Queues and Lévy Fluctuation Theory by Krzysztof Dębicki & Michel Mandjes

Author:Krzysztof Dębicki & Michel Mandjes
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Relying on standard results for the Brownian bridge, this equals

observe that this expression does not depend on d (why?). Then it is readily verified that

This gives us a way to sample − inf0 ≤ s ≤ t X s , conditional on the value of X t . As a result, we have found an efficient way to sample Q t .

Now return to the setting of X corresponding to . The idea is to iteratively simulate the workload at the jump epochs of the Poisson process. In the following pseudocode E(λ) stands for a sample from the exponential distribution with mean λ −1, and N(d, σ 2) for a sample from the normal distribution with mean d and variance σ 2; B refers to a sample from the distribution of the job sizes in the compound Poisson process. The pseudocode generates an exact sample of Q t in the case that X corresponds to the sum of a Brownian motion and a compound Poisson process.

Pseudocode 10.1

Input: Q 0 = x and t; ; . Output: Q t .

T: = 0;  Q = x;

while T < t do

s: = E(λ);  ;

if T < t then

z: = N(ds, σ 2 s);  ;  ;

else

; z = N(dr, σ 2 r);  ;

end (of ‘if’);

end (of ‘while’); return Q t : = Q.

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