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