0.Virtual Laboratories > The Poisson Process > 1 [2] 3 4 5 6 7 8
The basic assumption for the Poisson process is that the behavior of the process after an arrival should be independent of the behavior before the arrival and probabilistically like the original process (regeneration).
In particular, the general regeneration assumption means that the times between arrivals, known as interarrival times, must be independent, identically distributed random variables. Formally, the interarrival times are defined as follows:
X1 = T1, Xk = Tk - Tk-1 for k = 2, 3, ...
where Tk is the time of the k'th arrival. We will assume that
P(Xi > t) > 0 for each t > 0.
Now, we also want regeneration to occur at a fixed time t. In particular, if the first arrival has not occurred by time t, then the time remaining until the arrival occurs has the same distribution as the first arrival time itself. This is known as the memoryless property and can be stated in terms of a generic interarrival time X as follows
P(X > t + s) | X > s) = P(X
> t) for all s, t
0.
Let G denote the right-tail distribution function of X:
G(t) = P(X > t), t 0.
1. Show that the
memoryless property is equivalent to the law of exponents:
G(t + s) = G(t)G(s) for
all s, t 0.
2. Show
that the only solutions of the functional equation in Exercise 1, which are continuous
from the right, are exponential functions. Let c = G(1). Successively
show that
In the context of Exercise 2 let r = -ln(c). Then r > 0 (since 0 < c < 1) so
G(t) = P(X > t) = e-rt, t 0.
Hence X has a continuous distribution with cumulative distribution function given by
F(t) = P(X 
t)
= 1 - G(t) = 1 - e-rt,
t 0.
3. Show that the
density function of X is
f(t) = re-rt, t 0.
A random variable with this density is said to have the exponential distribution distribution with rate parameter r. The reciprocal 1 / r is known as the scale parameter.
4. Show directly
that the exponential density really is a probability density function.
5. In the
exponential experiment, vary r with the scroll bar and watch how the shape of the
probability density function changes. Now set r = 2, run the experiment 1000 times
with an update frequency of 10, and watch the apparent convergence of the empirical
density function to the probability density function.
6. In the
exponential experiment, set r = 1. Run the experiment 1000 times, updating after
each run. Compute the appropriate relative frequencies to empirically investigate the
memoryless property
P(X > 3 | X > 1) = P(X > 2).
7.
Show that the quantile
function of X is
F-1(p) = -ln(1 - p) / r for 0 < p < 1.
In particular, the median of X occurs at ln(2) / r, the first quartile at [ln(4) - ln(3)] / r, and the third quartile at ln(4) / r.
8. Suppose that
the length of a telephone call (in minutes) is exponentially distributed with rate
parameter r = 0.2.
9. Suppose that
the lifetime of a certain electronic component (in hours) is exponentially distributed
with rate parameter r = 0.001.
The following exercises give the mean, variance, and moment generating function of the exponential distribution.
10. Show that
E(X) = 1 / r.
11. Show that var(X)
= 1 / r2.
12. Show that E[exp(uX)]
= r / (r - u) for u < r.
The parameter r is known as the rate of the Poisson process. On average, there are 1 / r time units between arrivals, so the arrivals come at an average rate of r per unit time. Note also that the median is always smaller than the mean for the exponential distribution:
ln(2) / r < 1 / r.
13. In the
exponential experiment, vary r with the scroll bar and watch how the mean/standard
deviation bar changes. Now set r = 0.5, run the experiment 1000 times with an
update frequency of 10, and watch the apparent convergence of the empirical mean and
standard deviation to the distribution mean and standard deviation, respectively.
14. Suppose
that the time between requests to a web server (in seconds) is exponentially distributed
with rate parameter 2.
15. Suppose that
the lifetime X of a fuse (in 100 hour units) is exponentially distributed with
16. The position X
of the first defect on a digital tape (in cm) has the exponential distribution with mean
100.