Please cite this article as:
Mikhail Matalytski, Victor Naumenko, Analysis of networks with time-dependent transition probabilities and messages
bypass between the queuing systems, Scientific Research of the Institute of Mathematics and Computer Science,
2012, Volume 11, Issue 4, pages 93-104.
The website: http://www.amcm.pcz.pl/
Scientific Research of the Institute of Mathematics and Computer Science 4(11) 2012, 93-104
ANALYSIS OF NETWORKS WITH TIME-DEPENDENT
TRANSITION PROBABILITIES AND MESSAGES BYPASS
BETWEEN THE QUEUING SYSTEMS
Mikhail Matalytski 1, Victor Naumenko 2
1
Institute of Mathematics, Czestochowa University of Technology, Czestochowa, Poland
2
Grodno State University, Grodno, Belarus
[email protected], [email protected]
Abstract. The object of investigation is an open exponential network with a messages
bypass of systems in transient behavior. The purpose of the research is to find stationary
probabilities of states and the average characteristics of the network when the transition
probabilities between the messages and bypass systems of the network, parameters of the
incoming flow of messages and services are time-dependent. To find the state probabilities
and the characteristics of a network is used the apparatus for the multivariate generating
functions. The examples are calculated on a computer.
1. General information
The results of research of the above networks in stationary behavior are given in
[1-3]. In [4] investigated the network in a transition behavior, but when the probability messages bypass between systems, networks do not depend on network status
and time. To find the state probabilities of the network in a transition mode, we
used the method of multidimensional generating functions, which was previously
used to find the state probabilities of the other networks [5, 6]. In this paper we
consider the case when the transition probabilities and messages bypass between
the network systems depend on time.
2. Formulation of the problem
Consider an open exponential QN of arbitrary structure consisting of n QS,
enumerated by numbers from 1 to n. Messages have a chance to join the queue, and
with an additional probability to move immediately in accordance with the transition probability matrix of the other QS, or leave the network. The probability of
joining the QS depends on the state of the QS and the number of QS with which
the messages are sent to this QS. It is assumed that the incoming flow of applications to the network is simple. The results of studies of such networks in the steady
state are given in [1-4]. This paper describes a method of finding the timedependent state probabilities of the network of such a network in the transient state.
94
M. Matalytski, V. Naumenko
Let mi - number of identical service lines in the QS Si , I i - a vector of dimension n, consisting of zeros except the i-th component, which is equal to 1, i = 1, n ;
pij - the transition probability of the message after service in the system Si into
the system S j , i, j = 0, n , we assume the system S 0 is the external environment.
Let us consider the case when the parameters of the incoming flow of messages
and services depend on time, i.e. the time interval [t , t + ∆t ) in the network receives
an message with a probability λ (t )∆t + o(∆t ) , and if at the time t of service on the
line i-th QS is located an message, at the range [t , t + ∆t ) of its services will end
with a probability µ i (t )∆t + o(∆t ) , i = 1, n . The message is sent to the i-th QS with
n
probability p0i ,
∑ p0i = 1 . The message sent to this QS from the external envii =1
ronment at moment time t, with a probability f ( i ) ( k , t ) , when the network is in
a state (k, t ) , joins the queue, and the probability 1 − f ( i ) ( k , t ) is not attached to the
queue, regardless of handled (i.e., it’s time of service with a probability of 1 is
equal to zero). If the message has been served in the i-th QS, it is likely to be sent
immediately to the j-th QS with probability pij , and leaves the QN with the proban
∑ pij = 1 , i = 1,..., n .
j =0
k (t ) = (k , t ) = (k1 , k 2 ,..., k n , t )
bility pi 0 ,
Let
- the state vector of the network, where ki - the
number of messages at the moment t in the system Si , i = 1, n ; ϕ i (k , t ) - the conditional probability that the message is delivered to the i-th QS at time t, when the
network is in a state (k, t ) , will not be serviced by any of the QS; ψ ij (k , t ) - the
conditional probability that the message is delivered to the i-th QS outside at time t,
when the network is in state (k, t ) , the first time, a service in j-th QS; α i (k , t ) - the
conditional probability that the message, served in the i-th queuing system at time
t, when the network is in a state (k, t ) , will no longer be served in any of QS;
β ij (k , t ) - the conditional probability that the message, served in the i-th queuing
system at time t, when the network is in state (k, t ) for the first time then receive
services in the j-th QS, i, j = 1, n .
According to the formula of total probability, we obtain:
n


