chapter 3 delay models in data networks. section 3.2 little`s theorem
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![Page 1: Chapter 3 Delay models in Data Networks. Section 3.2 Little`s Theorem](https://reader033.vdocuments.us/reader033/viewer/2022061609/56649d565503460f94a34d19/html5/thumbnails/1.jpg)
Chapter 3
Delay models in Data Networks
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Section 3.2
Little`s Theorem
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3.2 Little`s Theorem : average number of customers
in system : mean arrival rate T:mean time a customer spends in
system
N
N
T
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Little`s Theorem Proof
N(t) = number of customers in system at time t
(t) = number of customers who arrived in interval [0,t]
Ti = time spent in system by the i-th customer
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Little`s Theorem
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Little`s Theorem
TN
take
t
T
t
tT
tdN
t
t
t
iit
ii
t
lim
)(
)(1)(
1
)(
1)(
10
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3.2.3 Application of Little`s Theorem
Ex3.1 : arrival rate in a transmission line NQ : average number of packets
waiting in queue W : average waiting time spent by a
packet in queue
NQ = W
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Application of Little`s Theorem
If = average Tx time =
: Average number of packets under Tx I.e. fraction of time that s busy utilization fact
or
XX
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Application of Little`s Theorem Ex3.2
N : average number packets in network T : average delay per packet
also Ti : average delay of packets arriving
at node i
n
iiTN
1
iii TN
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3.3 M/M/1 Queuing System M/M/1
First M : arrival , Poisson Second M : service , Exponential 1 : server number
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M/M/1 Queuing System Arrival Poisson process
A(t) : number of arrivals from 0 to time t
Number of arrivals that occur in disjoint intervals are independent
Number of arrivals in any interval of length is Poisson distributed with parameter , ,1,0,
!
)()()(
nn
entAtAP
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M/M/1 Queuing System Properties of Poisson process
1. Inter arrival times are independent and exponentially distributed with parameter
tn : time of the n-th arrival
2
1
1var,
1)( :
0,1
iancemeanePpdf
SeSP
tt
nn
Sn
nnn
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M/M/1 Queuing System
2. For every t0, 0
!2
)(1:
0)(
)(2)()(
)(1)()(
)(10)()(
2
0lim
eNote
owhere
otAtAP
otAtAP
otAtAP
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M/M/1 Queuing System3. A = A1+A2++AK is also Poisson with
rate = 1+ 2++ K
Poissonmerge
A1
A2
AK…
……
..
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M/M/1 Queuing System
4.
Poisson split
P
1-P Poisson with (1-P)
Also Poisson with
P
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M/M/1 Queuing System Service time : Exponential distribution
with parameter Sn : service time of n-th customer
nS
n
Sn
eSPpdf
SeSSP
)(:
0,1
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M/M/1 Queuing System Properties of Exponential : memoryles
s
0, ),(|
0, ),(|
trforrSPtStrSP
trforrPttrP
nnn
nnn
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Markov chain formulation Let's focus at the times,0,,2,…,k,…
Nk = number of customers in system at time k = N(k)
Where N(t) is continuous-time Markov Chain Nk is discrete-time
Let Pij : transition probabilities = P{Nk+1=j|Nk=i}
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Markov chain formulation
1,1, ),(
0),(
0),(
1),(1
)(1
,
1,
1,
00
iiijandioP
ioP
ioP
ioP
oP
ji
ii
ii
ii
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Markov chain formulation
Note During any time interval, the total number
of transitions from state n to n+1 must differ from the total number of transitions from n+1 to n by at most 1
I.e. frequency of transitions from n+1 to n = frequency of transitions from n to n+1
ntNPnNPP
iesprobabilitstatesteady
tk
kn
)(
limlim
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Markov chain formulation
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Markov chain formulation
Take ->0 Pn=Pn+1
Pn+1=Pn, n=0, 1, … 等比數列where = / utilization Pn+1= n+1P0, n=0,1,…
Since <1, and
)23.3,...(1,0)1(
110
0
nP
PP
nn
nn
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Markov chain formulation
1,
11
)25.3(1
)1(
)24.3(1)1(
1)1(
)1()(lim
2
2
00
WNW
NT
nnPtNEN
Q
n
n
nn
t
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M/G/1 System
