ISSC 2005, Dublin, Sep embe 1- 2
A No el Con e gence Algo i hm o he Hyb id S a egy
Model Packe Reduc ion Technique
Declan Delaney°
°°
°, Seamus McLoone*, Tomas Wa d*
*Dep . o Elec onic Enginee ing,
NUI Maynoo h,
Maynoo h, Co. Kilda e, I eland
E-mail: [email protected]
°
Dep . o Compu e Science,
NUI Maynoo h,
Maynoo h, Co. Kilda e, I eland
E-mail: decla[email p o ec ed]
__________________________________________________________________________________________
Abs ac – Se e al app oaches exis o main aining consis ency in Dis ibu ed In e ac i e
Applica ions. Among hese a e echniques such as dead eckoning which use p edic ion
algo i hms o app oxima e ac ual use beha iou and hus educe he numbe o upda e
packe s equi ed o main ain spa ial consis ency. The Hyb id S a egy Model ope a es in a
simila way, exploi ing long- e m pa e ns in use beha iou whene e possible. O he wise
i simply adop s a sho - e m model. A majo p oblem wi h hese echniques is he
econs uc ion o he local beha iou a a emo e node. Using he modelled dynamics
di ec ly can esul in unna u al and sudden jumps in posi ion whe e upda es occu .
Con e gence algo i hms a e hus equi ed o smoo hly econs uc emo e beha iou om
discon inuous samples o he ac ual local beha iou . This pape makes wo impo an
con ibu ions. P ima ily, i p oposes a no el con e gence app oach o he Hyb id S a egy
Model. Secondly, and mo e undamen ally, i exposes a lack o sui able and quan i iable
measu es o di e en con e gence echniques. In his pape he s anda d smoo hing
algo i hm employed by DIS is used as a benchma k o compa ison pu poses.
Keywo ds – Hyb id S a egy Model, Con e gence, Dis ibu ed In e ac i e Applica ions
__________________________________________________________________________________________
I INTRODUCTION
Techniques ha gene a e and ansmi app oxima e
models o local use beha iou ha e been de eloped o
educe he numbe o upda e packe s ha mus be
ansmi ed be ween pa icipan s o Dis ibu ed
In e ac i e Applica ions (DIAs). One such echnique
is dead eckoning, which educes ne wo k a ic by
p e en ing upda e packe s om being sen i he local
pa icipan ’s posi ion has no a ied om a model o
ha posi ion by a p e-es ablished h eshold (IEEE
1993; IEEE 1995). Howe e , dead eckoning igno es
in o ma ion pe aining o he s uc u e o he
en i onmen , he his o ical beha iou o use s wi hin
he en i onmen and he objec i es o he use s wi hin
he en i onmen . Consequen ly a echnique known as
he Hyb id S a egy Model o HSM has ecen ly been
de eloped which exploi s his in o ma ion by
iden i ying long- e m pa e ns in use beha iou . By
combining hese longe - e m models o s a egies wi h
a sho e m model, such as dead eckoning, a
educ ion in ne wo k a ic has been shown (Delaney
e al. 2003b; Delaney e al. 2003a; Ma shall e al.
2004a; McCoy e al. 2005).
An impo an issue a ising om his app oxima ion o
local use beha iou is he econs uc ion o a ai h ul
eplica ion a a emo e loca ion gi en his o en di e gen
model o he local beha iou . I he ecei ed upda es a e
ende ed di ec ly, he esul ing beha iou can be disjoin
and unna u al. To o e come his p oblem, echniques a e
employed o smoo hen he beha iou . These a e known
as con e gence echniques. In he case o dead eckoning
i s o de and cubic spline algo i hms a e used (IEEE
1993; Lin and Schab 1994).
This pape add esses he issue o a sui able con e gence
algo i hm o he HSM app oach. I p oposes a
con e gence echnique based on he weigh ing o use
beha iou al models using a sui able blending unc ion.
This echnique is isually compa ed wi h he s anda d
DIS smoo hing algo i hm (IEEE 1993). In ca ying ou
his wo k, i was disco e ed ha he e is a dea h o
objec i e measu es in e ec i ely e alua ing con e gence
algo i hms (o a leas he au ho s a e unawa e o any
exis ing objec i e c i e ia o e alua ing such algo i hms).
The pape will ocus h oughou on use en i y mo ion
dynamics, as such mo emen is esponsible o he
ISSC 2005, Dublin, Sep embe 1- 2
majo i y o upda e packe s gene a ed in Dis ibu ed
In e ac i e Applica ions.
