Broadband multimedia networks Definations Ppt
Transcript of Broadband multimedia networks Definations Ppt
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DATA NETWORKS (TELE4352)DATA NETWORKS (TELE4352)
White.Cell
Company Presentation To
Aurec Technologies
September 10, 2000
April 21, 20161
Queuing Theory Queuing Theory
bybyDr. Johnson I AgbinyaDr. Johnson I Agbinya
agbinya!big"on#.n$%.a& agbinya!big"on#.n$%.a&
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Contents
• Delay models in data networks• Queuing models
• Notation
• Little’s theorem
• M/M/x queues
• Multi-Server systems
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Delay Models In data Networks
• he essential delay in data networks !onsists o""our !om#onents
• #ro!essing delay
• queuing delay
• transmission delay
• #ro#agation delay
• Processing delay• this is the time a #a!ket is !orre!tly re!eived at the head node
o" the link and the time it is assigned to an outgoing linkqueue "or transmission
• $s inde#endent o" the tra""i! !arried %y the node #rovided#ro!essing #ower is not its limitation
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Delays
• Queuing delay• this is the time between when a packet is assigned to
a queue and the time it starts being transmitted
• Transmission delay • this is the time between when the frst and last bits o
the packet are transmitted
• Propagation delay• the time the last bit is transmitted at the head node o
the link and the time the last bit is recei!ed at the tailnode" This time is proportional to the length o theph#sical link between the transmitter and recei!er
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Delays
• &ro#agation delay is a "un!tion o" the !hara!teristi!s o" the
medium %etween the ' and ('
• $s inde#endent o" the tra""i! !arried %y the link
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Characteristics o links
• $s a %it #i#e with a "ixed transmission !a#a!ity )%its #er
se!ond*
• de#ends on the #hysi!al !hannel and the inter"a!e )eg+
Modems* - is the rate at whi!h the inter"a!e a!!e#ts %its "or
transmission• he link !a#a!ity may serve several tra""i! streams that are
multi#lexed onto the link
• $n statisti!al multi#lexing the streams are merged into a single
queue and transmitted on a "irst !ome "irst served ),,S*
%asis• $n statisti!al multi#lexing. the length o" time it takes
to transmit a #a!ket o" length L is L/. where is
the !a#a!ity o" the link in %its #er se!ond+
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Characteristics o "inks• TDM and FDM• m tra""i! streams
• link is su%divided into m #ortions. one #er tra""i! stream
• FDM: 0/m )where 0 is the !hannel %andwidth and m
!hannels*
• $n #ra!ti!e guard %ands are used. whi!h means the %andwidthallo!ated to a tra""i! stream will %e slightly less than 0/m
• transmission !a#a!ity o" ea!h !hannel is /m. where is the!a#a!ity that would %e availa%le i" the whole system %andwidthis allo!ated to a single !hannel
• TDM: time axis is divided into m slots o" "ixed length mse#arate links with !a#a!ity /m+ ransmission time when time
slots are short relative to #a!ket length is a%out Lm/ and a%out
L/ "or slots o" #a!ket length )m-1* #a!kets waiting
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$%e%in& Models• $n data !ommuni!ation systems. many 2o%s share
the same system resour!e )eg+ &3. disks.
#rinters. !hannel !a#a!ity et!*
• $n #rin!i#le only one 2o% or #a!ket !an use a resour!e at a
time• all other 2o%s waiting to use the resour!e wait in queues
• Queuing theory is used to model the time that
2o%s )#a!kets* s#end in various queues in the
system
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Notations• 4rrival #ro!ess
• Servi!e time distri%ution
• Num%er o" servers
• System !a#a!ity
• Servi!e dis!i#line
• 4rrival &ro!ess• Example: $n general students arrive "or a le!ture randomly
#a!kets in data !ommuni!ation system arrive at a node also in
a random manner
• $" #a!kets arrive at times. then the varia%les
• are !alled interarrival times• "orms a sequen!e o" $nde#endent and $denti!ally Distri%uted
)$$D* random varia%les
n
t t t ,,,21
1−−= nnn t t τ
nτ
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Notations 5
&oisson &ro!ess• he most !ommon arrival #ro!ess is the &oisson arrivals thismeans that the interarrival times are $$D and are ex#onentiallydistri%uted
• Service Time Distri%ution
• he time ea!h #a!ket needs to transmit is !alled the servi!etime• servi!e times are normally random varia%les and $$D distri%uted• ex#onential distri%ution is most !ommonly used to model
servi!e time
• Number o Servers• the transmission "a!ilities may have one or more !hannels )e+g+
!ir!uit swit!h* all o" whi!h are !onsidered #art o" the samequeuing system. and any !hannel may %e assigned to any !all)#a!ket*
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Notations• $" all the !hannel !a#a!ities are not identi!al. they are usually
divided into grou#s o" identi!al !hannels )servers* with se#aratequeues "or ea!h grou#
• in this !ase ea!h grou# is a queuing system
• System Capacity• he maximum num%er o" #a!kets )!alls* that !an stay may %e
limited due to s#a!e )%u""er* availa%ility and also to avoid longwaiting time. this num%er is !alled the system !a#a!ity
• !a#a!ity is "inite in most systems• $s easier to assume in"inite !a#a!ity "or the sake o" analysis• the system !a#a!ity in!ludes those waiting "or servi!e as well as
those %eing served )or re!eiving servi!e*
• Service Discipline• the order in whi!h the #a!kets )!alls* are served is !alled the
servi!e dis!i#line+ he most !ommon is ,,S
• 6ther #ossi%ilities are last !ome "irst served )L,S*
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(peciication o $%e%es
• he notation a!b!m!" is used to des!ri%e a
queuing system. where• a s#e!i"ies the ty#e o" arrival #ro!ess
• $" a is s#e!i"ied %y M. then the arrival #ro!ess is &oisson and
the interarrival times are $$D ex#onential random varia%les
• b denotes the servi!e time distri%ution
• i" % is given %y M. then the servi!e times are $$D ex#onential
)memory-less* random varia%les
• $" % is given %y 7. then the servi!e times are $$D a!!ording tosome general distri%ution
• m s#e!i"ies the num%er o" servers )!hannels*.
