CC BY
4
ВЕСТНИК ТОМСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА
2024 Управление, вычислительная техника и информатика № 68
Tomsk State University Journal of Control and Computer Science
Original article UDC 519.872.5 doi: 10.17223/19988605/68/4
Asymptotic analysis of a closed G-network with rewards Tatiana V. Rusilko1, Dmitry A. Salnikov2
12 Yanka Kupala State University of Grodno, Grodno, Republic of Belarus 1 tatiana. rusilko@gmail. com 2 [email protected]
Abstract. A closed exponential G-network with positive and negative customers is studied. In addition, the G-network under study generates a sequence of rewards or earnings associated with network transitions from one state to another. The total network reward is a random process governed by the probabilistic relations of the Markov process that determines the number of customers in the network nodes. The purpose of this paper is to asymptotically study the G-network with rewards under the assumption of a large number of customers. The main objective is to calculate the expected total reward of the G-network in the asymptotic case. It is proved that reward density function satisfies the generalized Kolmogorov backward equation. An ordinary differential equation for the expected reward that the G-network will earn in a time t if it starts in a given initial state, is derived.
Keywords: queueing network; G-network; network with rewards; asymptotic analysis method.
Acknowledgments: The research was supported by the State Program of Scientific Research of the Republic of Belarus "Convergence-2025" (sub-program "Mathematical models and methods", assignment 1.6.01).
For citation: Rusilko, T.V., Salnikov, D.A. (2024) Asymptotic analysis of a closed G-network with rewards. Vestnik Tomskogo gosudarstvennogo universiteta. Upravlenie, vychislitelnaja tehnika i informatika - Tomsk State University Journal of Control and Computer Science. 68. pp. 38-47. doi: 10.17223/19988605/68/4
Научная статья
doi: 10.17223/19988605/68/4
Асимптотический анализ замкнутой G-сети с вознаграждениями
Татьяна Владимировна Русилко1, Дмитрий Александрович Сальников2
12 Гродненский государственный университет им. Янки Купалы, Гродно, Беларусь
1 tatiana. rusilko@gmail. com 2 [email protected]
Аннотация. Исследуется замкнутая экспоненциальная G-сеть массового обслуживания с положительными и отрицательными заявками. Кроме того, исследуемая G-сеть порождает последовательность вознаграждений или доходов, связанных с переходами сети из одного состояния в другое. Полное вознаграждение такой сети является случайным процессом, управляемым вероятностными связями марковского процесса, определяющего число заявок в узлах сети. Цель данной работы - асимптотический анализ G-сети c вознаграждениями в случае большого числа обслуживаемых заявок и решение задачи прогнозирования полного ожидаемого вознаграждения G-сети. Доказано, что плотность вознаграждения удовлетворяет дифференциальному уравнению в частных производных второго порядка, являющемуся обобщением обратного уравнения Колмогорова. Получено обыкновенное дифференциальное уравнение для ожидаемого вознаграждения G-сети, которое сеть получит за определенное время, исходя из данного начального состояния.
Ключевые слова: сеть массового обслуживания; G-сеть; сеть с вознаграждениями; метод асимптотического анализа.
Благодарности: Работа выполнена в рамках государственной программы научных исследований Республики Беларусь «Конвергенция-2025» (подпрограмма «Математические модели и методы», задание 1.6.01).
© T.V. Rusilko, D.A. Salnikov, 2024
Для цитирования: Русилко Т.В., Сальников Д.А. Асимптотический анализ замкнутой G-сети с вознаграждениями // Вестник Томского государственного университета. Управление, вычислительная техника и информатика. 2024. № 68. С. 38-47. doi: 10.17223/19988605/68/4
Introduction
A Markov process is a mathematical model that is useful in the study of complex systems. The basic concepts of a Markov process are «system state» and «state transition». R. Howard proposed that a continuous-time Markov process receives a reward of rj dollars when the system makes a transition from state i to
state j, i Ф j ; the system earns a reward at the rate of ri dollars per unit time during all the time that it occupies state i [1. P. 99]. Note that rij- and ri have different dimensions. According to Howard's concept, it is not necessary that the system earn according to both reward rates and transition rewards [1]. He called rij the
«reward» or «earnings» associated with the process transition from i to j. The set of rewards for the process may be described by a reward matrix R with elements rij-. The rewards need not be in dollars, they could be
energy levels, units of production, or any other physical quantity relevant to the problem. In Howard's interpretation, the Markov process generates a sequence of rewards as it makes transitions from state to state and is called the «Markov process with rewards» [1]. The reward is thus a random process with a probability distribution governed by the probabilistic relations of the Markov process. In [1], the problem of finding the expected total earnings V (t) that the system will earn in a time t if it starts in the state i was solved. Note the term «expected» and notation Vt (t) are used in the sense of «prospective» or relating to the future time interval t if the initial state of the process is i.