The s uc u e o he pape is as ollows. Sec ion II
summa izes he Hyb id S a egy Model. Sec ion III
desc ibes he no el con e gence algo i hm, based on
he blending o use beha iou al models. The s anda d
con e gence algo i hm used in DIS is also in oduced
he e. Sec ion IV p esen s a se ies o g aphical
simula ion esul s, compa ing ou pu s om he HSM
algo i hm using (i) no con e gence, (ii) he s anda d
DIS con e gnce and (iii) he p oposed con e gence
echnique. Some concluding ema ks in addi ion o
u u e wo k a e p esen ed in sec ion V.
II THE HYBRID STRATEGY MODEL
Dis ibu ed In e ac i e Applica ions in ol e
po en ially housands o simul aneous pa icipan s. In
an e o o educe he numbe o upda e packe s
ansmi ed be ween use s, models o use dynamics
a e employed so ha only samples o he dynamics
need o be communica ed. Use dynamics can hen be
econs uc ed emo ely using he samples as inpu o
he model. The mos commonly used model is dead
eckoning. This ope a es by main aining a sho - e m
model o he beha iou o each en i y a each node.
When he modelled and he ac ual beha iou o an
en i y di e by a h eshold alue, an upda e packe
indica ing i s cu en s a e is sen o all o he nodes.
Be ween upda es he emo e nodes employ he same
sho - e m dead eckoning model.
Ou wo k o da e has shown ha long- e m beha iou
may also be modelled o a g ea e o lesse ex en and
ha he use o such a model would complemen he
sho - e m model (Delaney 2005). This mo i a ed he
de elopmen o he Hyb id S a egy Model (HSM) o
u he educe he numbe o upda e packe s ha need
o be ansmi ed. The hyb id model, M, o en i y
dynamics is o he ollowing o m:
Γ
−
+
=
)1( ppM
χ
(1)
whe e
χ
is any con en ional dead eckoning model, Γ
is one o se e al long- e m en i y s a egy models and
p is a bina y weigh ing ac o go e ned by:
p = 1 o
θ
≥Γ−M
= 0 o he wise (2)
whe e
θ
ep esen s a dis ance measu e h eshold
be ween he modeled beha io and he long e m
model Γ. In his way a long- e m model, Γ, based on a
p io i da a can be employed when he en i y mo emen
is ‘close’ o such ajec o ies. The pa icula model
used is dynamically chosen so ha imp o ed i s can be
ob ained o e single model app oaches. A any ins an in
ime he nea es model om he se a ailable is used in
equa ion (1).
Long- e m models can be gene a ed om his o ical da a
and knowledge o he en i onmen . He e, long- e m
models we e cons uc ed by eco ding ac ual use
dynamics in wo sepa a e en i onmen s. To do so, use s
na iga ed be ween a s a posi ion and a a ge posi ion
wi h he objec i e o doing so in he sho es ime
possible; each pa h aced ou by he use en i y is e e ed
o as a ajec o y. The op imum beha iou o ealise an
objec i e is e e ed o as a s a egy. A ajec o y ha
a emp s o mi o he s a egy is called a s eady-s a e
ajec o y. The sho - e m model employed in he HSM
is dead eckoning. In his pape he models all ela e o
use spa ial dynamics.
III THE NOVEL CONVERGENCE ALGORITHM
Each upda e packe gene a ed by he HSM echnique
indica es ha a ansi ion has occu ed om one model o
ano he . I his ansi ion is implemen ed ins an ly, he
esul ing use mo emen may appea unna u al and
disjoin . The con e gence algo i hm se es o smoo hen
he ansi ion, hence inc easing he playabili y o he
applica ion and imp o ing he ealism o he econs uc ed
use dynamics. This is illus a ed in Figu e 1(a)-(b).
Figu e 1(a)-(b): (a) A sudden jump in posi ion occu s when
he e is a ansi ion be ween models; (b) Con e gence
smoo hens he ansi ion, imp o ing mo emen ideli y.
A sudden jump in use
mo emen occu s
Upda e
packe
Use s a egy model 1
Use mo emen –
use ollows
s a egy 1 Use mo emen –
use ollows
s a egy 2
Use s a egy model 2
(a)
(b)
Con e gence makes
mo emen mo e na u al
Upda e
packe
Use s a egy model 1
Use mo emen –
use ollows
s a egy 1 Use mo emen –
use ollows
s a egy 2
Use s a egy model 2
ISSC 2005, Dublin, Sep embe 1- 2
Weigh ed model con e gence blends wo models, he
model ha was used up o he mos ecen upda e
packe and he new model ha bes desc ibes he
emo e use ’s beha iou . This is illus a ed in Figu e 2
whe e a ansi ion has jus occu ed om s a egy 1 o
s a egy 2. (x1,y1) is he poin on s a egy 1 a which
he en i y was loca ed when an upda e packe was
ecei ed. S a ing om his poin he weigh ed model
con e gence uses a sec ion o he s a egy 1 model up
o poin (x4,y4). Likewise, a sec ion o s a egy 2 om
poin (x3,y3) up o poin (x2,y2) is chosen. He e, (x3,y3)
is ound by an o hogonal p ojec ion o (x1,y1) on o
s a egy 2. Bo h sec ions a e equal in leng h and
con ain he same numbe o poin s, N. These wo
coo dina e da a se s a e hen blended acco ding o a
weigh ing scheme o ob ain a smoo hened pa h.
