Upload
pillan
View
44
Download
0
Embed Size (px)
DESCRIPTION
Generalized Jackson Networks (Approximate Decomposition Methods). The arrival and departure processes are approximated by renewal process and each station is analyzed individually as GI/G/m queue. - PowerPoint PPT Presentation
Citation preview
1
Generalized Jackson Networks(Approximate Decomposition Methods)
• The arrival and departure processes are approximated by renewal process and each station is analyzed individually as GI/G/m queue.
• The performance measures (E[N], E[W]) are approximated by 4 parameters . (Don’t need the exact distribution)
• Whitt, W. (1983a), “The Queuing Network Analyzer,” Bell System Technical Journal, 62, 2779-2815.
• Whitt, W. (1983b), “Performance of Queuing Network Analyzer,” Bell System Technical Journal, 62, 2817-2843.
• Bitran, G. and R. Morabito (1996), “Open queuing networks: optimization and performance evaluation models for discrete manufacturing systems,” Production and Operations Management, 5, 2, 163-193.
},,,{ 022
0 jjSa jjCC
2
Single-class GI/G/1 OQNS Notations:• n = number of internal stations in the network.
For each j, j = 1,…,n:• = expected external arrival rate at station j
[ ],• = squared coefficient of variation (scv) or variability of
external interarrival time at station j
{ = [Var(A0j)]/[E(A0j)2]},• = expected service rate at station i [j =1/E(Sj)]• = scv or variability of service time at station j
{ = [Var(Sj)]/[E(Sj)2]}.
j0
0 01/ [( )]j jE A
j
2
0 jaC
2
0 jaC
2
0 jSC2
0 jSC
3
Notations (cont)- For each pair (i,j), i = 1,…,n, j = 1,…,n:
= probability of a job going to station j after completing service at station i. *We assume no immediate feedback,
- For n nodes, the input consists of n2 + 4n numbers
• The complete decomposition is essentially described in 3 stepso Step 1. Analysis of interaction between stations of the
networks,o Step 2. Evaluation of performance measures at each
station,o Step 3. Evaluation of performance measures for the
whole network.
ijp
0iip
4
Step 1:• Determine two parameters for each stations j:
(i) the expected arrival rate j = 1/E[Aj] where Aj is the interarrival time at the station j.
(ii) the squared coefficient of variation (scv) or variability of interarrival time, = Var(Aj)/E(Aj)2.
For j
• can be obtained from the traffic rate equations:
for j = 1,…,n
where ij = piji is the expected arrival rate at station j
from station i. We also get = j/j, 0 1
j
n
iijjj
10
2
iaC
j j
5
Step 1: (cont.)• The expected (external) departure rate to station 0 from
station j is given by .
• Throughput = or
• The expected number of visits vj = E(Kj) = j/0.
The s.c.v. of arrival time can be approximated by Traffic variability equation (Whitt(1983b))
Where and
n
i jijj p10 1
0
n
j j1 0
n
j j1 0
n
ija
j
ijja wCwC
jj0
22 1
2
1
1 4(1 ) ( 1)jj j
wx
2
0
1j
n ij
ij
x
6
Step 1: (cont.)
= the interarrival time variability at station j from station i.
= the departure time variability at station j from i.
is interdeparture time variability from station j.
Note that if the arrival time and service processes are Poisson (i.e., = = 1), then is exact and leads to
= 1.
Note also that if j 1, then we obtain On the other hand if j 0, then we obtain
22222 )1(jjj ajSjd CCC
2
jiaC
2
jidC
jidjida pCpCCjjiji
1222
2
jdC
2
jSC2
jaC2
jdC
2
jdC
2
jdC 2
jSC2
jdC 2
jaC
7
Step 2:
• Find the expected waiting time by using Kraemer & Lagenbach-Belz formula [modified by Whitt(1983a)]
where if < 1
if 1
1//
22222222
][2
),,()(
)1(2
),,()()( MM
SajSa
jj
SajSaj
j WECCgCCCCgCC
WE jjjjjjjj
1
)(3
)1)(1(2exp
),,( 22
22
22
jj
j
jj Saj
aj
Saj CC
C
CCg
Step 3:• Expected lead time for an arbitrary job (waiting times +
service times) 1
( ) ( ( ) ( ))n
j j jj
E T v E W E S
2
jaC
2
jaC
8
Single-class GI/G/m OQNS with probabilistic routing
More general case where there are m identical machines at each station i.