ϕ i (k , t ) = (1 − f (i ) (k , t ) ) pi 0 + ∑ pij ϕ j (k , t )
j =1

 , i = 1, n ,
n
ψ ij (k , t ) = f ( i ) (k , t )δ ij + (1 − f (i ) (k , t ))∑ pilψ lj (k , t ) , i, j = 1, n ,
l =1
(1)
Analysis of networks with time-dependent transition probabilities and messages bypass …
n
αi ( k , t ) = pi 0 + ∑ pij φj ( k − Ii , t ) , , i = 1, n ,
95
(2)
j =1
n
βij ( k , t ) = ∑ pil ψlj ( k − Ii , t ) , i, j = 1, n ,
(3)
l =1
where δ ij - the Kronecker delta. We have the equalities
n
n
αi ( k , t ) = 1 − ∑ βij ( k , t ) , φi ( k , t ) + ∑ ψij ( k , t ) = 1 , i, j = 1, n .
j =1
j =1
From (1) and (3) we find
ψij ( k , t ) = f ( i ) (k , t )δij + (1 − f (i ) (k , t ) ) βij ( k − Ii , t ) , i, j = 1, n .
(4)
The probabilities of states of the network under consideration satisfy the difference-differential equations (DDE) is proved in [4]:
n
dP(k , t )
= −∑ [λ (t ) p 0i (1 − ϕ i (k , t )) + µ i (t ) (1 − β ii (k , t )) min (mi , k i )] P(k , t ) +
dt
i =1
n
n
+ λ (t )∑∑ p0iψ ij (k − I j , t ) u (k i ) P (k − I j , t ) +
i =1 j =1
n
+ ∑ µ i (t ) min (mi , k i + 1) α i (k + I i , t ) P( k + I i , t ) +
i =1
n
+
∑ µ i (t ) min(mi , k i + 1)βij (k + I i − I j , t )u (k j )P(k + I i − I j , t ) ,
(5)
i , j =1
i≠ j
1, x > 0
where u ( x ) = 
- is the Heaviside function.
0, x ≤ 0
3. Finding the state probabilities
Let mi = 1 , i = 1, n , and suppose that all network system operating in high load
mode, i.e. k i (t ) > 0 ∀t > 0 , i = 1, n , then the system (5) takes the form
n
dP(k , t )
= −∑ [λ (t ) p 0i (1 − ϕ i (k , t )) + µ i (t ) (1 − β ii (k , t ))] P (k , t ) +
dt
i =1
n
n
+ λ (t )∑∑ p0iψ ij (k − I j , t ) P (k − I j , t ) +
i =1 j =1
n
∑ µ i (t )α i (k + I i , t ) P(k + I i , t ) +
i =1
n
+ ∑ µi ( t ) βij ( k + Ii − I j , t ) P ( k + Ii − I j , t ) .
i , j =1
i≠ j
(6)
M. Matalytski, V. Naumenko
96
Note, that the number of equations in (6) is countable, when the network is
open, and of course, when it is closed.
We denote by Ψn ( z , t ) , where z = ( z1 , z 2 ,..., z n ) , n-dimensional generating function:
∞
∞
∞
Ψn ( z , t ) = ∑ ∑ ... ∑ P (k1 , k2 ,.., kn , t )z1k1 z2k2 ⋅ ... ⋅ znkn =
k1 =0 k2 =0
kn = 0
∞
∞
∞
n
kn =1
i =1
∑∑ ...∑ P(k , t )∏ z
k1 =1 k2 =1
ki
i
,
(7)
the summation is taken over each k i , from 1 to ∞ , i = 1, n , because the network is
operating in a high load mode.
Consider the case where the conditional probabilities ϕ i (k , t ), ψ ij (k , t ) , α i (k , t ) ,
βij (k , t ) do not depend on the state of the network. The expression for the generating function (7) can be rewritten as
n
Ψn ( z, t ) = C n ( z )exp− ∫ ∑ [λ (t ) p0 i (1 − ϕ i (t ) ) + µ i (t )(1 − βii (t ) ) +
 i =1
n
n
z j  
α (t )
+ λ (t ) p 0i z i ∑ψ ij (t ) + µ i (t ) i + µ i (t )∑ β ij (t )  dt  ,
zi  
zi
j =1
j =1

(8)
where the function C n ( z ) was defined in the proof of Lemma 2 in paper [4], from
the conditions, that at the initial time the network is able to ( x1 , x2 ,..., xn , 0), xi > 0 ,
i = 1, n , P ( x1 , x2 ,..., xn , 0) = 1 , P(k1 , k 2 ,..., k n , 0) = 0 , ∀xi ≠ k i , i = 1, n :
n
n
n
n

