues appea in he Howa d equa-
ions, we can se 00, so we will ha e a sol able linea sys em o equa ions wi h he same
numbe o equa ions as unknowns.

6 Jou nal o Applied Ma hema ics
3.2. Polynomial Fi ing
The adi ional unca ion se s qss0 o all s/∈
S,bu alue ex apola ion pe o ms a mo e
efficien unca ion. Basically, alue ex apola ion conside s he ela i e s a e alues ou side 
S
ha appea in he Howa d equa ions as an ex apola ion o some ela i e s a e alues inside

S. As we unca e he e ial o bi dimension beyond a alue Q, he alue ex apola ion ech-
nique uses he s a e alue o some s a es in 
S o app oxima e C, Q 1, which is expec ed
o imp o e he accu acy signi ican ly, as i is be e han igno ing hese ela i e s a e alues.
No e ha i ex apola ion yielded he exac alue o hose s a es ou side 
S, he esul s ob-
ained by sol ing he unca ed model would be exac . Also no e ha including alue ex a-
pola ion nei he inc eases he compu a ional cos no inc eases he numbe o Howa d equa-
ions, which emains equal o |
S| C1×Q1.
Summa izing, he objec i e o alue ex apola ion is o ind an ex apola ion unc ion
ha i s wi h some poin s in 
Sso ha i app oxima es also poin s ou side 
S. I is impo an o
choose a i ing unc ion ha makes he Howa d equa ions emain a closed sys em o linea
equa ions. The mos common i ing unc ions ha ul ill ha condi ion a e he polynomials.
We can use all he s a es in 
Sin o he i ing p ocedu e global i ingo , wha is mos com-
monly used, only a subse S o hem local i ing.
Fo he sake o simplici y, in he ollowing desc ip ion we will assume he e exis s a
mapping W om he wo-dimensional se o s a es in o a single-dimensional se , o example,
he eal numbe s: W:
S →R. Hence, below we deal wi h s a es as i hey we e eal alues
gi en as wWs. The speci ic mapping used o he model unde s udy is speci ied la e
on.
The choice o Wwill highly depend on he s a es we wan o ex apola e i s ela i e
s a e alue. No e also ha he unc ion wand he se 
S need o be chosen so ha he pa -
ame e s in wha e unambiguous alues, ha is, in he case o choosing a polynomial as
he i ing unc ion, he numbe o diffe en poin s in 
S has o be equal o g ea e han he
numbe o coefficien s in he polynomial. In gene al, he p ocedu e o compu e he coeffi-
cien s o he i ing polynomial aiconsis s in minimizing he leas mean squa ed e o
E
w∈W w− w2.3.6
Then he op imal alues o he ai’s can be compu ed by sol ing he equa ions
∂E
∂ai
0∀i. 3.7
In ou case, we a e using as many poin s as he numbe o pa ame e s o he i ing
polynomial, so he i ing p ocedu e is an o dina y polynomial in e pola ion and E0,
ha is, all he conside ed poin s will lie in he cu e o he polynomial. In his case,
he p oblem can be o mula ed as ollows. Gi en a se o n|W
S ||
S |poin s
w0, w0,...,wn−1, wn−1, whe e he e a e no wo iden ical wi, we can de e mine an
n−1- h deg ee polynomial so ha wi wi o i0,...,n−1, whe e
wa0a1wa2w2···an−1wn−1.3.8
Jou nal o Applied Ma hema ics 7
The in e pola ing polynomial sa is ies he ollowing nlinea equa ions:
wia0a1wia2w2
i···an−1wn−1
i wii0,...,n−1,3.9
which in a ma ix o m a e
Aa 
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1w0... wn−1
0
1w1... wn−1
1
.
.
..
.
.....
.
.
1wn−1... wn−1
n−1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
a0
a1
.
.
.
an−1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦

⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
w0
w1
.
.
.
wn−1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
b. 3.10
The ma ix o coefficien s o his sys em Ais a Vande monde ma ix, whose de e -
minan is non anishing and he e o e Ais in e ible. Thus, he e always exis s a unique
solu ion o he conside ed linea sys em o equa ions o , equi alen ly, he e exis s a unique
polynomial ha goes h ough all he npoin s. Howe e , Vande monde ma ices a e o en
badly condi ioned, specially i some wia e e y close, so he p ocedu e o compu e he i ing
polynomial is also badly condi ioned. I is impo an o no e ha he unici y o he i ing
polynomial does no mean ha i canno be w i en in a basis diffe en om he s anda d
basis. Mo e conc e ely in his wo k we ha e used he Lag ange basis.
Fo he conside ed in e pola ion p oblem, he polynomial in i s Lag ange se ing is a
linea combina ion
Lw
n−1

j0
wjjw3.11
o Lag ange basis polynomials
jw
n−1

i0
i/
j
w−wi
wj−wi
w−w0
wj−w0
··· w−wj−1
wj−wj−1
w−wj1
wj−wj1
··· w−wn−1
wj−wn−1
.3.12
Fo he unca ed p oblem o in e es and as shown in Figu e 3, we will ha e a Howa d
equa ion in which appea s C, Q 1, ha is a s a e alue o a s a e ha does no belong o

S. The e o e, we mus app oxima e he alue C, Q 1by using some ela i e s a e alues
o s a es belonging o 
S. I is impo an o emphasize ha o he ex apola ion o C, Q 1
we only use s a es om he las ow o he model shown in Figu e 3, ha is, s a es o he o m
sC, m, wi h a ying m. Wi h his choice, we de ine he mapping Was WC, mm.
8 Jou nal o Applied Ma hema ics
(0, Q)
···
···
···
···
(0, Q–1)
(1, Q)
(2, Q)
(1, Q–1)
(2, Q–1)
(0, 0)
(1, 0)(1, 1)(1, 2)
(2, 0)(2, 1)(
2, 2)
(0, 1)(0, 2)
μ
μ
μ
2μ
2μ
2μ
μλ μλ μλ μλ
λ2μλ2μλ2μ
λμ
λ2μλ2μ
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(C–1, 0)(C–1, 1)(C–1, 2)(C–1, Q–1)(C–1, Q)
(C, Q)
λCμ λCμ λCμ λCμ λ
λλλλ
Cμ
···
(C, 0)(C, 1)(C, 2)
μ Pi2μ PiQμ Pi
ꉱ
S
(C, Q–1)(C, Q+1)
Qμ
Qμ
Qμ
Figu e 3: T unca ed model and s a es ha appea in Howa d equa ions ou side he unca ed model.
Mo eo e , we use an n−1- h deg ee polynomial ha in e pola es he npoin s in S :{si
C, Q −i|i0,...,n−1}and hen WS {wiQ−i|i0,...,n−1}:
w0Q−→ w0 C, Q,
w1Q−1−→ w1 C, Q −1,
.
.
.
wjQ−j−→ wj C, Q −j,
.
.
.
wn−1Q−n−1−→ wn−1 C, Q −n−1.
3.13
This way, he gene al o m o he ex apola ion s a e when using an n−1- h deg ee
polynomial is
nC, Q 1LnQ1
n−1

j0
C, Q −jjQ1.3.14
Fo example, in he case o linea ex apola ion n2,weuseQ, C, Q and Q−1,
C, Q −1, ha ing
2C, Q 1L2Q1 C, Q0Q1 C, Q −11Q1
 C, QQ1−Q−1
Q−Q−1 C, Q −1Q1−Q
Q−1−Q
2 C, Q− C, Q −1.
3.15
Jou nal o Applied Ma hema ics 9
Table 1: Re enue unc ion de ini ion.
Blocking p obabili y Pb k,m1 o kC, o allm
k,m0 o he wise
Nonse ice p obabili y Pns k,mmμ Pi/λ o kC, o allm
k,m0 o he wise
Mean numbe o use s e ying N e k,mm o all k, o allm
P obabili y o being in s a e K, MπK, M k,m1 o kK,mM
k,m0 o he wise
P obabili y o ha ing Kbusy se e s BK k,m1 o kK, o allm
k,m0 o he wise
Following a simila p ocedu e, we can ob ain he nex ela ionship o n3andn4:
3C, Q 13 C, Q−3 C, Q −1 C, Q −2,
4C, Q 14 C, Q−6 C, Q −14 C, Q −2− C, Q −3.
3.16
In gene al, o n−1- h deg ee polynomials and using he Lag ange basis o educe
he complexi y o he p ocedu e, a simple closed- o m exp ession o he ex apola ed alue
can be ob ained by
nC, Q 1
n−1