Let Ci : customer I Wi = waiting time of Ci
Xi = service time of Ci
Ni = # of customers found waiting in queue when Ci arrives Ri = residual service time of the customer in service when Ci arrives
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M/G/1 System
Ci start serviceCi arrives
Ri
Ni
Xi-1Xi- Ni
)48.3(1
)()()()(
1lim
1
RW
XENEREWE
WRu
NRW
XRW
jiii
Qi
i
Nijjii
i
In steady-state,
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M/G/1 System
To calculate R, by graphical approach:
Time
Residual service time r()
Ci starts service
M(t)=# of service completion in [0, t]
tXM(t)
XM(t)
X2
X2
X1
X1
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M/G/1 System
Time avg of r() in [0, t]
)(
)(
2
1
2
11)(
1
)(
1
2
2)(
10
tM
X
t
tM
Xt
drt
tM
ii
i
tM
i
t
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M/G/1 System
)1(2
)(
1
)(2
1
)(lim
)(lim
2
1
)(1
lim
2
2
)(
1
2
0
XERW
XE
tM
X
t
tM
drt
R
tTake
tM
ii
tt
t
t
P-K Formula(3.53)
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Ex3.15
Consider a go back n ARQ:
Assume that error in the forward channel is p, return channel is error-free
Packet arrives as a Poisson process with rate packets/frame
1 2 3 … n-1 n 1
1 2 3
time
time
sender
receiver
Timeout (n-1) frames
Prop. delay
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Ex3.15
Service time X : from when a packet transmitted until it is successfully received
1 , if 1st tx is successful (1-p)X={ 1+n, if 1st tx is un- successful; 2nd is su
ccessful p(1-p)1+kn, if 1st k is un- successful;(k+1)th successful Pk(1-p)
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Ex3.15
0
22
0 0
0
)1()1(][
)1(
)1()1()(
k
k
k k
kk
k
k
ppknXE
kpnpp
ppknXE
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Ex3.15
WXETXE
XEW
p
ppn
p
npXE
p
npXE
)(,))(1(2
)(
)1(
)(
1
21][
11)(
2
2
222
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3.5.1 M/G/1 Queue with vacations
1. When the server has served all customers, it goes on vacation
2. If the system is still idle after a vacation interval, go on another vacation interval
3. If a customer arrives during a vacation, customer waits until the end of vacation. Chapter 1 section 1.3.1 page 34in Network or Transport Layer
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M/G/1 Queue with vacations
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M/G/1 Queue with vacations
Assume vacation intervals v1, v2… are iid and are independent of customers arrival & service times.→A customer must wait for the completion
of the current service or vacation interval, and then the service of all customers waiting before it.
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M/G/1 Queue with vacations
Where R is the mean residual time for completion of service or vacation when the customer arrives.
1
RW
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M/G/1 Queue with vacations
Let L(t) = # of vacations completed by tM(t) = # of services completed by t
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2)(
1
2)(
10 2
11
2
11)(
1i
tL
ii
tM
i
tV
tX
tdr
t
)(
)(
2
1
2
11
)(
1
2
2)(
1 tL
V
t
tLX
t
tL
ii
i
tM
i
)( 2VE
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)(2
)(
)1(2
)(
1
22
VE
VEXERW
)(
)()1(
2
1
2
)( 22
VE
VEXER
)(
1)(),(
)(
)1(lim
VEt
tLVE
tL
tt
Because Fraction of time occupied with vacation = 1-
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Ex3.16 : FDM, SFDM, TDM m streams of traffic with rate /m(Poi
sson) FDM system – Divide available bandwi
dth into m subchannels. Transmission time for a packet on each of these subchannels is m.
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FDM
mm
XE )(
2222 0)()( mmXEXVarXE
)56.3()1(2)1(2
mmWFDM
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Slotted FDM System Packet trans starts only at time m,
2m,…When the queue is idle, server takes a vacation of m. (if idle again after vacation, take another)
)57.3(2)1(2
mW
mW FDMSFDM
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TDM System
Look at SFDM queue, ->same queue WTDM=WSFDM
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Summary
mm
T
mm
T
mifbetterm
Tm
T
FDM
SFDM
FDMTDM
)1(2
)1(2
2),12
(1)1(2
Service time
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Reservations & Polling
Satellite
Collision -> solution:polling or reservation
S1 D1 D1 S2 D2
Cycle
S1 D1 D1 S2 D2…
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Reservation & Polling M Poisson traffic streams with rate /m1. Gated System – only those packets
which arrive prior to the user’s preceding reservation period are transmitted.