The state of a closed exponential queueing network is the vector k(t) with elements representing the number of customers at each node. The process k (t) is a continuous-time Markov process on the finite state space. The concept of Markov processes with rewards was first used by M. Matalytski as fundamental for defining exponential queueing networks that generate earnings [2, 3]. Queueing networks with earnings were called HM-networks (Howard - Matalytski).
G-networks are generalized queueing networks of queueing nodes with several types of customers: positive customers, negative customers and in some cases triggers. Negative customers and triggers are not serviced, so they are identified as signals. In this article, only negative customers are considered as signals. When a negative customer arrives at a node, one or a group of positive customers is removed or «killed» in a non-empty queue. G-networks were first introduced and studied by E. Gelenbe [4-6]. Their field of application is modelling computing systems and networks, evaluating their performance, modelling biophysical neural networks, pattern recognition tasks and others [7-11].
The purpose of this paper is the asymptotic analysis of a closed exponential G-network with rewards, which implies an approximation method of queueing network study under the critical assumption of a large but limited number of customers [10-11]. We are interested in the expected earnings of the G-network if it operates for a time t with a given initial condition.
1. Formulation of the problem
A closed exponential G-network of a finite number of nodes S0 , Sj, ..., Sn is the focus of this paper. A fixed number K of customers circulate between the nodes. Customers are homogeneous in terms of their service in the network nodes. The node S0 is an IS-node (Infinite Server) of K identical exponential servers capable of servicing all customers in the network in parallel. The IS-node plays the role of an «external environment» or a finite source of K customers. The node S0 generates a Poisson flow of customers with rate A,0k0 , where A,0 is the flow parameter, k0 is the number of customers in the node S0 . This arrival flow is divided into a flow of positive and negative customers. And the probability of a positive customer arriving at short
time interval At is X0k0p+At + o(At), while the probability of a negative customer arriving is kop-At + o(At), i =n, i (p+ + p-) = 1.
Each network node St, i = 1, n , is a queueing system with mi servers and unlimited waiting area for positive customers. The probability of completing the positive customer service at a node Si during a short time interval At is ^ min(mi, kt )At + o(At), where kt is the number of customers in the node Si; the probability of completing the service two or more customers is o(At). The completion of the customer service at different network nodes in a short time interval At are mutually independent events. Customers are served according to the FIFO rule. When a customer has completed service in the node Si it is instantly transferred
to the node Sf as positive with probability p+ or negative with probability p-, otherwise it is transferred to
n i - \ -
the IS-node S0 with probability p+ = 1 _i( p+ + p-), i j , i, j = 1, n . Negative customers arriving at
a node are not served by the node servers; therefore, they are considered as signals. A negative customer arriving at a node St, i = 1, n , removes one positive customer located at the same node and both of them are immediately transferred to the IS-node S0 as positive customers.
The state of the G-network under study can be described by a «-dimensional continuous-time Markov process on finite state space k(t) = (kj(t), k2(t),..., kB(t)), where kt(t) represents the number of customers
in the node Si at the moment t, 0 < kt (t) < K, i = 1, n , t e [0, +<x>). The number of customers in the IS-node
n
S0 is k0(t) = K - i kj (t). Let us assume that the G-network is in the state k if at some time t components
/=1
k,(t) = k,, i = 1,n, form a vector k = (kj, k2,..., kn).
Let us suppose that the G-network described above earns R+ conventional units (c.u.) when a positive customer makes a transition from the node Si to the node Sj and it earns R- c.u. when a negative customer makes the same transition, i ^ j , i, j = 0, n . We call R+ and R- the «reward» associated with the transition of a positive and a negative customer, respectively, from Si to Sj ; we call matrices R+ = {rjand R- = (rjthe reward matrices. Suppose further that the G-network receives a reward at the rate of R(k) c.u. per unit time during all the time that it occupies the state k . The question of interest is: what will be the expected total earnings of the G-network in a time t, t eT , if the network is now in the state k .
