So eal- ime communica ions o e Blue oo h unde
in e e ences om ISM de ices
J. L. Se illano, D. Cascado, F. Dı´ az del Rı´ o, S. Vicen e, G. Jimenez
and A. Ci i -Balcells
ETSII, Uni e sidad de Se illa, A . Reina Me cedes, s/n. 41012, Se illa, Spain
SUMMARY
Blue oo h is a sui able echnology o suppo so eal- ime applica ions like mul imedia s eams a he
pe sonal a ea ne wo k le el. In his pape , we analy ically e alua e he wo s -case deadline ailu e
p obabili y o Blue oo h packe s unde co-channel in e e ence as a way o p o ide s a is ical gua an ees
when ansmi ing so eal- ime affic using ACL links. We conside he in e e ence om independen
Blue oo h de ices, as well as om o he de ices ope a ing in he ISM band like 802.11b/g and Zigbee.
Finally, we show as an example how o use ou model o ob ain some esul s o he ansmission o a oice
s eam.
KEY WORDS: Blue oo h; co-channel in e e ence; so eal- ime; wo s -case deadline ailu e p obabili y
1. INTRODUCTION
Blue oo h is cu en ly one o he mos success ul s anda ds o low-powe , sho - ange wi eless
pe sonal a ea ne wo ks (WPAN). A wide ange o de ices a e being in oduced, wi h applica ions
suppo ing asynch onous and/o isoch onous da a ans e s. The mo i a ion o his wo k is ha
mo e and mo e wi eless de ices a e also in oduced ha ope a e in he same equency bands
han Blue oo h de ices ( he so-called ISM-Indus ial, Scien ific & Medical bands), including
IEEE 802.11, IEEE 802.15.4, Wi elessUSB, e c. Depending on he le el o sensi i i y o he
in e e ence om o he de ices, some ypes o affic may suffe om unp edic able delays o
packe losses due o hese in e e ences. In his pape , we e alua e he eal- ime beha iou o
Blue oo h communica ions, and show how in e e ences om o he ISM de ices can be aken
in o accoun .
A ypical Blue oo h sys em is composed o a small numbe o de ices ha o m a wi eless
ne wo k called a picone . Connec ions a e es ablished ad hoc by a Blue oo h uni ha becomes a
Con ac /g an sponso : Spanish Minis y o Science and Technology; con ac /g an numbe : He e o ed: TIC2001-
1868-C03-02
Co espondence o: Jose´ Luis Se illano, ETSII, Uni e sidad de Se illa, A . Reina Me cedes, s/n. 41012, Se illa, Spain.
E-mail: [email p o ec ed]
mas e so ha he o he uni s (sla es) synch onize wi h i . The Blue oo h channel is hen di ided
in o slo s o leng h 625 ms so ha ime slo s a e al e na i ely used by mas e and sla es ( ime
di ision duplex). Any uni may unc ion as a mas e o as a sla e ( his ole is main ained only
o he du a ion o he picone ), bu al hough i may pa icipa e as sla e in mul iple picone s,
i can only be a mas e in one picone .
Two ypes o connec ions can be es ablished in Blue oo h [1]: synch onous connec ion-
o ien ed (SCO) and asynch onous connec ionless links (ACL). In ACL links, he mas e o he
picone pe o ms a polling among all he sla es, al hough he polling policy is no specified by
he cu en Blue oo h specifica ion because he bes policy depends on he applica ion. Sla e
de ices can only ansmi i he mas e has eques ed hem in he p eceding slo . ACL packe s
equi e acknowledgemen and hey a e e ansmi ed in case o e o s using a as au oma ic
epea eques (ARQ) scheme un il hey a e success ully deli e ed. Se e al ypes o ACL packe s
a e defined depending on whe he hey use o wa d e o co ec ion (FEC) o no . Also, mul i-
slo packe ansmissions a e allowed ( h ee o fi e slo s packe s).
On he o he hand, SCO links a e based on a fixed and pe iodic p e-alloca ion o slo s (e e y
wo, ou o six slo s) o gua an eed ansmission o con inuous (audio– ideo) s eams. SCO
packe s a e always one slo leng h and hey a e ne e e ansmi ed. I e o s occu du ing
ansmission (no co ec ed by FEC) hey a e igno ed and he packe is deli e ed as i is
ecei ed. Finally, he mas e does no ha e o poll he sla es in SCO links, so a sla e may
ansmi an SCO packe wi hou a p e ious eques .
A ypical use o Blue oo h would be he suppo o so eal- ime applica ions like mul imedia
s eams. In hese applica ions, he use o SCO links monopolizes bandwid h and lea es e y
