Queueing theory primer - UC Davis: Networks Labnetworks.cs.ucdavis.edu/.../NDP14_QueueingTheory.pdf · Queueing theory primer Lecturer: Massimo Tornatore Original material prepared

Queueing theory

primer

Lecturer: Massimo Tornatore Original material prepared by:

Professor James S. Meditch

Typesetter: Dr. Anpeng Huang

Courtesy of: Prof. Biswanath Mukherjee 1

A change of focus

• So far we have investigated «static» problems

– Traffic requests are given and constant in time

• E.g., Multi commodity flow problem

• In general, mathematical programming, optimization and graph

theory, heuristics…

• Now we move to a class of dynamic problems

– Random or stochastic flow problems

– The times at which the demands arrive are uncertain

and also the size of the demands are unpredictable

• Queueing (in our case «traffic») theory

2

Source

• Notes taken mainly from – L. Kleinrock, Queueing Systems (Vol 1: Theory)

• Chapter 1 and 2

– L. Kleinrock, Queueing Systems (Vol. 2: Computer

Applications)

• Chapter 1

3

Delay and Congestion, why?

• The reason for this behaviour is the irregularity

(i.e, statistical distribution) of:

– Arrivals (i.e., interarrival times)

– Services (i.e., service times)

R C

If R>C, we expect congestion (intuitive)

If R<C, there might still be congestion (why?)

4

A. Notation and terminology

5

cn = customer n = arrival time of cn

tn = – = interarrival time →

wn = waiting time for cn →

xn = service time for cn →

sn = system time for cn →

sn = wn + xn

n

n

t~

1n

s~

x~

w~

*Note that these distributions do not depend by n (same distribution for all arrivals/services) 6

Arrival process Service process

kttE

ttE

dt

tdAta

ttPtA

])~

[(

1]

~[

)()(

]~

[)(

k

kxxE

xxE

dx

xdBxb

xxPxB

])~[(

1]~[

)()(

]~[)(

k

7

CDF

pdf

mean

kth moment

Laplace transform/moment generating fn

kk

sk

kk

xs

tsst

tds

sAdA

eEsB

tfEeEdtetasA

)1()(

)0(

][)(

)]([][)()(

0

*)(*

~*

~

0

*

8

Queueing system performance

Input variables: system defined by and

Output variables: performance defined by

1. N(t) no. of customers in system at time t

2. wn waiting time

3. sn system time

HP: (statistical equilibrium,

stationarity)

t~ x~

NtNt )(,

9

N w s

k

k

k

N

k

N

zPzQzE

NNE

kNPP

kNPkF

0

)(][

][

][

][)(

)(][

]~[

)()(

]~[)(

*~

sWeE

WwE

dy

ydWyw

ywPyW

ws

)(][

]~[

)()(

]~[)(

*~

sSeE

TsE

dy

ydSys

ysPyS

ss

10

Other performance variables:

I = idle period

D = interdeparture time

G = busy period

Nq = no. of customers in queue

11

Kendall’s notation for queueing systems

A/B/m A/B/m/K/M no. of users

no. of servers queue size

M exponential(Markovian) D deterministic

Er r-stage Erlangian G general

Hr R-stage hyperexponential

12

B. General results

1. Utilization factor

10

avg.) (on the usein capacity system offraction

workdo tosystem theofcapacity

arrivesork at which w rate avg.