for j = 1,…,n
Where
n
iijaja jj
CC1
22
n
iiiijijijajjj xqqpCpw
j1
220 ])1[()1(1
0
)1( 2iijijjij qpw
2
1
1 4(1 ) ( 1)jj j
wx
i
ijij
j
ijijij
j
iij pqpq
, and
i
Si
m
Cx i
)12.0,(max1
2
9
Single-class GI/G/m OQNS with probabilistic routing (cont.)
• Then the approximation of expected waiting time is similar to GI/G/1 case:
))//((2
)()(
22
jj
Sa
j mMMWECC
WE jj
10
General Job Shop (text sec. 7.7)
Rely on approximations for
These can be found from a sample path analysis of each machine center using
s.c.v. of interarrival times:
s.c.v. of service times:
Probability distribution of service times
1 1
and m m
i i ii iE N E N E T v E T
2
iaC2
iSC
iS iF t P S t
11
Single Machine, Poisson External Arrivals
From Chapter 3, if there is one server in each m/c center, the mean time spent at m/c center i by an arbitrary job can be approximated by using 4 parameters, as shown in equation 3.142, 3.143 and 3.144 in the text. If use
or
If use
22 ,,,ii Saii CC
][)1(2
(
1
)1(ˆ222
22
22
ii
Sia
Si
Sii SE
CC
C
CT ii
i
i
12 iaC
22 iaC
][)1(2
)2(
2
)1(ˆ222
2
22
ii
Siaii
Sii
Sii SE
CC
C
CT ii
i
i
][)1(2
)2(ˆ222
ii
Siaiii SE
CCT ii
(3.142)
(3.143)
(3.144)
12
Approximating Parameters
Four parameters can be approximated by solving linear system of equations below:
The first equation find . The second and third equation find s.c.v of arrival and departure time. Note that the first and second equations are the same as in Whitt(1983) while the third equation is different.
We’ll consider two extremes of job routing diversity.
22 ,,,ii Saii CC
i
m
ijjiai
i
ijidjijij
ia
Si
iaid
n
jjijii
miCpCppC
miCc
CC
mip
iji
iii
;1
222
22
222
1
,...,1)],1([)]1([1
,...,1),1()1)(1(1
,...,1,
13
Symmetric Job ShopA job leaving m/c center i is equally likely to go to any other
m/c next:
Then
So we can approximate each m/c center as M/G/1.
Then, can be replaced by mean waiting time in M/G/1 queue.
01 , ; 1 (leaves system)ij ip m j i p m
2
, 1,..., ;
1 as i
i
a
i m
C m
),,,(ˆ 22
ii saii CCW
)]1(2[
)1(
)1(2
][][
222
ii
Si
i
ii
iCSE
WE
14
Symmetric Job Shop
15
Uniform Flow Job ShopAll jobs visit the same sequence of machine centers. Then
In this case, M/G/1 is a bad approximation, but GI/G/1 works fine.
The waiting time of GI/G/1 queue can be approximated by using equation 3.142, 3.143 or 3.144 above and the lower and upper bound is
2
, 1,..., ;
but is unaffected by i
i
a
i m
C m
),,,(ˆ 22
ii saii CCW )1(2
)2( 222
i
Siaii iiCC
)1(2
)1( 222
i
Siiai iiCC
16
Uniform-Flow Shop
17
Job Routing Diversity
Assume high work load:and machine centers each with same number of machines,
same service time distribution and same utilization. Also,
Q: What job routing minimizes mean flow time?A1: If then uniform-flow job
routing minimizes mean flow time;A2: If then symmetric job routing
minimizes mean flow time.
1, 1,...,i i m
1, 1,...,iv i m
2 2 2 and 1a S SC C C
2 2 2 and 1a S SC C C
18
Variance of Flow Time
The mean flow time is useful information but not sufficient for setting due dates. The two distributions below have the same mean, but if the due date is set as shown, tardy jobs are a lot more likely under distribution B!
