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A '''Combined Linear Congruential Generator''' ('''CLCG''') is a [[Pseudorandomness|pseudo-random]] number generator [[algorithm]] based on combining two or more [[linear congruential generator]]s (LCG). A traditional LCG has a [[Periodic function|period]] which is inadequate for complex system simulation.<ref name="Banks, Carson, Nelson, Nicol-2010">Banks 2010, Sec. 7.3.2</ref> By combining two or more LCGs, random numbers with a longer period and better statistical properties can be created.<ref name="Banks, Carson, Nelson, Nicol-2010">Banks 2010, Sec. 7.3.2</ref>
The algorithm is defined as:<ref name="L'Ecuyer-1988">L'Ecuyer 1988</ref>


: <math>X_i \equiv \left( \sum_{j=1}^k (-1)^{j-1}Y_{i,j} \right)\pmod{(m_1 - 1)}</math>


where:
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: <math> m_1</math> — the "[[modulo operation|modulus]]" of the first LCG
: <math> Y_{i,j}</math> — the i<sup>th</sup> input from the j<sup>th</sup> LCG
: <math> X_i </math> — the i<sup>th</sup> generated random integer
 
with:
: <math> R_i \equiv \begin{cases}
  X_i/m_1  & \text{for }  X_i >  0 \\
  (m_1-1)/m_1 & \text{for } X_i=0
  \end{cases}
</math>
where <math>R_{i}</math> is a [[Uniform distribution (continuous)|uniformly distributed]] random number between 0 and 1.
 
== Derivation ==
If ''W''<sub>''i'',1</sub>, ''W''<sub>''i'',2</sub>, ..., ''W''<sub>''i'',k</sub> are any independent, discrete, random-variables and one of them is uniformly distributed from 0 to ''m''<sub>1</sub>&nbsp;&minus;&nbsp;2, then ''Z''<sub>''i''</sub> is uniformly distributed between 0 and ''m''<sub>1</sub>&nbsp;&minus;&nbsp;2, where:<ref name="L'Ecuyer-1988">L'Ecuyer 1988</ref>
:<math> Z_i= \left( \sum_{j=1}^k W_{i,j} \right) \pmod {(m_1-1)} </math>
 
Let ''X''<sub>''i'',1</sub>, ''X''<sub>''i'',2</sub>, ..., ''X''<sub>''i'',''k''</sub> be outputs from ''k'' LCGs. If ''W''<sub>''i'',''j''</sub> is defined as ''X''<sub>''i'',''j''</sub>&nbsp;&minus;&nbsp;1, then ''W''<sub>''i'',''j''</sub> will be approximately uniformly distributed from 0 to ''m''<sub>''j''</sub>&nbsp;&minus;&nbsp;1.<ref name="L'Ecuyer-1988">L'Ecuyer 1988</ref> The coefficient "(&minus;1)<sup>''j''&minus;1</sup>" implicitly performs the subtraction of one from ''X''<sub>''i'',''j''</sub>.<ref name="Banks, Carson, Nelson, Nicol-2010">Banks 2010, Sec. 7.3.2</ref>
 
== Properties ==
The CLCG provides an efficient way to calculate pseudo-random numbers. The LCG algorithm is computationally inexpensive to use.<ref name="Pandey-2008">Pandey 2008, Sec. 2.2</ref> The results of multiple LCG algorithms are combined through the CLCG algorithm to create pseudo-random numbers with a longer [[Periodic function|period]] than is achievable with the LCG method by itself.<ref name="Pandey-2008">Pandey 2008, Sec. 2.2</ref>
 
The period of a CLCG is dependent on the seed value used to initiate the algorithm. The maximum period of a CLCG is defined by the function:<ref name="Banks, Carson, Nelson, Nicol-2010">Banks 2010, Sec. 7.3.2</ref>
 
:<math> P=((m_1-1)(m_2-1)\cdots(m_k-1))/(2^{k-1}) </math>
 
== Example ==
The following is an example algorithm designed for use in 32 bit computers:<ref name="L'Ecuyer-1988">L'Ecuyer 1988</ref>
 