C

R

13

G/G/1

• Let us start with no assumptions on arrival and

service distribution and one single server

• It can be generally proven that:

sec / service of rate avg.

sec / arrivals no. avg.

x

1

where

x

14

G/G/m

• In case of multiple (m) servers:

)10 wherefor except (

10 requiresStability

serversbusy offraction avg.

servereach for timeservice avg. 1

,

D/D/m

mm

x

15

2. System time

WxT

wExEsE

wxs

]~[]~[]~[

~~~

16

3. Little’s result

WNTN

TN

q

,

The average number of customers in a queueing system

is equal to the average arrival time of customers to that

system, times the average time spent in that system

17

t)(0, interval theduring system in the customers of no. avg.

t)(0, during arrived who

customers allover averagedcustomer per timesystem

)area!grey the(i.e, t time toup system in thespent have

customers all time totalsec)-(customer )()(0

t

t

t

N

T

dssNt

0(t)- (t)N(t)

t)(0,in departures no.)(

t)(0,in arrivals no.)(

t

t

18

N

)(

)()(Ncan But we

t)(0,in systemin customers no. Avg. )(

N

t)(0,in customer per timesystem Avg. erssec/custom )(

)(

t)(0,in rate arrival Avg. sec

)()(

t

t

t

tt

t

T

t

t

t

t

t

t

t

tT

customers

t

tt

19

m , ://

get also weand

)(

:tproven tha becan it Similarly,

(q.e.d) then exist, lim and lim If

mNNmGG

WxT

NNNxN

WN

TNTT

q

qs

s

q

ttt t

20

C. Poisson process

])0(in arrivals[ Pr

,1,0,!

)()(

1,0,1]

~[

d,distributelly exponentia

tindependen : timesalInterarriv

,tk

kek

ttP

ttettP

t

tk

k

t

i

Pure birth system (see p. 60-63, 65,Vol. 1) 21

1. Derivation of the Poisson process

rate) (arrivalintensity process

t)o(t1

]in arrival 1Pr[1]in arrival 0Pr[

t)o(t]in arrival 1Pr[

ΔtΔt

Δt

)()()()()(

)()(]1)[()(

1

1

tottPttPtPttP

tottPttPttP

kkkk

kkk

22

1)0( that note )()(

)2(

)(]1)[()(

,2,1 )()()(

)1(

000

00

1

PtPdt

tdP

tottPttP

ktPtPdt

tdPkk

k

If I divide by «dt», I obtain:

Similarly for P0:

Solving the the first-order differential equation (2):

then inserting P0(t) in (1) for k=1:

then continuing by induction:

tetP )(0

0,0,!

)()( tke

k

ttP t

k

k

ttetP )(1

]~

[1)( )!on!distributi neg. exp.with (relation note Final* 0 ttPetP t

23

2. Properties of the Poisson Process

2

2

1

)1(

1

)1(

0

)(

2

)(

0

1

1

)s!(memoryles ddistributelly exponentia are timesalInterarriv

)(

1]~

[ iv.

),(

][)( iii.

ii.

rate arrival avg.

)()( i.

t

eta

ettPA(t)

tetdz

tzdQ

ezEztPQ(z,t)

t

ttkPtN

t

t

z

zt

z

zt

k

tNk

k

tN

k

k

24

Random process

• Poisson Process is a stochastic (or

random) process

• Now we will use some more advanced

stochastic processes (Markov chains)

• But… what is a stochastic process?

– It is «family» of random variables X(t)

indexed by time t

– A possible (intuitive) case is when the

random process represents the sum of

simple random variables at instant t 25

Example of a random process

S1(t), S2(t) S3(t)…. are

random variables

X(t) is random process