E[T]
A
B
Due
19
Flow Time of an Aggregate Job
Pretend we are following the progress of an arbitrary tagged job. If Ki is the number of times it visits m/c center i, then its flow time is (Ki is the r.v. for which vi is the mean)
Given the values of each Ki, the expected flow time is
and then, “unconditioning”,
th
1 1
, where is the time spent in on visitiKm
ij iji j
T T T i j
1
, 1,..., m
i i ii
E T K i m K E T
1 1
=m m
i i i ii i
E T E K E T v E T
20
Variance
In a similar way, we can find the variance of T conditional upon Ki and then uncondition.
Assume {Tij, i = 1,…,m; j = 1, …} and {Ki, i = 1,…,m} are all independent and that, for each i, {Tij, j = 1, …} are identically distributed.
Similar to conditional expectation, there is a relation for conditional variance:
YVar[ ] Var VarYX E X Y E X Y
21
Using Conditional Variance
Since T depends on {K1, K2, …, Km}, we can say
The first term equals
1 2
1 2
, ,..., 1
, ,..., 1
Var Var T ,...,
Var T ,...,
m
m
K K K m
K K K m
T E K K
E K K
1 2 1 2, ,..., 1 , ,...,1 1 1
1 1
Var ,..., Var
Var Var
i
m m
Km m
K K K ij m K K K i ii j i
m m
i i i ii i
E T K K E K T
E K T v T
22
Using Conditional Variance
The second term is
1 2 1 2, ,..., 1 , ,...,1 1 1
1 1
2
1 1
1
Var ,..., Var
Var 2 Cov ,
Var 2 Cov ,
Cov ,
i
m m
Km m
K K K ij m K K K i ii j i
m
i i i i j ji i j m
m
i i i j i ji i j m
m
i j i ji j
E T K K K E T
K E T K E T K E T
E T K E T E T K K
K K E T E T
1
m
23
Variance
The resulting formula for the variance is:
If arrivals to each m/c center are approximately Poisson, we can find Var[Ti] from the M/G/1 transform equation (3.73), p. 61.
But we still need Cov(Ki, Kj ).
1 1 1
Var Var + Cov ,m m m
i i i j i ji i j
T v T K K E T E T
2
22 2
2
0 0
, , Vari iT Ti i i i i
s s
dF s d F sE T E T T E T E T
ds ds
24
Markov Chain ModelThink of the tagged job as following a Markov chain with states 1,…,m for the machine centers and absorbing state 0 representing having left the job shop. The transition matrix is
1 12 11
2 21 21
1 21
1 0 0 ... 0
1 0 ...
1 0 ...
1 0
m
j mj
m
j mj
m
mj m mj
p p p
p p p
p p p
Ki is the number of times this M.C. visits state i before being absorbed;
let Kji be the number of visits given X0 = j.
25
with probability
with probability , l = 1,…,m
Where if j = i and 0 otherwise.
with probability
with probability , l =
1,…,m
jili
ji
ji KK
,
m
l jlp1
1
jlp
1ji
),)((
,
jrlrjili
jrji
jrji KKKK
m
l jlp1
1
jlp
)83.7(
)84.7(
26
Expectations
Take expectation of (7.83), we get
and take expectation of (7.84),
and
1
, 1,...,ji
j i mE K
I P
riKEKEKEKEKKE irjirijrjrji ],[][][][][
}1][2]{[][ 2 iijiji KEKEKE
27
1
m
i j jijE K E K
m
i jrjijri KEKEKKE1
][][][ Because
and , we obtain
and
where
Therefore
m
j jiji KEKE1
22 ][][
riKEKEKEKEKKE riririri ],[][][][][
}1][2]{[][ 2 iiii KEKEKE
[ ] [ ] [ ] [ ] [ ] [ ],
Cov[ , ][ ] 2 [ ] [ ] 1 ,
i ij j ji i j
i j
i ii i
E K E K E K E K E K E K i jK K
E K E K E K i j