Little's law

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In queueing theory, a discipline within the mathematical theory of probability, Little's result, theorem, lemma, law or formula[1][2] is a theorem by John Little which states:

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

Although it looks intuitively reasonable, it is quite a remarkable result, as the relationship is "not influenced by the arrival 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.[4] So in a bank, the customer line might be one subsystem, and each of the tellers another subsystem, and Little's result could be applied to each one, as well as the whole thing. The only requirements are that the system is stable and non-preemptive; this rules out 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 wait but relate the entire probability distribution (and moments) of the number in the system to the wait.[5]

History[edit]

In a 1954 paper Little's law was assumed true and used without proof.[6][7] The form L = λW was first published by Philip M. Morse where he challenged readers to find a situation where the relationship did not hold.[6][8] Little published in 1961 his proof of the law, showing that no such situation existed.[9] Little's proof was followed by a simpler version by Jewell[10] and another by Eilon.[11] Shaler Stidham published a different and more intuitive proof in 1972.[12][13]

Example[edit]

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

Entrance → Browsing → Counter → Exit

In a stable system, the rate at which people enter the store is the rate at which they arrive at the store (called the arrival rate), and the rate at which they exit as well (called the exit rate). By contrast, an arrival rate exceeding an exit rate would represent an unstable system, where the number of waiting customers in the store will gradually increase towards infinity.

Little's Law tells us that the average number of customers in the store L, is the effective arrival rate λ, times the average time that a customer spends in the store W, or simply:

L = \lambda  W \,

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

L = 10 \cdot 0.5 = 5

Now suppose the store is considering doing more advertising to raise the arrival rate to 20 per hour. The store must either be prepared to host an average of 10 occupants or must reduce the time each customer spends in the store to 0.25 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 notice that there are on average 2 customers in the queue and at the counter. We know the arrival rate is 10 per hour, so customers must be spending 0.2 hours on average checking out.

W =\frac{L}{\lambda} = \frac{2}{10} = 0.2

We can even apply Little's Law to the counter itself. The average number of people at the counter would be in the range (0, 1) since no more than one person can be at the counter at a time. In that case, the average number of people at 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 the arrival rate is much greater than the exit rate, the store will eventually start to overflow, and thus any new arriving customers will simply be rejected (and forced to go somewhere else or try again later) until there is once again free space available in the store. This is also the difference between the arrival rate and the effective arrival rate, where the arrival rate roughly corresponds to the rate at which customers arrive at the store, whereas the effective arrival rate corresponds to the rate at which customers enter the store. However, in a system with an infinite size and no loss, the two are equal.

Estimating parameters[edit]

To use Little's law on data formulas must be used to estimate the parameters as the result does not necessarily directly apply over finite time intervals, due to problems like how to log customers already present at the start of the logging interval and those who have not yet departed when logging stops.[14]

Applications[edit]

Software-performance testers have used Little's law to ensure that the observed performance results are not due to bottlenecks imposed by the testing apparatus. See:

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

Distributional form[edit]

An extension of Little's law provides a relationship between the steady state distribution of number of customers in the system and time spent in the system under a first come, first served service discipline.[17]

See also[edit]

Notes[edit]

  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.  edit
  4. ^ Serfozo, R. (1999). "Little Laws". Introduction to Stochastic Networks. pp. 135–154. doi:10.1007/978-1-4612-1482-3_5. ISBN 978-1-4612-7160-4.  edit
  5. ^ Keilson, J.; Servi, L. D. (1988). "A distributional form of Little's Law". Operations Research Letters 7 (5): 223. doi:10.1016/0167-6377(88)90035-1.  edit
  6. ^ a b Little, J. D. C.; Graves, S. C. (2008). "Little's Law". Building Intuition. International Series in Operations Research & Management Science 115. p. 81. doi:10.1007/978-0-387-73699-0_5. ISBN 978-0-387-73698-3.  edit
  7. ^ Cobham, Alan (1954). "Priority Assignment in Waiting Line Problems". Operations Research 2: 70–76. doi:10.1287/opre.2.1.70. JSTOR 166539.  edit
  8. ^ Morse, Philip M. (1958). Queues, inventories, and maintenance: the analysis of operational system with variable demand and supply. Wiley. 
  9. ^ Little, J. D. C. (1961). "A Proof for the Queuing Formula: L = λW". Operations Research 9 (3): 383–387. doi:10.1287/opre.9.3.383. JSTOR 167570.  edit
  10. ^ 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.  edit
  11. ^ 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.  edit
  12. ^ 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.  edit
  13. ^ 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.  edit
  14. ^ Kim, S. H.; Whitt, W. (2013). "Statistical Analysis with Little's Law". Operations Research 61 (4): 1030. doi:10.1287/opre.2013.1193.  edit
  15. ^ Little, J. D. C. (2011). "Little's Law as Viewed on Its 50th Anniversary". Operations Research 59 (3): 536–549. doi:10.1287/opre.1110.0940. JSTOR 23013126.  edit
  16. ^ Harris, Mark (February 22, 2010). "Little’s Law: The Science Behind Proper Staffing". Emergency Physicians Monthly. Retrieved September 4, 2012. 
  17. ^ Bertsimas, D.; Nakazato, D. (1995). "The Distributional Little's Law and Its Applications". Operations Research 43 (2): 298. doi:10.1287/opre.43.2.298. JSTOR 171838.  edit

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