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Model of the Nodes in the Packet Network Chapter 10. Queuing system. Kendall ’s notation (1). Classification of queuing system depends on: Structure: number of servers Arrival stream: interarrival time distribution Service stream:service time distribution - PowerPoint PPT Presentation
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Maciej Stasiak, Mariusz GłąbowskiArkadiusz Wiśniewski, Piotr
Zwierzykowski
Model of the Nodes in the Packet Network
Chapter 10
2
Queuing system
3
Kendall’s notation (1)• Classification of queuing system depends on:
o Structure: number of serverso Arrival stream: interarrival time distribution o Service stream: service time distribution o Queue: queue capacity, queuing
strategy
• Kendall’s notation: A / B / N / K / So A: interarrival time distribution o B: service time distribution o N: number of serverso K: queue capacity (number of waiting positions) o S: number of traffic sources (population size)
4
Kendall’s notation (2)• A / B / N / K / S• Interarrival (service) time distribution (example of
standard notation)o M: Markovian, i.e. exponential distribution of
interarrival (service) time;o D: Deterministic, i.e. constant time intervals; o G: General, i.e. arbitrary distribution of interarrival
(service) time. May include correlation; o GI: General Independent, i.e. arbitrary distribution
of interarrival (service) time without correlation;
o Ph: Phase type distribution of time intervals.
5
Kendall’s notation (3)• Service strategy (example of standard strategies)
o FCFS: First Come – First Served, i.e. ordered queue, waiting calls are serviced in successive order;
o LCFS : Last Come – First Served (also denoted as LIFO: Last In – First Out), i.e. reverse ordered queue, waiting calls are serviced in reverse successive order;
o SIRO: Service In Random Order (also denoted as RS: Random Selection), i.e. all waiting calls in the queue have the same probability of being chosen
for service;
6
Little’s Theorem• Basic system parameters:
o L the average number of calls in the system,o W mean holding time in the system per call,o Q the average number of calls in the queue,o T mean holding time in the queue per call.
L = l W
Q = l T
the average number of calls in the queue
= X the average call intensity
mean holding time in the queue per call
7
Little’s Theorem
• A(t) number of arrivals at the moment t,• B(t) number of calls outgoing from the system at the moment t,• Z(t) =A(t) - B(t) number of calls serviced in the system at the moment t,• ti holding time of call i, serviced in the system.
• Arrival and departure process in the queuing system
A(t)
B(t)
t1 t5
A(t), B(t)
t1234567
8
Little’s Theorem• Average number of calls serviced in the system
within the period (0,τ):
• Mean number of arrivals within the period (0,τ):
• Mean holding time of a call in the system:
Wti
i )(1l
l
ò
lll
0
)(1
11
)(1
WttdttZLi
ii
i
9
One server delay system with infinite queue M/M/1/∞
• One server available for any call if it is not busy,• Poisson arrival process with average intensity l,
o exponential service time with mean value 1/μ ,o Calls are waiting according to basic service strategy FIFO
(first in first out). • The queue is infinite. It means that carried traffic is
equal to offered traffic and calls are not blocked.
QUEUE = ∞
output streaminput stream (λ)
SERVER
μ
10
M/M/1/ ∞ system• State transition diagram
· state „0” - the server is free, · state „1” - one call is served, no call is waiting in the queue, · state „2” - one call is served and one call is waiting in the queue, · . . ., · state „n” - one call is served and n-1 calls are waiting in the queue. · . . .,
M/M/1/ ∞ system - Analysis• State transition diagram of M/M/1/∞ delay system
• Local balance equation for the M/M/1/∞ system
11
···
0
1
21
10
.1
,,
,,
kk
NN
P
PP
PPPP
l
ll
),1(1
11
),1(
112
0
0
AA
AAAP
AAPAP
k
kkk
solution
12
M/M/1/ ∞ system - characteristics• Average number of calls in the system:
• Mean holding time in the system per call (Little’s Theorem):
AAAkAkPL
k
k
kk
1)1(
11
.1 A)(AW
l
13
M/M/1/ ∞ system - characteristics• Average number of calls in the queue:
• Mean holding time in the queue per call (Little’s Theorem):
AAALPLPLLLQ
kkbusy
1)1(1
2
10
A)(AT
1
2
l
M/M/1/ ∞ system - characteristics• Average number of calls in the system – formula
derivation:
14
AAAkAkPL
k
k
kk
1)1(
11
1
1
1
1
11
)1()1(
)1()1(
k
k
kk
k
k
k
k
kk
dA
AdAA
dAdAAA
kAAAAkAkPL
21
)1(11AdA
AAd
dA
Adk
k
15
System with finite queue: M/M/1/N-1 system
• State transition diagram for M/M/1/N-1 system
QUEUE = N-1
output streaminput stream (λ)
SERVER=1
μ
16
M/M/1/N-1 system analysis
• Local balance equation for the system M/M/1/N
111
NN
N AAAPE
···
0
1
21
10
.1
,,
,,
kk
NN
P
PP
PPPP
l
ll
PAP kk
1
120 1
11
NN
AAAAAP
solution
17
System M/M/1/N-1 results
M/M/N/∞ system• N servers available for any call if are not busy,• Poisson arrival process with average intensity l,
o exponential service time with mean value 1/μ ,o calls are waiting according to basic service strategy FIFO
(first in first out). • The queue is infinite. It means that carried traffic is
equal to offered traffic and calls are not blocked.
