broadband definations ppt.ppt

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May 2, 2016 1

DATA NETWORKS (TELE4352)DATA NETWORKS (TELE4352)

White.Cell

Company Presentation To

Aurec Technologies

September 10, 2000

May 2, 20161

Queuing Theory Queuing Theory

bybyDr. Johnson I AgbinyaDr. Johnson I Agbinya

agbinya!big"on#.n$%.a& agbinya!big"on#.n$%.a&



May 2, 2016 2


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 ast bits o!

the packet are transmitted

• Propagation delay• the time the ast bit is transmitted at the head node o!

the ink and the time the ast bit is recei"ed at the tainode# This time is proportiona to the ength o! thephysica ink 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 of 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 of 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*



May 2, 2016 11


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*



May 2, 2016 12


&'ecification of "#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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• $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*



May 2, 2016 14



• ,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 =



May 2, 2016 16


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



May 2, 2016 17


)he M*M*1 "#e#e•

Single Server• &oisson 4rrival #ro!ess

• <x#onential Servi!e ime

• ,,S Queuing Dis!i#line

• *rrival Process + Poisson• the &oisson #ro!ess is used widely to model the %ehaviour o"

queues

• $n the M/M/1 system. !ustomers arrive a!!ording to &oisson

&ro!ess( )

( )

+n

en P

n τ λ τ λ −

=



May 2, 2016 1!


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 +

∞

==∑ e

nn

n

0 +



May 2, 2016 1%


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



May 2, 2016 20


'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



May 2, 2016 21


&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 s

n e s p µ µ −

=

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