• " denotes the maximum num%er o" #a!kets )!alls* allowed in
the system at any time
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(peciication o $%e%es
• $n general. "or data networks. we deal with the"ollowing queues
• M/M/1
• M/M/1/8
• M/M/m
• M/M/m/m and
• M/7/1
• i" the interarrival times are ex#onentially distri%uted. with mean
#! . the ex#e!ted time to the next arrival is always #!
regardless o" the time sin!e the last arrival• %ulk arrival and %ulk servi!e• arrival or servi!e !onsists o" a grou# o" 2o%s )!ustomers* is
denoted %y a su#ers!ri#t )ty#i!al o" how tourists are served at
many servi!e #oints*
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(peciication o $%e%es
• ,or exam#le. %ulk &oisson arrivals or servi!e are denoted %y• where x re#resents the grou# si9e. whi!h is generally a
random varia%le and its distri%ution needs to %e s#e!i"ied
se#arately
( ) x M
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"ittle)s "aw•
Little’s law #rovides the relationshi# %etween arrival rateand the mean res#onse time o" the system+ he law is:
• Mean num%er in the system ; arrival rate x mean
res#onse time
• this relationshi# a##lies to all systems or #arts o" systems in
whi!h the num%er o" !ustomers entering the system is equal
to those !om#leting servi!e e+g+:
5 tic$et %ueues at rail&ay stations
5 service at coee s'ops
5 service at supper mar$ets
5 arrivals and departures at normally unctioning airports
T N λ =
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"ittle)s "aw• Little’s law !an %e a##lied to any system or su%system. "or
exam#le. %y a##lying it to 2ust the waiting "a!ility. we get
mean number in %ueue ( arrival rate x mean &aiting time
similarly) or t'ose currently receiving t'e service) &e 'ave
Mean number in service ( arrival rate x mean service time
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*he M+M+1 $%e%e•
Single Server• &oisson 4rrival #ro!ess
•
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-oisson -rocess
• he average num%er o" arrivals within an interval
τ is λτ
=ut
here"ore
• 0e ex#e!t this to %e the !ase "or a #ro%a%ility "un!tion
( ) ( )
∑ ∑∞
=
−=0 n
n
nen P τ λ τ λ
( )∑∞
=
=0
1n
n P
( ) τ λ
τ λ +
∞
==∑ enn
n
0
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-oisson -rocess I
• Mean num%er o" arrivals in time τ
where n;k-1 as
we have
λ ; mean arrival rate )mean arrivals #er se!ond*
1/λ ; mean inter arrival time in se!onds
( ) ( ) ( ) ( )
∑ ∑∑∞
=
∞
=
−∞
=
===1 00 k n
n
k nek kpk kpk E
τ λ λτ τ λ
( ) τ λ τ λ +∞
=
=∑ enn
n
0
( ) λτ =k E
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'ro"$r%i$s o 'oisson 'ro$ss$s
• ,our #ro#erties are !onsidered here:
• Merging o" &oisson streams results to a &oisson stream
• S#litting o" a &oisson stream results to &oisson streams• De#arture "rom an M/M/1 queue are a &oisson #ro!ess
• De#arture "rom an M/M/m queue are a &oisson #ro!ess
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(er.ice (tatistics• ustomer )#a!ket* servi!e times have an ex#onential distri%ution
with #arameter +
∀ µ is !alled the servi!e rate
• i" sn is the servi!e time o" the nth !ustomer )#a!ket*
• the #ro%a%ility density "un!tion o" sn is
• and its mean and varian!e are 1/µ and 1/µ> res#e!tively
• the servi!e time sn are mutually inde#endent and also inde#endent o"
all inter arrival times
[ ] 0,1 ≥−=≤ − se s s P sn µ
( ) ,n sn e s p µ µ −=