This article is cited in
1 paper
On some generalizations of the Pollachek–Khinchine formula
D. V. Gusak (Husak) Institute of Mathematics, Ukr. Nat. Acad. Sci. Kyiv
Abstract:
For the skip-free Poisson process
$\xi(t) \ (t\geq0, \xi(0)=0),$ $$ \xi(t)=at+S(t), \ a<0, \ S(t)=\sum_{k\leq\nu(t)}\xi_k, \ \xi_k>0, \xi(0)=0, $$
where
$\nu(t)$ is a simple Poisson process with intensity
$\lambda>0,$ the moment generating function (m.g.f.) of
$\xi^+=\sup_{0\leq t<\infty}\xi(t)$ is defined by the well-known Pollachek–Khinchine formula under the condition
$m=E\xi (1)<0$ (see [1-3]).
For a homogeneous process
$\xi(t)$ with bounded variation, we establish prelimit and limit generalizations of this formula, which define the m.g.f. of
$$ \xi^+(\theta_s)=\sup_{0\leq t\leq\theta_s}\xi (t), \ \xi^+=\lim_{s\to0}\xi^+(\theta_s) \left(P\{\theta_s>t\}=e^{-st}, \ s>0\right). $$
These generalizations are essentially based on the condition
$P\{\tau ^+(0)= \gamma ^+(0)=0\}=0,$ where
$(\tau ^+(0),\gamma ^+(0))$ is the initial ladder point of
$\xi (t)\ (t\geq0, \xi(0)=0)$.
Some another relations for the m.g.f. of
$\xi^+(\theta_s)$ and
$\xi^+$ are established for the general lower semicontinuous process
$\xi(t)$ on the base of results in [3-5].
Keywords:
Semicontinuous processes, Pollachek–Khinchine formula.
MSC: Primary
60G50; Secondary
60K10
Language: English
Fulltext:
PDF file (171 kB)