Systems with Dynami Behavior
Wang Ke Prithwish Basu Thomas D.C. Little
Multimedia Communi ations Laboratory
Department of Ele tri al and Computer Engineering
Boston University,
8 Saint Mary's St., Boston, MA 02215, USA
Ph: 617-353-8042, Fax: 617-353-1282
fke, pbasu, td lgbu.edu
MCL Te hni al ReportNo. 07-20-99
Abstra t{Video-on-Demand (VoD) systems have unusuallyhigh bandwidthrequirements.
Sin ebandwidth isapre ious ommodityinourpresentday networks,itisextremely desir-
ablethat its onsumptionbeminimized. One way ofloweringbandwidth demandsinaVoD
system is to luster di erent users to be servi ed by a single data stream. This lustering
is done by exploiting the users' a ess patterns to videos of di erent popularity. S hemes
su hasbat hingandadaptivepiggyba kinghavebeenproposedtosavebandwidth. Inthese
s hemes, one problemthat is present is the need to de ide how mu h bandwidth should be
allo ated to servi e users' rst time requests, i.e., when they rst ome in the system and
request to wat h a video; and how mu h bandwidth to servi e users' intera tion requests,
i.e., when they are already in the system and request a VCR-like a tion on the stream. If
we regard the bandwidth available in terms of a limited number of hannels, the problem
we have is in how to allo ate the hannels so as to improve the Quality-of-Servi e of the
system (minimizethe user's waiting time for servi e ompletion, for instan e). To solve the
problem,we need tobuild a mathemati almodelof the VoD system.
In this paper, we present one queueing model of the VoD system with adaptive piggy-
ba kingunder simplifyingassumptions. We des ribethe a tual lustering pro ess, examine
we are making in order to have a tra table problem but whose solution is still useful. We
thenderive theexpressions that allowusto omputethe values needed forour model. With
the model, we an de ne mathemati ally the problem of hannel allo ation and treat the
problemwith known optimizationmethods.
Keywords: Video-on-demand, hannelallo ation
Intoday'sin reasinglynetworked world,Video-on-Demandisoneareathathasre eived sub-
stantialinterestfromboththeresear handapppli ationdomains. Fortheformer,Video-on-
Demandpresents hallengingresear hproblems,whilethelatterisinterestedinthe ommer-
ialviabilityoftheidea andthe pro tsthatmight omefromemploying omputernetworks
todelivermovies.
Sin e the on eption of this domain, the resear h ommunity has been busy solving
di erentproblemsthatarosefromtryingtodeliversyn hronizedvideoandaudiooverabest-
e ort delivery network. One glaring problem that be ame obvious is the in ompatibility of
the bandwidthdemand and the bandwidthavailableinpresent daynetworks. Itis intrying
toexploit behavior patterns for bandwidthredu tion that s hemes su h asbat hing[1℄ and
adaptivepiggyba king[6℄have been proposed.
Both inbat hing and adaptive piggyba king the fundamentalidea is in lustering users.
Userrequestsarriveatdi erenttime intervalsbutideallythey shouldbeservi edbyasingle
data stream. In bat hing s hemes su h lustering takes pla e beforethe data stream starts
whilein adaptivepiggyba kingthe lustering takes pla eatthe sametime the datastreams
are being transmitted. As in all Video-on-Demand servi es, sin e the physi al resour es
are limited,a poli yis adopted toregulate a ess to these resour es. An optimal poli y,in
termsofbandwidthand ustomersatisfa tion,istheone that anservi ethelargestnumber
of ustomers within ertain parameters of Quality of Servi e and within the bandwidth
onstraint.
To dis uss a poli y that is optimal we need a model that an translate the Video-on-
Demand system into mathemati al equations. Su h equations must be losely tied to the
lustering s heme adopted, so that its bene ts an be fully explored mathemati ally. In [1℄
one su h mathemati almodelwas proposed regarding astream bat hing s heme.