k0
−1kn
k1 C, Q −k,3.17
whe e nis he numbe o coefficien s aken o Lag ange polynomials.
3.3. Re enue Func ion
As pe o mance pa ame e s a e no compu ed om he s eady s a e p obabili ies as usual,
i is impo an o explain mo e ca e ully how hey a e compu ed. By de ini ion, sis he
e enue a e ob ained when he sys em is in s a e s. The e o e, we mus de ine he e enue as
he pe o mance pa ame e we wan o compu e. The effec o ha ac ion is ha he compu ed
will be he pe o mance pa ame e we a e looking o . Addi ionally, he inpu s sin he
Howa d equa ions mus be p ope ly se . Table 1 gi es se e al examples on how scan
be se in o de o ob ain ce ain pe o mance pa ame e s such as: blocking p obabili y Pb
P ob{KC}, mean numbe o use s in he e ial o bi N e EM, nonse ice p obabili y
Pns p obabili y o a use lea ing he sys em due o impa ience wi hou ob aining se ice,
p obabili y o being in a ce ain s a e πK, M, and p obabili y o ha ing Kbusy se e s
BK.
As an example, we ocus on he blocking p obabili y and we de ine he e enue unc-
ion o be 1 in hose s a es in which an a emp is blocked, ha is, when C, m1, o all m,
and 0 in he es o s a es, k,m0,k
/
C, o all m.
16 Jou nal o Applied Ma hema ics
Rela i e e o
FM
NR
VE
10 15 20 25 30 35
Q
10−2
100
10−4
10−6
10−8
10−10
10−12
10−14
Figu e 7: Rela i e e o in N e o diffe en echniques.
Time (s)
Rela i e e o
FM
NR
VE
10−2
100
10−4
10−6
10−8
10−10
10−12
10−14
10−1
10−2100101
Figu e 8: Rela i e e o in Pb e sus compu a ion cos .
ha eason, i is manda o y o de elop app oxima e echniques in o de o sol e hese sys-
ems. To he bes o ou knowledge, all he echniques s udied in he li e a u e o sol e hese
sys ems a e based on hei s eady s a e p obabili ies. In his pape we p opose an al e na i e
echnique based on a diffe en me ic: he ela i e s a e alues and he Howa d equa ions ha
ela e hem, ins ead o he balance equa ions. Wi h his echnique, unca ion o he s a e space
can be done in a mo e efficien way, as he s a e alues ou side he unca ed s a e space a e

Jou nal o Applied Ma hema ics 17
Time (s)
Rela i e e o
FM
NR
VE
10−2
100
10−4
10−6
10−8
10−10
10−12
10−1
10−2100101
Figu e 9: Rela i e e o in Pns e sus compu a ion cos .
Time (s)
Rela i e e o
FM
NR
VE
10−2
100
10−4
10−6
10−8
10−10
10−12
10−14
10−1
10−2100101
Figu e 10: Rela i e e o in N e e sus compu a ion cos .
ex apola ed om some known s a e alues. In o de o p ese e he linea i y o he esul ing
sys em o equa ions, we ha e only used polynomials as ex apola ion unc ions.
In a i s pa , we ha e s udied he use o diffe en o de s o he ex apola ion poly-
nomials. La e , we ha e compa ed he new echnique wi h wo well-known app oaches ap-
pea ed in he li e a u e 8,11in e ms o accu acy and compu a ional cos . Resul s show ha
he p oposed echnique highly imp o es he p e ious app oaches in e ms o compu a ional
cos and, specially, in e ms o accu acy, so i s use is highly ecommended.
18 Jou nal o Applied Ma hema ics
Acknowledgmen s
This wo k has been suppo ed by he Spanish go e nmen unde P ojec s TIN2010-21378-
C02-02 and TIN2008-06739-C04-02/TSI and by Comunidad de Mad id h ough P ojec S-
2009/TIC-1468.
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