18
M/M/N/∞ system
19
N
∞QUEUE=output streaminput stream
SERVERS= N
20
M/M/N/∞ system• State transition diagram of M/M/N/ delay system
• Local balance equation for the M/M/N/ ∞ system
N
N
N N N
NN
·········
0
1
1
10
.1
,,
,,
,,
kk
mNmN
NN
P
PNP
PNP
PP
l
l
l
,!!!!
,for!
,for!
11
0
1
0
1
00
0
0
ANN
NA
iA
NA
NA
iAP
NkPNA
NA
NkPkA
P
NN
i
ii
i
NN
i
i
NkN
k
k
solution
21
M/M/N/∞ system: Erlang C-formula• State transition diagram of M/M/N/ ∞ system
• Erlang C-formula (probability that an arbitrary arriving call has to wait in the queue) AN
NNA
iA
ANN
NA
PAE NN
i
i
N
NkkN
!!
!)( 1
0
,2
N
N
N N N
NN
22
M/M/N/∞ system - characteristics• average number of calls in the queue:
• average number of calls in the system:
o where: Lbusy is average number of calls served in the system.
)]/(1[)(
)]/(1[1
!
)/(
)/(
!!
,22
1
0
11
01
11
01
NANAEA
NANNAP
NAd
NAd
NNAP
NAk
NNAPkPQ
NN
k
kN
k
kN
kkN
)]/(1[)(
1)( ,2
11 NANAE
AAQPNPkQLQL N
Nkk
N
kkbusy
M/M/N/∞ system - characteristics• mean holding time in the queue per call (Little’s
Theorem):
• mean holding time in the system per call (Little’s Theorem):
23
)]/(1[)(
11/ ,2
NANAE
LW N
l
)]/(1[)(1/ ,2
NANAE
QT N
l
M/M/N/∞ system - characteristics• M/M/N/∞ system connection with M/M/N/0 system
(Erlang formula for full availability group)
24
,)1/()](/1[
)1/(1)(,1
,2 aaAEaAE
NN
where a=A/N
25
M/M/N/m system• N servers available for any call if its are not busy,• Poisson arrival process with average intensity l,
o exponential service time with mean value 1/μ ,o calls are waiting according to basic service strategy FIFO
(first in first out). • The queue is finite, limited to m calls
26
M/M/N/m system
27
M/M/N/m system: system with infinite queue• Blocking/waiting probability in the system M/M/N/m
)1/()1()](/1[)1/()1(
,1
1
/// aaaAEaaB m
N
m
mNMM
Waiting probability as a function of the queue capacity in the M/M/5/m system.
µ0
0,2
0,4
0,6
0,8
1
1,2
0 1 2 3 4 5 6 7 8 9 10 11
m=0
m=1
m=2
m=5
m=10
m=#
A
B
28
M/G/1/∞ system – Assumptions• One server available for any call if it is not busy• Poisson arrival process with average intensity l• Any service time distribution with mean value 1/µ
and variance σ2τ• Calls are waiting according to FIFO strategy (first in
first out)• The queue is infinite. Carried traffic is equal to
offered traffic
29
M/G/1/∞ system • Pollaczek-Khinchine’s formula
o average number of calls in the system:
o mean holding time in the system per call :
)1(21 222
AAW
l
l
variance of service time distribution.
