Little's law 1

Little's lawIn the mathematical theory of queues, Little's result, theorem, lemma, law or formula[1][2] is a theorem by JohnLittle which states:

The long-term average number of customers in a stable system L is equal to the long-term average effectivearrival rate, λ, multiplied by the (Palm‑)average time a customer spends in the system, W; or expressedalgebraically: L = λW.

Although it looks intuitively reasonable, it's a quite remarkable result, as the relationship is "not influenced by thearrival process distribution, the service distribution, the service order, or practically anything else."[3]

The result applies to any system, and particularly, it applies to systems within systems. So in a bank, the customerline might be one subsystem, and each of the tellers another subsystem, and Little's result could be applied to eachone, as well as the whole thing. The only requirements are that the system is stable and non-preemptive; this rulesout transition states such as initial startup or shutdown.In some cases it is possible to mathematically relate not only the average number in the system to the average waitbut relate the entire probability distribution (and moments) of the number in the system to the wait.[4]

HistoryIn a 1954 paper Little's law was assumed true and used without proof.[5][6] The form L = λW was first published byPhilip M. Morse where he challenged readers to find a situation where the relationship did not hold.[5][7] Littlepublished in 1961 his proof of the law, showing that no such situation existed.[8] Little's proof was followed by asimpler version by Jewell[9] and another by Eilon.[10] Shaler Stidham published a different and more intuitive proofin 1972.[11][12]

ExampleImagine a small store with a single counter and an area for browsing, where only one person can be at the counter ata time, and no one leaves without buying something. So the system is roughly:

Entrance → Browsing → Counter → Exit

This is a stable system, so the rate at which people enter the store is the rate at which they arrive at the store, and therate at which they exit as well. We call this the arrival rate. By contrast, an arrival rate exceeding an exit rate wouldrepresent an unstable system, where the number of waiting customers in the store will gradually increase towardsinfinity.Little's Law tells us that the average number of customers in the store L, is the effective arrival rate λ, times theaverage time that a customer spends in the store W, or simply:

Assume customers arrive at the rate of 10 per hour and stay an average of 0.5 hour. This means we should find theaverage number of customers in the store at any time to be 5.

Now suppose the store is considering doing more advertising to raise the arrival rate to 20 per hour. The store musteither be prepared to host an average of 10 occupants or must reduce the time each customer spends in the store to0.25 hour. The store might achieve the latter by ringing up the bill faster or by adding more counters.We can apply Little's Law to systems within the store. For example, the counter and its queue. Assume we noticethat there are on average 2 customers in the queue and at the counter. We know the arrival rate is 10 per hour, socustomers must be spending 0.2 hours on average checking out.

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We can even apply Little's Law to the counter itself. The average number of people at the counter would be in therange (0, 1) since no more than one person can be at the counter at a time. In that case, the average number of peopleat the counter is also known as the utilisation of the counter.However, because a store in reality generally has a limited amount of space, it cannot become unstable. Even if thearrival rate is much greater than the exit rate, the store will eventually start to overflow, and thus any new arrivingcustomers will simply be rejected (and forced to go somewhere else or try again later) until there is once again freespace available in the store. This is also the difference between the arrival rate and the effective arrival rate, wherethe arrival rate roughly corresponds to the rate of which customers arrive at the store, whereas the effective arrivalrate corresponds to the rate of which customers enter the store. In a system with an infinite size and no loss, the twoare however equal.

ApplicationsSoftware-performance testers have used Little's law to ensure that the observed performance results are not due tobottlenecks imposed by the testing apparatus. See:• Software Infrastructure Bottlenecks in J2EE by Deepak Goel [13]

• Benchmarking Blunders and Things That Go Bump in the Night by Neil Gunther [14]

Other applications include staffing emergency departments in hospitals.[15][16]

Notes[1] Alberto Leon-Garcia (2008). Probability, statistics, and random processes for electrical engineering (3rd ed.). Prentice Hall.

