1 Chapter 4. 2 Non-nearest neighbor MC’s Example 012 λ2λ μ 2μ.

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Transcript of 1 Chapter 4. 2 Non-nearest neighbor MC’s Example 012 λ2λ μ 2μ.

Page 1: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

1

Chapter 4

Page 2: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

2

Non-nearest neighbor MC’s

Example0 1 2

λ 2λ

μ 2μ

2

1

0

222

2

21

210

210

12

210

12

)(

2

1

22

2

P

P

P

PPP

PP

PPP

PPP

PP

PPP

Page 3: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

3

Erlang Distribution

μ

1 varienceoft coefficien

1

1

2

22

22

)x(

σc

μσ

μx

μs

μ(s)B

μeb(x)

b

b

b

*

μxr = 1

Page 4: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

4

Erlang Distribution

2

1 ,

2

1

2

1)

2

1(2

1

)2(2)(

)2

2(][)(

2

22

2

22

22

2

22)*(*

2

bb

μx

s

*μx

cc

μμσ

μx

eμxμxb

μs

μHsB

μs

μ(s)Hμeh(x)

b

r = 2

2μ2μ

Page 5: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

5

r-stage Erlang Distribution

rc

rc

rμrμrσ

rμrx

r

erμμrμxb

rμs

rμsB

μs

μ(s)Hμeh(x)

bb

rμμr

r

*μx

b

1 ,

1

1)

1(

1)

1(

)!1(

)()(

)()(

2

22

2

222

1

*

2

rμrμ rμ

1 2 r

Page 6: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

6

r = 1

2

b(x)

x

1

)1

()(lim

)1()1

1()(

0

*

xxb

er

s

rs

sB

r

srr

r = 1

2

b(x)

x

1

μe-μ r = ∞

Page 7: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

7

M/Er/1

(k, i)

λ departures

Queue1 2 r

rμ rμrμ

# in system

stage of service customer is in

0, 01, 1

1, 2

1, r

2, 1

2, 2

2, r

λ λ

γμ

γμ

γμ

γμ

γμ

γμ

3, 1

λ

λ

λ 3, r

γμ

Page 8: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

8

k-1, i

k-1, ik-1, i

k-1, i

k-1, i

γμ

γμ

λ λ

ystemcust. in s"owed" to f service # stages olet j

.-

)P(k,i)r(λ,i)λP(k)P(k,ir

)(dim1化為

11

計算較簡單

0 1

),(),( ,k

r

i

ikYZikPYZP解

r

ik ikP

1

),(

j-r

j-1 j j+1

λ

rμ rμ rμ rμ

λ λλ

j+r

rμ rμ rμ

λλ

0 1 2

rμ rμ rμ

r-1 r r+1

λ

λ λ

rμ rμ rμ

r+2

λ λ λλ

Page 9: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

9

,,, j λPrμμ)Pr(λ

, j PrλP

) )((

)( ,r,r, j λPrμμPr(λ

)( ,r-,, , j rμμPr(λ

, j PrλP

in system] of serv. P[j stagesLet P

)(r-i)r(k-j

rjjj

rjjj

jj

kr

)r(kjjk

j

21

0

32

31)

2121)

0

11

1

10

1

1

10

11

式合併

Page 10: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

10

rr

r

j

rjrj

r

j

jj

j

jj

j

jjj

λZZrμ

rμλ

]Z

[Pr

λZZrμ

rμλ

Pr]Zrμ

rμ[λPP(Z)

P(Z)λZZ]PP[P(Z)Z

rμ]P)[P(Z)r(λ

ZPλZZPZ

rμZP)r(λ

ZP(Z) and let P, for jif let P

11

00

010

100

11

11

1

0

Page 11: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

11

)(

)()1)(1()(1

111sin

0

0

ZD

ZN

ZZr

r

ZrZP

μ

λρ, where ρP

)P(Pce

r

jj

)Z

Z()