This p ocess o blending he wo long- e m models by
an app op ia e weigh ing scheme is desc ibed by he
ollowing equa ion:
21 )1(
Γ
−
+
Γ
=
ppW (3)
whe e
W is he model esul ing om he blending
o Γ1 and Γ2;
Γ
1 and
Γ
2 a e long- e m models;
p is a weigh ing ac o wi h 10
≤
≤
p.
The weigh ing ac o , p, may be gene a ed by a
sui able no malized weigh ing unc ion. A na u al
blending o models sugges s gi ing mo e weigh ing
when close o a s a egy model and co e ing he
in e media e dis ance quickly. In ui i ely we wan a
g adual smoo h change om he i s model, quickly
head o he second model and hen g adually
con e ge o he second model – see Figu e 1(b). This
desi ed beha iou sugges ed he use o a sigmoid
blending unc ion:
)(
1
1
),,( bxa
e
baxp −−
+
= (sigmoid) (4)
whe e
x ∈ [0,1];
a, b ∈ℜ.
A a e o con e gence is also speci ied. This de e mines
he speed wi h which he weigh ing unc ion begins using
he new model and o ge s he old model. I is exp essed
as a eal numbe be ween 0 and 1 and is an essen ial
ac o o he con e gence p o ocol. In he simula ion es
bed he x inpu alues o he con e gence unc ions
depends on he a e o he con e gence pa ame e alue.
Beginning om 0, x is inc emen ed by he a e o
con e gence pa ame e alue on each clock cycle o he
es bed (e e y 32ms). This con inues un il x eaches 1.
In his pape , ou con e gence echnique is compa ed wi h
he s anda d DIS con e gence app oach. In he la e case
an in e pola ion algo i hm usually based on cubic splines
is employed o in e pola e be ween he las modelled
posi ion o he en i y and new posi ion communica ed in
he upda e packe . The in e pola ion no mally speeds up
he en i y so ha i con e ges o a ealis ic posi ion a he
han jumping di ec ly o he posi ion indica ed in he
upda e packe .
IV SIMULATIONS AND RESULTS
In his sec ion he ope a ion o wo con e gence
algo i hms is examined: he s anda d DIS smoo hing
algo i hm and he no el weigh ed con e gence algo i hm
using a sigmoid blending unc ion. Da a was eco ded
om a numbe o use s o wo dis inc en i onmen s as
desc ibed in (Delaney e al. 2003b; Ma shall e al. 2004a;
Ma shall e al. 2004b). Each use na iga ed om a ixed
s a loca ion o a ixed end loca ion and he ajec o ies
hey aced ou we e sampled using bo h dead eckoning
and he HSM. These samples we e hen used o
econs uc he ajec o ies using bo h con e gence
algo i hms.
To acili a e a isual compa ison be ween he a ious
plo s p oduced, wo use ajec o ies (one om each o
wo di e en en i onmen s) a e used o illus a ion
h oughou : ajec o y T1 om es en i onmen 1 and
ajec o y T2 om es en i onmen 2. These we e
chosen because hey a e s eady s a e ajec o ies wi h
in e es ing en i y dynamics. They hus p o ide se e al
s a e upda e packe s and hence equi e he use o
con e gence h oughou he ajec o y leng h. T1 and T2
a e indica ed as dashed lines in all Figu es.
As a p elude o applying he con e gence algo i hm, HSM
upda e packe s a e gene a ed o he wo ajec o ies.
Figu e 2: Con e gence means ha he e is a smoo h
ansi ion om one s a egy o ano he by blending mul iple
poin s on each s a egy.