The main objective of the paper is to predict the expected total reward of the G-network. It is obvious that the G-network reward is governed by the Markov process k (t) and depends on both the initial network state k and the remaining observation time t, t eT.
2. Asymptotic analysis of a G-network with rewards
General purpose of asymptotic methods in queueing theory is to study servicing processes of queueing systems and networks by finding suitable approximations for them under some critical (limit, asymptotic) assumption. The scientific researches of Tomsk State University are widely known in the field of asymptotic methods [12, 13].
In this paper, a closed exponential G-network with rewards is studied under the asymptotic assumption of a large number of customers K. The passage to the limit from a Markov chain k(t) to a continuous-state Markov process %(t) is used. In contrast to discontinuous processes, continuous processes in any short time interval At ^ 0 have some small change in the state Ax ^ 0. The mathematical approach used in this paper
is based on a discrete model of a continuous Markov process described in many books on the theory of diffusion Markov processes [14, 15].
Let V (k, t) be the expected total G-network reward that the network will ern in a time t if it starts in the state k . Notation v(x, t) is used for reward density in the case of asymptotic approximation, x is the start state, t is remaining time. The concept of reward density is given in the proof of the following theorem.
Theorem 1. In the asymptotic case of a large number of customers K the reward density function v(x, t) provided that it is differentiable with respect to t and twice continuously differentiable with respect
2 2
to Xj, i = 1, n , satisfies up to o(s ) = o(1/ K ), where e = 1/ K, the generalized multidimensional Kolmogorov backward equation:
^ = -¿4(x, t)^ + e f Bj(x, t)^ + q(x), (1)
dt i=1 ox- 2 j, j=1 SXj SXj
n
where 4 (x t) = f ^j min(/j, xj)(pj- - p+ + Sj-) +
j=1
n - f n - \ + min(l! , Xi ) S Pu (1 - 0(xj )) - A 1 - X x I ( p + - p-i )
j=1 J j V /=1 r '
are drift coefficients,
n
Bii (X t) = X M j min(/j, Xj)( p + + p- + 5 ji) +
j=1
n - f n _ N
min(li , X) X p-(1 - 0(x,)) + A 1 - X X I ( p+i - p-i )
j=1 j j V i=1 r '
and
j = 1
Bj (x, t) = ^ min(lj,Xj ) (p- - p+ ), i ^ j , are diffusion coefficients;
f n i \ n
q(x) = r(x) + k^ X^imin(li,x)(p+ri+ + p-ri-(2-0(xj))j + X^imin(l, x)p+°r'+ + (2)
\
n f n
+ X A 1 - X x, I (p+ir°+ + p°,r°, )
i=1 V i=1 r v
is earning rate;
S jj is the Kronecker delta.
( i \
Proof. Let us introduce the following notations: Ii is a n-vector of the form Ii =
0,0,...,0,1,0,0,...,0
(1, x > 0,
9( x) = <! is the Heaviside step function. Let us assume that at the initial moment of time, Markov
[0, x < 0;
process k(t), which determines the G-network state, is in the state k. Taking into account the above assumptions, we consider all possible transitions of k(t) from the given initial state k at t = 0, and the corresponding network earnings in a short time interval At. In this way we can relate the expected total reward in a time t + At, V(k, t + At), to V(k, t) by an equation.
- A transition from the state k to the state k + Ij -1, with probability
^ min(rni, kf)p+ At + o(At), which means the customer was served at the node Si and joined the node Sj as positive, i, j = 1, n . In this case, the network would receive the reward R+ plus the expected total reward V (k +1 j -1,, t) to be made if it starts in the state k + Ij -1, with time t remaining, V (k +1 j -I, t).
- A transition from the state k to the state k - Ij -1, with probability
^ min (mt, kf) p-At + o(At),
which means the customer was served at S; and transmitted to Sj as negative, i, j = 1, n . The network reward is Rjj c.u. plus the reward V (k -1 j -I,, t) that the network would receive for the remaining time t if the initial state was k - -1,.
