Discrete-time stochastic processes

# This allows all the properties of homogeneous poisson

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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 &lt; s1 &lt; · · · , &lt; sn &lt; 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 ° ° ° ° 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 &lt; t with Xi &gt; 0 for 1 ≤ i ≤ n (see Figure 2.10). i=1 fX (n) |N (t) (x (n) | n) = n! tn for X (n) &gt; 0, Xn i=1 Xi &lt; 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 &lt; ui ≤ t, 1 ≤ i ≤ n. Both fS (n) and fU (n) are uniform over the volume of n-space where they are non-zero, 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 n-cube is partitioned into n! regions, one where u1 &lt; u2 &lt; · · · &lt; un . For each permutation π (i) of the integers 1 to n, there is another region9 where uπ(1) &lt; uπ(2) &lt; · · · &lt; uπ(n) . By symmetry, each of these regions has the same volume, which then must be 1/n! of the volume tn of the n-cube. 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 ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣° ♣ ° ♣ ♣ ♣♣ ♣ ♣ 2♣♣° f =2/t ♣ ♣ ♣♣ s2 ♣♣♣ ♣♣♣ ♣♣♣ ♣♣♣ ♣♣♣ ♣♣♣ ♣♣°° ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣° ♣ ♣♣ ♣♣ ♣° ♣° 0 0 s1 ♣♣ ♣♣ ♣ u2 ♣♣♣ ♣♣ ♣♣ ♣ 0 t 0 ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ f =1/t2 u1 ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ t Figure 2.11:...
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