Talk:Erlang distribution

'''this article should be kept, since Erlang is also used in domains that are less familiar with the general gamma distribution. Isn't \lambda the same as the standard deviation - a term less used in formal statistics but still used in other domains. What is its relationship to the Weibull distribution, now often used because it is available in spreadsheets as MS Excel. GioCM (talk) 19:54, 7 March 2014 (UTC)''' Mode is wrong (99% sure)

What it says: $$\scriptstyle \frac{1}{\lambda}(k \,-\, 1)\,$$ for $$\scriptstyle k \;\geq\; 1\,$$

What works for me: $$\scriptstyle \frac{1}{k}(\lambda \,-\, 1)\,$$ for $$\scriptstyle l \;\geq\; 1\,$$ (not sure if the constraint should be changed or not). — Preceding unsigned comment added by 213.37.13.104 (talk) 06:48, 13 August 2012 (UTC)

Fixed Entropy Expression Chris (talk) 08:10, 6 February 2009 (UTC)

Entropy of Erlang Distribution Looks Odd

The expression for the entropy of the erlang distribution is probably wrong. Consider the case of k=1 where the erlang reduces to a simple exponential distribution whose entropy is (1- ln (lambda)). The expression provided reduces to 1/lambda. Chris Rose crose@winlab.rutgers.edu

Chris (talk) 22:52, 8 January 2009 (UTC)

The discussions on this page have been reogranized. I have split previous comments up and duplicated the signature lines when spliting comments and added braces for clarity. Acuster 06:18, 19 August 2005 (UTC)

Importance of the topic
The redirect from the Erlang-B page, via the Erlang-C reference link, is completely incorrect. As already noted several times, this page presents the Erlang_k or restricted Gamma distribution. The B and C functions are usually treated separately from the the k-distribution because they arise in an entirely different context in queueing theory. The correct content for Erlang-C exists at http://de.wikipedia.org/wiki/Erlang_C. Unfortunately, it's in German but the math is correct. —Preceding unsigned comment added by RedRooz (talk • contribs) 22:15, 2 February 2008 (UTC)

We do not need a long article here since the Erlang distribution is basically the Gamma distribution. vignaux 02:06, 2004 Sep 20 (UTC)


 * I agree that this article could be merged into gamma distribution (unless it can be significantly expanded to include a detailed discussion of the phone traffic scenario). --MarkSweep 17:17, 7 Nov 2004 (UTC)


 * It is true that they are all actually probability distributions but the Erlang Distribution is a continuous distribution, a special case of the gamma but with some special characeristics - it can be modelled as a sum of exponentials, for example. I think that it unique enough that it should have its own page as well as being referred to in the gamma distribution page as a special case. --vignaux 07:44, 2004 Nov 9 (UTC)


 * Ok, then the present article should be kept and revised to reflect the situation you describe. --MarkSweep 00:34, 10 Nov 2004 (UTC)


 * The Erlang is widely used and named as its own distribution. It deserves a page of its own with explicit reference to the Gamma. Algebraically, it is often possible to generate Erlangs in computer programs where it would much harder to generate Gammas since the factorial is widely defined but the Gamma functions are less frequently available. Acuster 06:18, 19 August 2005 (UTC)


 * I've found this page to be useful. More useful than the page on the Gamma distribution. One thing I like about this page is that it is explained, in simple terms, where the Erlang distribution comes from. It is relatively easy to understand, since, in part 'k' is an integer (event number). But the meaning of the Gamma distribution, for which the corresponding parameter is real, is a mystery to me. I say keep this page, but fix the Gamma distribution page, which is lacking any kind of motivational explanation.

Plots
I wanted to put plots in this article, but the Gamma distribution plots would work just as well, if only the two articles agreed on whether to use &theta; or &lambda; as the parameter. It would also facilitate understanding the relationship between the two distributions. Is there any reason not to choose one and have both articles use the same parameter? PAR 09:48, 19 August 2005 (UTC)


 * There may be a reason. If I remember correctly when you do the math to derive the erlang from the exponential, you get one parameter. Then, the Gamma generally uses the other. I'll look into it. Acuster 05:29, 20 August 2005 (UTC)


 * I added plots from the Gamma distribution article, but the CDF does not appear in my browser. Does it look ok to everyone else?