)1(2)1(22 222222
AAA
AAAL
ll
2
30
M/G/1/∞ system • Pollaczek-Khinchine’s formula with residual service
time:
• Where: is the second moment of service time distribution
)1(
12
)1/(2
0 AEATT
l
2E
Service time
Residual service time T0 t
t0
2
2
0lET
31
System M/D/1/∞ - Assumptions • One server available for any call if it is not busy,, • Poisson arrival process with average intensity l,• Constant service time distribution with mean value
1/µ ,• Calls are waiting according to FIFO strategy (first in
first out).
• Characteristics of the system M/D/1/∞o Service time is constant, so its variance is equal to zero:
0σ τ 2
21
11 A
AL
M/M/1/∞ and M/D/1/∞ systems comparison• Average number of calls in the system
32
0
2
4
6
8
10
0 0,2 0,4 0,6 0,8 1A
L
M/M/1
M/D/1
33
M/G/R PS system – Assumptions• Poisson arrival process with average intensity l• Any service time with mean value 1/μ• Available resources are fairly divided between packet
streams x offered to the system • All the offered streams are serviced quasi-
simultaneously• Number of servers is equal to R
34
M/G/R PS system• M/G/R PS – special case of M/M/N/∞ system• A service of particular packet streams corresponds to
the operation of mechanisms implemented in TCP protocol
• Aspiration for assurance of equal access to a shared transmission channel
• Convergence of models describing M/G/R PS and TCP• Model M/G/R-PS is conventionally used for packet
network dimensioning
35
System M/G/R PS• Number of servers
• where: o V - capacity of the server (link)o Rmax - maximum bit rate of the traffic stream
max/ rVR
36
System M/G/R PS• Average time spent by a task (call) in the M/G/R PS
system:
• where: o fR - delay factor,o x - average length of task (call) x, for example, data file,o ρ - intensity of offered traffic to one server (from among R):o K - number of users.
1()(
1)( ,2
RAE
rxf
rxxW R
R
)]/(1[)(
11/ ,2
NANAE
LW N
l
system M/M/N/µ
jjj
K
jjj rxR
1
1
1 ,/ l
37
System M/G/R PS• Average time spent by a task (call) in the M/G/R PS
system:
• where: o A - total offered traffic intensity:
o E2,R(A) – Erlang’s C formula:
1()(
1)( ,2
RAE
rxf
rxxW R
R
1
1
,2
!!
!)(R
i
Ri
R
R
ARR
RA
iA
ARR
RA
AE
RA
38
System M/G/R PS• Average time spent by a task (call) in the M/G/R PS
system with taking into account the delay in access link:
• Delay in the access link:
o where ρa is the traffic offered to access link with bit-rate equal to r:
1()(
11)( ,2
total RAE
rxf
rxxW R
a
aa r
xfrxxW
a
11)(
AAAW
111
)1( l
system M/M/1/µ
lrx
a
39
M/G/R PS system dimensioning1. Determination of the initial value of the link capacity
V=r.
2. Determination of the transmission delay W(x)=f (ftotal).
3. Do the obtained delay values exceed required threshold ?
1. YES – increase capacity and go to step 2.2. NO – required capacity has been reached.
4. Terminate calculation.
40
System M/M/1/m – buffer dimensioning
• The capacity m of the buffer for traffic with assumed QoS parameters can be determined on the basis of the acceptable level of loss packet probability E:
21
1 11
mm
m AAAPE 11
1
N
NN A
AAPE
system M/M/1/N1
41
System M/M/1/m – buffer dimensioning
• Approach 1o The capacity m of the buffer for traffic with assumed QoS
parameters can be determined on the basis of acceptable level of loss packet probability E:
• Approach 2o The capacity m of the buffer for traffic with assumed QoS
parameters can be determined on the basis of average number of packet in queue Q:
1)( mAmnPE
AAQm
1
2
42
Example – comparison of buffer dimensioning methods• ATM links (150 Mbits/s)
o traffic sources 1000 CBR sources (64 kbits/s)
o required ATM packet intensity 166 700 packet/so packet service time 2.830 µso offered traffic intensity 0.472 Erl.
o Determine required buffer capacity for
810E
24 3.70 10-9M/M/1/N
0.42 1 0.22M/M/1/ (2)
24 7.06 10-9M/M/1/ (1)
mBModel
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