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

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

Asynchronous Transfer Lines (ATL) W2W3 TH B2B3 W1 TH B1M1 M2M3 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?

The G/G/1 model: A single-station Modeling Assumptions: Part release rate = Target throughput rate = TH Infinite Buffering Capacity one server Server mean processing time = t e St. deviation of processing time =  e Coefficient of variation (CV) of processing time: c e =  e / t e Coefficient of variation of inter-arrival times = c a TH B1M1

An Important Stability Condition Average workload brought to station per unit time: TH·t e It must hold: Otherwise, an infinite amount of WIP will pile up in front of the station. TH B1M1

Performance measures for a stable G/G/1 station 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: TH B1M1

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 c a and/or c e ; 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, c a =1.0, and the resulting expression for CT q is exact (a result known as the Pollaczek-Kintchine formula). The expression for c d 2 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.

Performance measures for a stable G/G/m station 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: M1 B TH M2 Mm

Analyzing a multi-station ATL TH 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.

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.

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

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. t o = E[X];  o 2 =Var[X]; c o =  o / t o. T = random variable modeling the effective processing time = where U i = 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. m r =E[U i ];  r 2 =Var[U i ]; c r =  r / m r Time between outages is exponentially distributed with mean m f. Availability A = m f / (m f +m r ) = percentage of time the system is up. Then, t e = E[T] = t o / A or equivalently r e = 1/t e = A (1/t o ) = A  r o

Breakdown Example Data: Injection molding machine has: 15 second stroke (t o = 15 sec) 1 second standard deviation (s o = 1 sec) 8 hour mean time to failure (m f = sec) 1 hour repair time (m r = 3600 sec) Natural variability c o = 1/15 = (which is very low)

Example Continued Effective variability: Which is very high!

Example Continued Suppose through a preventive maintenance program, we can reduce m f to 8 min and m r to 1 min Which is low! (the same as before)

Example:employing the developed theory for diagnostic purposes M1BM2 t o1 =19 min c o1 2 =0.25 m f1 =48 hrs m r1 =8 hrs MTTR ~ expon. t o2 =22 min c o2 2 =1.0 m f2 =3.3 hrs m r2 =10 min MTTR ~ expon. C a 2 = 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.

Diagnostics example continued: Capacity analysis based on mean values

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!

Example: ATL Design Need to design a new 4-station assembly line for circuit board assembly. The technology options for the four stations are tabulated below (each option defines the processing rate in pieces per hour, the CV of the effective processing time, and the cost per equipment unit in thousands of dollars).

Example: ATL Design (cont.) Each station can employ only one technology option. The maximum production rate to be supported by the line is 1000 panels / day. The desired average cycle time through the line is one day. One day is equivalent to an 8-hour shift. Workpieces will go through the line in totes of 50 panels each, which will be released into the line at a constant rate determined by the target production rate.

A baseline design:Meeting the desired prod. rate with a low cost

Reducing the line cycle time by adding capacity to Station 2

Adding capacity at Station 1, the new bottleneck

An alternative option:Employ less variable machines at Station 1 This option is dominated by the previous one since it presents a higher CT and also a higher deployment cost. However, final selection(s) must be assessed and validated through simulation.

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

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

Determining an optimized batch size Let u m  r a t. Then u = u m / k  k = u m / u. Substituting this expression for k in the expression for CT, we get: Recognizing that, we set and we get where To minimize CT, it suffices to minimize y(u). This can be achieved as follows: andwhich further implies that Remark: If c e 2  0, the term  in the original expression for u * will significant. In that case, we can set and obtain u * and k * as before.