- The process transitions from the state k to the state k -I,. in three cases. First, with the probability of
Mi min(mi, ki)p+0At + o(At), which means the customer was served at Si and transmitted to the IS-node S0 , i = 1,n . The network earning from this transition is R+ c.u. Secondly, with the probability of
K-iki 1 po-.At + o(At),
V i=i J
which is possible when a negative customer arrives to Sj from the IS-node S0 , i = 1, n . The network earning in this case is R- c.u. Thirdly, with the probability of
Mi min(mi, ki)p- (1 - 0(kj)) At + o(At), when a positive customer is transmitted as a negative from S; to an empty node Sj , i, j = 1, n . The network
earning is R j c.u. In each of the three listed cases, the expected reward V(k -11, t) is added to the mentioned network earnings.
- A transition from the state k to the state k +1 with probability
K -ik< ] P+At+ °(A ) '
1 -
which is possible when a positive customer arrives to S, from the IS-node S0 , i = 1, n . The network reward is R+i c.u. plus the expected total reward V(k +1,, t) that the network would earn in the remaining time t if it started in the state k +1.
- During time interval At the G-network remains in the state k with probability
£^ Mi min(m,ki) (l + p- (1 -Q(kj ))) + ï X^K - k,
which entails a reward R(k)At plus the expected reward that the network will earn in the remaining t units of time, V(k, t). The probabilities of other transitions and rewards are considered to be of order o(At).
Having regard to the listed above, let us sum up the product of probabilities and rewards over all options. Dividing both sides of the resulting equation by At, we take the limit as At ^ 0 . As a result, we have the following set of difference-differential equations that completely define V(k, t) :
Лt + °(At ),
8V (k, t) n ---= Z Mi min(m, k )p+
8t
Z Mi mln(m, kt )p+ (V (k +1 j -1, t) - V (k, t) ) +
i, J =1
+ Z Mi min(mi, k, ) p- (V (k -1J - Ii, t) - V (k, t) ) +
i, J=1
+Z Mi min(m,,k, )Pio (V(k -1,, t) - V(k, t)) +
i=1
+z ^^ к -Zk j Poi (V (k - i, , t ) - V (k, t) ) + + ^ м, mrn(m,,k, )p- (1 -0(ky )) (V(k - Ii, t) - V(k, t)) +
i, J =1
+:t ^^ K-:tkl j po+i (V(k +1,, t) - V(k, t)) + Q(k), where
Q(k) = R(k) + ^min(m,k)(p+R+ + PjR- (2 - 0(k;)))) +
n n f n \/ \
min(mi,k)p+0R+ + K-X(p-tR-i + P+X,). i=i i=i v i=i y
According to R. Howard, let us difine a quantity Q(k) as the «earning rate» of the network [1]. This earning rate is composed of combination of reward rates and transition rewards.
The set (3) cannot be solved for large K and n. In this regard, we will use the approximation and study the limiting behavior of the random process k(f) in the asymptotic case of a large number of customers K, K »1. We proceed to the limit from the Markov chain k(7) to the continuous Markov process
k(t)_(kx(t) k2(t) kn(t)
%(t ) = -
v K K K
K
as K tends to be large. The phase space of the vector \(t ) is
X = |x = (Xi, *2, •••, Xn): xt > 0, i = 1,n, 2 x < 1|-
The increment of (t) in the short time At ^ 0 is Axi = s = 1 / K, and Axi ^ 0 as K ^w. Therefore, the process ^(t) tends to be continuous as K ^w (s^0), and the vector i;(t) is a continuous-state Markov process on X. In the considered asymptotic case, the total reward of the G-network is a continuously changing process depending on the initial state x, xeX, and the upcoming time t.
In physics, the mass density (volumetric mass density or specific mass) is a substance's mass per unit of volume. Mathematically, density is defined as mass divided by volume:
mass m(x1 < x1 +s, x2 <^2 < x2 +s, ..., xn <^n < xn +s)
p =-= lim-- .
volume s^0 sn
In probability theory, probability density is the probability per unit volume. By analogy, let us introduce into
consideration the concept of «reward density», meaning the reward that the network erns per unit of state
space in time t based on the initial state at the point x:
. reward V(X <X1 + S x> <^2 <x2 + S ..., xn <^n <xn + S t)
v(x, t) =-= lim-- .
state space s^0 sn
Realizing the passage to the limit as K ^w, a n-dimensional lattice with vertices at discrete points
k
K
as K increases. It is required to take into account the continuous change in reward on X. Let us use the approximation V(xK, t) = v(x, t)sn for reward when xt < ^ < xt + s, i = 1, n . Similarly, based on reward rates and transition rewards, R(x) and R., we introduce parameters of earning rate per unit of state space X, r (x)
k
and ri. , into consideration. Therefore, r(x)sn , r.sn are earning parameters in case of small change in — .