S a egy 1 – S1
S a egy 2 – S2
S a egy 3 – S3
(x1,y1)
(x3,y3)
(x2,y2)
(x4,y4)
ISSC 2005, Dublin, Sep embe 1- 2
150 200 250 300 350 400 450 500 550 600 650
0
50
100
150
200
250
300
350
400
450
y
x
Con e gence Ra e: 0.015
These upda e packe s a e hen used o econs uc he
ajec o ies a emo e use would econs uc and ende
i he e was no con e gence. Figu e 3(a)-(b) shows
he upda e packe s gene a ed o T1 and T2 by he
HSM wi h a h eshold alue o 10 wo ld uni s and no
con e gence algo i hm. S a s indica e he upda e
packe s and he dashed line ep esen s he o iginal
en i y ajec o y.
The econs uc ed ajec o ies a e shown as solid lines
and i can be seen ha al hough hey main ain he
o e all shape o he o iginal use ajec o ies, hey a e
qui e jagged and in ol e sudden jumps in posi ion
each ime an upda e a i es. The sec ions p oduced
a e simply segmen s o he s a egy models ha ma ch
he o iginal ajec o y. The upda e packe s lie be ween
hese segmen s and indica e a ansi ion om one long-
e m model o ano he .
Clea ly, he esul ing ajec o y ende ed o he emo e
use is highly un ealis ic, wi h poo ideli y o he
o iginal use ajec o y, pa icula ly in e ms o
smoo hness. Hence he need o a con e gence
algo i hm.
The i s a emp a smoo hing he en i y’s emo e
ajec o y o make i mo e ealis ic was based on poin - o-
poin i s o de linea con e gence, he s anda d
smoo hing algo i hm speci ied in he IEEE DIS s anda d
(IEEE 1993; IEEE 1995). Applica ion o he con e gence
algo i hm o he upda e packe s gene a ed by he HSM is
shown in Figu e 4(a)-(b). The a e o con e gence was
se o 0.015 and he con e gence dis ance was 10 wo ld
uni s.
I can be seen ha he gene al cha ac e o he o iginal
local ajec o y indica ed by he dashed line is main ained
in he econs uc ions. The gaps in Figu e 3(a)-(b) ha e
been smoo hened o e . Howe e , he econs uc ion
shown by he solid line is s ill somewha jagged wi h
sudden, unna u al changes o di ec ion in se e al places.
In Figu e 4(b) he boxed a ea indica es an inaccu a ely
econs uc ed sec ion. This is a consequence o a cons an
con e gence a e being employed ha is slowe han he
eloci y o he en i y. This can be ixed by a ying he
con e gence a e h oughou he simula ion. This was no
implemen ed he e.
(a)
(b)
Figu e 3(a)-(b): Plo s o he emo e econs uc ion o (a) T1
and (b) T2 using upda e packe s gene a ed by he HSM,
wi h no con e gence algo i hm. The dashed line ep esen s
he o iginal ajec o y.
Figu e 4(a)-(b): Plo s o he emo e econs uc ion o (a) T1
and (b) T2 using upda e packe s gene a ed by he HSM, wi h
he poin - o-poin con e gence. The dashed line ep esen s
he o iginal ajec o y.
En i onmen 1
Use 1
T ajec o y 3
x
y
En i onmen 1
Use 1
T ajec o y 3
x
y
En i onmen 1
En i onmen 2
Use 1
T ajec o y 3
x
y
En i onmen 2
Use 1
T ajec o y 3
x
y
En i onmen 2
0 100 200 300 400 500 600
0
50
100
150
200
250
300
350
y
x
Con e gence Ra e: 0.015
(b)
(a)
ISSC 2005, Dublin, Sep embe 1- 2
The no el weigh ed con e gence app oach is now
conside ed. The HSM in ol es ansi ions be ween
di e en models and each ansi ion in ol es wo
models. The weigh ed sigmoid con e gence algo i hm
blends he wo s a egy models acco ding o a sigmoid
blending unc ion. Th ee pa ame e s can be a ied –
he a e o con e gence and he wo pa ame e s ha
de e mine he shape o he sigmoid cu e. Figu e 5(a)-
(b) illus a es a econs uc ion using a ypical sigmoid
blending unc ion wi h a = 15, b = 0.5 and he a e o
con e gence = 0.015. The local ajec o y is shown as
a dashed line.
Once again he esul ing cu es show ha he gaps in
Figu e 3(a)-(b) ha e been smoo hened o e . Howe e ,
he econs uc ion emains somewha jagged wi h
some unna u al mo emen in he o m o oscilla ions
as indica ed in Figu e 5(b). The pe o mance o he
con e gence algo i hm is dependan on he pa ame e s
o he sigmoid unc ion. In an e o o unde s and he
impac o he sigmoid pa ame e s (i.e. a and b) on he
econs uc ion, addi ional smoo hing es s we e ca ied
ou .
