1. Analysis and Design of Asynchronous Transfer Lines as a series of G/G/m queues: Overview and Examples

2. Topics
Modeling the Asynchronous Transfer Line as a series of G/G/m queues
Modeling the impact of preemptive, non-destructive operational detractors
Employing the derived models in line diagnosis
Employing the derived models in line design
The role of batching in the considered manufacturing systems
An analysis of a workstation involving parallel batching

3. Asynchronous Transfer Lines (ATL)W1 TH B1 M1 W2 W3 TH TH B2 M2 B3 M3
Some important issues:
What is the maximum throughput that is sustainable through this line?
What is the expected cycle time through the line?
What is the expected WIP at the different stations of the line?
What is the expected utilization of the different machines?
How does the adopted batch size affect the performance of the line?
How do different detractors, like machine breakdowns, setups, and maintenance, affect the performance of the line?

4. The G/G/1 model: A single-stationB1 M1
Modeling Assumptions:
Part release rate = Target throughput rate = TH
Infinite Buffering Capacity
one server
Server mean processing time = te
St. deviation of processing time = e
Coefficient of variation (CV) of processing time: ce = e / te
Coefficient of variation of inter-arrival times = ca

5. An Important Stability ConditionTH B1 M1
Average workload brought to station per unit time:
TH·te
It must hold:
Otherwise, an infinite amount of WIP will pile up in front of the station.

6. Performance measures for a stable G/G/1 stationTH B1 M1
Server utilization:
Expected cycle time in the buffer: (Kingman’s approx.)
Expected cycle time in the station:
Average WIP in the buffer: (by Little’s law)
Average WIP in the station:
Squared CV of the inter-departure times:

7. Remarks
For a station with variable job inter-arrival and/or processing times, utilization must be strictly less than one in order to attain stable operation.
Furthermore, expected cycle times and WIP grow to very large values as u1.0.
Expected cycle times and WIP can also grow large due to high values of ca and/or ce; i.e., extensive variability in the job inter-arrival and/or processing times has a negative impact on the performance of the line.
In case that the job inter-arrival times are exponentially distributed, ca=1.0, and the resulting expression for CTq is exact (a result known as the Pollaczek-Kintchine formula).
The expression for cd2 characterizes the propagation of the station variability to the downstream part of the line, and it quantifies the dependence of this propagation upon the station utilization.

8. Performance measures for a stable G/G/m stationTH M2 Mm
Server utilization:
Expected cycle time in the buffer:
Expected cycle time in the station:
Average WIP in the buffer:
Average WIP in the station:
Squared CV of the inter-departure times:

9. Analyzing a multi-station ATLTH
Key observations:
A target production rate TH is achievable only if each station satisfies the stability requirement u < 1.0.
For a stable system, the average production rate of every station will be equal to TH.
For every pair of stations, the inter-departure times of the first constitute the inter-arrival times of the second.
Then, the entire line can be evaluated on a station by station basis, working from the first station to the last, and using the equations for the basic G/G/m model.

10. Operational detractors: A primal source for the line variability
Effective processing time = time that the part occupies the server
Effective processing time = Actual processing time +
any additional non-processing time
Actual processing time typically presents fairly low variability ( SCV < 1.0).
Non-processing time is due to detractors like machine breakdowns, setups, operator unavailability, lack of consumables, etc.
Detractors are distinguished to preemptive and non-preemptive. Each of these categories requires a different analytical treatment.

11. Preemptive non-destructive operational detractors
Outages that take place while the part is being processed.
Some typical examples:
machine breakdowns
lack of consumables
operator unavailability

12. Modeling the impact of preemptive detractors
X = random variable modeling the natural processing time (i.e., without the delays due to the detractors), following a general distribution.
to = E[X]; o2=Var[X]; co=o / to .
T = random variable modeling the effective processing time = where
Ui = random variable modeling the duration of the i-th outage, following a general distribution, and
N = random variable modeling the number of outages during a the processing of a single part.
mr=E[Ui]; r2=Var[Ui]; cr = r / mr
Time between outages is exponentially distributed with mean mf.
Availability A = mf / (mf+mr) = percentage of time the system is up.
Then,
te = E[T] = to / A or equivalently re = 1/te = A (1/to) = A ro

13. Breakdown Example Data: Injection molding machine has:
15 second stroke (to = 15 sec)
1 second standard deviation (so = 1 sec)
8 hour mean time to failure (mf = sec)
1 hour repair time (mr = 3600 sec)
Natural variability co = 1/15 = (which is very low)

14. Example Continued
Effective variability:
Which is very high!

15. Example Continued
Suppose through a preventive maintenance program, we can reduce mf to 8 min and mr to 1 min
(the same as before)
Which is low!

16. Example:employing the developed theory for diagnostic purposesB M2
to1 =19 min
co12=0.25
mf1=48 hrs
mr1=8 hrs
MTTR ~ expon.
to2 =22 min
co22=1.0
mf2=3.3 hrs
mr2=10 min
Ca2=1.0 20
parts
Desired throughput is TH = 2.4 jobs / hr but practical experience has shown that it is not attainable by this line. We need to understand why this is not possible.

17. Diagnostics example continued: Capacity analysis based on mean values

18. Diagnostics example continued: An analysis based on the G/G/m model
i.e., the long outages of M1, combined with the inadequate capacity of the interconnecting buffer, starve the bottleneck!

19. Lot Sizing
If affordable, a lot-for-lot (L4L) policy will incur the lowest inventory holding costs and it will maintain a smoother production flow.
Possible reasons for departure from a L4L policy:
High set up times and costs => need for serial process batching to control the capacity losses
Processes that require a large production volume in order to maintain a high utilization (e.g., fermentors, furnaces, etc.) => need for parallel process batching
Selection of a pertinent process batch size
It must be large enough to maintain feasibility of the production requirements
It must control the incurred
inventory holding costs, and/or
part delays (this is a measure of disruption to the production flow caused by batching)
Move or transfer batches: The quantities in which parts are moved between the successive processing stations.
They should be as small as possible to maintain a smooth process flow

20. Optimal Parallel Batching: A factory physics approach
Model Parameters:
k: (parallel) batch size B: maximum batch size
ra: arrival rate (parts/hr) ca: CV of inter-arrival times
t: batch processing time (hrs) ce: CV for effective batch processing time
Then
CT = WTBT + CTq+t
From the above,
Remark: Notice that CT as u1 but also as u0 !

21. Determining an optimized batch size
Let um  rat . Then u = um / k  k = um / u . Substituting this expression for k in the expression for CT, we get:
and we get
Recognizing that
, we set
where
To minimize CT, it suffices to minimize y(u). This can be achieved as follows:
and
which further implies that
Remark: If ce2  0, the term  in the original expression for u* will significant. In that case, we can set
and obtain u* and k* as before.