li le oom o o he ACL links. Se e al au ho s [2, 3] ha e p oposed ha ACL links could be
used o ca y oice and o he isoch onous affic. These applica ions equi e QoS gua an ees in
e ms o delay, delay a ia ion and loss a e [4], so se e al a emp s ha e been made o p o ide
hese gua an ees in ACL links [5, 6]. Howe e , hese wo ks igno e he possibili y o in e e ences
om o he de ices. When using wi eless connec ions we should always keep in mind ha a
sys em wi h se e al in e e ing de ices is p obabilis ic in na u e. In he case o ACL links,
al hough he polling pe o med by he mas e p o ides de e minism wi hin a picone , he e is
always he possibili y o a collision wi h o he Blue oo h de ices connec ed o diffe en logical
channels, as well as wi h o he in e e ing de ices ope a ing in he ISM band: WLANs like IEEE
802.11, IEEE 802.15.4 ne wo ks [7], Wi elessUSB [8], mic owa e o ens, e c.
The p oblem o coexis ence wi h o he de ices has ecei ed conside able a en ion in he
li e a u e [9–14]. Howe e , hese s udies a e limi ed o he compu a ion o pa ame e s like bi
e o a es (BER) o he p obabili y ha a gi en packe suffe s a collision. To he bes o ou
knowledge, hey ha e no been used o p o ide eal- ime gua an ees o Blue oo h connec ions
aking in o accoun co-channel in e e ences. Maybe he only excep ion is [15] whe e he au ho s
compu e he p obabili y ha a packe does no mee i s deadline o wo-way ansac ions unde
in e e ences om o he Blue oo h de ices. Howe e , hey do no ake in o accoun he effec o
he polling algo i hm o he in e e ence be ween SCO and ACL packe s. Fu he mo e, hei
sou ces o in e e ence a e only o he independen Blue oo h de ices.
In his pape , we p esen an analysis ha allows us o ob ain he p obabili y ha a Blue oo h
packe exceeds i s deadline, aking in o accoun he effec o he polling wi hin he picone , he
p esence o SCO links and he in e e ences om o he ISM de ices, including 802.11b/g,
Blue oo h and Zigbee. Ou app oach was ou lined in a p e ious pape [16], whe e we p oposed
he use o schedulabili y es s like hose ha ha e been used o p o ide gua an ees on message
delays in dis ibu ed eal- ime sys ems [17]. These analyses use he wo s -case ansmission imes
o bound message esponse imes, and o assess he schedulabili y o he sys em. When packe
ansmission imes canno be uppe -bounded, as is he case wi h Blue oo h picone s unde co-
channel in e e ences, hen a p obabilis ic analysis is needed. One possibili y is s a is ical a e
mono onic scheduling (SRMS) [18]. A simple app oach is he p obabilis ic ime-demand
analysis used o single-p ocesso sys ems wi h semi-pe iodic asks ( asks eleased pe iodically
bu wi h andom compu a ion imes) in Re e ence [19]. Howe e , any o hese app oaches a e
difficul and compu a ionally expensi e. In his pape , we would a he e alua e he wo s -case
deadline ailu e p obabili y (WCDFP), conside ing collisions om o he ISM de ices in a
simila ashion o ha used o ake in o accoun ansmission e o s in con olle a ea ne wo k
(CAN)-based sys ems in Re e ence [20]. To illus a e ou me hodology, we show some esul s
o he ansmission o a oice s eam, ‘...one o he mos sough -a e applica ions o IEEE
802.15.1 [i.e. Blue oo h] de ices, and hey a e mos suscep ible o in e e ence’ [9]. Howe e , ou
app oach is use ul in any ime-c i ical applica ion.
The pape is o ganized as ollows. In he nex sec ion, we fi s compu e he wo s -case
esponse imes igno ing in e e ences. Then in Sec ion 3 we discuss how o include he effec o
in e e ences, and in Sec ion 4 we p opose an unified model o include in e e ences om
diffe en ISM de ices, pa icula ly 802.11b/g, Blue oo h and Zigbee. In Sec ion 5, we e alua e as
an example he ansmission o a oice s eam h ough an ACL link unde in e e ences. Finally,
we p esen ou conclusions.
2. WORST-CASE RESPONSE TIME
Conside a picone composed o a mas e and up o se en sla es, wi h se e al SCO and ACL
links. SCO packe s ha e p e-assigned slo s and always p e-emp ACL packe s. Fu he mo e,
hey a e ne e delayed, so hei esponse imes emain cons an ega dless o possible
in e e ences. Tha is, he packe may be co up ed o no , bu i s esponse ime is no affec ed