Westudyinthispapers hemesthatemployadaptivepiggyba king. Spe i ally,adaptive
piggyba kingreferstothete hniqueofsettingdi erentdisplayratestodi erentdatastreams
sothat their temporaldistan e an be eliminatedgradually. Thus, if one request arrivesfor
movie A, a new stream, let's all it s
1
, is started to servi e the request. Then if another
requestforthesamemovieAarrivesT minuteslater,ase ondstreams
2
isstartedtoservi e
this se ond request. Stream s
1
is then given a slower display rate while stream s
2
is given
a faster display rate until they ome to the same position within the movie stream and an
hara terizesservi eaggregation[5℄. This2-streams enario anbeextrapolatedtohundreds
ofusersandifweaddintera tionrequests,itbe omesa omplexsystemwithno learoptimal
poli y. But it isin this omplex system that we must de ne Quality-of-Servi e onstraints.
One issue that appears is the hannel allo ation problem. Sin e the bandwidth is limited,
if we servi e all in oming rst-time requests, then there might not be enough bandwidth
left for intera tion of the users. On the other hand, reserving too mu h bandwidth for user
intera tion may result inawaste ofbandwidth resour es. Of ourse,\enough" and \waste"
must beunderstoodin ontext, anditbe omes learthen thatwemust establisha riterion
under whi h those 2 words an be applied justly. One su h riterion is the probability that
theusersquitthesystembe auseoflongdelaysininitiatingtheservi ethey haverequested.
Inotherwords,thenumberof hannelsallo atedforea hpurposein uen esthewaitingtime
usersexperien euponrequesting anewmovieoraVCR-likeintera tion. Ifthe waitingtime
istoolong, users may hoose to quitthe system (a behaviorknown asreneging). We would
like then to see an allo ation s heme that would minimize this reneging behavior. But
su h minimization an only be arried out if we an su essfully translate both the system
behavior and the user behavior into relatedmathemati alfun tions.
Attempts have been made to analyze video-on-demand s hemes when ombining both
bat hing and adaptive piggyba king [2℄, but a mathemati al analysis of hannel utilization
when employing adaptive piggyba king under a poli y other than \simple merging" [6℄ is
still la king. We propose here an analyti al model of hannel utilization when employing
RSMA algorithm[4℄ appliedto the adaptive piggyba king s heme.
The organization of the remainder of this paper is as follows. In se tion 2 we des ribe
our Video-on-Demand system and the modelweare proposing toadopt in order toanalyze
it. Then we omment on the RSMA algorithmbrie y and the e e ts it has on the streams
and howthese e e ts are related tothe modelwe are proposing. Afterwards, we pro eedto
dis uss ndingand omputingthe parametersthe modelneeds. Inse tion3and4wederive
the equations that allowus to ompute the parameters determined previously. In se tion 5
we on lude presenting how a small model an be built, notes on the omputation pro ess
and omment onfurther resear h dire tions.
In this paper we use anapproa h proposed by Dan et al. [1℄ tomodelthe VoD servi e asa
queueing system. In this queueing system, the servers are the hannels, and the ustomers
are the usersof the VoD system. The\servi e time"isthe \ hannel utilization"time. Thus
whenauser rstmakesrequests, theserequests arepla edonqueues,waitingtobeservi ed
by a hannel. Andthe samehappens when a user initiatesan intera tion request.
Our modeling problem is then on how to nd the proper values that will hara terize
the queues and obtain the waiting time distribution fun tion of the queueing system. A
ommon approa h has been to assign Poisson arrival pro ess as the arrival pro ess of the
requests [1, 7℄. Then the queueing system we are studying is an M=G=n system, where n
indi ates the number of hannels available. The next unknown fun tion is the servi e time
distribution fun tion. At present we restri t ourselves to obtaining the mean servi e time
for modeling purposes. While the mean servi e time is de nitely an in omplete value, it is
stilla measure of the system hara teristi that in uen e the waitingtime distribution. We
onsider it as an initial parameter that will be omplemented futurely should it be shown
insuÆ ientfor satisfa tory modeling.
