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

More documents

Recommendations

Info

$Øvelse 1: Få LATEX op at stå Øvelse 2: At skrive matematiske ...$

Claim 2. � ∞ 0 γ N(h) dh < ∞ if and only if EK 2 < ∞. The divergence of the integral is often taken is an indication of long range dependence. In this sense, presence or absence of long memory in the arrivals of a cluster process depends only on the cluster size K, but not on the incluster interarrival time X. To know more about the rate of decay of the covariance function one does need to have information about the in-cluster interarrival times. 5
If EK 2 = ∞ then the actual rate of decay of the covariance function depends on the tail of K, and on in-cluster interarrival time distribution. Here is one possible situation. Theorem 1. Assume that P(K > k) is regularly varying with index α ∈ (1,2) or α = 1 and EK < ∞. Assume also that X has a nonarithmetic distribution and EX < ∞. Then γ N(h) ∼ λ(EX) α−2 � ∞ h F K(y) dy α−2 1 ∼ λ(EX) α − 1 h F K(h) , if α > 1 as h → ∞. 6
Page 1 and 2: Poisson Cluster process as a model
Page 3 and 4: Most of the models considered are b
Page 5: Claim 1. A necessary and sufficient
Page 9 and 10: On the other hand, if EK 2 = ∞, t
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 ...

Create successful ePaper yourself

Delete template?

Save as template?