2. Exhaustive system – all packets are transmitted including those that arrive during this data period
3. Partially gated – all packets that arrive up to the beginning of the data interval.
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Single-User
Gated system:
S D D S D D … D
m=1
time
Di arrives Di startsWi
Ri
Vl(I)
Di ends tx
l(i)-th reservation intervalNi : # of packets arrive in front of Di
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Single-User A reservation(vacation)starts when the
system has served all packets which arrive prior to the preceding reservation interval.
A vacation(M/G/1 queue with vacation) starts when the system has served all packets which have arrived.(corresponds to exhaustive system)
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Single-User
WXWEXENE
WNEN
VE
VEXER
itake
RE
i
i
i
)()()(
)(
)(2
)()1(
2
)(
)(
22
與以前一樣
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Single-User
exhaustiveVE
VEXEW
VE
VE
VEXEW
)(2
)(
)1(2
)(
)61.3(1
)(
)(2
)(
)1(2
)(
22
22
Single-user
gated
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Multi-User
Ni is redefined as # of packets which must be transmitted before packet i
S D D D … S D D time
Packet i arrives Packet i startsWi
Ri
Ni
Pakcet i ends
… S D D
Sum=Yi
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Multi-user
Where Yi : includes all reservation intervals packet I must want for.
)63.3()()()()()( iiii YEXENEREWE
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Multi-User If number the users 0, 1, 2,…,m-1, the
l-th reservation interval is used to make reservation for user l mod m
1
0
1
0
22
)(2
)()1(
2
)(m
ll
m
ll
VE
VEXE
R
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Multi-user
?),65.3(1
YYR
W
YWRW
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Multi-user For an exhaustive systemLet lj=E ( Yj | packet i arrives in user l’s
reservation or data intervals and belongs to user (l+j) mod m)
0,...
0,0
mod)(mod)2(mod)1( jVVV
j
mjlmlml
Packet i belongs to each user with same prob. = 1/m
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Multi-user
)66.3(
...
1
1
1
1
mod)(
mod)1(mod)2(mod)1(
mod)2(mod)1(
mod)1(
1
1
1
1
m
j
mjl
mmlmlml
mlml
ml
m
jlj
m
jlj
Vm
jm
VVV
VV
V
m
m
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Multi-user
All users have equal average data length in steady state.
P(packet i arrives during user l’s data interval)
P(packet i arrives during user l’s reservation interval)
m
1
0
)1( m
k
k
l
V
V
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Multi-user
1
1
m
j
E (Yi|pkt i arrives in user l’s data or reser. int.) X P(pkt i arrives in user l’s reser. Or data int. )
1
0
1
0
2
1
0
1
11
0
mod)(
1
)68.3(2
)1(
2
)(
))1(
(
m
l
l
m
ll
m
l
m
jm
k
k
l
mjl
Vm
Vwhere
Vm
VVm
V
V
mV
m
jm
![Page 61: Chapter 3 Delay models in Data Networks. Section 3.2 Little`s Theorem](https://reader033.vdocuments.us/reader033/viewer/2022061609/56649d565503460f94a34d19/html5/thumbnails/61.jpg)
Multi-user
If Vl’s have same dist.