C n ( z ) = exp Λ(0 )∑ p 0i − ∑ p 0i Φ i (0) + ∑ M i (0) − ∑ Β ii (0) +
i =1
i =1
i =1
i =1

n
n
n n z
n
n

1
j
+ ∑ p 0i z i ∑ Υij (0 ) + ∑ Α i (0) + ∑ ∑ Β ij (0)∏ z lxl .
j =1
i =1 z i
i =1 j =1 z i
i =1
 i =1
(9)
Let us introduce the following notations:
Λ (t ) = ∫ λ (t ) dt , M i (t ) = ∫ µ i (t ) dt , Φ i (t ) = ∫ λ (t )ϕ i (t )dt , Υij (t ) = ∫ λ (t )ψ ij (t )dt ,
Α i (t ) = ∫ µ i (t )α i (t )dt , Β ij (t ) = ∫ µ i (t )β ij (t )dt .
(10)
Using the notation (10), expression (8) can be rewritten as
n
n
n
n
 
Ψn ( z , t ) = C n ( z )exp−  Λ(t )∑ p0 i − ∑ p0i Φ i (t ) + ∑ M i (t ) − ∑ Β ii (t ) +
i =1
i =1
i =1
i =1
 
n n z
n
n
n
 
1
j
+ ∑ p0 i zi ∑ Υij (t ) + ∑ Α i (t ) + ∑ ∑ Β ij (t )  .
i =1 j =1 z i
j =1
i =1
i =1 z i


(11)
Analysis of networks with time-dependent transition probabilities and messages bypass …
97
Then from (9) and (11) that
n
n
n

Ψn ( z , t ) = exp−  (Λ(t ) − Λ(0))∑ p0 i − ∑ p0i (Φ i (t ) − Φ i (0 )) + ∑ (M i (t ) − M i (0 )) −
i =1
i =1
i =1
 
n
n
n
n
1
− ∑ (Β ii (t ) − Β ii (0)) + ∑ p 0 i z i ∑ (Υij (t ) − Υij (0)) + ∑ (Α i (t ) − Α i (0 )) +
z
i =1
i =1
j =1
i =1 i
n
n
+ ∑∑
i =1 j =1
zj
zi

(Β (t ) − Β (0))∏ z
ij
ij
n

 i =1
xl
l
.
This expression can be rewritten as
n
n

Ψn ( z, t ) = a0 (t )exp∑ p0 i z i ∑ (Υij (t ) − Υij (0)) ×
j=1
 i =1

 n n zj
n
n 1

× exp∑ (Α i (t ) − Α i (0 )) exp∑ ∑ (Β ij (t ) − Β ij (0))∏ z lxl
 i =1 z i

 i =1 j =1 z i
 i =1
n
a0 (t ) = exp− ∑ [((Λ(t ) − Λ(0)) − (Φ i (t ) − Φ i (0 ))) p0i +
 i =1
+ (M i (t ) − M i (0 )) − (Β ii (t ) − Β ii (0 ))]}.
(12)
Transform (12) to a form suitable for finding the state probabilities of the network,
expanding its member exponential in a Maclaurin series. From (7) and (12) that
n
Ψn ( z, t ) = a 0 (t )a1 ( z , t )a 2 ( z, t )a3 ( z , t )∏ z lxl ,
l =1
where
n
n
 n n
a1 ( z , t ) = exp∑ p0i z i ∑ (Υij (t ) − Υij (0 )) = ∏∏ exp{p0 i zi (Υij (t ) − Υij (0 ))} =
j =1
 i =1
 i =1 j =1
l
li
n
n
n ∞ p z (Υ (t ) − Υ (0 ))
∞
∞ n
p0 i z i (Υij (t ) − Υij (0)) i
0i i
ij
ij
= ∏∏ ∑
= ∑ ... ∑∏∏
=
li !
li !
l1 = 0 ln = 0 i =1 j =1
i =1 j =1 li =0
[
]
∞
∞
l1 =0
ln = 0
= ∑ ...∑
[
p01l1 ⋅ ⋅ ⋅ p0lnn z1l1 ⋅ ⋅ ⋅ z nln
l1!l2 !⋅ ⋅ ⋅ln !
n
]
n
∏∏ [(Υij (t ) − Υij (0))]
li
,
i =1 j =1
n 1
 n
1