by possible in e e ences. Al hough he beha iou o SCO links could also be e alua ed using
he p obabili y ha he packe is co up ed as a measu e o pe o mance, se e al easons make
ACL links mo e in e es ing. In coexis ence scena ios, ha is, si ua ions whe e he main eason
o packe d op is due o he in e e ence om o he de ices and no due o noise o ange, FEC
p o ec ion will cause he Blue oo h uni o send mo e packe s o a gi en amoun o da a, and
his esul s in highe in e e ences [9]. DHx (Da a High a e) ACL packe s a e he e o e
p e e ed since hey a e no FEC encoded. On he o he hand, i has been shown ha i ACL
links a e used o isoch onous affic (like oice) ins ead o SCO links, o he ACL links like TCP
beha e be e , wi h he oice quali y only sligh ly affec ed [3]. Fo hese easons, in his pape we
conside he use o ACL links o suppo eal- ime ansmissions, and as a esul we ocus only
on he compu a ion o wo s -case esponse imes o ACL packe s, while SCO packe s a e
conside ed simply as a sou ce o in e e ence.
The e o e, conside N(N47) sla es wi h ac i e ACL links. In his pape , we assume o
simplici y ha each ACL packe occupies a single slo . Only a single ACL link can exis be ween
he mas e and e e y sla e [21], and he polling is pe o med only wi h sla es wi h ACL links.
Ou going ACL packe s a e queued whene e hey suffe delays om h ee possible sou ces:
*The polling mechanism. An ACL packe has o wai o he poll when i is he o he
nodes’ u n.
*P e-emp ion om SCO packe s.
*In e e ences om o he ISM de ices. These in e e ences may des oy he packe s so hey
in oduce addi ional delays.
In his sec ion, we p esen an analysis o compu e wo s -case esponse imes o ACL packe s
including he delays due o he fi s wo sou ces o in e e ences, while in he ollowing sec ion
we ex end his analysis o include in e e ences om o he ISM de ices.
In ou model, an SCO packe jis cha ac e ized by i s pe iod (T
SCOj
) and i s cons an
ansmission ime (C
SCO
). Al hough he Blue oo h channel is di ided in o slo s o leng h
625 ms, due o he ime di ision duplex mechanism we should coun wo slo s pe ans-
mission. In o he wo ds, e e y ansmission includes one slo o he packe and ano he one
o he acknowledgemen . The e o e, we conside o ou disc e e- ime model a ime uni
dslo ¼2n625 ms;so ha CSCO ¼1d slo .
On he o he hand, an ACL packe iis cha ac e ized by i s pe iod (T
i
),
z
deadline (D
i
)and
ansmission ime (C
i
). In wha ollows, we assume ha ACL packe s ansmission imes do no
depend on he p io i y le el ( he Blue oo h MAC laye does no suppo p io i ies excep o he
p io i y o SCO o e ACL packe s). As a esul , we w i e Ci¼C; o all i. We also assume ha
packe s mus be ecei ed be o e he end o he pe iod o he sending ask ( ha is, D
i
5T
i
), o
a oid packe s om successi e in oca ions o he sending ask o delay each o he .
Se e al in a-picone polling algo i hms ha e been p oposed [22]. In his pape , we conside a
pu e ound- obin (one packe pe isi ), because i is he mos used in cu en Blue oo h
connec ions [5]. Fu he mo e, i has been shown ha PRR has a good pe o mance a high
loads [22]. Anyway, o he polling schemes could also be conside ed in ou model.
We a e in e es ed in he compu a ion o he wo s -case esponse ime o an ACL packe ha
may be delayed by o he ACL/SCO packe s. Wi h he PRR algo i hm, we can dis inguish
be ween wo cases:
*Packe s queued a a gi en node. Since only a single ACL link can exis be ween he mas e
and e e y sla e, he e is no in e e ence on a gi en ACL packe mdue o local (wi hin he
node) highe p io i y packe s excep o SCO packe s.
*Packe s queued a o he nodes (ACL links be ween he mas e and he o he sla es). While
ou e e ence ACL packe mis wai ing o he poll, hese packe s a e being ansmi ed.
They can be modelled as a single ‘high p io i y packe ’ wi h pe iodici y equal o he polling
pe iod T
POLL
[17].
The in e e ence due o packe s in o he nodes is gi en by he numbe o imes his ‘highe
p io i y packe ’ wi h pe iod equal o T
POLL
is scheduled o ansmission du ing he wo s -case
packe esponse ime:
Qm
TPOLL

