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1
Content available The potentials method for a closed queueing system with hysteretic strategy of the service time change
100%
Zhernovyi Y. , Kopytko B.
Journal of Applied Mathematics and Computational Mechanics
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2015
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tom Vol. 14, nr 2
131--143
EN
We propose a method for determining the characteristics of a single-channel closed queueing system with an exponential distribution of the time generation of service requests and arbitrary distributions of the service times. In order to increase the system capacity, two service modes (the main mode and overload mode), with the service time distribution functions F ( x ) and F ( x ) respectively, are used. The overload mode starts functioning if at the beginning of service of the next customer the number of customers in the system ξ(t ) satisfies the condition ξ (t ) > h2. The return to the main mode carried out at the beginning of service of the customer, for which ξ (t ) = h1, where 1 ≤ h1 < h2. The Laplace transforms for the distribution of the number of customers in the system during the busy period and for the distribution function of the length of the busy period are found. The developed algorithm for calculating the stationary characteristics of the system is tested with the help of a simulation model constructed with the assistance of GPSS World tools.
2
Content available Steady-state characteristics of three-channel queueing systems with Erlangian service times
100%
Kopytko B. , Zhernovyi K.
Journal of Applied Mathematics and Computational Mechanics
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2016
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tom Vol. 15, nr 3
75--87
EN
We propose a method of study the M/E2/3/∞ queueing systems: standard system and systems with the threshold and hysteretic strategies of the random dropping of customers in order to control the input flow. Recurrence relations to compute the stationary distribution of the number of customers and the steady-state characteristics are obtained. The developed algorithms are tested on examples using simulation models constructed with the assistance of the GPSS World tools.
3
Content available The system M2θ/G/1/m with threshold control of the arrival rate and service time
100%
Zhernovyi Y. , Kopytko B.
Journal of Applied Mathematics and Computational Mechanics
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2014
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tom Vol. 13, nr 2
149--163
EN
We consider a M2θ/G/1/m queueing system with arrival of customer batches, which uses a threshold control mechanism of the service time and arrival rate. The system receives two independent flows of customers, one of which is blocked in an overload mode (under the condition that the number of customers in the system exceeds a given threshold value h). Full blocking of the input flow is carried out from the moment when the queue length reaches the number m until the beginning of the service of the first customer, for which the number of customers in the system does not exceed h. From the beginning of the service of the first customer during the excess of number of customers in the system of h until the completion of full blocking the time of service of customer is distributed under the law of F(x) (an increased service rate is used). Rest of the time the system applies the normal service rate with the distribution function F(x) of service time. Laplace transforms for the distributions of the number of customers in the system during the busy period and for the distribution function of the busy period are found. The average duration of the busy period is obtained. Formulas for the stationary distribution of the number of customers in the system, for the probability of service and for the stationary characteristics of the system are established. The obtained results are verified with the help of a simulation model constructed with the assistance of GPSS World tools.
4
Content available remote Feller semigroup for diffusion process piesewise-constant generalized diffusion matrix and generalized drift vector
100%
Kopytko B. , Novosyadlo A.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2012
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tom Vol. 11, nr 4
75-84
EN
We obtain an integral representation of the classical solution of the conjugation problem for the second order parabolic equation with derivatives with respect to tangent variables at the conjugation conditions. Using this solution, we construct the Feller semigroup to which there corresponds a diffusion process with a piecewise-constant generalized diffusion matrix and a generalized drift vector.
5
Content available Recurrence relations for two-channel queueing systems with Erlangian service times and hysteretic strategy of random dropping of customers
100%
Zhernovyi Y. , Kopytko B.
Journal of Applied Mathematics and Computational Mechanics
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2018
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tom Vol. 17, nr 2
93--103
EN
This article proposes a method of study the M/Es/2/m and M/Es/2/∞ queueing systems with a hysteretic strategy of random dropping of customers. Recurrence relations are obtained to compute the stationary distribution of the number of customers and steadystate characteristics. The constructed algorithms were tested on examples with the use of simulation models constructed with the help of GPSS World.
