Little’s law

Inventories and queue lengths It is often important to know where inventories and queues are distributed through the system. The average total flow time and the average total queue length are connected by Little s law ... 

A fundamental relationship that holds in a broad range of environments states that the average time to process an order at a workstation is a highly nonlinem function of the workload in the system and that as the workload approaches a nominal capacity both the mean and variance of the lead time increase exponentially, as illustrated in Figure 1. A second fundamental relationship is Little s law (Hopp and Spearman 1996), which states that the average work-in-process inventory (WIP) level and the average time to process a job through the system (i.e., lead time or cycle time) are directly proportional. 

The mathematics yields insights and general principles, such as Little s law (see Section 5.1), which cannot be readily discovered from simulations. 

This is Little s law. Illustrations of it may be seen in (39) and (41), once the ergodic relations (48) are taken into account. To justify (49) heuristically, following Ross (1997), if jobs arrive at rate A and on average spend time W in the system, then demand fqrjoccupancy of the system arrives at a rate of XW, and so occupancy must be supplied at a rate of XW also, meaning that XW jobs must be present on average. 

This simple fact may be applied to various systems. For example, in the context of a queue, the system could be taken to mean the waiting area or the service area rather than the entire system. If applied to the service area of a single-server queue, Little s law yields... 

To see that this is a special case of (49), note that 1 - P(L = 0) is not only the steady-state probability that the server is busy, it is also the steady-state expected number of jobs in service, which in turn, by an ergodic property, is the time-average number of jobs in the system in question. Of course, 1 / /X is the expected time in service and plays the role of W in (49). When Little s law is applied to the waiting area, it tells us that the expected number of jobs waiting is the product of the arrival rate and the average waiting time. 

To explain, the rightmost sum is the expected total number in the system by (39). This is divided by the overall exogenous arrival rate to get the average time-in-system, using Little s law (49). 

Note that supply chains are replenishment systems, thus the average amount shipped should match the average demand. If the transport lead time were L, the average in transit inventory is LD. Notice that this is just Little s law from queueing theory, which says that the average work in process inventory is equal to the demand rate times the average lead time through the system. 

Notice that the equation above is Litde s law, which states that the average number of units in a queue is the demand rate times the average lead time. Little s law links lead time to work in process inventory. In our case, the inventory in transit is work-in-process inventory between the component plant and the assembly plant where it is required. Once we get the average in-transit inventory, we merely have to multiply it with the holding cost per unit per unit time to get the holding cost associated with the in-transit inventory. 

Another example of using Little s Law is to examine the flow through the manufacturing or operations function. For operations, if the throughput of orders is 100/week and its throughput is 1 week for each order, then there will be inventory equal to 100 orders in the operations system (i.e., I = 100 orders/week 1 week = 100 orders). If this company is a make-to-order company, then the orders will be shipped when they are finished and sent to shipping by operations. 

From Little s Law as shown above, we can see that there is a relationship between the amount of inventory that we have in stock and the flow time (F). This relationship is F = 1/T. So, if we have an inventory of 1,000 units and a throughput of 100 units/week, then the average time it takes for inventory to flow through our system is F = (1,000 units)/(100 units/week) = 10 weeks. 

All inventory management systems answer two questions How much to order, and when to order. The rules established to answer these two questions must be continuously reviewed and revised as appropriate. As demonstrated using Little s Law earlier in the chapter, the answer to these questions depends on other characteristics in the system. For example, as the flow time of an item becomes shorter, the system will need less inventory, so the two questions of when and how much need to be answered again. 

Inventory also has a significant impact on the material flow time in a supply chain. Material flow time is the time that elapses between the point at which material enters the supply chain to the point at which it exits. For a supply chain, throughput is the rate at which sales occur. If inventory is represented by /, flow time by T, and throughput by >, the three can be related using Little s law as follows ... 

For example, if an Amazon warehouse holds 100,000 units in inventory and sells 1,000 units daily, Little s law tells us that the average unit will spend 100,000/1,000 = 100 days in inventory. If Amazon were able to reduce flow time to 50 days while holding throughput constant, it would reduce inventory to 50,000 units. Note that in this relationship, inventory and thronghpnt mnst have consistent units. 

Lot sizes and eyele inventory also influence the flow time of material within the supply ehain. Recall from Little s Law (Equation 3.1) that... 

Average inventory = cycle inventory -I- safety inventory = 5,000 -I- 1,000 = 6,000 B M thus carries an average of6,000 phones in inventory. Using Little s law (Equation 3.1), we have Average flow time = average inventory/throughput = 6,000/ 2,500 = 2.4 weeks Each phone thus spends an average of 2.4 weeks at B M. 

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