ðTPOLL CÞ
whe e Q
m
is he queuing ime o ou e e ence ACL packe m, ha is, he ime packe mspends
in he queue ( ime om being queued o he ime ansmission begins). In he PRR case, nodes
a e only allowed o ansmi a single packe pe isi . The e o e, TPOLL ¼NnC:E e y ime his
z
I he affic in he ACL link is no isoch onous, hen he pe iod should be in e p e ed as he minimum ime be ween
successi e packe s.
‘packe ’ in e e e a gi en packe m, we ha e o wai TPOLL C¼ðN1ÞnC:The e o e, he
o al in e e ence on a gi en packe mis
Qm
NnC

ðN1ÞC
We also ha e o add he pe iodic and fixed in e e ence om SCO packe s, which does no
depend on he node hey may be gene a ed. The mas e can suppo up o h ee simul aneous
SCO links while sla es can suppo wo o h ee SCO links. The e o e, he in e e ence om
hese packe s can be ound by calcula ing he numbe o imes successi e ins ances o SCO
packe s could be scheduled o ansmission in on o a gi en ACL packe m:
X
8j2SCOS
Qm
TSCOj

CSCO
whe e SCO
S
is he se o SCO links in he picone . We assume ha he ji e ( he maximum ime
a iabili y be ween subsequen packe s o an SCO link) is null. The queuing ime can now be
ound i e a i ely [17]:
Qnþ1
m¼Qn
m
NnC

ðN1ÞCþX
8j2SCOS
Qn
m
TSCOj

CSCO ð1Þ
The o al esponse ime o a gi en ACL packe min he wo s case is hen:
Rm¼QmþCð2Þ
I in e e ences om o he ISM de ices a e igno ed, hen wha e e polling scheme is used
collisions do no occu wi hin a picone . The e o e, he packe ansmission ime is a cons an .
Since we a e conside ing single-slo packe s, hen C¼1d slo o 2n625 ms ( ime di ision
duplex).
3. CONSIDERING INTERFERENCES
I in e e ences a e conside ed, we say ha a collision occu s when he packe is ecei ed
inco ec ly due o hese in e e ences. The sende is no ified o his e o in he slo di ec ly
ollowing he unsuccess ul ansmission using a as -ARQ scheme [1]. The packe is hen
e ansmi ed a he nex oppo uni y (in al e na e slo s) un il i is success ully ecei ed.
The e o e, he ansmission ime Ccanno be conside ed a cons an anymo e. Ins ead, i
becomes a (disc e e) andom a iable, so Cand Q
m
a e andom a iables. The e o e, a
p obabilis ic ime-demand analysis should be used analogous o ha used o single-p ocesso
sys ems wi h semi-pe iodic asks ( asks eleased pe iodically bu wi h andom compu a ion
imes) in Re e ence [19]. Howe e , his app oach is e y difficul and compu a ionally expensi e
because we ha e o compu e he p obabili y densi y unc ion o Q
m
om he p obabili y densi y
unc ion o andom a iable C. Ins ead, since we a e only in e es ed in de e mining he
p obabili y ha a packe mee s i s deadline, we will use a diffe en app oach.
Le us deno e by P
S
he p obabili y o success ul ansmission. In a picone wi h ACL links,
he was ed ime uni s due o collisions co espond o a sequence o Be noulli ials wi h
p obabili y o success P
S
. The o e head o e e y collision is jus a packe e ansmission C.
The e o e, he effec o in e e ences can be included in ou model o Sec ion 2 by adding a

collision o e head E(X) ha is a unc ion o he numbe o collisions du ing a ime pe iod X.I
SCO packe s a e igno ed, EðXÞ¼knC;wi h kbeing he numbe o collisions du ing any ime
pe iod X. Howe e , we ha e o ake in o accoun ha SCO packe s a e no e ansmi ed in case
o collisions (in ha case, he packe is simply co up ed bu deli e ed ‘as is’). The e o e, k
should be compu ed as he numbe o collisions occu ing in any ime pe iod X excluding he
ansmission pe iod o SCO packe s. We will ake his in o accoun in due cou se.
Anyway, conside ing co-channel in e e ences, Equa ion (1) becomes
Qnþ1
m¼EðQn
mþCÞþ Qn
m
NnC

