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 linesStation 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)

Approximate Mean Value Analysis: The underlying ideasI. Model a CONWIP line with its WIP level set to W as a CQN with W jobsII. 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.

M1 M2 Mn

Notationn = 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]

(b) E[number of jobs at station Sj] WIPj(W-1)

(c) E[number of jobs in service] uj(W-1)

(d) uj(W-1) = TH(W-1) te(j)

u j (W 1)te ( j)(ce

2( j) 1)

2 (Kleinrock)

Deriving the algorithm iteration (cont.)Combining the results of the previous slide:

But then,

Obviously, for W=0, CT(0) =TH(0) = WIPj(0) = 0Furthermore, application of the above formulae for W=1 gives:

CT j (W ) te

2( j)

2(ce

2( j) 1)TH(W 1) (WIPj (W 1) 1)te ( j)

CT(W ) CTj (W )j1

n

TH(W ) W

CT(W )

WIPj (W ) TH(W )CTj (W )

CTj (1) te ( j)

CT(1) te ( j) To

j1

n

(Raw Process Time)

TH(1) 1

CT(1)

1

To

WIPj (1) 1

To

te ( j) u j (1)

(from Little’s law)

The W-TH(W) space

rb

TH(W)

W1 Wo=rbTo

1/To

1/To

Ideal Operational Point

Deriving the upper bounds for TH(W)

• For W, TH(W)rbminj{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:

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

Wo rbTo 1

tb

te ( j) u j (rb ) nj1

n

j1

n

TH(W ) W

To

Example: Attaining the upper bound for TH(W) with balanced, deterministically

paced line

W=Wo=rbTo=5

W=6

W=4

t1 t2 t3 t4 t5= = = = = 1.0

TH=rb=1

CT=To=5

TH=rb=1

CT=To+tb=6

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 non-idling policy that moves all W jobs as a single batch from station to station. Indeed, for that policy

CT(W ) W To and TH(W ) W

W To

1

To

Example: Attaining the lower bound for TH(W) through batching

T = 0

T = 6

T = 12

T = 18

t1 t2 t3 t4= = = = 2.0W=3

TH = W / (W To) = 3 / 24= 1 / 8

T = 24

The W-CT(W) Space

To

Wo1

CT(W)

W

To 1/rb

Ideal Operational Point

Bound derivation for the W-CT(W) space

Upper Bounds in W TW (W ) :

W Wo : TH(W ) W

To

CT(W ) W

TH(W )To

W Wo : TH(W ) rb CT(W ) W

rb

Lower Bounds in W TH(W ) :

TH(W ) 1

To

CT(W ) W

1 To

To W

The depicted bounds are derived from the bounds obtained for the W-TH(W) space through Little’s law, as follows:

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:

• A performance that is worse than the above is a strong indication of systematic mismanagement.

CT j (W ) W 1

nt t (1 W 1

n)t (from symmetry)

CT(W ) n(1W 1

n)t nt (W 1)t To

W 1

rb

(notice the linear dependence on W )

TH(W ) W

CT(W ) W

To (W 1) /rb

W

Wo /rb (W 1) /rb

W

Wo W 1rb

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

TH(W)

WWo1

TH

W

1/To1/To

rb

rb’

1/To’

1/To’

Wo’W’

Notice that

rb rb' tb tb

' To To' 1/To 1/To

'

Wo' rb

' te ( j) 1 rb' te ( j)

jb

j

1 rb te ( j)jb

Wo

W ' W

Increase rb (by adding capacity or making more effective use of the existing capacity at the line bottleneck(s))

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)

WWo1

TH

W

1/To1/To

rb=rb’

1/To’1/To’

Wo’W’

Notice that

j b : te ( j) te' ( j) To To

' 1/To 1/To'

rb rb'

Wo' 1 rb

' te' ( j) 1

jb

rb te ( j)jb

Wo

W ' W

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 CTW/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,

TH(W ') W '

CT ' W W

CT W

rb

W CT xW CT

1 x rb W CT

TH(W ) xTH(W )

1 x

rb

TH(W )

0.85rb 0.8rb x 0.8rb

1 x

rb

0.8rb

x 0.4166

i.e., the necessary increase is almost 42% of the original WIP!