:<math> k=2 </math>
LCGs are used with the following properties:
:<math> a_1=40,014</math>
:<math>m_1=2,147,483,563</math>
:<math>a_2=40,692</math>
:<math>m_2=2,147,483,399 </math>
:<math>c_1=c_2=0</math>
 
The CLCG algorithm is setup as follows:
 
1. The seed for the first LCG, <math> Y_{0,1} </math>, should be selected in the range of [1, 2,147,483,562].
: The seed for the second LCG, <math> Y_{0,2} </math>, should be selected in the range of [1, 2,147,483,398].
: Set: <math> i=0 </math>
 
2. The two LCGs are evaluated as follows:
:<math> Y_{i+1, 1} = 40,014 \times Y_{i,1}\pmod {2,147,483,563}</math>
:<math> Y_{i+1, 2} = 40,692 \times Y_{i,2}\pmod {2,147,483,399}</math>
 
3. The CLCG equation is solved as shown below:
:<math> X_{i+1}= (Y_{i+1, 1} - Y_{i+1, 2})\pmod{2,147,483,562} </math>
 
4. Calculate the random number:
: <math> R_{i+1} = \begin{cases}
  X_{i+1}/2,147,483,563 & \text{for }  X_{i+1} >  0 \\
  2,147,483,562/2,147,483,563 & \text{for } X_{i+1}=0
  \end{cases}
</math>
 
5. Increment the counter (i=i+1) then return to step 2 and repeat.
 
The maximum period of the two LCGs used is calculated using the formula:.<ref name="Banks, Carson, Nelson, Nicol-2010">Banks 2010, Sec. 7.3.2</ref>
:<math>(m-1)</math>
This equates to 2.1x10<sup>9</sup> for the two LCGs used.
 
This CLCG shown in this example has a maximum period of:
:<math> (m_1-1)(m_2-1)/2 = 2.3 \times 10^{18}</math>
This represents a tremendous improvement over the period of the individual LCGs. It can be seen that the combined method increases the period by 9 orders of magnitude.
 
Surprisingly the period of this CLCG may not be sufficient for all applications:.<ref name="Banks, Carson, Nelson, Nicol-2010">Banks 2010, Sec. 7.3.2</ref> Other algorithms using the CLCG method have been used to create pseudo-random number generators with periods as long as 3x10<sup>57</sup>.<ref name="L'Ecuyer-1996">L'Ecuyer 1996</ref><ref name="L'Ecuyer-1999">L'Ecuyer 1999</ref><ref name="L'Ecuyer, Simard, Chen, Kelton-2002">L'Ecuyer 2002</ref>
 
==See also==
* [[Linear congruential generator]]
 
==References==
<references/>
* Banks, Jerry., Carson, John S., Nelson, Barry L., Nicol, David M., (2010). ''Discrete-Event System Simulation'', 5th edition, Prentice Hall, ISBN 0-13-606212-1.
* {{cite journal | author = P. L'Ecuyer |title=Efficient and Portable Combined Random Number Generators |journal=[[Communications of the ACM]] |year=1988 |volume=31 |pages=742–749, 774}}
*{{cite web|last=Pandey|first=Niraj|title=IMPLEMENTATION OF LEAP AHEAD FUNCTION FOR LINEAR CONGRUENTIAL AND LAGGED FIBONACCI GENERATORS|url=http://www.cs.fsu.edu/research/projects/nirajproject.pdf|publisher=FLORIDA STATE UNIVERSITY|accessdate=13 April 2012}}
* {{cite journal | author = P. L'Ecuyer |title=Combined Multiple Recursive Number Generators|journal=[[Operations Research]] |year=1996 |volume=44 |pages=816–822}}
* {{cite journal | author = P. L'Ecuyer |title=Good Parameters and Implementations for Combined Multiple Recursive Random Number Generators|journal=[[Operations Research]] |year=1999 |volume=47 |pages=159–164}}
* {{cite journal | author = P. L'Ecuyer, R. Simard, E.J. Chen, W.D. Kelton |title=An Object-Oriented Randon-Number Package with Many Long Streams and Substreams|journal=[[Operations Research]] |year=2002 |volume=50 |pages=1073–1075}}
 
