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Discrete & Continuous Models & Applied Computational Science 2023, 31 (2) 105-119
ISSN 2658-7149 (online), 2658-4670 (print) http://journals-rudn-ru/miph
Research article
UDC 519.872
PACS 07.05.Tp, 02.60.Pn, 02.70.Bf DOI: 10.22363/2658-4670-2023-31-2-105-119 EDN: VUBLKP
Heterogeneous queueing system with Markov renewal arrivals and service times dependent on states of arrival
process
Evgeny P. Polin1,2, Svetlana P. Moiseeva1, Alexander N. Moiseev1
1 National Research Tomsk State University, 36, Lenin Avenue, Tomsk, 634050, Russian Federation
2 National Research Tomsk Polytechnic University, 30, Lenin Avenue, Tomsk, 634050, Russian Federation
(received: April 14, 2023; revised: April 25, 2023; accepted: June 26, 2023)
Abstract. In the proposed work, we consider a heterogeneous queueing system with a Markov renewal process and an unlimited number of servers. The service time for requests on the servers is a positive random variable with an exponential probability distribution. The service parameters depend on the state of the Markov chain nested over the renewal moments. It should be noted that these parameters do not change their values until the end of maintenance. Thus, the devices in the system under consideration are heterogeneous. The object of the study is a multidimensional random process — the number of servers of each type being served with different intensities in the stationary regime. The method of asymptotic analysis under the condition of equivalent growing of service times in the units of servers is applied for the study. The method of asymptotic analysis is implemented in the construction of a sequence of asymptotic of increasing order, in which the asymptotic of the first order determines the asymptotic mean value of the number of occupied servers. The second-order asymptotic allows one to construct a Gaussian approximation of the probability distribution of the number of occupied servers in the system. It is shown that this approximation coincides with the Gaussian distribution.
Key words and phrases: queuing system, random environment, Markov renewal process, asymptotic analysis method
1. Introduction
Queueing theory is a field of applied mathematics that deals with the study and analysis of processes in various service, production, management, and communication systems in which homogeneous events are repeated many times. Examples of such systems include consumer services; systems for
© Polin E.P., Moiseeva S.P., Moiseev A.N., 2023
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Q 0®
receiving, processing, and transmitting information, automatic production lines, telecommunication systems, and others [1].
The independence of processes in queueing systems is generally assumed when developing queueing models. However, real systems often involve several process dependencies, and failure to consider these can lead to a serious under errors in the estimation of the performance measures. Semi-Markov processes are used in modeling stochastic control problems arising in Mar-kovian dynamic systems where the sojourn time in each state is a general continuous random variable. They are powerful, natural tools for the optimization of queues, production scheduling, reliability/maintenance [2, 3]. For example, in a machine replacement problem with deteriorating performance over time, a decision-maker, after observing the current state of the machine, decides whether to continue its usage, initiate maintenance (preventive or corrective) repair or replace the machine.
Semi-Markov Processes include renewal processes and continuous-time Markov chains as special cases. In a semi-Markov process similar to Markov chains, state changes occur according to the Markov property, i.e., states in the future do not depend on the states in the past given the present. However, the sojourn time in a state is a continuous random variable with distribution depending on that state and the next state. A renewal process is a generalization of a Poisson process that allows arbitrary holding times. Its applications include such as planning for replacing worn-out machinery in a factory. A Markov renewal process is a generalization of a renewal process that the sequence of holding times is not independent and identically distributed. Their distributions depend on the states in a Markov chain. The Markov renewal processes were studied by Pyke in the 1960s [4, 5].
In the proposed work, we consider a heterogeneous queueing system (QS) with a Markov renewal process (MRP) for the process of its arrival and an unlimited number of servers. The service time for requests have an exponential probability distribution. Parameter of the service depends on the state of the Markov chain nested over the renewal moments. It should be noted that these parameters do not change their values until the end of maintenance. Thus, the devices in the system under consideration are heterogeneous.This problem for the Queueing System M\M\m in a Markov Random Enviroment was addressed in [6-8].
The objects of the study are the number of servers of each type being served in the stationary regime. Such a QS can be attributed to the class of non-homogeneous QS operating in a random environment.
Currently, a significant part of the information, telecommunication, and other systems operate in a changing environment. The impact of a random environment can be expressed, for example, in a change in the parameters of the functioning of the system. In this regard, questions arise about the stability of such systems to external influences. Therefore, the study of systems operating in a random environment is an urgent task. In various works devoted to the study of systems in Markov and semi-Markov random environments, various variants of the system's response to a change in the state of the external environment were considered in [9-11].
