Transcript Modeling and Analysis of CONWIP-based Flowlines as Closed Queueing Networks

Modeling and Analysis of
CONWIP-based Flowlines as
Closed Queueing Networks
Topics
• Modeling CONWIP Flowlines as Closed
Queueing Networks
• Implications of the derived results for the
performance of the line
–
–
–
–
Performance bounds
Factors that affect the line performance
Strategies for enhancing the line performance
Explaining the “inertia” of CONWIP lines
CONWIP-based production lines
Station 1
Station 2
Station 3
FGI
Some important issues:
• What is the throughput attainable by a certain selection of
target WIP level?
• What is the resulting cycle time?
• How do we select the target WIP level that will attain a desired
production rate?
• How do the various operational detractors affect the
performance of this line?
CONWIP flowlines as
Closed Queueing Networks (CQNs)

M1

M2

Mn
Approximate Mean Value Analysis: The underlying ideas
I.
Model a CONWIP line with its WIP level set to W as a CQN with W jobs
II. The “PASTA” effect of CQNs with W jobs and exponential processing times:
The expected number of jobs observed at the various stations by a job arriving at
some station Sj, is equal to the expected number of jobs observed at any random
time at the same stations when the system is operated with W-1 jobs in it.
III. Assume that the “PASTA” effect holds in an approximate sense for general
distributions and develop an algorithm that will compute the performance
measures of interest iteratively, for various W levels, starting with W=0.
IV. The resulting method will work well for lines with single-server stations.
Notation
n = number of stations
te(j) = mean effective processing time at station j
ce2(j) = SCV for effective processing time at station j
TH(W) = the line throughput when operated with WIP level W
CT(W) = expected job cycle time through the line
CTj(W) = expected job cycle time at station j when the WIP level is W
WIPj(W) = expected WIP level at station j when the WIP level is W
uj(W) = utilization of the server at station j when the WIP level is W
Deriving the algorithm iteration
CTj(W) = E[remaining processing time for the job at the server of Sj] +
(E[number of jobs at station Sj]-E[number of jobs in service])te(j) +
te(j)
But
(a) E[remaining processing time for the job at the server of Sj] =
Prob(Server of Sj busy)E[remaining process time | busy] =
uj(W-1)E[remaining process time | busy] 
t e ( j)(c e2 ( j)  1)
u j (W 1)
(Kleinrock)
2
(b) E[number of jobs at station Sj]  WIPj(W-1)
 E[number of jobs in service]  uj(W-1)
(c)
(d) uj(W-1) = TH(W-1) te(j)
Deriving the algorithm iteration (cont.)
Combining the results of the previous slide:
t e2 ( j) 2
CTj (W ) 
(c e ( j) 1)TH(W 1)  (WIP j (W 1)  1)t e ( j)
2
But then,

n
CT(W )   CTj (W )
j1
W
CT(W )
WIP j (W )  TH(W )  CTj (W )
TH(W ) 
(from Little’s law)
Obviously, for W=0, CT(0) =TH(0) = WIPj(0) = 0
Furthermore, application of the above formulae for W=1 gives:

CTj (1)  t e ( j)
n
CT(1)   t e ( j)  To (Raw Process Time)
j1
1
1

CT(1) To
1
WIP j (1)  t e ( j)  u j (1)
To
TH(1) 
The W-TH(W) space
TH(W)
Ideal Operational Point
rb
1/To
1/To
1
Wo=rbTo
W
Deriving the upper bounds for TH(W)
• For W, TH(W)rbminj{1/te(j)}, the bottleneck rate of the line.
• rb can also be achieved with finite WIP in a deterministic setting, i.e., in a line
with ce(j) = 0, j, and synchronized with pace tb=1/rb.
• However, by Little’s law, a line with raw process time To, in order to produce at
rate rb, will need a WIP level of Wo=rbTo; this WIP level is known as critical WIP.
• An interpretation of Wo is given by the following formula:
n
1 n
W o  rb To   t e ( j)   u j (rb )  n
t b j1
j1
i.e., Wo is the level of WIP that we must maintain in the system in order to maintain
the bottleneck utilization at 100%. Otherwise, the bottleneck will starve.