K
Let ej = I i s .
Thus, in the asymptotic case under study, equation (3) can be represented as the following partial differential equation:
dv (x t)=k 2 ^min(sm,, xi) p+ (v(x+e} - ei, t) - v(x, t))+
ihL KL
K ' K '"'' K
transforms into set of points x = (xj, x2, •.., xH)eX, the «point density» increases
i, j=1
+K £ min(sm,, x, )p- (v(x - e j - e,, t) - v(x, t) )+ (4)
', j=1
n
+K min(sm,, Xi )p,0 (v(x - e,, t) - v(x, t)) + ;=1
+K i 1 -txt 1 p- (v(x - e,, t) - v(x, t)) +
1=1 V 1=1 J
+K i Mi min(emi, xt)p- (1 -0(Xj)) (v(x - et, t) - v(x, t)) +
^ j=1
+Ki1 -iXi]p+ (v(x + e;, t)-v(x, t)) + q(x),
i=1 V i=1 J
where q( x) is earning rate on state space X and it is defined by (2).
If v(x, t) is a twice continuously differentiable function with respect to xt, then we can use the Taylor series to second order about the point x for functions v(x + ej - ei, t), v(x - ei, t), v(x + ei, t), i, j = 1, n . Equation (4) becomes:
0v(x, t) n \ +
---= E minora,, X,-)p+
dt iJ=i 2
((
Cv(x, t) Cv(x, t)
cX,-W J
cX
\ f
8 2
С2v(x, t) ^ C2v(x, t) i С2v(x, t)
cX,-
V J
cXj cX2
cX /
//
f f
+ E M, min(8m,, X,)p-
Cv(x, t) Cv(x, t)
dx, V J
dx.
\ f
8 2
a2v(x, t) a2v(x, t) i a2v(x, t)
+Em, min(8m,, x, )рг+0 ,=i
^ Cv(x, t) 8 a2v(x, t
¿к,-
2 ax,2 ,
dxt
V J
■E^l 1 -E x, |Po0 ,=i V ,=i
OXj dxt
0x2
//
^ Cv(x, t) 8a2v(x, t
dx.
2 С ,
+ E M, min(8m,,x,)pj (1 -0(Xj))
f av(x, t) 8a2v(x, t)^
,,j=i
+E^li -E x, |p+i ,=i V ,=i
0X
f6v(x, t) 8S 2v(x, t)^
Ox
2 Ox,2 ,
2 Ox, 2 ,
+ q(x) + o(82).
Having grouped first-order and second-order partial derivatives of a function v(x, t) in the resulting equation, we conclude that the compact mathematical expression (1) is valid up to o(s2) = o(1 /K2). The equation (1) differs from the well-known multidimensional Kolmogorov backward equation only by the earning component q(x). The theorem is proved.
3. Mathematical model for the mean expected reward of the G-network
It is obvious that the diffusion coefficients Bj (x, t) of the equation (1) are of order s, then the term
£ n Q 2v(x t) 2 2
— i Bj (x, t)--— on the right side of (1) is 0(£ ). In this regard, up to terms of order 0(£ ), the re-
2 i, j=1 j dx, Qxj
ward density is given by the following equation:
Cv(x, t) n . . . Cv(x, t) . . —= -E A (x, t) —(x-^ + q(x). at ,=i dXj
(5)
Integrating the density v(x, t) within a «-dimensional region D, D cX, we get the expected total reward that the network will earn in time t if it starts in state x e D:
Vd(t) = ||...Iv(x, t)dx .