In this paper, we consider the case assuming that the service mode of claims does not change until they leave the system. The method of asymptotic analysis under the condition of equivalent growing of service times in the
units of servers is applied for the study. This asymptotic condition means proportional growth of the average service times in both service units and it is taken from practice. The method of asymptotic analysis is implemented in the construction of a sequence of asymptotic of increasing order, in which the asymptotic of the first order determines the asymptotic mean value of the number of occupied servers. The second-order asymptotic allows to construct an approximation of the probability distribution of the number of occupied servers in the system. It is shown that this approximation coincides with the Gaussian distribution.
2. Markov renewal process
A renewal process is a generalization of a Poisson process that allows arbitrary waiting time between events. Its applications include such as planning for replacing worn-out machinery in a factory. A Markov renewal process is a generalization of a renewal process that the sequence of holding times is not independent and identically distributed. Their distributions depend on the states in a Markov chain. The Markov renewal processes were studied by Pyke [4, 5] in 1960s.
2.1. Mathematical model of the Markov renewal process
Consider a two-dimensional homogeneous Markov random process {^(n),T(n)} with discrete time n = 1,2,3,..., where ^(n) takes values from some discrete set ^(n) = k = 1,2,3,... and r(n) takes on non-negative values.
We denote
F(k2, X; ki ,y) = P{$(n + 1) = ¿2, r(n + 1) < x\$(n) = A^, r(n) = y} =
= F(^2, x; ki) = Pk1k2 Ak2 (x).
A random stream of homogeneous events t1 < ... < tn < tn+1 < ... will be called the Markov renewal flow or MR-flow given by the matrix of transition probabilities P and functions Ak(x) distribution of interval lengths rn+1 = tn+1 — tn, for which the equalities rn+1 = rn hold.
To study the MR-flow, we define the process z(t) as the length of the interval from the time t to the time tn+1 of the next event in the considered flow and the process
k(t)=i(n), tn < t<tn+i,
that is, the process k(t) on the interval tn ^t < tn+1 retains the value that it received at the beginning of this interval and which coincides with the value £(n) of the embedded Markov chain.
For a Markov renewal flow, the three-dimensional process {k(t), z(t),m(t)} is Markov, therefore, for its probability distribution
P(k, z, m, t) = P{k(t) = k, z(t) < z, m(t) = m}
by the formula of total probability we obtain the equality
P(k,z — At,m,t + At) =
= P(k, z, m, t) — P(k, At, m,t) + ^ P(u, At, m — l,t)Pvk Ak (z) + o(At)
V
from which it follows that the probability distribution P(k, z, m, t) is a solution to the Kolmogorov equations
dP(k, z, m, t)
Ft =
dP(k,z,m,t) dP(k,0,m,t) sr^ dP(u,0,m — l,t) ^ t . . ( , , , ) ( , , , ) --L^ Ak (z). (1)
V
By defining the functions
to
H(k, z, u,t)=Y, eJUmp(k, z, m, *),
m=0
the equations (1) can be rewritten as
( , , , ) ( , , , ) ( , 0, , ) ( , 0, , )
dt dz dz d~Z 6 PvkAk (Z).
V
The basic equation for a semi-Markov flow has the form
dh(z,u,t) = dh(z,u,t) + dhiOUit),
dt = dz + dz {e PA(Z) —1), (2)
where P is the matrix of transition probabilities, A(z) = diag [Ak(z)], I is identity diagonal matrix. To find its particular solution, we define the initial condition in the form
h(z, u, 0) = r(z),
where r(z) — stationary probability distribution of the values of a two-dimensional random process {k(t),z(t)}.
2.2. Finding the distribution r(z) Vector r(z) is a solution to the equation obtained from (2) ^ + 1) = 0, therefore it can be written as
r(z)=i ^^(I — PA(x))dx. (3)
Jo Vz
Since r(k, z) = P{k(t) = k, z(t) < z) then r = r(<). Therefore, we obtain
r = — PA(x))dx. (4)
Jo dz
By virtue of the necessary condition for the convergence of the improper integral, we can write down the equality to zero of the integrand at x ^ <, we obtain the system of equations
^*-P)=> (5)
dr(0)
for ———, where P = A(<).
oz
Since the system (4) coincides with the system of Kolmogorov equations for the stationary probability distribution r of values of the embedded Markov chain, then
9r(°) X (6)
= Xr, (6)
where A is some multiplicative constant, the value of which is found as follows. Substituting (6) into (4), we obtain
/to
r (P — A(x)) dx.
