CODEWITHSUNDEEP
javapoisson
Alex
Alex

Reputation: 73

Simulating Poisson Waiting Times

I need to simulate Poisson wait times. I've found many examples of simulating the number of arrivals, but I need to simulate the wait time for one arrival, given an average wait time.

I keep finding code like this:

public int getPoisson(double lambda) 
{   
    double L = Math.exp(-lambda);   
    double p = 1.0;   
    int k = 0;   

    do 
    {    
        k++;     
        p *= rand.nextDouble(); 
        p *= Math.random(); 
    } while (p > L);   

    return k - 1; 
}

but that is for number of arrivals, not arrival times.

Efficieny is preferred to accuracy, more because of power consumption than time. The language I am working in is Java, and it would be best if the algorithm only used methods available in the Random class, but this is not required.

Upvotes: 7

Views: 7457

Answers (2)

amit
amit

Reputation: 178441

Time between arrivals is an exponential distribution, and you can generate a random variable X~exp(lambda) with the formula: 

-ln(U)/lambda` (where U~Uniform[0,1]).

More info on generating exponential variable.

Note that time between arrival also matches time until first arrival, because exponential distribution is memoryless.

Upvotes: 6

jdbertron
jdbertron

Reputation: 741

If you want to simulate earthquakes, or lightning or critters appearing on a screen, the usual method is to assume a Poisson Distribution with an average arrival rate λ.

The easier thing to do is to simulate inter-arrivals:

With a Poisson distribution, the arrivals get more likely as time passes. It corresponds to the cumulative distribution for that probability density function. The expected value of a Poisson-distributed random variable is equal to λ and so is its variance. The simplest way is to 'sample' the cumulative distribution which has an exponential form (e)^-λt which gives t = -ln(U)/λ. You choose a uniform random number U and plug in the formula to get the time that should pass before the next event. Unfortunately, because U usually belongs to [0,1[ that could cause issues with the log, so it's easier to avoid it by using t= -ln(1-U)/λ.

Sample code can be found at the link below.

https://stackoverflow.com/a/5615564/1650437

Upvotes: 0

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