a 2 ( z , t ) = exp∑ (Α i (t ) − Α i (0)) = ∏ exp (Α i (t ) − Α i (0 )) =
 i =1 z i
 i =1
 zi

n
∞
=∏∑
[(Α (t ) − Α (0))z ]
i
−1 qi
i
i
qi !
i =1 qi = 0
∞
=
q1 = 0
n
∞
∞
q1 = 0
qn = 0
n
∞
∑ ... ∑∏
qn = 0 i =1
[(Α (t ) − Α (0))z ]
i
i
qi !
∏ [Α i (t ) − Α i (0)]
∑ ... ∑ i =1
qi
q1!⋅... ⋅ q n !
z1− q1 ⋅ ⋅ ⋅ z n− qn ,
−1 qi
i
=
M. Matalytski, V. Naumenko
98
 n n zj

a3 ( z , t ) = exp ∑∑ (Β ij (t ) − Β ij (0 )) =
 i =1 j =1 z i

[
−1
zj
 n n ∞ (Β ij (t ) − Β ij (0))z j z i
= ∏∏ exp (Β ij (t ) − Β ij (0)) = ∏∏ ∑
ri !
i =1 j =1
 zi
 i =1 j =1 ri =0
n
∞
n
n
∞
n
= ∑ .. ∑∏∏
[(Β (t ) − Β (0))z z ]
ij
ij
n
∞
= ∏∏ ∑
i =1 j =1 ri = 0
[(Β (t ) − Β (0))z z ]
ij
−1 ri
i
ri !
r1 = 0 rn = 0 i =1 j =1
n
j
ij
j
−1 ri
i
∞
ri
=
n
n
zj

=∏∏ exp (Β ij (t ) − Β ij (0)) =
i =1 j =1
 zi

∞
n
n
= ∑ .. ∑∏∏
ri !
]
[(Β (t ) − Β (0))z z ]
ij
ij
j
−1 ri
i
ri !
r1 =0 rn =0 i =1 j =1
=
ri
 n

 ∏ (Β ij (t ) − Β ij (0))
∏
∞
∞
j =1
i =1
 z r1 + r2 +...+ rn z r1 + r2 +...+ rn ⋅ ⋅ ⋅ z r1 + r2 +...+ rn z − r1 z − r2 ...z − rn =
= ∑ ... ∑ 
n
1
2
1
2
n
r1!⋅ ⋅ ⋅rn !
r1 = 0 rn = 0
n
ri
 n

 ∏ (Β ij (t ) − Β ij (0 ))
∏
∞
∞
j =1
i =1
 z R− r1 ⋅ ⋅ ⋅ z R − rn
= ∑ ... ∑ 
1
n
r1!⋅ ⋅ ⋅rn !
r1 = 0 rn = 0
n
n
Multiplying a0 (t ) , a1 ( z , t ) , a 2 ( z, t ) , a3 ( z , t ) and
∏ zlx
l
obtain the expression
l =1
for the generating function (7) has the form:
∞
∞
∞
∞
∞
∞
Ψn ( z , t ) = a 0 (t ) ∑ ... ∑ ∑ ... ∑ ∑ ... ∑
l1 = 0
l n = 0 q1 = 0
q n = 0r1 = 0
rn = 0
 p0lii
∏  l !q !r ! ×
i =1  i
i i
n
li
ri


 n

 n
q
×  ∏ (Υij (t ) − Υij (0 )) (Α i (t ) − Α i (0 )) i ×  ∏ (Β ij (t ) − Β ij (0 )) z ixi + li − qi − ri + R  , (13)

 j =1

 j =1

n
where R = ∑ ri .
i =1
4. Finding the average number of messages in the network systems
It is known that the expectation of the с-th component of a multidimensional
random variable can be found by differentiating the generating function (13) and
putting on z i = 1, i = 1, n . Therefore, the average number of messages in system S c
will use the relation:
99
Analysis of networks with time-dependent transition probabilities and messages bypass …
∞
∞ ∞
∞ ∞
∞
∂Ψn ( z , t )
= a 0 (t ) ∑ ... ∑ ∑ ... ∑ ∑ ... ∑ ( xc + l c − qc − rc + R ) ×
∂z c
l1 = 0 ln = 0 q1 = 0 qn = 0r1 = 0 rn = 0
ri
li
n 
n
p li  n

 
q 
× ∏  0 i  ∏ (Υij (t ) − Υij (0)) (Α i (t ) − Α i (0)) i  ∏ (Β ij (t ) − Β ij (0 ))  ×
i =1  l i !qi !ri !  j =1

