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Unformatted text preview: ing process is equivalent to the Erlang density for
the successive arrival epochs. Specifying the probability density for S1 , S2 , . . . , as Erlang
speciﬁes the marginal densities of S1 , S2 , . . . ,, but need not specify the joint densities of
these rv’s. Figure 2.4 illustrates this in terms of the joint density of S1 , S2 , given as
fS1 S2 (s1 s2 ) = ∏2 exp(−∏s2 ) for 0 ≤ s1 ≤ s2 and 0 elsewhere. The ﬁgure illustrates how the joint density can be changed without
changing the marginals. ✑
✑
✑
✑
s1
✑
✑
✑
✑
✑
✑ 0 fS1 S2 (s1 s2 ) > 0 s2 Figure 2.4: The joint density of S1 , S2 is nonzero in the region shown. It can be
changed, while holding the marginals constant, by reducing the joint density by ε in
the upper left and lower right squares above and increasing it by ε in the upper right
and lower left squares. There is a similar eﬀect with the Bernoulli process in that a discrete counting process for
which the number of arrivals from 0 to t, for each integer t is a binomial rv, but the process
is not Bernoulli. This is explored in Exercise 2.5.
The next deﬁnition of a Poisson process is based on its incremental properties. Consider
e
the number of arrivals in some very small interval (t, t + δ ]. Since N (t, t + δ ) has the same 68 CHAPTER 2. POISSON PROCESSES distribution as N (δ ), we can use (2.15) to get
n
o
e
Pr N (t, t + δ ) = 0
= e−∏δ ≈ 1 − ∏δ + o(δ )
n
o
e
Pr N (t, t + δ ) = 1
= ∏e−∏δ ≈ ∏δ + o(δ )
n
o
e
Pr N (t, t + δ ) ≥ 2
≈ o(δ ). (2.17) Deﬁnition 3 of a Poisson process: A Poisson counting process is a counting process
that satisﬁes (2.17) and has the stationary and independent increment properties.
We have seen that Deﬁnition 1 implies Deﬁnition 3. The essence of the argument the other
way is that for any interarrival interval X , FX (x + δ ) − FX (x) is the probability of an arrival
in an appropriate inﬁnitesimal interval of width δ , which by (2.17) is ∏δ + o(δ ). Turning this
into a diﬀerential equation (see Exercise 2.7), we get the desired exponential interarrival
intervals. Deﬁnition 3 has an intuitive appeal, since it is based on the idea of independent
arrivals during arbitrary disjoint intervals. It has the disadvantage that one must do a
considerable amount of work to be sure that these conditions are mutually consistent, and
probably the easiest way is to start with Deﬁnition 1 and derive these properties. Showing
that there is a unique process that satisﬁes the conditions of Deﬁnition 3 is even harder,
but is not necessary at this point, since all we need is the use of these properties. Section
2.2.5 will illustrate better how to use this deﬁnition (or more precisely, how to use (2.17)).
What (2.17) accomplishes, beyond the assumption of independent and stationary increments, in Deﬁnition 3 is the prevention of bulk arrivals. For example, consider a counting
process in which arrivals always occur in pairs, and the intervals between successive pairs
are IID and exponentially distributed with parameter ∏ (see Figure 2.5). For this process,
n
o
n
o
e
e
Pr N (t, t + δ )=1 = 0, and Pr N (t, t+δ )=2 = ∏δ + o(δ ), thus violating (2.17). This
process has stationary and independent increments, however, since the process formed by
viewing a pair of arrivals as a single incident is a Poisson process. N (t)
4
✛ 2
✛ 0 X1 X2 ✲3 ✲1 S1 S2 Figure 2.5: A counting process modeling bulk arrivals. X1 is the time until the ﬁrst
pair of arrivals and X2 is the interval between the ﬁrst and second pair of arrivals. 2.2. DEFINITION AND PROPERTIES OF THE POISSON PROCESS 2.2.5 69 The Poisson process as a limit of shrinking Bernoulli processes The intuition of Deﬁnition 3 can be achieved in a much less abstract way by starting with
the Bernoulli process, which has the properties of Deﬁnition 3 in a discretetime sense. We
then go to an appropriate limit of a sequence of these processes, and ﬁnd that this sequence
of Bernoulli processes converges in various ways to the Poisson process.
Recall that a Bernoulli process is an IID sequence, Y1 , Y2 , . . . , of binary random variables
for which pY (1) = q and pY (0) = 1 − q . We can visualize Yi = 1 as an arrival at time i and
Yi = 0 as no arrival, but we can also ‘shrink’ the time scale of the process so that for some
integer j > 0, Yi is an arrival or no arrival at time i2−j . We consider a sequence indexed
by j of such shrinking Bernoulli processes, and in order to keep the arrival rate constant,
we let q = ∏2−j for the j th process. Thus for each unit increase in j , the Bernoulli process
shrinks by replacing each slot with two slots, each with half the previous arrival probability.
The expected number of arrivals per time unit is then ∏, matching the Poisson process that
we are approximating.
If we look at this j th process relative to Deﬁnition 3 of a Poisson process, we see that
for these regularly spaced increments o...
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.
 Spring '09
 R.Srikant

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