The b alue de e mines he poin a which bo h models
a e gi en equal weigh ing in he ansi ion. High alues
o b mean ha his poin is u he om he s a o he
ansi ion and so he old s a egy cu e will be dominan
in he blending p ocess. Lowe alues o b mean ha he
new s a egy will domina e o mo e o he ansi ion. A
alue o 0.5 esul s in bo h being weigh ed equally when
he ansi ion is hal way comple e. Simula ions showed
ha o he en i onmen s examined, he mos sui able b
alue is 0.5. In mos applica ions i can be en isaged ha
he in luence o he old long- e m model will dec ease and
he new s a egy will inc ease a a cons an a e, wi h bo h
models con ibu ing equally when he ansi ion is 50%
comple e.
The a alue de e mines he slope o he sigmoid – he
g ea e he a alue he g ea e he slope. Simula ions
show ha his alue has a di ec e ec on he oscilla ions
no ed in Figu e 5(a)-(b). Low alues o a se e o
Con e gence Ra e: 0.015
a = 15
b = 0.5
x
y
Con e gence Ra e: 0.015
a = 15
b = 0.5
x
y
oscilla ions
(a)
(b)
(b)
(a)
Con e gence Ra e: 0.015
a = 5
b = 0.5
x
y
Con e gence Ra e: 0.015
a = 5
b = 0.5
x
y
Figu e 6(a)-(b): Plo s o he emo e econs uc ion o (a) T1
and (b) T2 using upda e packe s gene a ed by he HSM.
Sigmoid weigh ed con e gence is used.
Figu e 5(a)-(b): Plo s o he emo e econs uc ion o (a) T1
and (b) T2 using upda e packe s gene a ed by he HSM.
Sigmoid weigh ed con e gence is used.

ISSC 2005, Dublin, Sep embe 1- 2
a enua e he oscilla ions and as a inc eases, he
se e i y o he oscilla ions inc eases. A high alue
indica es a s eep slope and only se es o speed up he
con e gence p ocess. Ea lie , in igu e 4 i was no ed
ha he a e o con e gence was oo slow esul ing in
poo accu acy in pa s o he ou pu . He e, he
con e gence a e is he same bu he a alue can
e ec i ely inc ease his a e and hence, he oscilla ions
ob ained. Clea ly, a comp omise be ween model
accu acy, ealism o a playe ’s mo emen and he
game playabili y is equi ed o choose sui able alues
o bo h pa ame e s a and he con e gence a e.
Simula ions showed ha alues o a = 5 and b = 0.5
p oduced good esul s o a con e gence a e o 0.015.
The associa ed plo s a e shown in Figu es 6(a)-(b).
Visually compa ing hese plo s wi h hose in Figu es
4(a)-(b) would seem o indica e ha he o me ob ains
a smoo he , mo e na u al ou pu a he expense o
model accu acy. Howe e , om he emo e poin o
iew his is an accep able ade-o in a wo ld whe e
playabili y and he end use pe cep ual expe ience a e
mo e impo an han achie ing absolu e consis ency.
V CONCLUSIONS
This pape has desc ibed a no el con e gence
algo i hm based on he blending o long- e m
beha iou al models using a sigmoid unc ion. The
echnique was e alua ed by isually compa ing i o
bo h he o iginal use en i y ajec o y and o he
smoo hing algo i hm used in he IEEE DIS s anda d.
I was ound ha he smoo hness o he econs uc ed
beha iou using he weigh ed sigmoid con e gence
algo i hm depends on he a e o con e gence, wi h a
high a e o con e gence esul ing in a jagged
econs uc ion and a low a e o con e gence esul ing
in he loss o beha iou al de ail. The esul s also
depend on he pa ame e s employed in he sigmoid
unc ion. Reducing he a alue inc eases he
na u alness o he econs uc ed cu e. A cen al b
alue o 0.5 was ound o be mos sui able.
The HSM echnique and he con e gence algo i hms
in ol e a numbe o inpu pa ame e s ha de e mine
he e icacy o he algo i hms. Ou wo k on
con e gence algo i hms has highligh ed he lack o
objec i e c i e ia o e alua ing he e ec i eness o
such algo i hms. The e is he e o e a need o u he
s udies o in es iga e possible objec i e e alua ion
measu es in e ms o smoo hness and ideli y o he
o iginal beha iou . In addi ion, he use o psycho-
pe cep ual da a will be examined o measu e he
impac o di e en con e gence algo i hms on he end
use s.
ACKNOWLEDGEMENT
This wo k is suppo ed by Science Founda ion I eland and
En e p ise I eland unde g an s no. SC/2002/129/ and
IRCSET/SC/04/CS0289.
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