If W<Wo, then in a deterministic setting we can pace the jobs through the system in
such a way that CT=To. Hence, the maximal line throughput will be
W
TH(W ) 
To
Example: Attaining the upper bound for
TH(W) with balanced, deterministically
paced line
t1
W=Wo=rbTo=5
=
t2
=
t3
=
t4
=
t5
= 1.0
TH=rb=1
CT=To=5
W=6
TH=rb=1
CT=To+tb=6
W=4
TH=W/CT
=4/5
CT=To=5
Deriving the lower bound for TH(W)
• Clearly, 1/To is a lower bound to TH(W) under global non-idleness, since this is
the rate of a line with only one job in it, and therefore, no parallelism.
• This bound is also achievable under any other finite WIP level W, by a nonidling policy that moves all W jobs as a single batch from station to station.
Indeed, for that policy
W
1
CT(W )  W  To and TH(W ) 

W  To To

Example: Attaining the lower bound for
TH(W) through batching
t1
W=3
=
t2
=
t3
T=0
T=6
T = 12
T = 18
T = 24
TH = W / (W To) = 3 / 24= 1 / 8
=
t4
= 2.0
The W-CT(W) Space
CT(W)
To
To
1/rb
Ideal Operational Point
1
Wo
W
Bound derivation for the W-CT(W)
space
The depicted bounds are derived from the bounds obtained for
the W-TH(W) space through Little’s law, as follows:
Upper Bounds in W  TW (W ) :
W
W
W  W o : TH(W ) 
 CT(W ) 
 To
To
TH(W )
W
W  W o : TH(W )  rb  CT(W ) 
rb
Lower Bounds in W  TH(W ) :
1
W
TH(W ) 
 CT(W ) 
 To  W
To
1 To
Practical Considerations
• The “ideal” performance is attained in an
optimized, deterministic setting.
• Usually, the actual performance of the line will be
compromised by
– the variability inherent in the system operation
– the impact of the applied control policies (e.g., the
batching policy that provides the lower bound for
TH(W))
A “benchmark” case
• In a COWIP line with
• Single-machine stations
•
•
Exponential processing times
te(j) = t, j
all feasible states are equiprobable.
• Hence, we have:
W 1
W 1
t  t  (1
)t (from symmetry)
n
n
W 1
W 1
CT(W )  n(1
)t  nt  (W 1)t  To 
(notice the linear dependence on W )
n
rb
W
W
W
W
TH(W ) 



rb
CT(W ) To  (W 1)/rb W o /rb  (W 1)/rb W o  W 1
CTj (W ) 

• A performance that is worse than the above is a strong
indication of systematic mismanagement.
Improving the System Performance
The problem: Given a line operating at a desired throughput rate, TH,
what are some possible mechanisms to reduce the expected cycle time
through the line, CT (and through Little’s law, the line WIP, W) ?
The key idea: We need to “pull” the curve describing the line
performance in the W-TH(W) space to the left.
Mechanism I
Increase rb (by adding capacity or making more effective use of
the existing capacity at the line bottleneck(s))
TH(W)
rb ’
Notice that
 rb  rb'  t b  t b'  To  To'  1/To 1/To'
1/To’
rb
TH
1/To’
1/To
 W o'  rb'  t e ( j)  1 rb'  t e ( j) 
j
jb
1 rb  t e ( j)  W o
1/To
jb
1
W’ Wo Wo’
W
W

 W ' W
Mechanism II
Add capacity to some non-bottleneck station(s) (this addition
essentially enables the better catering to the bottleneck needs, but it can
help only to a limited extent)
TH(W)
rb=rb’
TH
1/To’
1/To
Notice that
 j  b : t e ( j) t e' ( j)  To  To'  1/To 1/To'
1/To’
 rb  rb'
 W o'  1 rb'  t e' ( j)  1rb  t e ( j)  W o
1/To
1
Wo’W’Wo
W
W

jb
 W ' W
jb
Mechanisms III and IV
Reduce the inherent variability at the different stations; the
corresponding reduction of the station CVs will “pull” the
performance curve in the W-TH(W) space closer to the curve
characterizing the upper bound.
Increase the line flexibility, which essentially enables the better
utilization of the bottleneck capacity (and takes us back to item (i)
above).
Demonstrating the “inertia” of
CONWIP lines
Problem: Consider a CONWIP line operated at 80% of its bottleneck
rate. Furthermore, the performance of the line compares favorably to
that of the “benchmark” case, and W>>Wo so that CTW/rb.
Compute the relative increase W/W that will increase the line
throughput to 85% of its bottleneck rate.
Demonstrating the “inertia” of
CONWIP lines (cont.)
Solution: Let W=W’-W=xW. Then,
W'
W  W
W CT  xW CT TH(W )  xTH(W )
TH(W ') 




'
W
x
CT CT 
1 x rb W CT 
1 TH(W )
rb
rb
0.8rb  x  0.8rb
 0.85rb 

x
1 0.8rb
rb
 x  0.4166
i.e., the necessary increase is almost 42% of the original WIP!