X(t,w) = Number of servers

busy at time t of realization

w of the process= one

realization of the process

Assumptions

Stationarity

• E[to,to+][X(t, w)] = A(t0, w)

= A (w)= A(w)

Ergodicity

• A(w) = A

S1

S2

S3

S4

X(t)

4

3

2

1

0

´

´

´

´

´

´

´

´

H 1

H 2

H 3

H 3

H 6 H 1

H 5 H 4

H 5 H 2

H 4 H 8

H 7

t 0

H 7

H 3 H 4 H 7 H 6 H 5

H 6 H 7 H 8

H 8

t 0 + t

26

D. Markov chains

• A random process is called a “chain” if the state space

is discrete (i.e., if X(t) can assume only discrete

values)

– Note: S = {0, 1, . . ., K} finite state

S = {0, 1, . . .} infinite state (countable)

• A chain can be

– Discrete-time

• If t can assume only discrete values, i.e., if X(t) can change values on at

discrete instants in time

– Continuous-time

• If t can assume any continuous value

• A chain is a Markov chain if … see next slide

27

D.1. Discrete-time Markov chains

][

],,,[

11

001111

nnn

nnn

iXjXP

iXiXiXjXP

28

NNSSXn },...,,1,0{,1) It’s a chain since:

3) It’s a Markov

Chain since:

2) It’s a discrete-time chain since the time («x-axis») is slotted

State probabilities

i

(n)

i

)(

1

)(

0

)()(

1

][ ][ define usLet

nnn

n

n

i iXP

(One-step) transition probabilities

j

n

ij

n

ij

n

nn

n

ij

niPPP

iXjXPP

, 1 ][

][ define usLet

)()1()1(

1

)1(

29

State equations

given ,2,1 )0(

)1()1()(

n

P nnn

Homogeneous chain

j

ijij

nnij

ij

n

ij

iPPP

niXjXPP

nPP

each for 1 ][

][

timeoft independen i.e, ,

1

)(

30

1

)state"steady (" iesprobabilit mequilibriu

thenexists, lim If

1 given

,2,1

)(

i

)(

i

)0(

)0()(

)1()(

i

i

n

n

nn

nn

P

P

nP

n

31

Proof of π(n) = π(n-1) P

32

Proof- Contd.

...

...]......[

......

2

1

0

)1()1(

2

)1(

1

)1(

0

)1(

2

)1(

21

)1(

10

)1(

0

)(

si

i

i

i

n

s

nnn

si

n

s

i

n

i

n

i

nn

i

P

P

P

P

P

PPP

i-th column of P

33

Example

Discrete-time birth-death (arrival-departure) process with no

waiting room.

P(1 arrival at any time n) = a

P(1 departure at any time n) = d

s = {0,1} π = [π0 π1]

34

Example – Contd.

101

100

)1()(

)1(

)1(

1

1

da

da

Pn

P

dd

aaP

nn

35

Example – Contd.

da

a

da

d

d

a

d

aad

10

010

0101

1)1(1

36

D.2. Continuous-time Markov chains

NNSStX },...,,1,0{,)(

37

1) It’s a chain since:

3) It’s a Markov Chain since:

2) It’s a continuos-time chain since the times t can assume any real value

])()([])(...,,)()([

...0.

1111

121

nnnnnn

nn

itXjtXPitXitXjtXP

ttttanyandnallForDef

t1 t2 t3…..

Finite or

countable

Real

Time spent in a state

r.v. geometric a is state ain

spent Chain time Markov time-discrete ain :Note

exp.) (negative 1][

..

Chain Markov time-continuos ofProperty

t

i

i

ietP

withvraisSEstateintime

38

Neg. Exp. Is «Memoryless»

39

tT

ee

ee

tT

tTttT

tT

ttTttTttT

t

t

ttt

Pr

111

11

Pr

PrPr

Pr

Pr Pr

0

00

0

00

0

0000

Pr T t t 0 T t0 Pr T t

t 0

et

e t t0

D.2.a. General Case

D.2.a.1. Transition probabilities