The system studied in [1℄ has a xed number of hannels dedi ated to popular movies.
Thusifthe movie'slengthisLminutesand therearen hannelsdedi atedtothe movie,the
movie is shown every L=n minutes over the n di erent hannels. User requests that arrive
are bat hed together and wait until the new transmission starts. User requests that arrive
for unpopular movies are servi ed by a pool of free hannels alled on-demand hannels.
Requests for new hannels due to intera tions from old users (already in the system) are
servi ed froma se ond pool of free hannels alled ontingen y hannels.
The s heme that we propose to study in this paper di er from the above model in the
following points. Firstly there are no dedi ated hannels for any single movie. All requests
are servi ed by the 2 hannel pools. Se ondly and fundamentally the system's behavior
is di erent. While in [1℄ all streams are displayed with one rate, streams in our system
have di erent rates, they undergo dynami servi e aggregation (merging) and often arry
more than one user. Thuswhile the approa h taken and the nature of the problemare the
same, i.e., to onsider the Video-on-Demand system as a queueing system and to ompute
its parameters, the systems themselves are 2 ompletely di erent systems with their own
unique hallenges for modelingand parameter omputation.
followingaZip andistributionfun tion. Weareassumingthatthe requestarrivalpro essis
Poissonwithtotalrate
TOT
. Thei-thmoviehasthusarequestarrivalrateof
TOT
P
Zipf (i).
Theintera tionisalsoaPoissonarrivalpro esswhoserateis
INT
. Fornowwearestudying
only pause-type intera tions. The duration of the pause follows anexponential probability
density fun tion.
The VoD server has 2poolsof free hannels. The total numberof free hannels (sum of
the2pools)islimitedbythebandwidth onstraint. Oneofthepools ontainstheon-demand
hannels, and the other holds the ontingen y hannels.
On-demand hannels servi e users that make their rst time requests at the Video-on-
Demand system. A user that is already in the system and being servi ed by an on-demand
streamandintera tsispla edonaqueuetobeservi edbya ontingen y hannel. Itmustbe
stressed here that both on-demandand ontingen y hannelsare stri tly the same in terms
of physi al hara teristi s. Their only di eren e liesin their utilizationpurpose within the
VoDsystem. A hannelisfreedifthe lastuserintera ts,themoviestreamendsoramerging
pro esso urs. A userwho ispla edonaqueue an hoose toquittheservi e ifthe waiting
time is longer than the user's patien e. Su h behavior is modeled as a reneging probability
density fun tion, whi h in reases with the waiting time.
Now, all new requests are servi ed immediately uponavailabilityof a hannel. And the
streamisgivenaframedisplayrateofS
max
. Atperiodi alintervalsoflengthSW theVideo-
on-Demandsystem looksat allthe streams thatare \wat hing" the samemovie, determine
theirframepositionswithinthemoviestream,andapplytheRSMAalgorithmsoastomerge
all the streams with minimum bandwidth onsumption. This is the \snapshot" algorithm.
After having applied the RSMA algorithm, ea h stream is assigned either a frame display
rateof S
max
orof S
min
. Thisrate iskeptuntileitherthe endofthe moviestreamisrea hed,
allusersinthestreamintera t,oranewintervalSW haspassed,whentheRSMAalgorithm
is applied again on all the streams. A user who intera ts in between the snapshots quits
the hannel and waits tobeservi ed by afree ontingen y hannel. If the user wasthe last
ustomer servi edby the hannel, the hannel isfreed and theresour es returned tothefree
hannel pool. After the intera tion interval has ended (a resume after a pause interval, for
instan e), the user is still servi ed by the ontingen y hannel with display rate S
max . And
this goes onuntil the movie ends orthe stream the user isin ismerged with anotherone.
