Simulating Poisson Waiting Times

Question

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.

Hebraist

The easiest thing is to simulate inter-arrivals: https://mcmap.net/q/832273/-how-do-i-generate-discrete-random-events-with-a-poisson-distribution – Soloist

You should for sure rely on well known implementations. Check this out: http://maths.uncommons.org/api/org/uncommons/maths/random/PoissonGenerator.html. – Croze

Answer 1

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.

Ogilvie

Thanks, but something isn't right; When I call this with 1/100, I get values of Infinity every time. The code is public static double waitingTime(double lambda) { return -Math.log(rand.nextDouble())/lambda; } Shouldn't it be valid to say that less than one arrival is expected per unit time? Or do I put in 1 and multiply by the expected wait time? – Hebraist

@Alex: I am only 99% sure, but I think in exponential distribution, lamda is number of occurences per time unit, if you have your average waiting time, you should set lamda=1/average waiting time, could that be the problem? – Ogilvie

Nevermind, I always make this mistake. I called the method with 1/100 instead of 1.0/100.0 This answer works. Thank you so much, I've been reading stackoverflow for a long time, and this is my first time posting, and I'm amazed at how fast and accurate your answer was. – Hebraist

Answer 2

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://mcmap.net/q/832273/-how-do-i-generate-discrete-random-events-with-a-poisson-distribution

Soloist

Hi, I need to generate random numbers in Poisson interval rate using java.. I tried using your function and method poissonRandomInterarrivalDelay always returns zero for any value of lambda. – Sudderth