# Little’s Law ## The Idea in Brief Little’s Law is a foundational principle in queueing theory and operations management. It states that the average number of items in a system equals the average arrival rate multiplied by the average time each item spends in the system. Despite its simplicity, it is powerful because it holds under very general conditions. The law provides managers, engineers, and analysts with a reliable way to link flow, time, and capacity in any stable process, from factory lines to computer networks. --- ## Key Concepts ### The Formula - **Little’s Law**: $L=λW $ where: - $L$: average number of items in the system (inventory, customers, jobs). - $\lambda$: average arrival (or throughput) rate. - $W$: average time an item spends in the system (waiting + processing). ### Assumptions - The system is **stable**: arrivals and departures balance over time. - Averages are well defined (not infinite or erratic). - The law applies broadly regardless of how arrivals occur or how items are served. --- ## Implications 1. **WIP and Lead Time** - If throughput ($\lambda$) is fixed, any increase in work-in-progress ($L$) lengthens lead time ($W$). - This means excess inventory or overloading a system directly delays delivery times. 2. **Throughput and Capacity** - To reduce lead time without cutting WIP, a system must raise its processing rate. - Conversely, bottlenecks that restrict throughput will increase waiting times even if WIP stays constant. 3. **Management Trade-offs** - Managers can only optimise two of the three variables (WIP, throughput, lead time) at once; the third is determined by Little’s Law. - For example, lean manufacturing strategies focus on reducing WIP to shorten lead times. 4. **System Stability** - Persistent growth in WIP indicates instability (arrivals exceed completions). - Recognising this early can help prevent breakdowns in production or service quality. ### Uses - **Operations management**: helps firms balance inventory, throughput, and cycle time. - **Manufacturing**: clarifies the link between work-in-progress (WIP), production rate, and lead time. - **Services**: useful for call centres, hospitals, or retail to estimate customer waiting times. - **Computer science**: applied in network traffic, server queues, and performance modelling. --- ## See in Field Notes - [Capacity and Flow](https://www.anishpatel.co/capacity-and-flow/) — Little's Law explains why adding WIP slows everything: each extra item lengthens the wait for everything else - [Expensive Yes](https://www.anishpatel.co/expensive-yes/) — The hidden queue: before saying yes, ask what it does to lead time