The RSMAalgorithmtakesinto onsiderationthe urrentpositionsofthe streams, om-
putes the binary tree that yields the lowest ost in terms of bandwidth onsumption for
max min
puting the binary tree, a maximum \ at h-up" window size of length CW is determined a
priori. This \ at h-up" window size de nes the maximum distan e a leading stream i an
be ahead of a trailing stream j and both of them still be joined together by the RSMA
algorithm. Spe i ally, su h distan e d
max
isgiven by
d
max
=
1 S
min
S
max
CW: (1)
In other words, if the leadingstream i isd
max
ahead of the trailingstream j, then the 2
streamswill meetat p
j
+CW, wherep
j
is the streampositionof j when the merge pro ess
started. In this paper, whenever a \group" of streams is mentioned, it refers to a set of
streams that ismerged together under RSMA algorithm.
The rst part of the modeling that we did an then be summarized as: user requests
arriveasaPoissonarrivalpro essand su hrequestsare pla edonaqueue tobeservi edby
on-demand hannels. Uponapause-intera tion,auserispla edonaqueuetobeservi edby
the ontingen y hannels. The waiting time ofthe queues is dependent onthe mean servi e
time. Users submitted to the queue quit a ording to adistribution fun tion dependent on
the waiting time. Sin e we have 2 hannel pools, it is ne essary to nd the di erent mean
servi e time for the 2 di erent poolsto hara terize the entire system.
3 On-Demand Channel Utilization
ConsiderthatthemovieshavelengthL. WeassumethatLisanintegermultipleofSW. We
alsoassumethatCW issmallerthanLsothattheoverall hara teristi of hannelutilization
isdetermined by the hannelutilizationstatisti within CW. Therefore, forrequests to the
on-demand pool, if T is the hannel utilizationtime, then:
E[T℄=E[T
SW +T
M1 +T
INT +T
M2 +T
M3
+℄; (2)
whereT
SW
isthe time untilanew SW omputationwindowstarts, T
M
j
is the timeinterval
between the (j-1)-th merge and j-th merge and T
INT
is the time the rst intera tion in
an interval happens. This is a very generi des ription of the expe ted time of hannel
utilization. A more expli it way of des ribing E[T℄ would be:
SW M1 SW M1
E[T
INT
j1 intera tion during T
SW +T
M
1
℄+
E[T
M2 jN
W
(1) intera tion inW =T
M2
;
nointera tion during T
SW +T
M
1
℄+
E[T
INT
j More than 1intera tion inW =T
M2
;
nointera tion during T
SW +T
M
1
℄+
E[T
M3 jN
W
(2) intera tions inW =T
M2 +T
M3
;
nointera tion during T
SW +T
M
1
;
N
T
M
2
(1) intera tion duringT
M2
℄+℄ (3)
N
W
(n) indi atesthat the numberof intera tions in time W isless thanor equalto n.
Clearly the omplexity grows and it is in reasingly diÆ ult to ompute the value. One
reasonforthisdiÆ ultyliesinthefa tthatthebinarytreebuiltbythedynami programming
has unknown statisti alproperties. More spe i ally, suppose that theinitialpositionsofN
streamsp
i
are separated byrandomintervalsof lengthgiven by anexponentialdistribution.
How dowe ompute the expe tedvalueof the mergingtimefor thebinary tree? Ifweknow
their initial positions, then a ording to the DP algorithm, we know their paths until all
streamsaremergedintoone. Buthowdowe expe t hangesonthese pathsifwe hange the
initialpositions? Averypra ti alquestionis,ifwehaveonlytheprobabilitydensityfun tion
of the distan es that separate the streams at the beginning of the merging pro ess, what
then is the average time it takes for one stream tobemerged with another? This expe ted
value would give us ameasure of hannel utilizationtime that an be used in modeling the
system as a queueingsystem.