 j =1
 

n
× zcxc +lc −qc −rc + R−1 ∏ zixi +li −qi −ri + R .
i =1,
i ≠c
It follows that the average number of applications in the system Sc is determined by the formula
N c (t ) =
∂Ψn ( z , t )
∂z c
∞
∞
∞
∞
∞
∞
= a0 (t ) ∑ ... ∑ ∑ ... ∑ ∑ ... ∑ (xc + lc − qc − rc + R ) ×
l1 = 0
z = (1,1, ...,1)
ln = 0 q1 = 0
qn = 0r1 = 0
rn = 0
ri
n
 p0lii  n

 
qi 
×∏
 ∏ (Υij (t ) − Υij (0 )) (Α i (t ) − Α i (0 ))  ∏ (Β ij (t ) − Β ij (0 ))  . (14)
i =1  l i ! q i ! ri !  j =1

 j =1
 

li
n
We make in (14) change of variables
lc = k c − xc + qc + rc − R and
∞
∞
∞
∞
∞
N c (t ) = a0 (t ) ∑ ... ∑ ∑ ... ∑
q1 = 0
qn = 0r1 = 0
k c = xc + lc − qc − rc + R , then
∞
∑
...
rn = 0 k1 = x1 − q1 − r1 + R
∑ kc
×
k n = xn − qn − rn + R
ki − xi + qi + ri − R

p0lii
 n

×∏
×
 ∏ (Υij (t ) − Υij (0 ))
i =1  (k i − xi + qi + ri − R )!qi !ri !  j =1


n
× (Α i (t ) − Α i (0 ))
qi
ri
 n
 
 ∏ (Β ij (t ) − Β ij (0 ))  .
 j =1
 
Because the network operating system under high
k i = xi − qi − ri + R ≥ 1 and, therefore, qi ≤ xi − ri + R − 1 , thus
∞
∞
∞
∞
N c (t ) = a0 (t ) ∑ ... ∑ k c ∑ ... ∑
k1 =1
k n =1
r1 = 0
rn = 0
x1 − r1 + R −1
∑
q1 = 0
...
load
mode
xn − rn + R −1
∑
×
qn = 0
ki − xi + qi + ri − R
n 
p0lii
 n

×∏
×
 ∏ (Υij (t ) − Υij (0 ))
i =1  (k i − xi + qi + ri − R )!qi !ri !  j =1


ri
n
 
qi 
× (Α i (t ) − Α i (0))  ∏ (Β ij (t ) − Β ij (0))  .
 j =1
 
(15)
M. Matalytski, V. Naumenko
100
5. Example 1
Let the intensity λ (t ) = λ t , µ i (t ) = µ i [cos(ω i t ) + 1] , i = 1, n . In this case
 sin(ω i t ) 
λ 2
+ t  , M i (0) = 0 , i = 1, n . Suppose, that
t , Λ(0) = 0 , M i (t ) = µ i 
2
 ωi

the probability of adherence message to the queue at time t is given by
f ( i ) (t ) = 1 − e − it . From (10) follows that
Λ(t ) =
Φ i (t ) = ∫ λ (t )ϕ i (t )dt = ∫ ϕ i (t )dΛ(t ) = ϕ i (t )Λ(t ) − ∫ ϕ i′(t )Λ(t )dt ,
Υij (t ) = ∫ λ (t )ψ ij (t )dt = ∫ψ ij (t )dΛ(t ) = ψ ij (t )Λ(t ) − ∫ψ ij′ (t )Λ(t )dt ,
Α i (t ) = ∫ µ i (t )α i (t )dt = ∫ α i (t )dM i (t ) = α i (t ) M i (t ) − ∫ α i′ (t ) M i (t )dt ,
Β ij (t ) = ∫ µ i (t )β ij (t )dt = ∫ β ij (t )dM i (t ) = β ij (t ) M i (t ) − ∫ β ij′ (t ) M i (t )dt ,
and we get that
 n λ
a0 (t ) = exp− ∑  t 2 − ϕ i (t )t 2 + ∫ ϕ i′(t )t 2 dt p0i +
 i =1  2
(
)
 sin(ω i t )
 sin(ω i t ) 
 sin(ω i t )  
+ µ i 
+ t − βii (t )
+ t  + ∫ βii′ (t )
+ t dt  =
 ωi

 ωi
 
 ωi

 n  λ
 sin(ω i t )
 
