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

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The ergodicity of the stationary point process N immediately implies that the number of arrivals N(t) grows roughly linearly with t: N(t) t a.s. → λ(EK + 1) , t → ∞ However, the deviations of the number of arrivals from the straight line may look differently depending, mostly, on the cluster size distribution. Theorem 2. Assume EK2 < ∞. Then N satisfies the functional central limit theorem ⎛ ⎜N(rt) − λrt(EK + 1) ⎝ � λrE[(K + 1) 2 ] , 0 ≤ t ≤ 1 ⇒ (B(t) , 0 ≤ t ≤ 1) , as r → ∞ in terms of convergence of the finite-dimensional distributions, where (B(t) ,0 ≤ t ≤ 1,) is the standard Brownian motion. ⎞ ⎟ ⎠ 7
On the other hand, if EK 2 = ∞, then the tail of K affects the limit, and, as usual, the regular variation of the tail is associated with a stable limit. Theorem 2. Assume that P(K > k) is regularly varying with index α ∈ (1,2). Assume also that EX < ∞. Then N satisfies the functional central limit theorem: � � N(rt) − λrt(EK + 1) , 0 ≤ t ≤ 1 Θ(r) ⇒ (Lα(t) , 0 ≤ t ≤ 1) , as r → ∞ in terms of convergence of the finite-dimensional distributions, where Θ : (0, ∞) → (0, ∞) is a nondecreasing function such that lim r P(K > Θ(r)) = 1 r→∞ and (Lα(t) ,0 ≤ t ≤ 1,) is a spectrally positive α-stable Lévy motion with Lα(1) ∼ Sα(σα,1,0), with σα > 0. 8
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: If EK 2 = ∞ then the actual rate
Page 11 and 12: Perhaps more natural in application
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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