In thispaper,weattemptanalysisinabroadsense wheneverpossible,butdue tomathe-
mati al omplexity,wewillhavetorestri tour problemtoaparti ulardistributionfun tion
toobtain parameters for simulationpurposes.
The rst step we take isin attemptingto ompute E[T
SW
℄. Letp
T
(i)bethe probability
of i arrivalsina T interval:
p
T (n)=
e
A T
(
A T)
n
n!
(4)
A INT
time and if wehave onlyone arrival,the expe tedtime untila new re omputation startsis:
E[T
SW
j1 arrival℄= Z
SW
0 e
A t
A
(SW t)e
INT (SW t)
dt (5)
The integrand is the result of multiplying the interarrival time distribution, the proba-
bility that thereis nointera tion and the amount of timeuntil anew re omputation starts.
Likewise,if we onsider allpossible arrivals,the expe ted value E[T
SW
℄ isgiven by:
E[T
SW
℄ = X
n=1 p
SW (n)
Z
SW
0 Z
SW
t1
Z
SW
tn
1 e
A P
n
i=1 (ti ti
1 )
n
A
e
INT P
n
i=1
(SW ti)
1
n n
X
i=1
(SW t
i )dt
n
dt
1
(6)
with t
0
=0.
Followingthe omputation of the term above, we pro eed for the expe ted time of rst
merge. i.e., the averagetime the streamstakeuntilthey rst mergewith anotherstream. It
is here that we fa e the major problem des ribed previously. If we assume Poisson arrivals
with exponential interarrival time, then it is too omplex to obtain the statisti s for this
\ rst merge" time, sin e the DP algorithm an determine ea h stream's rate only after
omputing the wholerange ofpossibleminimum ostpaths. And \translating"this pro ess
intostatisti al properties remainsto be done. This \ rst merge" time isimportantbe ause
untilthe rst merge,ea h stream is arrying onlyone user, and any intera tion would have
terminated the hannel utilization.
A simplifying assumption we are for ed to take at this moment is to onsider that the
streamsareallequidistant. Even thoughthisnarrowsthe possibleappli ationsofour model,
still there are reasons to study the system under su h onditions. If we want to s hedule
se ondary ontent tothe users in the system, then the whole system must be syn hronized
insu haway thatthe beginningofthe servi ewillalways fallona dis rete\grid"[3℄. Thus
it is highly likely that for the most popular movies the system willhave streams whi h are
equidistant fromone another.
Consider nowd
max
,the maximumdistan ethat anseparate2streamsinasinglegroup,
E[T
M
1
j2 users℄= d
max
S
max S
min
"
e
INT
dmax
Smax S
min
#
2
: (7)
If the number of users n in the system is odd, then there will be n 1 users with the
samemergingtimebut 1userwith doublethemergingtime. Letd
i
=d
max
=[(S
max S
min )i℄,
then
E[T
M1
℄ =
X
ieven
i2 d
i 1 p
dmax (i)
h
e
INT d
i 1 i
i
+
X
iodd
i2 d
i 1 p
d
max (i)
i 1
i h
e
INT d
i 1 i
i 1
+ 2
i h
e
INT 2d
i 1 i
: (8)
These values already in lude the probability that no intera tion will o ur during the
interval onsidered. Anadditional ommentmust beadded here. Tests of onvergen e have
not been in luded be ause due to syn hronization issues on real movieseventuallywemust
onsider individual, and thus dis rete and nite, frames. In other words there is a nite
number offrames that separates the leadingfromthe trailing streamwithinone group, and
this givesan upper bound onthe maximum numberof streams that a tree may have.
