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Unformatted text preview: esult is (2.34).
Proof 2: This alternative proof derives (2.34) by looking at arrivals in very small increments
of size δ (see Figure 2.9). For a given t and a given set of n times, 0 < s1 < · · · , < sn < t, we
calculate the probability that there is a single arrival in each of the intervals (si , si + δ ], 1 ≤
i ≤ n and no other arrivals in the interval (0, t]. Letting A(δ ) be this event,
✲δ✛ 0 ✲δ✛ ✲δ✛ s1 s2 s3 t Figure 2.9: Intervals for arrival density. Pr {A(δ )} = pN (s1 ) (0) pN (s1 ,s1 +δ) (1) pN (s1 +δ,s2 ) (0) pN (s2 ,s2 +δ) (1) · · · pN (sn +δ,t) (0).
e
e
e
e The sum of the lengths of the above intervals is t, so if we represent pN (si ,si +δ) (1) as
e
∏δ exp(−∏δ ) + o(δ ) for each i, then
Pr {A(δ )} = (∏δ )n exp(−∏t) + δ n−1 o(δ ).
The event A(δ ) can be characterized as the event that, ﬁrst, N (t) = n and, second, that
the n arrivals occur in (si , si +δ ] for 1 ≤ i ≤ n. Thus we conclude that
fS (n) N (t) (s (n) ) = lim δ →0 Pr {A(δ )}
,
δ n pN (t) (n) which simpliﬁes to (2.34).
The joint density of the interarrival intervals, X (n) = X1 . . . , Xn given N (t) = n can be
found directly from Theorem 2.6 simply by making the linear transformation X1 = S1 2.5. CONDITIONAL ARRIVAL DENSITIES AND ORDER STATISTICS °
°
°
°
s2
°
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° 79 ❅
❅
❅
x2
❅
❅
❅
❘
❅
❅
❅
❅ s1 x1 Figure 2.10: Mapping from arrival epochs to interarrival times. Note that incremental
cubes in the arrival space map into parallelepipeds of the same volume in the interarrival
space. and Xi = Si − Si−1 for 2 ≤ i ≤ n. The density is unchanged, but the constraint region
P
transforms into n Xi < t with Xi > 0 for 1 ≤ i ≤ n (see Figure 2.10).
i=1
fX (n) N (t) (x (n)  n) = n!
tn for X (n) > 0, Xn i=1 Xi < t. (2.36) It is also instructive to compare the joint distribution of S (n) conditional on N (t) = n with
the joint distribution of n IID uniformly distributed random variables, U (n) = U1 , . . . , Un
on (0, t]. For any point U (n) = u (n) , this joint density is
fU (n) (u (n) ) = 1/tn for 0 < ui ≤ t, 1 ≤ i ≤ n.
Both fS (n) and fU (n) are uniform over the volume of nspace where they are nonzero, but
as illustrated in Figure 2.11 for n = 2, the volume for the latter is n! times larger than the
volume for the former. To explain this more fully, we can deﬁne a set of random variables
S1 , . . . , Sn , not as arrival epochs in a Poisson process, but rather as the order statistics
function of the IID uniform variables U1 , . . . , Un ; that is
S1 = min(U1 , . . . , Un ); S2 = 2nd smallest (U1 , . . . , Un ); etc.
The ncube is partitioned into n! regions, one where u1 < u2 < · · · < un . For each
permutation π (i) of the integers 1 to n, there is another region9 where uπ(1) < uπ(2) < · · · <
uπ(n) . By symmetry, each of these regions has the same volume, which then must be 1/n!
of the volume tn of the ncube.
Each of these n! regions map into the same region of ordered values. Thus these order
statistics have the same joint probability density function as the arrival epochs S1 , . . . , Sn
conditional on N (t) = n. Thus anything we know (or can discover) about order statistics
is valid for arrival epochs given N (t) = n and vice versa.10
9 As usual, we are ignoring those points where ui = uj for some i, j , since the set of such points has 0
probability.
10
There is certainly also the intuitive notion, given n arrivals in (0, t], and given the stationary and
independent increment properties of the Poisson process, that those n arrivals can be viewed as uniformly
distributed. It does not seem worth the trouble, however, to make this precise, since there is no natural way
to associate each arrival with one of the uniform rv’s. 80 CHAPTER 2. POISSON PROCESSES t ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ t ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣°
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 Spring '09
 R.Srikant

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