Z

Z)(

Z

Z--Z)((

)ZZλ(ZrμZ)(D(Z)

r

r

1111

][1

21

2

r,Z,,ZZZ

ll uniquer roots, a

21

r

i

jiij

n

i

r

inn

r

i

i

i

ZAPZP

ZZ

ZZ

AZP

1

,1i

1

)()1()(

1

1A ,

1)1()(

Page 12: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

12

Er/M/1

二維 or

一維 : j = stages of arrival completed by all customers in system

plus 〃 all customers in arrival box

0 1 2 r-1 r r+1

rλ rλ rλ rλ rλ rλ rλ

μ μ μ μμμ

j-r j-1 j j+1

μ μ μ μ

j+r

rλ rλ rλ rλ rλ rλ rλ

μ μ μ μ

departures

Queue

μ

1 2 r

rλ rλrλ

Arrival box

Page 13: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

13

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

)(

)(

1,j , )(

1-r,1,2,j ,

0j ,

0

1

10

1

1

1

11

1

11

1)1(

0

1

1

0

r

j

jjr

r

j

jj

j

jrjr

j

jj

r

j

jj

j

jj

kr

rkjjk

j

j

rjjj

rjjj

r

ZPZPZ

ZZPrZPPZPr

ZPZ

ZPZrZPZPr

PP

PjZZP

rrPPrPr

PPrPr

PPr

Page 14: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

14

1

0

0

1

IAppendix see

1

1

1

0

0j

)1)(1(

1)1(

1|| 1

1|| 1:. '

1)1()(

X where,1)1(

)1(

)(

1)1(1P since

r

j

jj

rr

rr

rr

r

j

jj

r

ZPK

ZZ

Z

ZrZr

Zinroots

ZinrootsrThmsRpuche

ZrZrZDLet

ZrZr

ZPZ

ZP

P

分母 (r+1) roots 中有一個根 at

Z=1 Im

Re

Z=1

Page 15: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

15

0

0

0

0

0

0

0

0

1

1

1

1

)1(

)1)(1(

)1

1)(1(

)(

)1)(1(

)1

1)(1(

)(

11

1)1(

)1)(1(

)1()(

ZZ

rZ

ZrZ

ZZ

Zr

ZZ

ZP

ZZ

Zr

ZZ

ZD

Z

rKP

ZZ

ZK

ZZP

r

r

r

r

01

01

0111

1

011

00

011

0

0

0

0

11

0

10

1

10

, k)Zρ(Z

, kρ PP

r)Zr(r rρris root of, Zμ

λ, where ρ

r , j)Zρ(Z

rj ) , Z(rP

j ,

) , jZ(r, where fffP

rkr

)r(k

rkjjk

jr

j

j

j

jrjjj

Page 16: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

16

Bulk Arrival System

M/M/1 Bulk Arrival Bulk size = r

與 M/Er/1 比較 , 把 M/Er/1 中的 rμ 改成 μ

ρ 改成r

0 1 2

μ μ μ

r-1 r r+1

λ

μ

λ λ

μ μ μ

r+2

λ λ λλ

μ

Page 17: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

17

註 : Bulk Service

M/M/1 Bulk Service Bulk size = r

與 Er/M/1 比較 , 把 Er/M/1 中的 rλ 改成 λ

ρ 改成

0 1 2 r-1 r r+1

λ λ λ λ λ λ λ

μ μ μ μμμ

r

Page 18: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

18

Bulk Arrival System

M/M/1 Bulk Arrival Bulk size = random gk = P[Bulk size = k]

與 M/M/1 Bulk Arrival, bulk size = k 比較

1

)(

k

kk Zg

ZG

0 1 2

μ μ μ

k-2 k-1 k

rgk-1

μ μ μ μ

k+1

μ

rgk-2rg2rg1

rgk

k+1

μ

rg1rg1rg1

rg2

rk

rkgk ,0

,1

Page 19: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

19

0 1

1 0 1

1

1

1

1

11

11

1

1

01

10

)(

321)(

0

j jk

jkjk

jj

k j jk

k

j

k

k

j

jjkjjk

k

kk

k

kk

k

jjjkkk

ZgZP

ZZPgZPZ

ZP

,,,, kPPPP

, kPP

)( 1

ZGZgi

ii

Page 20: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

20size]E[Bulk (1)G'