ieachfortq

jit

tttptq

t

tttptq

matrixratetransitiont

ItPtQ

tttptP

stisXjtXPtsp

j

ij

ij

ij

iiii

ij

ij

0)(

),(lim)(

1),(lim)(

)(lim)(

)],([)(

])()([),(

If these limits do not exist, we do not have a continuous-time Markov chain

Δt→0

Δt→0

Δt→0

40

D.2.a.2 State Probabilities

t

j

j

N

j

dssQt

tgiven

tQtdt

td

Ntttt

SjjtXPt

0

10

])(exp[)0()(

1)()0(

)()()(

)],(...)()([)(

])([)(

41

D.2.b. Homogenous case

tQ

ij

ijij

ijij

etQtdt

td

qQ

constqtq

sofindeptssptp

)0()()()(

][

.)(

.),()(

42

Homogenous case – Contd.

1

0

:)0(

)(lim

)(

j

jj

Q

oftindependen

t

existsitifstateSteady

t→∞

43

Models used in this course Birth-death processes

• We want to apply the previously seen random processes

to model queues in telecom equipment

• The state changes only

– When a packet arrives (birth)

– When a packet leaves (death)

• Basically you can only move between adjacent states

– State k State k+1 (birth)

– State k State k-1 (death)

• Continuous time

– Packet arrive and depart at any time!

• Mathematical treatment follows in the next slides

44

E. Continuous-time birth-death processes aka arrival-departure processes

(1) Continuous-time Markov chain dealing with a population

of size N at time t

LLS

SkktNPtPk

}...,,1,0{

])([)(

(2) System state changes by at most one (up or down, or no

change) in Δt

45

1j-k if 0,

,1

,1

jk

kk

kkk

p

pdt

pdt

(3) Births and deaths independent

– Follows from Markovianity

(4) Transitions

ratedeath

totktNtttindexactlyP

totktNtttindexactlyP

ratebirth

totktNtttinbexactlyP

totktNtttinbexactlyP

k

k

k

k

k

k

)(1])(),(0[

)(])(),(1[

)(1])(),(0[

)(])(),(1[

Contd.

46

Transitions - State Eqns.

)(1)( )3

0 )()()()()( )2

1 )()(

)()()()()( )1

0

1010000

,11

,11,

LtP

ktptPtptPttP

ktptP

tptPtptPttP

L

k

k

kkk

kkkkkkk

47

- 3 (set of) equations regulate the birth-death process

State Eqns. – Contd.

)]()[()](1)[()(

)]()[()]()[(

)](1)][(1)[()(

11000

1111

tottPtottPttP

tottPtottP

tottottPttP

kkkk

kkkk

48

- Solving of previous equations

*Similar derivation as seen in the Poisson process

State Eqns. – Contd.

0

0)()()(

1

)()()()()(

11000

1111

t

ktPtPdt

tdP

k

tPtPtPdt

tdPkkkkkkk

k

49 *Solving these equations is quite complex, so, in practice, the approach used

is the one shown in the next slide (inspection over a state diagram)

State-Transition Rate Diagrams

tPtPE kkkkk 1111 into Flow

flowy probabilit ofNotion

tPE kkkk ofout Flow

... 1, 0,k

E ofout Flow - E into Flow

k11k11-k

kk

tPtPtP

dt

tdP

kkkk

k

Note: a simple «inspection» technique to find the same equations

50

1,2,...k

:

P Hence, 0dP

um)(equilibri balancedboundary across flows want weif

11

11

kk

k

k

kk

kkkk

k

PP

PP

Note

Ptdt

t

51

(Generic M/M/1) Queue

...,1,00)(

,

sec/

sec/

kdt

tdP

tbehaviorEquilibrum

jobsrateservice

jobsratearrival

k

k

k

52

System Representation

1

1

0 1

0

1

0 1

0

0