==External links==
* [https://www.msu.edu/~amthoral/random.ppt An overview of use and testing of pseudo-random number generators ]
 
{{DEFAULTSORT:Combined Linear Congruential Generator}}
[[Category:Pseudorandom number generators]]
[[Category:Modular arithmetic]]

Latest revision as of 09:58, 24 December 2014


What Is A Timeshare Home?

Many timeshare proprietors are inquiring them selves this question - "How do I sell my timeshare?". The real truth is there are no straightforward solutions. You have to preserve your anticipations realistic as it is a customer's industry out there and you would be really blessed to get a very good price for your gw timeshare services. But if you really feel the need to have to get rid of your timeshare as soon as attainable, right here are some things you must spend focus to.

If you contemplate trying to market your 7 days on e-Bay be aware that there are so a lot of weeks outlined for practically practically nothing that it will most likely cause the price tag of your 7 days to be dragged down. Set a reserve as low as you are inclined to promote for; if you don't have a reserve you may find you have marketed the week for $1!

First I went on the Web and searched all over the place. I tried to locate an simple way to Groupwise, but there just was not these kinds of a point. Tons of places marketed an effortless area to market a timeshare, but it simply wasn't correct. So I experienced to find an additional way. An company.

You'll hear individuals notify you how challenging it is to offer a timeshare in this hard industry. You may possibly uncover it's less complicated than you think if you start with a sensible marketing cost and a good deal of good promoting. There are numerous free true estate internet site on the web like Craigslist or Zillow. You could even try the auction web sites like eBay. Speak to other owners at your vacation resort they might discover a purchaser for you. With a little persistence you need to be in a position to market the timeshare in just a short time.

And you don't have to vacation much for this getaway. You can travel regional and maintain the fees down. Locate a resort within 300 miles and push to it. Established aside a specific quantity to devote on vacation and adhere to it. With timeshare trade you can trade your timeshare for one more timeshare at above five,000 resorts around the world. No matter exactly where you reside you'll find a quality vacation resort nearby. Possibly it may possibly not have been your initial option this calendar year but go ahead and consider something new. You can exchange your timeshare Groupwise online and it won't expense you anything at all. Give this a consider. Verify out TimeshareJuice for the cost-free alternative on-line or Redweek ($100 exchange fee).

I am likely to presume that you are reading this due to the fact you are fed up with timeshare ownership. You were probably pushed to buy a timeshare when you had been on getaway with your loved ones or close friends and they promised you a free of charge breakfast and the possibility to get tickets to a Groupwise sizzling, nearby attraction. But now the economic climate sucks and you're caught with one thing that you can't afford to go to or hold. Correct?

They appear at all varieties of market conditions and what other buyers are seeking for. They can effortlessly look for through their databases and find the proper buyer for your timeshare.

If you're seeking to offer your timeshare, the greatest location to begin is by seeing exactly how considerably money you can get out of your house. I worked with an Groupwise industry specialist to decide what my time share was well worth and how much cash I could get out of it. If you have a timeshare, this information is offered to you for totally free! Get your exact Timeshare Funds Price for one hundred%twenty five Cost-free. That's what I did! Then you can make the choice to possibly hire your timeshare or basically promote it.

Throughout the very last decade, organizations like these have established their value, many moments over. They have been in a position to get tens of millions of timeshare homeowners and promote their qualities in file quantity of time. They can attain this due to the fact of how they commit millions on advertising, every single 12 months, to entice individuals who are interested in getting timeshares all in excess of the globe. In change, they take your timeshare, demand the customer a retail commission. Like closing expenses on a residence and market it.

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