0
Since re = 1 then
A = 1 = 1 (7)
ff r (P — A(x)) edx f~ (1—F(x))dx- ()
Equalities (7), (6) and (3) solve the problem of finding the probability distribution r(z).
3. Mathematical model
Consider a queueing system MRPIMI« with an unlimited number of servers of different types, operating in a semi-Markov random environment (see the figure 1). Arrivals are determined as Markov renewal process, interarrival periods have cumulative distribution functions A1(x),A2(x),...AK(x) and the matrix of transition probabilities P = [pij], i,j = 1,2,... ,K — embedded in the moments of occurrence of events Markov chains with a finite number of states k(t) = 1,2,... ,K. The service discipline is defined as follows: if the embedded Markov chain is in the state k(t) = i, then the incoming customer will be serviced on the i-th type server during a random time, exponentially distributed Fi (x) = 1 — .
The problem is to study a multidimensional random process — numbers occupied servers of different types in the system at time t, which is denoted by i(t) = [i1 (t),i2(t),... ,iK(£)]. The process i(t) is not Markov. For clarity,
consider the case when the external environment takes only 2 different states. We define a four-dimensional Markov random process {k(t),z(t),i1 (t),i2(t)}, where z(t) is the length of the interval from the time t to the time of the next event in the stream Markov renewal, k(t) is a Markov chain embedded with respect to renewal times.
H,
MRP
A(z),P
Hz
hit)
■4(0
Figure 1. Queueing system M RPlMl< in a semi-Markov random environment
For research, we will obtain some characteristics for the number of events occurring in the MR stream. For the probability distribution
P(k, z, i-i, %2, t) = P{k(t) = k, z(t) < z, i-i(t) = ii, %2(t) = 12} we write down the Kolmogorov system of differential equations:
dP(1,z,ii,i2,t) dP(1,z,ii ,i2 ,t) dP(1,0,i-,i2 ,t)
dt
dz
dz
- (ii Pi + h V2 )P(1, Z, ii, ir2, t) + dP(1,0,tQz ^ Pii Ai (Z) + + dP(2,0,Hi -1,t2,t]P2iAi(z) + P(1,Z,ii + 1,I2,t)(ii + 1)»i+
dz
+ P(1,Z,ii,l2 + 1,t)(i2 + 1)^2,
dP(2,z,ii,i2 ,t) dP(2,z,ii,i2 ,t) dP(2,0,i1,i2 ,t)
dt
dz
dz
- (iliH + ir2 P2 )P(2, Z, i\, ir2, t) + dP(2, ^ -— P22 ^2 (z) +
+ dP(1,0,t^2 -1,t) Pl 2 A2 (z) + P(2, z,ii + 1, i2 ,t)(h + 1)iH +
+ P(2,Z,i1,%2 + 1,t)(%2 + 1)^2-For a stationary probability distribution, we write this system in the form
dP(1,z,ix,i2) dP(1,0,i1,i2)
dz
dz
. . . , dP(1,0,u — 1,i2)
- (h^i + l2V2)P(1,z, H, l2) +---Q~z-PiiAi(z)+
+ dP(2,0,Qz 1,12 }P21 A (z) + P(1,Z,ii + 1, )(ii + 1)^1 +
+ P(1,Z,ii ,%2 + 1)(l2 + 1)^2 =0,
dP(2,Z,l1,12 ) dP(2,0,l1,12 ) — dz dz
. dP(2,0,i1 ,i2 — 1) — (h V1 + 1>2 V2 )P(2, Z,%1 ,%2 ) +--^-P22 A-2 (Z) +
+ 9P(1,0,qz ,%2 — 1)P12 A (z) + P(2, Z,i1 + 1, )(h + 1)»1 +
+ P(2,z,i1 ,t2 + 1)(t2 + 1)/i2 =0.