21

1020

1

01

0

1100

1111

1

1

...,,

1

00

1,)(0

)(;.1

k

k

i i

i

k

i i

ik

k

k

kkkkkkk

kkkk

p

pp

pppp

p

kpp

kppp

ptPanddependentState

53

Classical M/M/1

1

0

0

1

1

1

,,

k

k

k

k

kk

p

kpp

kall

54

k k+1

k-1

10 2

Classical M/M/1 – Contd.

0)1(

)1(1

10

1

11 ,1

0

01

kp

pp

If

k

k

k

k

k

k

k

k

55

2

2

0

22

20

00 0

1

0 0

)1(

)(

1

)1(1

1

)(

)1(

N

k

kN

k

k

k

k

k k

kk

k k

k

k

pNk

N

d

d

d

d

d

dkk

kpkN

56

Contd.

1

1

TTN

NW

WWxT

1

1

/

1

/11

57

Contd.

10

][

)1(][

1

2

iffvalidareresultsAbove

kNP

pkNP

NWN

k

ki

i

ki

i

qq

58

M/M/m Queue Model

1

m

mkm

mkkk

0

Equivalently,

59

Use state-dependent birth-death model to determine 0, kpk

320

3

13

210

2

12

10011

3!3

22

ppp

ppp

ppp

02

2

12

0

1

11

01

!

!

!

pmm

p

pmm

p

pm

p

m

m

m

m

m

m

1

0

11

0 1

11

0

1

1)()(

1

1

1

1

m

k

mk

m

k

mk

ρm!

mρ

k!

mρ

/mρρ

/mρm!

ρ

k!

ρp

mkpm

m

mkpk

m

pkm

k

k

0

0

!

0!

)(

60

M/M/m Queue system characteristics

mkpm

m

mkpk

mp

pkm

k

k

0

0

!

0!

)(

1

0

0

1

1)()(

1m

k

mk

ρm!

mρ

k!

mρp

mk

kpqueueingp ][

0)1(!

)(][ p

m

mmp

m

mNpN

pmN

smq

m

1

1

1 sq N

xN

WN

T

Where

mp

61

62

Backup slides

M/G/1 Queue

Poisson arrival process: customers/sec

General service process: kxxxB ,),(

queueing discipline is FCFS

1. Pollaczek-Khinchin (P-K) relations

)1(2

)1(2

1][][

22

2

22

xNWxN

WxTx

W

xxExxEx

63

Ex.

sec33.633.42

53.2

)3

13(4.08.0

sec33.4

3

13

)2.0(2

)3

13)(4.0(

)1(2

8.0)2)(4.0(

sec3

13

2

1

sec2

sec/4.0

2

2

3

1

22

TWxT

N

WN

W

xW

x

dxxx

x

cust

64

2. P-K transform relations

a.

zs

sx

k

k

k

k

sBzB

where

zzB

zzBzQ

dxexbxbLsB

zQZpzpzQ

|)()(

)(

)1)(1()()(

)()]([)(

)]([,)(

**

*

*

0

*

1

0

b.

)(yw prob. density fn. of waiting time

0)],([)(

)(

)1()(

)]([)(

*1

*

*

*

ysWLyw

sBs

ssW

ywLsW

)(ys prob. density fn. of system time

)]([)(

)()(

)1()(

)]([)(

*1

*

*

*

sSLys

sBsBs

ssS

ysLsS

65

yeys

s

s

ss

s

sss

ssS

)(

2

*

)()(

)1(

)(

)1(

)/(

)1()(

66

3. Modeling

a. Imbedded Markov Chain

nq no. customers left behind by nC

nv no. customers which enter during nx

,...1,0,

0

01

1

1

1

nq

qv

qvqq

n

nn

nnn

n

Cont. –time Markov chain

b. Tagged job

)1/(

)1(

)(

)(

RW

RW

WRWxRW

R

xWRW

Residual work (residual life time)

67

Residual Life - R

x

z

Random Arrival

Service process:

b x x x( ), , 2

F x P Z xZ ( ) [ | system busy at time of arrival]

r = E[Z | system busy]

Intuition: r =x

2 ? Wrong!

68

Chances are higher that x arrives during longer periods

f x density fn. for length of service period

during which arrival occurs, given system busy

f x x) Kx b(x) (x)

f (x) d(x) K x b(x) dx = kx = 1, K =1

x

f (x) = x b(x)

x

r = x

2

x b(x)

x d(x) =

x

2x

Z

Z

Z00

Z

0

2

( )

( ) (

Residual Life - Cont’d

69

R = E[Z | system busy] E[system busy]

