1Comments on Little’s Law T. Gutowski Oct 31, 2008 1. Little’s Law (from queuing theory) applies to any system over the long term. L = λ W average parts average arrival average time in the system in system rate Proof by Little in 1961 uses probability theory and discrete mathematics. Using continuum theory for any conserved quantity λin Æ L Æ λout outindtdLλλ−=∴ L is not defined unless, λin = λout (steady-state). Hence for long time averages, λ Æ L Æ λ Now if you stop the input, the time to empty the system is: W = L/λ ∴ L = λW 2. Other arrangements (deterministic) let λ = throughput rate Single process λ Æ 1 Æ λ 1 = λW, W = 1/λ Parallel process λ Æ 1 Æ λ L = 2 2λ Æ Æ 2λ 2 = 2λW λ Æ 1 Æ λ W = 1/λ ex. Multicavity injection molding Serial process L = 3 λ Æ Æ 1 Æ 1 Æ 1 Æ Æ λ 3 = λW λ λ λ W = 3/λ ex. transfer line2Different process rates (this applies to infinite buffers or a deterministic system for which buffers are irrelevant) λ = min (λ1,λ2,λ3)Æ Æ Æ Æ Æ λ λ1 λ2 λ3 λ = min (λ1,λ2,λ3) = λ bottleneck 3. Now consider a system with a single server of rate μ. And arrival rate λ λ Æ L Look inside the system λ Æ μ Service rate is μ ≥ λ, so for a deterministic system there is no waiting (i.e. no queue inside the system). Deterministic: part arrives at t = 0 (no variation) part comes out μ1=t part arrives at λ1=t part comes out at μλ11+=t So the time interval between parts = λ1. Time to process = μ1 Fraction of time part is being processes, or Utilization is μλλμ=1134. “Push” Vs “Pull” for a serial process with N Steps (deterministic) 1 2 N Æ 1 Æ 1 …… 1 Æ L =N λ λ λ Ν=λW W= Ν/λ “Push System” Release order of the front of the system, produced in batch size B, system initially empty. Time to get batch B out of the system = λBN “Pull System” like the supermarket you pull a part out at the end of the system. The system has WIP inventory at each process Time to get one part out = λ1 Time to get any number of parts, say B, out = λB The advantage of the Pull System is that parts come out quickly, on demand. The disadvantage is that the system is tied up with work in progress (WIP) inventory. 5. Actual Systems are usually much more complicated than the simple systems just described. Real systems may be making multiple part types at the same time. This generally complicates the system, increases set up time and reduces the system production rate λ in terms of parts per hour. See Figure 2-15 on page 46 (Chapter 2) of J.T. Black’s book. “The Design of a Factory with a Future” for the range of production rates for various production systems. Real systems can be made up of collections of subsystems that extend beyond the ability of a single person to easily grasp. For example, real systems can extend to operations in different buildings, different locations and even different companies. Recall Dan Whitney’s example of the chain of companies involved that control the gap between the fender and the hood of the vehicle. One methodology that has developed to address this problem is called Value Stream Mapping (VSM). Basically you get all the people involved in making a product together to map out where the product goes and how long it stays there. Once you have this map you can then go about identifying problems and improving the system. The first step in applying Little’s Law is to be very clear about drawing the system
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