D
Applying this transformation to both parts of (5), using the rules of integration, Leibniz integral rule, the linearity of coefficients A, (x, t) in x and the boundary condition A(x, t)v(x, t) = 0, x e T(D), where T(D) is the reflecting boundary of the region D [15], we obtain a first order ordinary linear differential equation for the expected reward VD (t):
d n OA-(x t)
fVD(t) = 1 ^-¿.lll. vd(t) + JJ...|q(x)dx. (6)
dt i=i oxi D
The differential equation (6) is a mathematical model of the expected total reward of the G-network that the network will earn in time t if its initial state belongs to set D. Now we have a linear differential equation that completely define VD (t) when VD (0) is known.
By analogy with a center of gravity of a material body D in physics, we can find the equilibrium point of the expected reward, when x e D:
ED(t) = ^ii...i x,-v(x, t)dx , i = M .
VD (t) D
Example. Consider a closed exponential G-network, including four nodes Si, i = 1,4, and IS-node S0. Customers are transmitted between network nodes with the following non-zero probabilities: p+1 = 0,98, p-1 = 0,02, p+2 = 0,6, p- = 0,02, p+o = 0,38, p+ = 0,75 , p- = 0,1, p+o = 0,15 , P3+4 = 0,9, p-4 = 0,1, p+1 = 0,1, p+0 = 0,9. The structure of this network is presented graphically in Fig. 1. The node S0 generates the Poisson flow of customers routed to the node S1, the flow parameter is X = 0,1 customers per unit of time. The number of customers circulating in the network is K = 10 000.
Fig. 1. G-network structure
We assume that each node Si has four identical servers m, = 4, i = 1,4, the service rates are ^ = 0,017, ^ 2 = 0,001, = 0,005, ^4 = 0,002 respectively. The network reward is determined by the following non-zero parameters: r0+1 = 4, r0- =-0,5, r1+) = 1, r1+2 = 5, r- =-0,5, r2+, = 2, r2+3 = 3, r— =-0,5, r3+4 = 6, r3"4 =-0,5, r4+0 = 4, r4+1 = 0,2, r4-1 =-0,2. A reward rate is a linear function r(x) = 250 + 2x + 3x2 - 7x3 - 5x4 ; the given coefficients determine the contribution of each queue to the reward rate. We are interested in the expected G-network reward if the network starts in the state x e D with time t remaining, where
D = jx = (x1, x2, x3, x4) : 50s < xi < 450s, i = 1,4J. The numerical solution of (6) when VD (0) = 0 is presented graphically in Fig. 2.
Fig. 2. Earnings of G-network in monetary units with remaining time
The resulting model allows us to predict the expected G-network reward VD (t) in monetary units as a function of remaining time t, when the initial state of the network is known, x e D. There is a tendency that the expected network reward exponentially increases or decreases, which depends on the ratio between earnings and losses from transmissions of a large number of customers between network nodes. In addition, calculations allowed us to conclude that the expected network reward is sensitive to the total number of network customers K, remaining time t, the volume of region D and nodal service parameters.
Conclusions
In this paper, a closed exponential G-network with rewards is studied. The earning sequence created by the network occurs through customer transmissions between nodes. The purpose of the study is to develop the asymptotic method for analyzing G-network reward in the case of a large number of customers. The paper presents the results of mathematical modeling that were used to predict the expected total reward of the G-network as a function of remaining time t, when the initial state of the network is known. The numerical example is presented.
Queueing networks are widely used in modeling complex systems and processes [16]. G-networks with rewards allow, along with the analysis of the probabilistic-time characteristics of modeled objects, to calculate their earnings associated with transmissions positive and negative customers. Real systems typically handle a large number of customers. In this regard, the results obtained in the asymptotic case of a large number of customers will be relevant with high accuracy.
It is planned to continue the study of queueing networks with rewards and to develop a mathematical model for the earned network reward by the time t, if its state at the moment t is known. The research carried out allows us to conclude that the generalized Fokker - Planck - Kolmogorov forward equation will serve as a mathematical model for the density of such reward.
References
1. Howard, R.A. (1960) Dynamic Programming and Markov Processes. Cambridge: Massachusetts Institute of Technology.
2. Matalytsky, M.A. & Pankov, A.V. (2004) Probabilistic analysis of income in banking networks. Vestnik BSU. Series 1. Physics,
Mathematics, Computer Science. 9. pp. 79-92.
3. Matalytski, M. & Pankov, A. (2003) Analysis of the stochastic model of changing of incomes in the open banking network.