=x

x . x =

1

2 x

W =

1

2x

x

22

22

2

1 2 1

( ) ( )

Markovian queueing Networks

N-node interconnection of queueing systems

queueing and service at each node

Exponential service times at each node

Model multi-service processes

Residual Life - Cont’d

70

1

3

4

2

r11

r12

r21

r31

r32

r22

r23

r44

r43

r34

1 1 43 44 ( )r r

1 31 32 34 ( )r r r 3

1. Open Networks

N nodes

Each node – single queue, servers

Exponential service time

mi

xi

i

1

sec

71

External arrivals – Poisson process (indep.) jobs/second

= P[that a job which completes service at node will proceed next

to node ]

P[that a job which completes service at node will leave

the network]

Avg. arrival rate at node from both external sources ands other

nodes (including itself)

i

rij ij

11

rijj

N

ii

i

rii

r i1

r i2

rNi

i

i

1

2i

N i i j jij

Nr

1

Flow Equations:

= [ 1 2 N

....... ]

[ ....... ]

[ ]

1 2 N

jiR r

R

72

Jackson’s theorem (1957)

Let p(k k k p[k jobs at node 1, k jobs at node 2, ....., k jobs at node N]

and

p k p[k jobs at node i]

Then

p(k k k p k p k p k

Each node in the network can be treated as an

M / M / m queue, m 1

1 2 N 1 2 N

i i i

1 2 N 1 1 2 2 N N

i i

, ....., )

( )

, ....., ) ( ) ( )........... ( )

73

1

2

N-1

N

……………

N = N ii=1

N

= ii=1

N

T =N

74

Example:

1 2 3

r12 r23

I/O I/O CPU

r11 r22

r11 02 . r12 08 .

r22 04 . r23 06 .

101

1 . sec

10 05

2 . sec

10 9

3 . sec

1 job / sec

75

Node 1

1 11 1 1 1

11

1

1

1

1 0 2 125

01251

01429

r . .

. . N1

Node 2

2 12 1 22 2 2 2

22

2

2

2

08 125 0 4 16667

0 08331

0 0909

r r . ( . ) . .

. . N2

Node 2

3 22 2

33

3

3

3

0 6 16667 1

0 91

0 9

9 2338

9 23

r . ( . )

. .

.

. sec

N

N = N + N + N

T =N

3

1 2 3

76

1

4

3

2 r12

r21

r14

r42

r22

r23

r33

r43

2. Closed Networks

N nodes; K jobs circulating through the network

No external arrivals or departures

Each node – single queue, servers Exponential service time mixi

i

1

sec

r24

r32

i ij

N

r

0 1

1 2

for each i

K K R = [r

(flow vector)

R

j=1

N

ii=1

N

ij ]

[ .......... ]

77

Gorjon and Newell (1967)

p(k k k 1

G(K)

x

k

{x satisfy

x x i = 1,2,....., N

G(K) =x

k

with k = (k ,k ,......., k ) and

A = set of all k vectors for which k + k +.....+k = k

(3) kk

1 2 Ni

k

i i

i

i i j

i

k

i ik

1 2 N

1 2 N

i i

i

i

, ....., )( )

( ) }

( )( )

( )

i

N

j jij

N

i

N

A

where

r

1

1

1

1

2

i i i

i i

k m

i i

ii

i

, k m

m m , k m

Utilization at node i: x

m i = 1,2, ..... , N

i i

!

!

78

Example: (Application-interactive computing)

Multi-Processor

.

.

.

.

.

.

. . . . . . . .

Node 1

Nodes 2,3, …….N

K terminals k jobs

N = avg. " think" time at each terminal (exp. distribution think times)

T =K

seconds

T = avg. responce time

1

79

),( M/M/1 2. kk

1k

1k

P

1,2,...k

P

mEquilibriu

k

k

P

P

0

k

0

0

2

12

01

1P

subject to

P

P

toleads This

k

k

k PP

PP

P

80