ðN1ÞCþX
8j2SCOS
Qn
m
TSCOj

CSCO
¼knCþQn
m
NnC

ðN1ÞCþX
8j2SCOS
Qn
m
TSCOj

CSCO ð3Þ
No e ha Cis again a cons an C¼1d slo ¼2n625 ms ( o single slo packe s) because all
he effec s o collisions a e included in unc ion E().
The esponse ime o packe mwi h kcollisions can now be compu ed i e a i ely. Fi s
we i e a e Equa ion (3) wi h k¼0;ob aining a fi s esul o Q
m
, and h ough Equa ion (2)
o R
m
. Then we inc emen kand epea he i e a i e compu a ion o ob ain new esul s o Q
m
and R
m
. We epea his algo i hm un il R
m
>D
m
(o he i e a ion does no con e ge). In his
way we ob ain K
m
, which we define o be he maximum numbe o collisions o which
R
m
4D
m
(since he final alue k
makes R
m
exceed he packe deadline, we ha e Km¼k 1).
Le Q
MAX
be he las ob ained alue o Q
m
, ha is, he queuing ime ob ained h ough he
ollowing i e a ion:
Qnþ1
m¼KmnCþQn
m
NnC

ðN1ÞCþX
8j2SCOS
Qn
m
TSCOj

CSCO ð4Þ
and also le R
MAX
be he las ob ained alue o R
m
:
RMAX ¼QMAX þCð5Þ
so i ep esen s he wo s -case esponse ime when he maximum ole able numbe o collisions
K
m
occu .
Once K
m
and R
MAX
a e ob ained, we a e now able o calcula e he wo s -case deadline ailu e
p obabili y (WCDFP) simply as he p obabili y ha mo e han K
m
collisions occu du ing
R
MAX
. Since we assume ha ime is slo ed, wi h a ime uni d slo ¼2n625 ms; hen he
p obabili y ha a collision occu s in a gi en ime uni is 1P
S
(i.e. he p obabili y ha he
ansmission is no success ul). The e o e, we ha e ha he p obabili y ha kcollisions occu
du ing a ime pe iod Xis
X
k
!
ð1PSÞkPXk
Sð6Þ
In ou case, he ime pe iod is R
MAX
bu ansmission o SCO packe s should be excluded.
The e o e,
X¼RMAX X
8j2SCOS
QMAX
TSCOj

CSCO ð7Þ
Now, he WCDFP o an ACL packe mis gi en by
WCDFPm¼1X
Km
k¼0
X
k
!
ð1PSÞkPXk
Sð8Þ
wi h Xgi en by Equa ion (7).
4. PROBABILITY OF SUCCESSFUL TRANSMISSION
The analysis de eloped in Sec ion 3 p o ides he WCDFP o an ACL packe as a unc ion o
he p obabili y o success ul ansmission. In his sec ion, we show how his p obabili y can be
ob ained when he e a e diffe en ISM de ices in e e ing wi h ou e e ence ACL packe .
We begin wi h he case o in e e ences due o o he independen Blue oo h de ices. In o de
o educe in e e ences (among o he easons), Blue oo h uses F equency Hopping, wi h a
pseudo- andom hopping sequence. We can hen assume ha a Blue oo h de ice ansmi s
using andomly chosen equencies. The e o e, he e is a possibili y ha se e al indepen-
den (bu in e e ing) Blue oo h de ices coexis in he same a ea and ha hey choose he same
hop ca ie .
We can assume ha hops a e e enly dis ibu ed o e 79 diffe en equency bands [23]. I can
be shown ha i M
B
is he numbe o in e e ing picone s (including he e e ence picone ) and
is he no malized load o e e e y picone (assuming a homogeneous affic, so is he same o
all he picone s) hen he p obabili y o success ul ansmission is [15]
PSjBB¼2ð1sÞ1
79