Computer Science. 3(5). pp. 19-29.
4. Gelenbe, E. (1991) Product form queueing networks with negative and positive customers. Journal of Applied Probability. 28.
pp. 656-663. DOI: 10.2307/3214499
5. Gelenbe, E. (1993) G-networks with triggered customer movement. Journal of Applied Probability. 30. pp. 742-748.
D0I:10.2307/3214781
6. Gelenbe, E. (1993) G-networks with signals and batch removal. Probability in the Engineering and Informational Sciences. 7(3).
pp. 335-342. DOI: 10.1017/S0269964800002953
7. Caglayan, M.U. (2017) G-networks and their applications to machine learning, energy packet networks and routing: introduction
to the special issue. Probability in the Engineering and Informational Sciences. 31(4). pp. 381-395. DOI: 10.1017/S0269964817000171
8. Zhang, Y. (2020) G-Networks and the performance of ICT with renewable energy. SN Computer Science. 1. Art. 56. DOI:
10.1007/s42979-019-0056-2
9. Zhang, Y. (2021) Optimal energy distribution with energy packet networks. Probability in the Engineering and Informational
Sciences. 35(1). pp. 75-91. DOI: 10.1017/S0269964818000566
10. Rusilko, T. (2022) Asymptotic analysis of a closed G-network of unreliable nodes. Journal of Applied Mathematics and Computational Mechanics. 21(2). pp. 91-102. DOI: 10.17512/jamcm.2022.2.08
11. Rusilko, T.V. (2023) The G-network as a stochastic data network model. Journal of the Belarusian State University. Mathematics and Informatics. 2. pp. 45-54. DOI: 10.33581/2520-6508-2023-2-45-54
12. Nazarov, A.A. & Pavlova, E.A. (2022) Study of SMO type MMPP|M|N with feedback by the method of asymptotic analysis. Vestnik Tomskogo gosudarstvennogo universiteta Upravlenie vychislitel'naya tekhnika i informatika - Tomsk State University Journal of Control and Computer Science. 58. pp. 47-57. DOI: 10.17223/19988605/58/5
13. Moiseeva, S.P., Bushkova, T.V. & Pankratova, E.V. (2022) Asymptotic analysis of resource heterogeneous QS under equivalently increasing service time. Automation and Remote Control. 83(8). pp. 1213-1227. DOI: 10.1134/S0005117922080057
14. Tikhonov, V.A. & Mironov, M.A. (1977)Markovskieprotsessy [Markov processes]. Moscow: Sovetskoe radio.
15. Gardiner, K.V. (1986) Stokhasticheskie metody v estestvennykh naukakh [Stochastic Methods in Natural Sciences]. Translated from English. Moscow: Mir.
16. Rusilko, T.V. & Pankov, A.V. (2024) Queueing network model of a call center with customer retrials and impatient customers. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 24(2). pp. 287-297. DOI: 10.18500/1816-9791-2024-242-287-297.
Information about the authors:
Rusilko Tatiana V. (Candidate of Physical and Mathematical Sciences, Associate Professor, Associate Professor of Fundamental and Applied Mathematics Department, Yanka Kupala State University of Grodno, Grodno, Republic of Belarus). E-mail: tatiana. [email protected]
Salnikov Dmitry A. (Master's student of the Department of Mathematical and Information Support of Economic Systems, Yanka Kupala State University of Grodno, Grodno, Republic of Belarus). E-mail: [email protected]
Contribution of the authors: the authors contributed equally to this article. The authors declare no conflicts of interests.
Информация об авторах:
Русилко Татьяна Владимировна - доцент, кандидат физико-математических наук, доцент кафедры фундаментальной и прикладной математики Гродненского государственного университета им. Янки Купалы (Гродно, Беларусь). E-mail: tatiana. [email protected]
Сальников Дмитрий Александрович - магистрант кафедры математического и информационного обеспечения экономических систем Гродненского государственного университета им. Янки Купалы (Гродно, Беларусь). E-mail: [email protected]
Вклад авторов: все авторы сделали эквивалентный вклад в подготовку публикации. Авторы заявляют об отсутствии конфликта интересов.
Received 22.05.2024; accepted for publication 03.09.2024 Поступила в редакцию 22.05.2024; принята к публикации 03.09.2024