We introduce partial characteristic functions of the form
to TO
H(k,z,u1 ,u2) = eJU2%2P(k,z,i1 ,i2), where j = V—1.
i1=0 i2=0
Let us write the system of differential equations for the partial characteristic functions
dH(1,z,U1 ,U2) dH(1,0,U1 ,u2)+
dz dz
+ jßl (1 - e-^) dH(1fui'u2) + Jß2 (1 - e-ju2) dH(l,z,Ul,u2) +
+ dH(1,0,Ui,U2) ^ ^ ^ {Z) + dH{2,0,Ui,w) ^ p2iMz) =
dH(2,z,Ui ,U2 ) dH(2,0,Ui ,u2)+
dz dz
+ jßl (1 - e--i) dH(2fui,u2) + jß22 (1 - e-^2) dh(2'*'ui,u2) +
OUi ÖU2
+ M^-»2 ) Pl2A2(z) + 'U2) ^ A.2 (z) = 0
with initial conditions
H(k- z-0'0) = r(k- z). In vector-matrix form, this system will take the form
^^ + dH0-Ui'U2) (pA(,)B(u) - I) + dz dz
+ № (1 - ) Sh(Z^'"2) + m2 (1 - ^) ^^'"2) = 0, (8)
with initial conditions
h(z, 0, 0) = r(z),
where
h(z, ux, U2) = [H(1, z, UX,U2), H(2, z, ux, u2)],
ejui 0 1 0
, I =
0 e>u* 0 1
The resulting system of equations (8) is the main one for further research. Since it is not possible to find an explicit form of a solution to the problem (8), we will seek the solution under the asymptotic condition of equivalent growing of service times in the units of servers. This asymptotic condition means proportional growth of the average service times in both service units and it is taken from practice.
4. Asymptotic analysis of the first order
We denote ^ = e, ^2 = qt, q = const (e is an infinitesimal quantity). Then we can write the asymptotic condition of equivalent growing of service times in the units of servers in the form ^ 0. In (8) we perform the
replacements
h(z,u1,u2) = f(z,w1,w2,e), ux = ew1, u2 = ew2, we obtain the matrix equation for f(z, wx,w2, t)
+ W>,Wi,W2,t) (PA(,)B(u, e) — I) +
+j (1 — e-^i) ^1^2,<0 + jq {l — e-J€W2) df(z,w w2,e) = o^
ow1 ow2
(9)
Theorem 1. The limiting solution for e ^ 0 to the equation (9)
f(z,w1,w2 ,e) has the form
f(z, w1,W2, e) = r(z) exp {j\ (r1w1 + ^^)} , (10)
where r(z) = [r1(z),r2(z)} is the vector of the probability distribution of the values of the embedded Markov chain, r = [r1,r2} is vector of stationary probability distribution of the values of the embedded Markov chain.
Proof. In the equation (9) we carry out the passage to the limit for e ^ 0, we obtain that f(z, wx,w2) is a solution to the equation
^^^ + (pa(z) - I) = 0,
oz oz
which defines the vector function r(z), therefore we will seek the function f(z, w1,w2, e) in the form of the expansion
f(z, wi,W2, e) = r(z)$(wi,W2) + o(e). (11)
In the equation (9) we carry out the passage to the limit as z ^ <x>, multiply this equation by the unit column vector e, expand the exponents in a Maclaurin series up to the first order. In the resulting expression, we substitute the expansion (11), divide by e and carry out the passage to the limit at e ^ 0, we obtain the equation for the function &(w1 ,w2)
d^(w1,w2) d^(w1 ,w2) .dr(0)^„T _
fc(0) x 1
where —— = Ar, rp = r, re =-1, X = --, W =
OZ J (1 — rA(x)e)dx
The solution will have the following form
w1 0
0 W2
&(wi,W2) = exp {jX (riWi + )}
Substituting the obtained solution into (11), we get (10). The theorem is proved.
By substitution and equality (3), we write down the approximate (asymptotic) equality
h(z, ui, u2) & f(z, wi, w2) = r(z) exp {jA wi +—2—2)} =
ri ui , r2 u2
= r(z) exp { jX ( +
I V Vi V2
Let us define the characteristic of the process [ii(t), i2(t)} in the stationary mode
Vi Ui T2 U2
h(ui,U2) = exp { jX (--+
I V Vi V2
which we will call the first-order asymptotics of the characteristic functions of the number of occupied servers in the system.
5. Asymptotic analysis of the second order
In the equation (8) we replace
, \ -\i Ti ^i T2 ^2 h(z, Ui, U2) = h2 (z, Ui, U2) exp { jX --+
Vi V2
we obtain the equation for h2 (z, ui,u2)
—2( , i, 2 +--^ ' 1, 2 (PA(z)B(u) - I
oz oz
(1 - e-^1) dh2(Z,U1,U2) + (1 - e-^2) dh2,"2)- (12) 0Ui 0U>2
-Xri (1 - e-jUl) h2 (z, Ui,U2)-Xr2 (1- e-^2) h2 (z, Ui,^2) =0.
