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