1 chapter 4. 2 non-nearest neighbor mc’s example 012 λ2λ μ 2μ
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1
Chapter 4
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
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
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
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
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 = ∞
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
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
rμ
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
rμ
λ λ
rμ rμ rμ
r+2
λ λ λλ
rμ
9
,,, j λPrμμ)Pr(λ
, j PrλP
) )((
)( ,r,r, j λPrμμPr(λ
)( ,r-,, , j rμμPr(λ
, j PrλP
Pπ
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
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
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()(
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
rλ
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
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
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
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
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
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
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
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
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
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
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
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)( :
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
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
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
27
1
))(()(
))(()(
2
22
222
b
i
ii
i
i
i
ii
i
iiii
ii
c
b
aiLet
baba
,bfor a
nequalitySchwartz ICauchy
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
* )()(
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()(
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?
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
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:
33
Feed forward
λ
p
1-p
pλ
(1-p)λ
1
2
3
λ λ
pλ
(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
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
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
36
K L
K L(L)
0(1963) force the system to always have K customers
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 ,!
)( //
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
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
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