 * Looks great to me, thanks. Acuster 21:53, 21 August 2005 (UTC)

The Poisson process
You should better explain Poisson process. vignaux 02:06, 2004 Sep 20 (UTC)

Erlang-B and Erlang-C
The material on the Erlang Formulae should be moved to another page-- it is not the Erlang Distributiion. vignaux 02:06, 2004 Sep 20 (UTC)

{However, w}{W}hat exactly do you mean by Erlang formulae vs. Erlang distribution? Would you care to elaborate? --MarkSweep 17:17, 7 Nov 2004 (UTC)


 * The Erlang-B and Erlang-C formulae are queue theory results giving the discrete probability distributions of the numbers in steady-state queues with certain characteristics (blocked -customers-cleared, for example). They are not usually referred to as Erlang Distributions. --vignaux 07:44, 2004 Nov 9 (UTC)


 * Ok, then the present article should be kept and revised to reflect the situation you describe. --MarkSweep 00:34, 10 Nov 2004 (UTC)


 * {t}The Erlang-B and Erlang-C formulae are mentioned in the Erlang Unit page. I will modify this page when I have time. --vignaux 04:01, 2004 Nov 12 (UTC)

Factorial Function
In the section "Probability density function" the article currently says that the gamma is a generalization using the gamma function instead of the factorial function. While this is true in a very real sense, the formula we provided for the Erlang already uses the gamma function in its denominator, and not a factorial at all. This could cause confusion for people who aren't familiar with these functions/distributions. I propose altering the pdf formula to use a factorial function, which is mathematically appropriate because the parameter is always integral for the Erlang distribution. It is also more approachable to talk about factorials than the gamma function, which is rather elaborately defined. For an integer k, it is always the case that Gamma(k) = (k-1)!. — Preceding unsigned comment added by Kruggable (talk • contribs) 14:36, 31 March 2014 (UTC)

It is worth pointing out the need for at least a minor correction. Under "Properties" if indeed the denominator is gamma(k) then since that is (k-1)! then saying that is the Poisson Distribution is incorrect, since the denominator for the Poisson Distribution is k!, not (k-1)!. (Observation by ecsd on 3 May 2014.) — Preceding unsigned comment added by 208.76.28.94 (talk) 11:03, 3 May 2014 (UTC)

The explanation of the waiting time interpretation uses imprecise language
I think that, "the waiting times between k occurrences of the event are Erlang distributed," should be changed to "the waiting time until the kth occurrence of the event," which I would say is equivalent to "the waiting time between k+1 occurrences of the event".

More specific definition of cumulative would be welcome
I assume the posted formula for the cdf is over a range from 0 to x. Is this correct? I've found this integral difficult to perform, hence my question.

Please define constants in equation
Please define the constants in the equation. I would but I don't know what they are!


 * Huh? The parameter n is explained in the preceding text, x is defined by function abstraction, and e is Euler's number. Am I missing something here? --MarkSweep 03:47, 11 Jan 2005 (UTC)


 * sorry, I just couldn't find them.Pdbailey 23:04, 12 Jan 2005 (UTC)

Probability density function
The two equivalent formulations of the Probability density function are so trivially equivalent that one of them should be deleted. —Preceding unsigned comment added by 134.96.243.196 (talk) 18:09, 10 November 2010 (UTC)

Cumulative Density Function
The Cumulative Density Function is wrong. It should be 1 - ... — Preceding unsigned comment added by 86.87.53.219 (talk) 11:38, 22 April 2016 (UTC)
 * There is no such thing as a cumulative DENSITY function. The letters "CDF" stand for "cumulative distribution function". The words "cumulative" and "density" contradict each other. Michael Hardy (talk) 18:43, 23 February 2019 (UTC)

Graphs
The example plots of the PDF and CDF refer to $$\mu$$ instead of $$\beta$$. Either the variable names in the legends or the variable names in the text need to be changed. — Preceding unsigned comment added by 92.72.161.224 (talk) 14:57, 16 November 2021 (UTC)