Virtual Laboratories > The Poisson Process > 1 2 3 4 5 6 7 [8]
With the method used in this chapter, all of the random variables in the Poisson process on [0, 
)
are ultimately constructed out of a sequence of independent random variables, each having
the exponential distribution with parameter r. Thus, to
simulate this process, we just need to know how to simulate independent exponential
variables using random numbers.
Recall that if F is the distribution function of a random variable X, then F-1 is the quantile function. Moreover, if U is uniformly distributed on the interval (0, 1), (thus, U is a random number) then F-1(U) has the same distribution as X. This quantile method of simulating X requires, of course, that we be able to compute the quantile function F-1 in closed form. Fortunately, this can be done for the exponential distribution.
1.
Show that if Uj, j = 1, 2, ... is a
sequence of random numbers, then the sequence below simulates independent
random variables, each having the exponential distribution with rate parameter r.
Xj = -ln(1 - Uj) / r, j = 1, 2, ...
Thus, these variables simulate the interarrival times for a Poisson process
on [0, ). Therefore, the arrival times are
simulated as
Tk = X1 + X2 + ··· + Xk for k = 1, 2, ...
and the counting variables are simulated as
Nt = #{k: Tk 
t} for t > 0.
We can also simulate a Poisson variable directly. The general method in the following exercise is also is a special case of the quantile method discussed above.
2. Suppose that f
is discrete density function on {0, 1, 2, ...}. If U is uniformly distributed on
(0, 1) (a random number), show that variable defined below has density f:
N = j if and only if f(0) + ··· + f(j -
1) < U f(0) + ··· + f(j).
Now we can use the result in Exercise 4 to simulate a Poisson process in a region D of Rk. We will illustrate the method with the rectangle D = [a, b] × [c, d] in R2 where a < c and b < d. First, use a random number U to simulate a random variable N that has the Poisson distribution with parameter r(b - a)(d - c), as in the previous exercise. Next, if N = n, let U1, U2, ..., Un and V1, V2, ..., Vn be random numbers and define
Xi = a + (b - a)Ui, Yi = c + (d - c)Vi for i = 1, 2, ..., n.
3. Show that the
random points of the Poisson process with rate r on D are simulated by
(Xi, Yi), i = 1, 2, ..., n.
For more information about Poisson processes and their many generalizations, see
2.8.
Let X denote the call length.
2.9.
Let T denote the lifetime
2.14.
Let T denote the time between requests.
2.15.
Let X denote the lifetime.
2.16.
Let X denote the position of the first defect.
3.4. 0.1991
3.5. 0.1746
3.10. 2, 0.6325
3.11. r = 1 /
10, k = 4
3.16. 0.5752
3.20.
r = hits 6.67 per minute
4.6. 0.7798
4.7. 0.8153
4.12. 32, 5.657
4.20. 0.8818
4.23. 0.6
4.26. 0.9452
4.30.
r = 5.7 per minute
5.6. 0.5814
5.11.
6.10. 0.7350
6.13.
7.3.
7.4.
7.5.
7.12. 0.0491
7.15. 0.2146
7.17.