þð2s1Þ1
79

2

2ðMB1Þ
ð9Þ
whe e s¼366=625 ¼0:5856 is he a io be ween he ac ual ansmission ime o a single slo
packe and he Blue oo h slo du a ion. Recall ha his exp ession is based on he ac ha a
success ul ansmission occu s i he e a e no collisions in wo consecu i e (and independen )
slo s: one o he packe and he o he one o he acknowledgemen (ACK). I he ACK is no
ecei ed, he packe is e ansmi ed as i a collision had occu ed in he packe i sel . This
exp ession can be simplified i we neglec he e m ( /79)
2
50.00016:
PSjBBffi12s
79

2ðMB1Þð10Þ
Since we a e conce ned wi h he wo s -case beha iou (in o de o p o ide QoS o eal- ime
gua an ees), hen his pessimis ic app oxima ion can be conside ed a lowe bound on he
p obabili y o success ul ansmission, and also he case o ully loaded picone s should be used,
ha is, ¼1:Fu he mo e, al hough a ime and equency coincidence does no always des oy
he packe , depending on he s eng h o he in e e ence signal a i ing o he ecei e [24, 25],
we also assume he wo s -case: in e e ence o jus one bi is enough o des oy he whole packe .
This p e ious model can be easily in e p e ed as he p obabili y ha none o he o he M
B
1
picone s ansmi wi h he same ca ie han ou e e ence picone in wo consecu i e slo s. The
ac s ha he ac ual packe ansmission ime is only o du a ion sslo s and ha independen
picone s a e no synch onized a e aken in o accoun h ough he ac o 2s. We will use his
model o show how o include in e e ences om o he independen Blue oo h de ices in he
compu a ion o he wo s -case esponse ime o ACL packe s. Howe e , many o he mo e
complex (and no so pessimis ic) in e e ence models could also be used [25], including he mul i-
slo case discussed in Re e ence [11].
O he non-Blue oo h de ices may also p oduce in e e ences. The mos impo an sou ce o
in e e ence a e 802.11 de ices, and hei effec on Blue oo h ansmissions a e well s udied in
he li e a u e [10,12–14]. In his pape , we use he model desc ibed in Re e ence [14] which
assumes he wo s case ha a ime and equency coincidence always des oys he packe . Le H
and Wbe he ansmission imes o a single slo Blue oo h packe (H¼366 ms) and o an IEEE
802.11 packe , espec i ely, and also le Lbe he dwell pe iod o an IEEE 802.11 packe . Finally,
i we define Gas
G¼dH=Le*LWH
hen i can be shown ha he p obabili y o a success ul Blue oo h ansmission unde
in e e ences om 802.11 de ices is [14]
}
PSjBW¼1jGj
L

1B
79

dH=Le
þjGj
L
1B
79

dH=LeG=jGj
"#
2MW
ð11Þ
M
W
is he numbe o in e e ing 802.11 de ices, and Bis he channel wid h o he 802.11
sys em, ha is, B¼22 MHz o 802.11b and B¼16:5 MHz o 802.11g (we only conside di ec
sequence sp ead spec um}DSSS sys ems). Diffe en om Re e ence [14], we ake again in o
accoun ha a success ul ansmission occu s i he e a e no collisions in wo consecu i e (and
independen ) Blue oo h slo s. Also no e ha we igno e coexis ence mechanisms like adap i e
equency hopping (AFH), al e na ing wi eless medium access (AWMA), e c. [10].
An in e es ing obse a ion is ha Equa ions (9) and (11) a e essen ially based on he same
easoning. Pa icula ly, al hough Equa ion (11) was ob ained in Re e ence [14] only o he case
o in e e ences om 802.11 de ices, in ac Equa ion (9) can be ob ained as a pa icula case o
Equa ion (11). Indeed, i we ake W¼H( he in e e ing de ice is also a Blue oo h packe ) and
s¼H=L51 in Equa ion (11), we ha e G¼ðL2HÞ50:Finally, since o a F equency
Hopping sys em like Blue oo h B¼1;we ob ain Equa ion (9) o he case ¼1:
The same app oach could also be used o o he ISM de ices. Fo ins ance, an IEEE 802.15.4
(Zigbee) ansmission using he ISM band will in e e e wi h Blue oo h in app oxima ely 3 ou
o he 79 hops [7, Annex E]. Usually, in a senso ne wo k he p obabili y o ac i i y o e e y
node is e y low, and also he ansmission ime o an 802.15.4 packe is much la ge han he
du a ion o a Blue oo h slo , so we could use Equa ion (11) wi h B¼3 and L4W4H:In his
case, G¼ðLWHÞ>0:The e o e, he p obabili y o a success ul Blue oo h ansmission
unde in e e ences om 802.15.4 de ices could be exp essed as
PSjBZ¼13
79 u