In the ase there is an intera tion within the interval onsidered, then the hannel uti-
lization is determined by the moment that the intera tion took pla e. Consider the ase of
a single streamand let L bethe durationof the movie, we have:
E[T
INT
j1 arrival℄= Z
SW
0 Z
SW+L
t
e
A t
A
e
INT (s t)
INT
e
INT (s t)
INT
(s t) 2
dsdt: (9)
The integrand above is the multipli ation of the distribution of the arrival time, the
distributionof the intera tion requestarrivaltime, the probabilityof havingone intera tion
and the time o urred between the beginning of the request and the intera tion. In ase of
2 arrivals, the equationbe ome:
E[T
INT
j2arrivals℄ = Z
SW
0 Z
SW+d1
t
1
Z
t1+dmax
t
1
Z
SW+d1
t
2
e
A t
1
A
e A 2
A
e
INT 1 1
INT
e
INT 2 2
INT
e
INT (s1 t1)
INT (s
1 t
1 )e
INT (s2 t2)
INT (s
2 t
2 )
(s
1 t
1 )+(s
2 t
2 )
2
ds
2 dt
2 ds
1 dt
1
: (10)
Noti e that we need onsider only se ond arrivals that arrive within d
max
of the rst
arrival. Andthatthe intervalweneedto onsider forintera tion isSW+d
1
,whi hin ludes
the beginning part of the mergingpro ess. This leads to the eventual equationfor T
INT :
E[T
INT
℄ = 1
X
n=1 p
SW (n)
Z
SW
0 Z
SW+d
n 1
t
1
Z
t
1 +dmax
t
1
Z
t
1 +dmax
t
n 1 Z
SW+f(n)d
n 1
t
n
e
A P
n
i=1 (t
i t
i 1 )
n
A
e
INT
2 P
n
i=1 (s
i t
i )
n
Y
i=1 (s
i t
i )
P
n
i=1 (s
i t
i )
n
ds
n dt
n
ds
1 dt
1
(11)
where
f(n)= (
1 ; neven
2 ; nodd
(12)
d
0
=L and t
0
=0.
As anbeseenbythesefewattempts,the omplexityofpre iseanalysisin reasestremen-
dously and here we are dealing only with the expe ted value until re omputation time and
the average rst merging time. For the subsequent average merging times, we resort to
dynami programming.
In previous work [3℄, we have established the dynami programmingalgorithm that an
determine the merging point p(i;j) of 2 streams i, j within one group and the optimal k
whi hyieldsthe minimum ostC(i;j)for themerge,whereC(i;j)=C(i;k
)+C(k
+1;j).
Knowing these,the time taken forthe wholemerging pro ess an be al ulated asfollows:
De ne:
T(i;i) = 0
T(i;j) = T(i;k
)+T(k
+1;j)+T (i;j;k
)+T (i;j;k
)
T
MAX (i;j;k
) = (
p(i;j) p(i;k
)
S
MAX q
MAX(i;j;k
) (N
INT
<k
i+1);i6=k
0 ;i=k
(13)
T
MIN (i;j;k
) = (
h
p(i;j) p(k
+1;j)
S
MIN i
q
MIN(i;j;k
) (N
INT
<j k
);j 6=k
+1
0 ;j =k
+1
(14)
wherethetermq
MAX(i;j;k
)
indi atestheprobabilitythatinT
MAX
=[p(i;j) p(i;k
)℄=S
MAX
there will be less than k
i+1 intera tions, and the term q
MIN(i;j;k
)
indi ates the same
for the T
MIN
=[p(i;j) p(k
+1;j)℄=S
MIN
interval. If we onsider Poisson arrivalpro ess,
q
MAX(i;j;k)
= k i
X
m=0 e
INT T
MAX
(
INT T
MAX )
m
m!
(15)
and analogously for q
MIN
. We assign the value of zero for i = k
and j = k
+1 be ause
they would haveyielded values forthe \ rst mergingtime",and those have been previously
omputed.
We an then take the expe ted value of mergingtime tobe:
E[T
M
℄=E[T
M
1
℄+ N
X
n=3 1
n
T(1;n)p(n) (16)
where p(n) indi ates the probability of having n streams during d
max
. This is of ourse a
rst approximation. A omplete mathemati al analysis would have totake intoa ount all
possible ombinations of the maximum number of intera tions along ea h path. It should
be noti ed that the method above yields the \exa t" value for the merging time given the
initialpositions. It is only when we know that the streams will be equidistant that we an
equate the value omputed tothe expe tedvalue.