6
Content available Recurrence relations for a multi-channel closed queueing system with Erlangian service times of second order
100%
Zhernovyi Y. , Kopytko B.
Journal of Applied Mathematics and Computational Mechanics
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2017
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tom Vol. 16, nr 3
123--128
EN
We propose a method for determining the steady-state characteristics of a multichannel closed queueing system with exponential distribution of the time generation of service requests and the second order Erlang distributions of the service times. Recurrence relations to compute the steady-state distribution of the number of customers are obtained. The developed algorithms are tested on examples using simulation models constructed with the assistance of the GPSS World tools.
7
Content available remote On constructing a multidimensional diffusion process with a membrane located on a given hyperplane and acting in an oblique direction
100%
Portenko M. , Kopytko B.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2012
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tom Vol. 11, nr 3
137-147
EN
Using the methods of the theory of classical potentials, we have constructed a Feller semigroup of linear operators that generates a multidimensional diffusion process whose diffusion matrix is given by a sufficiently regular function and whose drift vector is given by a generalized function of the type of a δ - function concentrated on a given hyperplane. Such process can serve as a mathematical model for describing the motion of a diffusing particle in a medium where a membrane is located on the hyperplane. The particle is receiving "a pulse of infinite intensity" at those instants of time when it is hitting the hyperplane. The direction of those pulses are determined by a vector field given on that hyperplane. It is important to emphasize that the trajectories of the process constructed are continuous.
8
Content available Diffusions in one-dimensional bounded domains with reflection, absorption and jumps at the boundary and at some interior point
100%
Kopytko B. , Shevchuk R.
Journal of Applied Mathematics and Computational Mechanics
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2013
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tom Vol. 12, nr 1
55--68
EN
By the method of classical potential theory, we obtain the integral representation of the two-parameter operator semigroup that describes the inhomogeneous Feller process on a closed interval [ r1,r2 ] that is a result of pasting together two diffusion processes given on ( r1,r, ) and ( r, r2 ), respectively, where - ∞ < r1 < r < r2 <∞ .
9
Content available remote Stationary characteristics of M θ /M/1 queue with switching of service modes
100%
Kopytko B. , Zhernovyi K.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2011
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tom Vol. 10, nr 2
117-128
EN
For an M θ /M/1 queue with a threshold switching of service modes at the start of the service of the next customer an algorithm for determining the stationary distribution of the number of customers and stationary characteristics (average queue length, average waiting time, variance of queue length) is proposed. In the case the minimum number of incoming customers in the group is comparable to threshold value h, the stationary characteristics are found in an explicit form. The results are verified by simulation models constructed with the assistance of GPSS World.
10
Content available Solution of one initial-boundary Wentzel problem for a parabolic equation with discontinuous coefficients by the boundary integral equation method
100%
Kopytko B. , Tsapovska Z.
Journal of Applied Mathematics and Computational Mechanics
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2017
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tom Vol. 16, nr 1
51--62
EN
In this article we consider the question of existence in the Holder class of the solution of the initial-boundary problem for a linear parabolic second-degree equation with discontinuous coefficients in noncylindrical domain. This domain is bounded of the smooth elementary surfaces of the Holder class H 2+λ, (2+λ )/2. The boundary conditions and the conjugation condition of the Wentzel type are given to external and internal boundaries of a domain respectively. We use the potential method to solve this problem.
11
Content available Recurrence relations for two-channel closed queueing systems with Erlangian service times
100%
Kopytko B. , Zhernovyi K.
Journal of Applied Mathematics and Computational Mechanics
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2018
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tom Vol. 17, nr 1
37--48
EN
This paper proposes a method for determining the steady-state characteristics of two-channel closed queueing systems with an exponential distribution of the time generation of service requests and the Erlang distributions of the service times. Recurrence relations for computing the steady-state distribution of the number of customers in the system are deduced. The obtained algorithms are tested on examples using simulation models in the GPSS World environment.
