Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.
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Transcript of Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.
![Page 1: Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.](https://reader035.fdocument.pub/reader035/viewer/2022062313/56649d555503460f94a31cf8/html5/thumbnails/1.jpg)
Chapter 5
The Queue M/G/1
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2
M/G/1
Arrival
Service
GeneralPoisson
teta )(
1 srver
arbitraryxb )(
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3
Bus Paradox
If uniform
最少 0 分,最多 10 分;平均 5 分鐘
If Poisson(or Exponential)
人任意時刻來到 bus stop, 平均要 wait 多久 ?
(depends on bus 來的 distribution)
min 10xbusstop
10
1
10 minbus arrival
customerarrival
10 min
10 minbus arrival
customerarrival 10 min 10 min
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4
X: “special” interarrival time we pick Y: residual life(waiting time as above) Let f(x) be the pdf of interval lengths (mn = nth monent)
fx(x) be the pdf of the interval we randomly pick
f(x) be the pdf of residual life Y
X(life)
Y(residual life)age
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5
1
1010
00
0 x
)()()(
)(m )(
)(1)(
1)(f
n)(assumptio )()(
)( , )(
m
xxfxkxfxf
kmdxxxfkdxxxflet
dxxxfkdxxkxf
dxx
xkxfxf
xfxxf
x
x
x
成正比會跟
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61
**
1
*
1
11
1
)(1)(11)(ˆ
)(1)(ˆ
)(ˆ])(1[)(
)(
0for ],[]|[
],[
0 ,]|[
Sm
SF
S
SF
SmSF
m
yFyf
dyyfyFm
dydx
m
xxf
x
dy
dxm
xxf
x
dy
xydxxXxPdxxXxdyyYyP
dxxXxdyyYyP
xyx
yxXyYP
y
yyx
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7
1
2
1
1
n
n
2
using1
life residual theofmonent th -n is r
length internal theofmonent th -n is mLet
m
mtimeresidualmeanr
l rule) L'Hospita, ()m(n
mr
n
nn
102
,102
10m10
1
lExponentia
52
,10
10 ticDeterminis
:
1
21
22
1
1
212
2
1
m
mr
m
m
mr
m
m
example
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8
Method of Embedded(Imbedded) Markov
Chain
dk = P[departure leaves behind k in system]
rk = P[arrival finds k in system upon his arrival]
pk = P[k in system at random point of time]
time
departure instants
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9
Example: D/D/1
If arrivals are Poisson, then pk = rk
If system states change by ± 1, then dk = rk
For M/G/1 pk=rk=dk
Deterministic
time
N(t)
1
departure
p r d
,,, k
kx
k,
p
,,, k
k, r
,,, k
k, d
k
k
k
320
1,
01
210
01
210
01
x
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10
Proof
1. Let A(t, t + △ t) = arrival occurs in interval (t, t + △ t)
)()(
])([
],([
])([])(|),([lim
],([
)],( ,)([lim
)],(|)([lim
0
0
0
tPtR
ktNP
tttAP
ktNPktNtttAP
tttAP
tttAktNP
tttAktNP(t)P
time t] system atfinds k inP[arrival (t)R
e t]tem at timP[k in sys(t)P
kk
t
t
tk
k
k
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11
2.
rk=dkk k+1
balance
k customers
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12
M/G/1GeneralPoisson
kkk
t
drpSBxb
eta
)()(
)(*
1 srver
1,,, 2 xxx b 2
kk
kk
d, then rte by change staif system
r then ps Poisson,if input i
1
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13
x
x
m
mr
)m(n
mr
)life (r residual monent ofn
x
B(x)(x)b
(x)bife pdf residual l
nn
nth
22
1
1ˆ
ˆ
2
1
21
1
1
21
2
srvice]in him findshat t
arrivalan delay willservice
foundcustomer a that E[timeLet W0
xλρ)(Oρ
x
x
2 2
Delay 的時間有人被 served
沒被 serve 的機率
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14
Imbeded Markov Chain
Let cn be the nth customer to enter system
qn be the number of customers left behind by the departure of cn
time
cn-2 cn-1 cn cn+1
qn left behind
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15
]n x arrival iP[ P
]n x arrival iP[ P
me of cservice ti where x
]xal during P[no arriv, PGiven q
d,Pdd
i]j|qP[qNeed P
][PP],,,d[dd
imediscrete tte state, discreov Chain. is a Markq
)rp(dk]P[q
k]P[q
n
n
nn
nn
kk
nnij
ij
n
kkknn
n
102
101
100
0
1
10
2
1
0
1
lim
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16
Let vn = number of arrivals during xn
P[vn+1 = k] = αk
10
210
210
210
00
0
P
)()(
)(!
)(
)(]~|~[
]~[
~
*
0
)1(
0
0
ZBdxxbe
dxxbk
ex
dxxbxxkvP
kvP
vtate s, nndent of s is indepev
xZ
xk
k
n
![Page 17: Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.](https://reader035.fdocument.pub/reader035/viewer/2022062313/56649d555503460f94a31cf8/html5/thumbnails/17.jpg)
17
)(B
)(
)()ek!
x)((
))(!
