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We solve main two-boundary problems for a random walk. The generating function of the joint distribution of the first exit
time of a random walk from an interval and the value of the overshoot of the random walk over the boundary at exit time is
determined. We also determine the generating function of the joint distribution of the first entrance time of a random walk
to an interval and the value of the random walk at this time. The distributions of the supremum, infimum, and value of a random
walk and the number of upward and downward crossings of an interval by a random walk are determined on a geometrically distributed
time interval. We give examples of application of obtained results to a random walk with one-sided exponentially distributed
jumps.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1485–1509, November, 2007. 相似文献
2.
For a Poisson process with exponentially distributed negative component, we obtain integral transforms of the joint distribution
of the time of the first exit from an interval and the value of the jump over the boundary at exit time and the joint distribution
of the time of the first hit of the interval and the value of the process at this time. On the exponentially distributed time
interval, we obtain distributions of the total sojourn time of the process in the interval, the joint distribution of the
supremum, infimum, and value of the process, the joint distribution of the number of upward and downward crossings of the
interval, and generators of the joint distribution of the number of hits of the interval and the number of jumps over the
interval.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 7, pp. 922–953, July, 2006. 相似文献
3.
Several two-boundary problems are solved for a special Lévy process: the Poisson process with an exponential component. The
jumps of this process are controlled by a homogeneous Poisson process, the positive jump size distribution is arbitrary, while
the distribution of the negative jumps is exponential. Closed form expressions are obtained for the integral transforms of
the joint distribution of the first exit time from an interval and the value of the overshoot through boundaries at the first
exit time. Also the joint distribution of the first entry time into the interval and the value of the process at this time
instant are determined in terms of integral transforms. 相似文献
4.
For a homogeneous process with independent increments, we determine the integral transforms of the joint distribution of the
first-exit time from an interval and the value of a jump of a process over the boundary at exit time and the joint distribution
of the supremum, infimum, and value of the process.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 10, pp. 1359–1384, October, 2005. 相似文献
5.
We investigate boundary functionals of a semicontinuous process with independent increments on an interval with two reflecting boundaries. We determine the transition and ergodic distributions of the process, as well as the distributions of boundary functionals of the process, namely, the time of first hitting the upper (lower) boundary, the number of hittings of the boundaries, the number of intersections of the interval, and the total sojourn time of the process on the boundaries and inside the interval. We also present a limit theorem for the ergodic distribution of the process and asymptotic formulas for the mean values of the distributions considered. 相似文献
6.
In this paper we solve a two-sided exit problem for a difference of a compound Poisson process and a compound renewal process. More specifically, we determine the Laplace transforms of the joint distribution of the first exit time, the value of the overshoot and the value of a linear component at this time instant. The results obtained are applied to solve the two-sided exit problem for a particular class of stochastic processes, i.e. the difference of the compound Poisson process and the renewal process whose jumps are exponentially distributed. The advantage is that these results are in a closed form, in terms of resolvent sequences of the process. We determine the Laplace transforms of the busy period of the systems M ? |G δ |1|B, G δ |M ? |1|B in case when δ~exp?(λ). Additionally, we prove the weak convergence of the two-boundary characteristics of the process to the corresponding functionals of the standard Wiener process. 相似文献
7.
In this article, we determine the integral transforms of several two-boundary functionals for a difference of a compound Poisson
process and a compound renewal process. Another part of the article is devoted to studying the above-mentioned process reflected
at its infimum. We use the results obtained to study a G
δ
|M
ϰ
|1|B system with batch arrivals and finite buffer in the case when δ∼ge(λ). We derive the distributions of the main characteristics of the queuing system, such as the busy period, the time of the
first loss of a customer, the number of customers in the system, the virtual waiting time in transient and stationary regimes.
The advantage is that these results are given in a closed form, namely, in terms of the resolvent sequences of the process. 相似文献
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