12
Content available One-dimensional diffusion processes in half-bounded domains with reflection and a possible jump-like exit from a moving boundary
100%
Kopytko B. , Shevchuk R.
Journal of Applied Mathematics and Computational Mechanics
|
2016
|
tom Vol. 15, nr 4
71-82
EN
By the method of the classical potential theory, we construct the two-parameter Feller semigroup of operators associated with such a diffusion phenomenon on a half-line with a moving boundary where either a reflection or jump phenomenon occurs at a boundary point.
13
Content available One-dimensional diffusions in bounded domains with a possible jump-like exit from a sticky boundary
100%
Kopytko B. , Shevchuk R.
Journal of Applied Mathematics and Computational Mechanics
|
2014
|
tom Vol. 13, nr 3
101--114
EN
Using analytical methods we obtain the integral representation of a two-parameter Feller semigroup on a closed interval [r1, r2] corresponding to such a diffusion phenomenon that sticking, partial reflection, absorption and jump phenomena occur at the endpoints r1, r2 and at some interior point r ∈ (r1,r2).
14
Content available The queueing system M2X/M/n with hysteretic control of the input flow intensity
100%
Zhernovyi Y. , Kopytko B.
Journal of Applied Mathematics and Computational Mechanics
|
2014
|
tom Vol. 13, nr 1
149--161
EN
We consider a multi-channel queueing system with unlimited queue and with exponentially distributed service time and the intervals between the arrival of customers batches, which uses a hysteretic control mechanism of the input flow intensity. The system receives two independent flows of customers, one of which is blocked in an overload mode. An algorithm for finding the stationary distribution of the number of customers and stationary characteristics (the mean queue length, the mean waiting time in the queue, the probability of customers loss) is proposed. The obtained results are verified with the help of a simulation model constructed with the assistance of GPSS World tools.
15
Content available The potentials method for the M/G/1/m queue with customer dropping and hysteretic strategy of the service time change
100%
Zhernovyi Y. , Kopytko B.
Journal of Applied Mathematics and Computational Mechanics
|
2016
|
tom Vol. 15, nr 1
197--210
EN
We propose a method for determining the probabilistic characteristics of the M/G/1/m queueing system with the random dropping of arrivals and distribution of the service time depending on the queue length. Two sets of service modes, with the service time distribution functions Fn (x) and Fn (x) respectively, are used according to the twothreshold hysteretic strategy. The Laplace transforms for the distribution of the number of customers in the system during the busy period and for the distribution function of the length of the busy period are found. The developed algorithm for calculating the stationary characteristics of the system is tested with the help of a simulation model constructed with the assistance of GPSS World tools.
16
Content available On characteristics of the Mθ/G/1/m and Mθ/G/1 queues with queue-size based packet dropping
80%
Zhernovyi Y. , Kopytko B. , Zhernovyi K.
Journal of Applied Mathematics and Computational Mechanics
|
2014
|
tom Vol. 13, nr 4
163--175
EN
We study the Mθ/G/1/m and Mθ/G/1 queuing systems with the function of the random dropping of customers used to ensure the required characteristics of the system. Each arriving packet of customers can be rejected with a probability defined depending on the queue length at the service beginning of each customer. The Laplace transform for the distribution of the number of customers in the system on the busy period is found, the mean duration of the busy period is determined, and formulas for the stationary distribution of the number of customers in the system are derived via the approach based on the idea of Korolyuk’s potential method. The obtained results are verified with the help of a simulation model constructed with the assistance of GPSS World tools.
17
Content available The busy period for the Mθ /G/1/m system with service time dependent of the queue length
80%
Zhernovyi Y. , Kopytko B. , Zhernovyi K.
Journal of Applied Mathematics and Computational Mechanics
|
2013
|
tom Vol. 12, nr 4
127--133
EN
We consider the Mθ/G/1/m system wherein the service time depends on the queuelength and it is determined at the beginning of customer service. Using an approach basedon the potential method proposed by V. Korolyuk, the Laplace transforms for the distribution f the number of customers in the system on the busy period and for the distribution function of the busy period are found.
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