)1('

,1

1)1(

))(1()1(

)1()(

)()(

)1(

)(

)1(

)(

)()(

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

0

0

0

00100

100

G

where

P

PfromPFind

ZGZZ

ZPZP

ZZGZ

ZP

ZGZ

ZP

ZGZ

PPPZZP

ZGZPZPPZPZ

PZP

rBulk size b

P

ρZ

ρ

ZGZZ

ZPZP

)(M/M/ Bulk size acheck

kk

)

)1(

1

1

)(1()1(

)1()(

11 )

0

Page 21: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

21

Bulk Service System

M/M/1 Bulk Service Bulk size = r

與 Er/M/1 比較 , 把 Er/M/1 中的 rλ 改成 λ

ρ 改成

0 1 2 r-1 r r+1

λ λ λ λ λ λ λ

μ μ μ μμμ

r

Page 22: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

22

Suppose less than r customers can be served immediately (no need to wait until full bulk = r)

0 1 2 r-1 r r+1

λ λ λ λ λ λ λ

μ μ μ μμμ

μ

μ

Page 23: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

23

r

r

k

kkr

r

k

kkr

k

krkr

k

kk

k

kk

r

rkkk

ZZ

ZPZ

PZP

ZPZPZ

ZZPPZP

ZPZ

ZPZZP

PPPP

PPP

00

00

1

1

1

11

1

210

1

)()(

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

)(

0k , )(

1,2,k , )(

r

kkPPNote

00)( :

Page 24: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

24

)1)(1(

1)1(

1

)(

1|| 1

1|| 1:. '

1)1(

)()(Let

)(

)()(

0

10

0

10

10

ZZ

Z

ZrZr

Z

ZZPK

ZZZinroots

ZinrootsrThmsRpuche

ZrZr

ZZPZP

r

ZZ

ZZPZP

rr

r

k

rkk

rr

r

k

rkk

rr

r

k

rkk

kk

P

ZZP

ZZZ

ZP

ZZ

KZP

)1

)(1

1(

)1(

)1

1(

)(

)1(

1)(

00

0

01)1(

0

Page 25: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

25

r-stage Erlang dist (Er)

r = 1

2

b(x)

x

1

11

varof Coef r

.

rμrμ rμ

1 2 r

μ2μ1 μr

1 2 r

1C

2C

Page 26: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

26

Hyperexponential Dist (HR)

μ2

μ1

μR

α1

α2

αR

1)

1(

)1

(

)(

)(

)(

)1

(

1

)(

)(

1

2

1

2

2

2

2

22

1

2

1

1

1

*

1

ii

R

k kk

bb

R

k kk

R

k kk

R

k

xkk

R

k k

kk

R

ii

x

xx

xc

x

x

exb

SSB

k

2

2

Page 27: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

27

1

))(()(

))(()(

2

22

222

b

i

ii

i

i

i

ii

i

iiii

ii

c

b

aiLet

baba

,bfor a

nequalitySchwartz ICauchy

Page 28: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

28

Series-Parallel Stage-Type device

α1

αk

αR

r1μ1r1μ1 r1μ1

1 2r1

rkμkrkμk rkμk

1 2 rk

rRμRrRμR rRμR

1 2 rR

service facilityone customer at one time

iononal functneral ratiTotally ge

R

k

r

kk

kkk

k

rS

rSB

1

* )()(

Page 29: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

29

Coxian Stage-Type Device

β1

1-β1

μ1

β2

1-β2

μ2

βr

1-βr

μr

r

i j

ji

jiii S

SB1 1

121* )1(1)1()(

Page 30: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

30

Networks of Queues

Paul Burke(1954) Burke’s Theorem:

The only(FCFS) queuing systems which give Poisson out for Poisson in is M/M/1/n

PoissonPoisson?