Having omputed all the above, we an take the expe ted value of hannel utilization
E[T℄for the on-demand hannelsto be:
E[T
OD
℄=E[T
SW
℄+E[T
INT
℄+E[T
M
℄: (17)
4 Contingen y Channel Utilization
To hara terizethe ontingen y hannelutilization, onsiderthepauseintera tion. When-
the pause. Now suppose the duration is su h that the trailing stream is still \behind" the
pausedstream'spositionwhenitresumes. Supposingon emoretheequidistantassumption,
givennstreamsseparatedbyd
max
,numberingthetrailingstreamstream\0"andtheleading
stream stream\n 1", the maximum pause time stream i an have is
T
P
=
id
max
S
max
(n 1)
(18)
Therefore, assumingthat thepause durationisexponentialwith mean (
P )
1
,the prob-
ability that the paused streamwillresume \within"the group is:
p
i
=
Z idmax
Smax(n 1)
0
e
P t
P
dt: (19)
If,uponresumingfromapause,thestreamisgivendisplayrateofS
min
,thentheexpe ted
utilizationtime of the ontingen y hannel untilit mergeswith another hannelis:
E[T
IN
℄= 1
n n 1
X
i=1
Z idmax
Smax(n 1)
0
e
P t
P
t+ i
n 1 d
max S
max t
S
max S
min
!
dt: (20)
Regarding the above term, the summation is over the n streams (ex luding the trailing
stream), and the se ond term in the parenthesis indi ates the hannel utilization starting
fromthe pause intera tion untilthe streammerges with the trailingstream.
Ifthe stream'spausetakesitout ofitsformer group,thena tually theexpe ted hannel
utilizationtimeisthe expe tedtimeuntilare omputationstartsand the rst mergeo urs.
Thus, let
q = n 1
X
i=0 1 p
i
(21)
The expe ted value of hannel utilizationfor ontingen y hannels is:
E[T
C
℄=E[T
IN
℄+q(E[T
SW
℄+E[T
M
1
℄) (22)
We have thus ome with equations that enable us to hara terize fully our queueing
system. From the arrival pro ess to the servi e time for both queues, in whi h the servi e
time in luded spe i aspe ts of the system under RSMA algorithm. Withthe parameters
above, we an start by studying the queueing system waiting time distribution. On e we
have an idea of the distribution for the waiting time, we an integrate over the reneging
probability density fun tion and we will have a per entage of users that renege in our VoD
system. The optimality of the system omes in minimizingthe number of on-demand and
ontingen y hannelssubje t to anupperbound onthe per entage of reneging users.
Regarding all the equations that appeared, it is important to note that these omputa-
tions need be arried out only on e for the system. It is only during the initialsetting up
of the system and later maybe for ne tuning that su h omputations are made, to give a
measure ofthe servi etime ofthe on-demandand ontingen y hannels. Furthermore,there
isnoneed tosolvethe integrals in lose format. A numeri almethodthat givesthe value of
the integral is enoughfor our system hara terization purposes.
Our next step is in exploring the system through simulation, whi h will enable us to
determineifwe an treatthesystem asaM=M=nqueueorwemustdelvefurtherto onsider
thesystem asaM=G=nor,even moregenerally,aG=G=nqueue. Withtheseotherqueueing
models we an obtain even more re ned distribution fun tions for the waiting time of the
system and thusapproximateeven loser withthe real-world model. Spe i tasksthat an
be targeted futurely in lude determining the statisti al hanges with the average merging
time without the equidistant assumption; hara terizing more omplex intera tion types,
su h as Fast Forward and Rewind and studying the trade-o s of assigning the number of
hannels inea h pool dynami ally,a ording tothe state in the VoD system
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