Poisson Cluster process as a model for teletraffic arrivals and its ...

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How long is the length of each cluster? The answer to this question is of importance for various structural properties of the cluster process, in particular for its dependence structure. Obviously, S K = K� Xi . i=1 Under what conditions is S K regularly varying? We start with the case when X has heavier tail than K. Proposition 1. Assume that P(X > x) is regularly varying for some α > 0, EK < ∞ and P(K > x) = o(P(X > x)). Then, as x → ∞, P(S K > x) ∼ EK P(X > x) . 9
Perhaps more natural in applications to communication networks is the situation where the tail K has heavier tail than X. Proposition 2. Assume P(K > k) is regularly varying with index β ≥ 0. If β = 1, assume that EK < ∞. Moreover, assume that EX < ∞ and P(X > x) = o(P(K > x)). Then, as x → ∞, P(S K > x) ∼ P(K > (EX) −1 x) ∼ (EX) β P(K > x) . Interestingly, the statement of the proposition fails, in general, in the case β = 1 and EK = ∞, but will still hold under even stronger assumptions on X. 10
Page 1 and 2: Poisson Cluster process as a model
Page 3 and 4: Most of the models considered are b
Page 5 and 6: Claim 1. A necessary and sufficient
Page 7 and 8: If EK 2 = ∞ then the actual rate
Page 9: On the other hand, if EK 2 = ∞, t
Page 13 and 14: What happens if the tail of X is su

Poisson Cluster process as a model for teletraffic arrivals and its ...

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