Page 31: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

31

M/M/1: λ μ

departures

Queue

d(x)(S)D*

timestureinterdeparfor pdf thebe d(x)Let

Server

Queue

Cn Cn+1 Cn+2

Cn Cn+1 Cn+2

Cn-1 Cn Cn+1 Cn+2

Wn Wn+1 Wn+2=0

XnXn+1 Xn+2idle

Page 32: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

32

d(t)λeλS

λ

)λSμλ

μ(

μS

μλ)

μ

λ(

μS

μ

λS

λ

μ

λ

μS

μ(S)D

μS

μ

λS

λ(S)| D

leaveschen/after arrives wc

μS

μ(S)| D

leavesefore c arrives bc

λt

*

empty*

nn

empty non*

nn

11

1

1

1Case 1:

Case 2:

Page 33: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

33

Feed forward

λ

p

1-p

(1-p)λ

1

2

3

λ λ

(1-p)λ

4

network in the as same inpit thewith

M/M/man osolution t theis )(kp where

)()()()(),,,(

iii

443322114321 kpkpkpkpkkkkp

??M/M/mi

Page 34: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

34

J.R.Jackson(1957) = External arrival rate to node i (Poisson)

mi = Number of parallel servers in node i (Exponential) with mean service (1/μi) sec.

rij = P[node j next after node i] P[leave network after service in node i] = λi = Total traffic handled by node i

(sum of external + internal arrivals)

N

ijir

1

1

iiii

NNN

N

jijjii

t λ with inpi an M/M/molution to) is the s(kwhere p

)(kp)(k)p(kp),k,,kp(k

lution unique so rλ λ

221121

1

i

rij

i j

i

Page 35: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

35

Nji kkkkk ,,,,,,, 21

Nji kkkkk ,,,,,,, 1121

jiiii rmk

),min(

μPoisson 1-p

p

λ

λ

NOT Poisson

Poisson

p-1

,)1()( 1

,

wherekPp

p k

Page 36: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

36

K L

K L(L)

0(1963) force the system to always have K customers

Page 37: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

37

Gordon and Newell(1967) mi = # servers in node i (Exponential) with mean service (1/μi) sec.

rij = P[node j next after node i].

11

N

jjir

emd" in systrs "trappeK customekN

ii

1

utionunique solnot rλλN

jijjii

1

rij

i j

i

ii

ii

ii μ

λx let ior stationn factor futilizatio(relative)

μm

λlet P ,

Kk ii

ki

N

i

iimk

ii

iii

ii

i

i

ii

k

xKG

mk, mm

mk, ! kk

)( )(

!

)(

1

wherek

x

KG),k,,kp(k

ii

ki

N

iN

i

, )()( 1

21

on freedistributiek

xpGMM x

k

k ,!

)( //

Page 38: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

38

Closed Queuing Networks

,N,,, im

x

m

x

μm

λ

m

x ρ

μ

λx

i

i

ii

i

i

ii

i

ii

32

such that stationsReorder

1

1

1K

Result JACKSON332232

lim

,lim

k

)(kp)(k)p(kp),k,,kp(k ) is the i(kiwhere pNNNK

Bottlenecknode

Page 39: 1 Chapter 4. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ.

39ss)B(x) (cla

ss) (clar

s) (clasμ

smer classeForm custo ij

i

G ,SharingProcessor Robin -Round 4.

G (1),LCFS 3.

G servers, # 2.

lExponentia FCFS, 1.

MM

outPoisson inPoisson

(RRPS)

preemptive

K

K

SWAPPINGDEVICE

CPU

A

B

A

0

B

1

A

1

B

0

1

1

不同 class of job have different probability to go

terminal