UNDERSTANDING SYSTEM BEHAVIOR AND MODEL VERIFICATION

11.4 UNDERSTANDING SYSTEM BEHAVIOR AND MODEL VERIFICATION

The generic packaging line model of Section 11.3 may be abstracted as a sequence of workstations (or servers) having fixed processing times per product unit, and with the proviso that the first workstation is never idle. Thus, the throughput of the system (production rate) coincides with that of the slowest workstation in the sequence.

In the example of Section 11.3, the labeling workstation (resource Labeler) is the slowest, with 8 seconds of unit processing time. Therefore, the utilization of the labeling workstation is expected to be 100%, since all upstream workstations have shorter (fixed) unit processing times. Since the filling and capping workstations are both faster than the labeling workstation, the labeling buffer is sure to fill up eventually, thereby blocking the capping workstation and later on the filling workstation, thereby giving rise to significant blocking probabilities in these workstations. Furthermore, every departure from the labeling workstation finds the downstream workstations (sealing

238 Modeling Production Lines and carton packing) in the idle state, since they too have shorter processing times than

the labeling workstation. In consequence, the labeling and sealing workstations are never blocked. Accordingly, a product unit departs from the labeling workstation every

8 seconds—which is also the interdeparture time in downstream machines—resulting in system throughput of 1/8 ¼ 0.125 units per second. In fact, the throughput of each

workstation in the system should also be 0.125. Indeed, using Eq. 8.7, which expresses o as the ratio of utilization to average unit processing time, the through- o ¼ 0.813 / 6.5 ¼ 0.125, while the throughput of the o ¼ 0.625 / 5 ¼ 0.125. As one facet of simulation model verification, we next verify that the average flow time and the average number of product units obeys Little’s formula (8.8), which is further discussed in Section 11.10. Define the subsystem S to be the sequence of processes from the Filling Process module to and including the Packing Process module (the raw material storage in front of Filling Process is included in S). The average flow time through S is measured from the arrival of a unit entity at queue Filling Process.Queue to the unit entity’s arrival at module Interdeparture Time. The replication estimated this average

N s , is just the sum of the corresponding average buffer contents (see Figure 11.8) plus the average number of units in service (average time in busy state; see Figure 11.11), that is,

N s ¼ 15:6448 þ 4:9860 þ 4:9551 þ 1:0 þ 0:9995 þ 0:9999 þ 0:6249 þ 0:7498 ¼ 29:66; (11 :1)

which is close to the theoretical expected value 30 (note that this is a closed system with

30 circulating unit entities). Thus, in this case, Little’s formula becomes the relation

(11 :2) N s

o is the throughput of S as discussed above

F s is the average flow time through S. Using the average flow time and the throughput in Figure 11.11 to calculate the right side of Eq. 11.2 yields

(11 :3) Because the right sides of Eqs. 11.1 and 11.3 are approximately the same, this fact tends

to verify the replication’s estimates. In fact, the discrepancy between Eqs. 11.1 and 11.3 becomes progressively smaller, as the replication run length increases.

Next, we perform sensitivity analysis by studying the impact of buffer capacities on system behavior. Such analysis will aid us in gaining insight into blocking—a pheno-

menon stemming from mismatch in the service rates and finite buffer capacities. Figure

11.13 displays three performance measures as functions of simultaneously increasing buffer capacities in Labeling Process and Capping Process. From left to right, the plotted measures are the throughput of S, the WIP level at Labeling Process, and the average flow time through S, excluding the Filling buffer. (If flow times were to include time in the Filling buffer, then they would remain constant. Why?)

Interestingly, the throughput of S is largely unaffected, while the average WIP level at Labeling Process and the mean flow time through S increase as the buffer capacities are increased. To understand these observations, recall that the system under study has fixed processing times. Recall further that buffers are placed in production lines in order

Modeling Production Lines 239 0.15 15 250

Through put 0.11 3 100

Mean Flow Time

0.1 Labeling Buffer Content 0 50 0 5 10 15 0 5 10 15 0 5 10 15

Buffer Capacity

Buffer Capacity (a)

Buffer Capacity

(b)

(c)

Figure 11.13 Impact of simultaneously increasing the capping and labeling buffer capacities on (a) throughput of S, (b) average number in labeling buffer, and (c) average flow time.

to absorb product flow fluctuations due to randomness in the system (e.g., random processing times, random downtimes, etc.). Such events introduce variability that tends to slow down product movement. But in systems with no variability (such as ours), the placement of buffers would have no impact on the throughput, as evidenced by Figure 11.13(a). However, placing larger buffers upstream of bottleneck workstations simply gives rise to larger WIP levels there, and consequently, to longer flow times. In contrast, buffers at workstations downstream of the bottleneck are always empty, regardless of capacity.

To summarize, our sensitivity analysis has revealed some valuable and somewhat unexpected design principles for deterministic production lines, namely, that larger buffers actually can have an overall deleterious effect on the system. On the one hand, increasing buffer sizes will not increase the throughput. On the other hand, such increases will result in larger on-hand inventories, thereby tying up precious capital in inventory. Worse still, it would take jobs longer to traverse the system. These effects can

be economically detrimental (think of the penalty of longer manufacturing lead times). Reasonably sized buffers are necessary, however, to absorb the effects of variability in product flow, in the presence of randomness. In Section 11.6, we will discuss the beneficial effect of placing buffers when machine failures and repairs are introduced.