2MZ
ð12Þ
whe e M
Z
is he numbe o in e e ing 802.15.4 de ices and u¼ðWþHÞ=LffiW=Lcan be
in e p e ed as he p obabili y o ac i i y o an IEEE 802.15.4 node. Recall ha in a ypical
senso ne wo k MZ4and u5:
}
In case he M
W
802.11 de ices ha e diffe en alues o W and L, ha is, he affic is no homogeneous, hen ob iously
Equa ion (11) becomes a p oduc o M
W
e ms.
To sum up, we ha e shown ha Equa ion (11), o iginally de eloped only o he case o
in e e ences om 802.11 de ices, p o ides us wi h a gene ic model ha can be used o model
in e e ences om o he ISM de ices. Redefining pa ame e s like B,H,Lo W, we can ob ain
he p obabili y o a success ul Blue oo h ansmission unde in e e ences om diffe en ISM
de ices. In he gene al case, he p oduc o he PSjBX e ms should be used. Fo ins ance, i
802.11, Zigbee, and independen Blue oo h de ices a e p esen , hen:
PS¼Y
X¼ B;W;Zg
ðPSjBXÞ
An impo an conside a ion is ha hese models assume he wo s case ha a ime and
equency coincidence always des oys he packe . Al hough we hink hey a e s ill use ul o he
compu a ion o wo s -case, eal- ime gua an ees, we should be ca e ul because many o hei
esul s may be oo pessimis ic. Fo ins ance, IEEE 802.15.4 ne wo ks a e ailo ed o
applica ions wi h low powe , so he p obabili y o in e e ences due o hese de ices is ela i ely
low e en in case o ime and equency coincidence. In o he wo ds, pa ame e s M
X
should no
be in e p e ed as nodes ope a ing in he same a ea, bu as nodes ha ha e sufficien powe o
cause a collision.
5. EXAMPLE: VOICE OVER ACL LINKS UNDER CO-CHANNEL INTERFERENCE
The analysis de eloped in his pape is use ul o any ime-c i ical applica ion. Only o
illus a ion pu poses, le us conside as an example he ansmission o a oice s eam in se e al
configu a ions. Fi s , conside a configu a ion composed o a picone o 3 sla es, whe e 2 sla es
use ACL links. We also assume ha he o he node is using an SCO link in o de o show how
ou model is able o include his addi ional sou ce o in e e ence. Howe e , in o de no o
monopolize he bandwid h we assume ha his SCO link uses HV3 packe s, ha is, one-slo
packe s ha a e sen e e y six slo s.
We assume ha one o he ACL links is used o ca y a oice s eam. Voice can be
implemen ed as 8 kbps coded speech, wi h oice packe s o 20 by es e e y 20 ms [3]. DH1
packe s (28 by es maximum) can be used when ca ying his affic o e ACL links. We assume
ha he deadline o hese packe s is equal o he ime gap be ween a i al o wo packe s, ha is,
Ti¼Di¼20 ms:
Now, le us apply he model desc ibed in Sec ions 2–4 o compu e he WCDFP o hese oice
packe s in he p esence o in e e ences om he o he ACL and SCO links o he picone , as
well as om o he independen Blue oo h de ices. In his configu a ion, 2 sla es use ACL links
so N¼2:Also, C¼CSCO ¼1d slo because only DH1 and HV3 packe s a e used. Finally,
TSCO ¼3d slo s o HV3 packe s, so Equa ion (3) becomes:
Qnþ1
m¼kþQn
m
2

þQn
m
3

We now i e a e wi h k¼0;and ob ain a final alue Qm¼2d slo s, and using Equa ion (2)
Rm¼3d slo s ¼3:75 ms;which is less han he deadline Dm¼20 ms:We hen inc ease k,and
epea he p ocess. This way, we ob ain a final alue o Km¼2;QMAX ¼12 d slo s and
RMAX ¼13 d slo s ¼16:25 ms;which mee s he deadline Dm¼20 ms:I mo e han wo
collisions occu hen his iming equi emen is no me .