m
VVwhere
V
VmXW
m
l
ll
V
V
1
0
2
2
22
)(
2)1(2
)(
)1(2
2
Exhaustive system(3.69)
![Page 62: Chapter 3 Delay models in Data Networks. Section 3.2 Little`s Theorem](https://reader033.vdocuments.us/reader033/viewer/2022061609/56649d565503460f94a34d19/html5/thumbnails/62.jpg)
- The partially gated system is the same as the exhaustive system except that if a packet arrives in its own user’s data interval (with prob. /m), it is delayed an extra cycle of reservation periods(mV)
Y is increased by
Multi-user
Vm
Vmm
0)1(
![Page 63: Chapter 3 Delay models in Data Networks. Section 3.2 Little`s Theorem](https://reader033.vdocuments.us/reader033/viewer/2022061609/56649d565503460f94a34d19/html5/thumbnails/63.jpg)
Multi-user- The fully gated system is the same as
partially gated system except if a pkt arrives during a user’s own reservation interval (prob. (1-)/m)
- It is delayed by an additional mV
- Y is increased by Vmm
)1
(
![Page 64: Chapter 3 Delay models in Data Networks. Section 3.2 Little`s Theorem](https://reader033.vdocuments.us/reader033/viewer/2022061609/56649d565503460f94a34d19/html5/thumbnails/64.jpg)
Priority Queuing N classes of customers class i
arrives a Poisson process with rate I
service time
Each class joins a separate queue
2,
1i
i
i Xu
X
1
2
Server
![Page 65: Chapter 3 Delay models in Data Networks. Section 3.2 Little`s Theorem](https://reader033.vdocuments.us/reader033/viewer/2022061609/56649d565503460f94a34d19/html5/thumbnails/65.jpg)
Priority Queuing Single server will server customers from
the highest priority queue first1. Non-preemptive
- a lower priority customer, once started, is allowed to finish, when a high priority customer arrives.
2. Preemptive resume- Service for a low priority customer is interrupted when a high priority customer arrives and is resumed from the point of interruption when all higher priority customers have been served
![Page 66: Chapter 3 Delay models in Data Networks. Section 3.2 Little`s Theorem](https://reader033.vdocuments.us/reader033/viewer/2022061609/56649d565503460f94a34d19/html5/thumbnails/66.jpg)
Non-preemptive
Let NQk=avg. # in queue for priority k
Wk= avg. queueing time for priority k k = k/k = system utilization for
priority k R = mean residual service time.
![Page 67: Chapter 3 Delay models in Data Networks. Section 3.2 Little`s Theorem](https://reader033.vdocuments.us/reader033/viewer/2022061609/56649d565503460f94a34d19/html5/thumbnails/67.jpg)
Non-preemptive
2112
2
1
12
11
111
11
111
1
1
WNNRW
RW
WRNRW
Q
Where 1W2 is the avg. # of higher priority customers that arrives while you are waiting
![Page 68: Chapter 3 Delay models in Data Networks. Section 3.2 Little`s Theorem](https://reader033.vdocuments.us/reader033/viewer/2022061609/56649d565503460f94a34d19/html5/thumbnails/68.jpg)
Non-preemptive
)...1)(...1(
)1)(1(
21121
2112
kkk
RW
RW
Similarly,
![Page 69: Chapter 3 Delay models in Data Networks. Section 3.2 Little`s Theorem](https://reader033.vdocuments.us/reader033/viewer/2022061609/56649d565503460f94a34d19/html5/thumbnails/69.jpg)
Non-preemptiveR=the residual time
2
1
)(2
1X
n
ii
Where =2nd moment of the service time avg. over all priority
2X
221
1 ... nn XX
![Page 70: Chapter 3 Delay models in Data Networks. Section 3.2 Little`s Theorem](https://reader033.vdocuments.us/reader033/viewer/2022061609/56649d565503460f94a34d19/html5/thumbnails/70.jpg)
Non-preemptive
)81.3()(2
1 2
1
XRn
ii
代入
)83.3(1
)82.3()...1)(...1(2 21121
2
1
kkk
kk
i
n
ii
k
WTand
XW
![Page 71: Chapter 3 Delay models in Data Networks. Section 3.2 Little`s Theorem](https://reader033.vdocuments.us/reader033/viewer/2022061609/56649d565503460f94a34d19/html5/thumbnails/71.jpg)
Preemptive
k
kT 1
Note that Tk will not be affected by customers from class k+1 to n
Unfinished work of Class 1 to k (A)
Work due to class 1to k-1 who arrives whenthis customer is waiting (B)
)84.3(2
)1
(Re...1
)(
1
2
1
k
iii
k
k
k
XRwhere
RWcall
RA
![Page 72: Chapter 3 Delay models in Data Networks. Section 3.2 Little`s Theorem](https://reader033.vdocuments.us/reader033/viewer/2022061609/56649d565503460f94a34d19/html5/thumbnails/72.jpg)
Preemptive
)86.3()1(
)1(1
,1
)87.3()...1)(...1(
)...1(1
...1
1
1)(
1
111
1
111
11
1
11
11
RTkfor
RT
TR
T
TTB
kk
kkk
k
k
iki
k
k
kk
k
iki
k
iki
i