We denote ^i = e2, ^2 = qt2, in (12) we replace
h2 (z,Ui ,U2) = f2 (z,Wi ,W2 ,t), Ui = €Wi, U2 = tW2, we obtain the equation for f2 (z, wi ,w2, t)
df2 (Z,Wi ,W2 ,e) df2(0,Wi ,W2 ,e)
-d~z-+-^-(PA№(w, t) - I) +
- e-jeWi ) f (Z,Wi,W2,6) + (1 - e-jeW2 ) df2(Z, Wi ,W2, 6)
dwi dw2
-Xri (1 - e-*™1) f2 (z, wi ,W2 ,e)-Xr-2 (1 - e-^2) f2(z, Wi ,W2,t) = 0.
(13)
Theorem 2. The limiting solution for t ^ 0 to the equation (13)
f2(z,wi,w2) has the form
n , , , , (j2 / 2 w"2\
f2(z, Wi ,W2) = r(z) exp { — I X I Ti wi + r2 — ) +
i
q+1 ' '2 q
+K[rjwl + 4r1 r2+ r2 — ))}' (14)
"OO
where k = X2 I (rA(x) — r(x)) edx.
J0
Proof. We will obtain the solution of the equation (14) in the following form
f2 (z,Wi,W2 ,e) = <$>(Wi ,W2 ) (r(z)+je(ri Wi +T2 W2 )f2 (z))+o2 (t), (15)
where f2(z) satisfies the condition f2(œ)e = 0. Substitute (15) into (13) and expand the exponents in a series up to the first order. Considering that
^rM + (PA(z) —1) = 0
dz + dz (PA(Z) I) = 0, we obtain the equation for finding the function f2 (z)
e— Aer(z) + e^^ (PA(^) — I) + XA(z) = 0. (16)
Oz Oz
From the equation (16) we find that
f (0)
»00
= Kr, where k = X2 I (rA(x) — r(x)) edx.
dz JQ
Substitute (15) into (13) and expand the exponents in a series up to the second order. Multiply by e and perform the passage to the limit z ^ <x>, we obtain the equation for finding the function $(w1,w2)
d$(w1,w2) d$(w1,w2)
wi-a--+ w21-a- =
owx ow2
= $(W1,W2 ) (-X (r^wl + r^w"2) - K (r1w1 + r-2 W2 )2 ) . (17)
The solution of the equation (17) has the form
$(w1,w2) =
= exp (A (riW2 + r2 ^L (r2w2 + 4^2 + r2 ))} (18)
Substituting the solution (18) into (15) and performing the passage to the limit e ^ 0, we obtain (14).
The theorem is proved. □
Due to the change, as well as the equality (14) for the function h2(z, ux,u2) we can write down the approximate (asymptotic) equality
h2(z,ux,u2) & f2(z,wx,w2) =
■2 / / 2 2 \ r U 1.. K , u2\
= r(Z) exp + ) +
22 ( o Ui U-\ Un n Un
+K \r2 — + 4r\r2—^^ + r2,— V Vl V1+V2 V2
Thus, the characteristic function of the number of occupied servers in the system under consideration has the form
*u *u I2
h2 (u-1, U2)= exp {jX(^ + ) + J-
ßi ^2 J 2
U2 vj2
X ( Vi-1 + T2— ) + Pi ß2.
22 , o Ul U~\ Un o Un
+K (r2— + 4r1 r2—+ r2 — Pi Pi + V2 ß2
■ (19)
6. Numerical example
Let us consider a numerical example where we can illustrate the accuracy of approximating formula (19). Consider queueing system with MRP arrivals,
where the Markov renewal process is given by matrices
P
0.3 0.7 0.6 0.4
A(x) = diag[A-i (x),A2 (x)}.
Here Ai (x) and A2 (x) are gamma distribution cdf-s with shape and rate parameters a and ft that have the following values:
ai =0.5, = 0.25, a2 = 1.5, = 1.5. The service times are exponentially distributed with service rates
Pi = 1- e, p2 = 2- £
for the the first and the second types of arrivals respectively. Here parameter e will be varied to establish the accuracy of approximation (19) accordingly to the asymptotic condition e ^ 0.