ISBN 0-13-147122-8.[2] Allen, Arnold A. (1990). Probability, Statistics, and Queueing Theory: With Computer Science Applications. Gulf Professional Publishing.

p. 259. ISBN 0120510510.[3] Simchi-Levi, D.; Trick, M. A. (2011). "Introduction to "Little's Law as Viewed on Its 50th Anniversary"". Operations Research 59 (3): 535.

doi:10.1287/opre.1110.0941.[4] Keilson, J.; Servi, L. D. (1988). "A distributional form of Little's Law" (http:/ / dspace. mit. edu/ handle/ 1721. 1/ 5305). Operations Research

Letters 7 (5): 223. doi:10.1016/0167-6377(88)90035-1. .[5] Little, J. D. C.; Graves, S. C. (2008). "Little's Law" (http:/ / web. mit. edu/ sgraves/ www/ papers/ Little's Law-Published. pdf). Building

Intuition. International Series in Operations Research & Management Science. 115. pp. 81. doi:10.1007/978-0-387-73699-0_5.ISBN 978-0-387-73698-3. .

[6] Cobham, Alan (1954). "Priority Assignment in Waiting Line Problems". Operations Research 2: 70–76. doi:10.1287/opre.2.1.70.JSTOR 166539.

[7] Morse, Philip M. (1958). Queues, inventories, and maintenance: the analysis of operational system with variable demand and supply. Wiley.[8] Little, J. D. C. (1961). "A Proof for the Queuing Formula: L = λW". Operations Research 9 (3): 383–320. doi:10.1287/opre.9.3.383.

JSTOR 167570.[9] Jewell, William S. (1967). "A Simple Proof of: L=λ W". Operations Research 15 (6): 1109-1116. doi:10.1287/opre.15.6.1109.

JSTOR 168616.[10] Eilon, Samuel (1969). "A Simpler Proof of L=λ W". Operations Research 17 (5): 915-917. doi:10.1287/opre.17.5.915. JSTOR 168368.[11] Stidham Jr., Shaler (1974). "A Last Word on L = λW". Operations Research 22 (2): 417–421. doi:10.1287/opre.22.2.417. JSTOR 169601.[12] Stidham Jr., Shaler (1972). "L = λW: A Discounted Analogue and a New Proof". Operations Research 20 (6): 1115–1120.

doi:10.1287/opre.20.6.1115. JSTOR 169301.[13] http:/ / www. onjava. com/ pub/ a/ onjava/ 2005/ 01/ 19/ j2ee-bottlenecks. html[14] http:/ / arxiv. org/ abs/ cs/ 0404043[15] Little, J. D. C. (2011). "Little's Law as Viewed on Its 50th Anniversary" (http:/ / www. informs. org/ content/ download/ 255808/ 2414681/

file/ little_paper. pdf). Operations Research 59 (3): 536–549. doi:10.1287/opre.1110.0940. JSTOR 23013126. .[16] Harris, Mark (February 22, 2010). "Little’s Law: The Science Behind Proper Staffing" (http:/ / www. epmonthly. com/ subspecialties/

management/ littles-law-the-science-behind-proper-staffing/ ). Emergency Physicians Monthly. . Retrieved September 4, 2012.

Little's law 3

External links• A Proof of the Queueing Formula L = λ W (http:/ / www. columbia. edu/ ~ks20/ stochastic-I/ stochastic-I-LL.

pdf), Sigman, K., Columbia University

Article Sources and Contributors 4

Article Sources and ContributorsLittle's law  Source: http://en.wikipedia.org/w/index.php?oldid=528048447  Contributors: Angela, Avatar, Bezenek, Bihco, Cybercobra, Cyclist, Danielx, David Haslam, DeadEyeArrow,Doctormatt, Dysprosia, Elwikipedista, Eranb, EyeSerene, Frosted14, Gareth Jones, GregorB, Gsinghglakes, HisashiKobayashi, Isheden, JA(000)Davidson, JoergenB, Jonathanwf, Ketil3,LachlanA, Larry V, Lawrence Cohen, Life of Riley, LimoWreck, MKen, Magioladitis, MartinHarper, Melcombe, Michael Hardy, Myleslong, Nickpowerz, Peripitus, Perkelton, Raul654,Rgclegg, Saeed.moazzen, Shreevatsa, Spakin, Tagishsimon, Tassedethe, Telrod, The Anome, The wub, Ttgamage, Uruiamme, Wfeidt, Wikinewbie123, Winterst, Xaven, Yellowdesk, 62anonymous edits

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