)(()(
~
*
0
)1(
0
x-
0k
k
00
00
Z
dxxbe
dxxbZ
Zdxxbk
exZV
k]ZvP[ZαV(Z)Let α
xZ
k
k
kxk
k
k
k
kk
transformZ
k
xZe
![Page 18: Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.](https://reader035.fdocument.pub/reader035/viewer/2022062313/56649d555503460f94a31cf8/html5/thumbnails/18.jpg)
18
xv
xx
dZ
Zd
Zd
ZdB
dZ
dV(Z)v E[v]
)(B) v(Z
λZ):(λBcheck V(Z)
ZZ
*
*
)()(
)(
)(
)(
1011
1
*
1
arrival rate
mean service time
進入的個數
222)2()0(
*
1
2
2
*2
1
2
22
)(B
)(
)(
)()(
x
dZ
Zd
Zd
ZBd
dZ
ZVdvv
ZZ
xxv 222
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19
server
queue
cn cn+1
τn
cn cn+1
τn+1
cn cn+1
tn
xnxn+1
qn+1qn
vn+1
nnk
nnk
nn
nnn
nn
thn
y ct behind bs left lef# customerqd
during x arrivals # customervα
me f cservice tixb(x)
time)erarrival (ττta(t)
ime of c:arrival tτ
customer:nc
int1
![Page 20: Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.](https://reader035.fdocument.pub/reader035/viewer/2022062313/56649d555503460f94a31cf8/html5/thumbnails/20.jpg)
20
11
02
nn
n
vq
: qCase
server
queue
cn+1
cn cn+1
cn+1
xn+1
qn+1qn=0
vn+1
0
0
1
11
1 if qn v
if qvqq
,n
n, nn
n
0
01
111
1
nnn-n
n
n
n
, if qvqq
rve c
, departure c
: qCase
se 會被便馬上被只要
當第 cn+1 離開時發現
原來的 進來的
![Page 21: Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.](https://reader035.fdocument.pub/reader035/viewer/2022062313/56649d555503460f94a31cf8/html5/thumbnails/21.jpg)
21
ise, otherw
,,,, klet
0
3211Δk
k
△k
-2 -1 0 1 2 3
1
ρvs busy] P[server is busy]P[system i
]qP[]qP[]qP[]E[Δv
vv]vE[
dk]q: P[q vΔqq
)rd (Nkdkqq
vΔqq
vΔqq
kpkd
q
k
qn
kkk
nn
nqnn
nqnn
n
n
~
~
01
11
11
0~0~10~0
~~
~~~
lim
![Page 22: Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.](https://reader035.fdocument.pub/reader035/viewer/2022062313/56649d555503460f94a31cf8/html5/thumbnails/22.jpg)
22
)1(2
)1(2)1(2
2
222
222
222
) (
2
2222
)(2d222
222
112
122
1
112
1222
1
vNq
vvq
vqvqvqq
vqvqvqq
vqvqvqq
nletq
xvvdZ
ZV
qqtake n
nqnnnnqnn
nqqnnnnqnn
nn
nnn
趨於取期望值兩邊平方求
nq nq
![Page 23: Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.](https://reader035.fdocument.pub/reader035/viewer/2022062313/56649d555503460f94a31cf8/html5/thumbnails/23.jpg)
23
1
1
1
2/
)1(2
0
0
2
22
WW
WxT
xxT
TN
xNq
Pollaczek-Khinchin
MEAN-VALUE Formula
P-K Mean-Value Formula
![Page 24: Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.](https://reader035.fdocument.pub/reader035/viewer/2022062313/56649d555503460f94a31cf8/html5/thumbnails/24.jpg)
24][)()(
][][][)(
][][][
)(
][]~[)()(lim
][)(][
)(][
1
1
11
~
0
0
11
111
11
nqn
nnqnnnqn
nnqnnnqnn
nnqnn
n
n
q
n
vqvq
n
vqvqq
vqq
nqnn
kkk
q
k
kn
n
qn
k
kn
nn
ZEZVZQ
ZEZEZZEZQ
ZZEZEZE
ZZ
vqq
ZQdpr
ZEZkqPZQZQ
ZEZQZkqP
ZQkqP
kd
)(ZV
![Page 25: Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.](https://reader035.fdocument.pub/reader035/viewer/2022062313/56649d555503460f94a31cf8/html5/thumbnails/25.jpg)
25
Z
qPZQqPZVZQ
nlet
Z
qPZQqPZVZQ
qPZQZ
qP
kqPZqP
kqPZqPZZE
n
nnnn
nnn
kn
kn
kn
kkn
qnqn
]0~[)(]0~[)()(
1)(
]0[)(]0[)()(
]0[)(1
]0[
][]0[
][]0[][
1
1
1
1
0 0
![Page 26: Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.](https://reader035.fdocument.pub/reader035/viewer/2022062313/56649d555503460f94a31cf8/html5/thumbnails/26.jpg)
26
kdZZB
ZZBZQ
-ρty]ver P[empSingle ser
)Q(her ] from eitqP[
ZBZV
ZZV
qPZZV
ZZV
ZqP
ZV
)(
)1)(1()()(
1
110~
)()(
)(
]0~[)1()(
)(1
)1
1](0~[)(Q(Z) Q(Z),
**
*
解
Pollaczek-Khinchin Transform equ.
