# Kadankov V. F.

### Two-boundary problems for a random walk

Kadankov V. F., Kadankova T. V., Yezhov I. I.

Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1485–1509

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.

### Two-boundary problems for a Poisson process with exponentially distributed component

Kadankov V. F., Kadankova T. V.

Ukr. Mat. Zh. - 2006. - 58, № 7. - pp. 922–953

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.

### On the distribution of the time of the first exit from an interval and the value of a jump over the boundary for processes with independent increments and random walks

Kadankov V. F., Kadankova T. V.

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1359–1384

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.

### System $G|G^κ|1$ with Batch Service of Calls

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 447-465

For the queuing system *G*|*G* ^{κ}|1 with batch service of calls, we determine the distributions of the following characteristics: the length of a busy period, the queue length in transient and stationary modes of the queuing system, the total idle time of the queuing system, the output stream of served calls, etc.

### Main Probability Characteristics of the Queuing System $G^k|G|1$

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1343-1357

For the queuing system *G* ^{κ}|*G*|1 with batch arrivals of calls, we present the distributions of the following characteristics: the length of a busy period, queue length in transient and stationary modes of the queuing system, total idle time of the queuing system, virtual waiting time to the beginning of the service, input stream of calls, output stream of served calls, etc.

### Boundary Functionals for the Difference of Nonordinary Renewal Processes with Discrete Time

Ukr. Mat. Zh. - 2000. - 52, № 10. - pp. 1345-1356

For the difference of nonordinary renewal processes, we find the distribution of the main boundary functionals. For the queuing system *D* _{η} ^{δ} |*D* _{ξ} ^{κ} |1, we determine the distribution of the number of calls in transient and stationary modes.

### On the Distribution of the Number of Calls in the Queuing System $D_{η}|D_{ξ}^{\kappa}|1$

Ukr. Mat. Zh. - 2000. - 52, № 8. - pp. 1075-1081

For the queuing system *D* _{η}|*D* _{ξ} ^{k}|1, we determine the distribution of the number of calls in transient and stationary modes.

### On the generating function of the time of first hitting the boundary by a semicontinuous difference of independent renewal processes with discrete time

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 553-561

For a semicontinuous difference of two independent renewal processes, we find the generating function of the time of first hitting the boundary.

### On the distribution of the maximum of the difference of independent renewal processes with discrete time

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1426–1432

We find the distribution of the maximum of the difference of two renewal processes with discrete time that is semicontinuous in discrete topology.

### Limit functionals for a semicontinuous difference of renewal processes with discrete time

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1710–1712

For a difference, semicontinuous in discrete topology, of two renewal processes with discrete time, the distribution of principal limit functionals is found.