To establish the accuracy of the approximation, we use its comparison with the results of simulation modeling of the corresponding system. For the error estimation (difference between the results), we use the Kolmogorov distance
A= max l^approx^i 2 ) - ^sim (}i,^2)\,
where Fapprox(ii,i2) is a cdf of Gaussian distribution (19) and Fsim(ii,i2) is a cdf built basing on the results of the simulation. The results of the comparison is presented in the table 1. We see that the Kolmogorov distance decreases with decreasing of parameter e, so, approximation (19) becomes more accurate for small values of this parameter.
Table 1
Kolmogorov distance A between the approximation and distribution based on the simulation results for various values of asymptotic parameter e
£ 0.1 0.05 0.01 0.005 0.001 0.0005 0.0001
A 0.1137 0.0501 0.0371 0.0323 0.0253 0.0226 0.0197
For example, if we suppose that error A < 0.05 means that the approximation is accurate enough then we can conclude that for the considered example, Gaussian approximation (19) is applicable for values £ < 0.05.
7. Conclusions
In this paper, the method of asymptotic analysis is used to study a mathematical model of the MRlMlœ system functioning under the condition of a changing environment. The case is considered when a semi-Markov random environment has 2 different states. It is proved that the asymptotic characteristic function of the number of occupied servers of each type in the considered
system is Gaussian with the vector of mathematical expectations
a =
X—, A— . »1 »2.
and the covariance matrix
K =
. r\ rt A— + k—
4k-
fi r.
1' 2
VX+V2
4k-
r r
11 2
X r2 r22
X— + k— V2 »2-
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For citation:
E. P. Polin, S. P. Moiseeva, A. N. Moiseev, Heterogeneous queueing system with Markov renewal arrivals and service times dependent on states of arrival process, Discrete and Continuous Models and Applied Computational Science 31 (2) (2023) 105-119. DOI: 10.22363/2658-4670-2023-31-2-105-119.
Information about the authors:
Polin, Evgeny P. — Assistant of Department of Probability Theory and Mathematical Statistics, National Research Tomsk State University (e-mail: [email protected], phone: +7(923)4480077, ORCID: https://orcid.org/0000-0002-0250-2368)
Moiseeva, Svetlana P. — Doctor in Physics and Mathematics, Professor at Department of Probability Theory and Mathematical Statistics, National Research Tomsk State University (e-mail: smoiseeva@ mail.ru, ORCID: https://orcid.org/0000-0001-9285-1555,
Scopus Author ID: 56436490300)
Moiseev, Alexander N. — Doctor in Physics and Mathematics, Head of the Department of Software Engineering, National Research Tomsk State University (e-mail: moiseev. [email protected], ORCID: https://orcid.org/0000-0003-2369-452X,
ResearcherID: N-7189-2014, Scopus Author ID: 55646953800)
УДК 519.872
PACS 07.05.Tp, 02.60.Pn, 02.70.Bf DOI: 10.22363/2658-4670-2023-31-2-105-119
Гетерогенная система массового обслуживания с входящим потоком марковского восстановления и временем обслуживания, зависящими от состояний вложенной цепи Маркова
Е. П. Полин1'2, С. П. Моисеева1, А. Н. Моисеев1
1 Национальный исследовательский Томский государственный университет,
пр. Ленина, д. 36, Томск, 634050, Россия
2 Национальный исследовательский Томский политехнический университет,
пр. Ленина, д. 30, Томск, 634050, Россия
Аннотация. В работе рассматривается гетерогенная система массового обслуживания с входящим потоком марковского восстановления и неограниченным числом серверов. Время обслуживания запросов на серверах является положительной случайной величиной с экспоненциальным распределением вероятностей. Параметры обслуживания зависят от состояния цепи Маркова в моменты восстановления. Следует отметить, что эти параметры не меняют своих значений до окончания обслуживания. Таким образом, устройства в рассматриваемой системе являются неоднородными (гетерогенными). Объектом исследования становится многомерный случайный процесс — количество серверов каждого типа, обслуживаемых с разной интенсивностью в стационарном режиме. Для исследования применён метод асимптотического анализа при условии эквивалентно долгого времени обслуживания. Метод асимптотического анализа реализуется при построении последовательности асимптотик возрастающего порядка, в которой асимптотика первого порядка определяет асимптотическое среднее значение числа занятых серверов. Асимптотика второго порядка позволяет построить гаус-совскую аппроксимацию распределения вероятностей числа занятых серверов в системе.
Ключевые слова: система массового обслуживания, случайная среда, поток марковского восстановления, метод асимптотического анализа