![Page 27: Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.](https://reader035.fdocument.pub/reader035/viewer/2022062313/56649d555503460f94a31cf8/html5/thumbnails/27.jpg)
27
example
kk
x
d
Z
ZZZ
Z
ZZZ
Z
ZZ
Z
ZZQ
eS
SB
)1(
1
1
)1()1(
)1)(1()1)(1(
)1)(1()(
)(
2
*
1// MM
![Page 28: Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.](https://reader035.fdocument.pub/reader035/viewer/2022062313/56649d555503460f94a31cf8/html5/thumbnails/28.jpg)
28
P-K Mean-
Value
Formula
1
2/
1
2/
1
2/
1
22
2
20
xNq
xxWxT
xWW
2
2
2
2
0x
xx
xW
time waiting)(yw
timesystem )(ys
kd
![Page 29: Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.](https://reader035.fdocument.pub/reader035/viewer/2022062313/56649d555503460f94a31cf8/html5/thumbnails/29.jpg)
29
example
1
1
)1
1(1
1212
1
2 ,
1
1
/1
2
22
T
xx
T
1// MM
)1(2
)1(
)1(
)()1(2
2
22
22
2
2222
2
b
b
cxW
xx
xx
xxxx
xW
![Page 30: Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.](https://reader035.fdocument.pub/reader035/viewer/2022062313/56649d555503460f94a31cf8/html5/thumbnails/30.jpg)
30
)-2(1
xW
0
:1//
1
-1
xW
1
:1//
2b
2b
c
DM
x
c
MM
02
1
1//
1//1//
DD
MMDM
W
WW
W
ρ
M/M/1
M/D/1
D/D/1
![Page 31: Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.](https://reader035.fdocument.pub/reader035/viewer/2022062313/56649d555503460f94a31cf8/html5/thumbnails/31.jpg)
31
)()(
)()(
FIFO) (assume
customer nfor timesystem:
)()(
*
*
th
*
ZSZQ
ZSZQ
s
ZBZV
n
n
n
server
queue
cn
cn
xn
vn::Poisson λcn
server
queue
cn
cn
sn
qn left behind
Poisson λcnfirst come first serve
![Page 32: Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.](https://reader035.fdocument.pub/reader035/viewer/2022062313/56649d555503460f94a31cf8/html5/thumbnails/32.jpg)
32
SZ
ZSLet
ZZB
ZZBZQZS
1
)(
)1)(1()()()(
***
)(
)1()()(
1)(
)1)(11()()(
***
*
**
SBS
SSBSS
SSB
S
SBSS
![Page 33: Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.](https://reader035.fdocument.pub/reader035/viewer/2022062313/56649d555503460f94a31cf8/html5/thumbnails/33.jpg)
33
)(
)1()(
)()()(
)()()(
~~~
**
***
SBS
SSW
SWSBSS
ywybys
wxs
wxs nnn
![Page 34: Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.](https://reader035.fdocument.pub/reader035/viewer/2022062313/56649d555503460f94a31cf8/html5/thumbnails/34.jpg)
34
example
yeys
SS
S
S
SSS
SSB
)1(
*
*
)1()(
)1(
)1()1()(
)(
1// MM
y
s(y)
ye )1()1(
![Page 35: Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.](https://reader035.fdocument.pub/reader035/viewer/2022062313/56649d555503460f94a31cf8/html5/thumbnails/35.jpg)
35
yeyuyw
SSW
S
S
SSS
SS
SS
SSW
)1(0
*
2*
)1()()1()(
)1(1)1()(
)1(
))(1()1)(()1()(
system idle
system busy
w(y)
ye )1()1( 1
y
![Page 36: Chapter 5 The Queue M/G/1. 2 M/G/1 Arrival Service GeneralPoisson 1 srver.](https://reader035.fdocument.pub/reader035/viewer/2022062313/56649d555503460f94a31cf8/html5/thumbnails/36.jpg)
36
0)(
0
*
*
*
****
)(ˆ)1()(
))(ˆ()1(
)(ˆ1
1)(
)(11
1
))(1
(
)1(
)(
)1()(
kk
k
k
k
ybyw
SB
SBSW
xSSB
xSB
xS
S
SBS
SSW
ice timeidual servpdf of res
(x)b(S)B* ˆˆ
convolved itself k times
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37
Stages of method
1. Series/Parallel
2. d = dP
3. Supplementary Variables
4. Imbedded Markov Chain
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38
U(t) = unfinished work in system at time t
= Time required to empty the system measured from time t, if no new customers are allowed to enter after tU(t)
t0 τ1τ1+x1
x1
x2
WAVG
Virtual Waiting Time (only for FCFS)
Busy Periody
Idle Period
I
WFCFS=WLCFS=WX