= exp − ∑  t 2 − ∫ ϕ i (t )tdt p 0 i + µ i 
+ t − ∫ β ii (t )(cos(ω i t ) + 1)dt   ,

 ωi
 
 i =1  2

(
)
Φ i (0) = 0 , Υij (0 ) = 0 , Α i (0) = 0 , Β ij (0 ) = Β ii (0) = 0 , Β ij (0 ) = 0 , i, j = 1, n . Conditional probabilities ϕ i (t ) , ψ ij (t ) , α i (t ) and β ij (t ) , according to (1)-(3), found from
the relations
n
(i )
(i )
n
pilψ lj (t )

 ψ ij (t ) = f (t )δ ij + (1 − f (t ))l∑
=1
ϕ i (t ) = (1 − f (t ) ) pi 0 + ∑ pijϕ j (t ) ,
,
j =1


(i )
n
n
α i (t ) = pi 0 + ∑ pijϕ j (t ),
j =1
βij (t ) = ∑ pilψ lj (t )
l =1
, i, j = 1, n .
(16)
Solving the systems of linear equations (16) in the package Mathematica, analytical solutions can be obtained, but they are cumbersome already at n = 3 . For
example, expression for the conditional probability of a time-dependent ϕ 1(t ) , at
n = 3 has the form:
Analysis of networks with time-dependent transition probabilities and messages bypass …
101
ϕ 1(t ) = ((e − t p13 (e −2 t p 22 − 1) − e −3t p12 p 23 )(e −4t p13 p30 − e − t p10 (e −3t p33 − 1)) −
− (e −3t p13 p 20 − e −3t p10 p 23 )(e −4t p13 p32 − e − t p12 (e −3t p33 − 1)))
((e
−t
p13 (e −2 t p 22 − 1) − e −3t p12 p 23 )(e −4 t p13 p 31 − (e − t p11 − 1)e − t (e −3t p33 − 1)) −
− (e −3t p13 p 21 − e −2 t (e − t p11 − 1) p 23 )(e −4 t p13 p 32 − e − t p12 (e −3t p33 − 1))) ,
and for the probability ψ
ψ
(t ) :
21
(t ) = (e 4t p20 − e 3t p11 p20 − e 3t p21 + e 4t p21 + e 3t p10 p21 + p13 p21 p30 + e t p23 p30 −
21
− p11 p 23 p 30 − p13 p 20 p 31 − p 23 p 31 + e t p 23 p 31 + p10 p 23 p 31 − e t p 20 p 33 + p 21 p 20 p 33 +
+ p 21 p33 − e t p 21 p33 − p10 p 21 p33 ) (− e 6 t + e 5t p11 +
+ e 3t p12 p 21 + e 4t p 22 − e 3t p11 p 22 + e 2 t p13 p31 − p13 p 22 p31 + p12 p 23 p31 + p13 p 21 p32 +
+ e t p 23 p 32 − p11 p 23 p 32 + e 3t p 33 − e 2 t p11 p 33 − p12 p 21 p 33 − e t p 22 p 33 − p11 p 22 p 33 ) .
Expression (13) takes the form
∞
∞
∞
∞
∞
∞
Ψn ( z , t ) = a 0 (t ) ∑ ... ∑ ∑ ... ∑ ∑ ... ∑
l1 = 0
l n = 0 q1 = 0
q n = 0r1 = 0
rn = 0
 λ
×  ∏  ψ ij (t )t 2 − ∫ψ ij′ (t )t 2 dt
 j =1  2
n
(
n
 p li
∏  l !q0!i r ! ×
i =1
) 


i
i
i
li
×
qi
 
 sin(ω i t )   
 sin(ω i t ) 
+ t dt  ×
×  µ i  α i (t )
+ t  − ∫ α i′(t )

 
ω
ω
i
i
  



 
ri
 n 
  xi +li −qi −ri + R 




sin(
)
sin(
)
ω
ω
t
t
i
i
,
×  µ i ∏  βij (t )
+ t  − ∫ βij′ (t )
+ t  dt   zi

 j =1 

ω
ω
i
i








after simplification we obtain:
∞
∞
∞
∞
∞
∞
Ψn ( z , t ) = a0 (t ) ∑ ... ∑ ∑ ... ∑ ∑ ... ∑
l1 = 0
ln = 0 q1 = 0
 n

×  ∏ ∫ψ ij (t )tdt 
 j =1

li
qn = 0r1 = 0
rn = 0
 λ 2  li µ iqi + ri p0lii
∏  2 t  l !q !r ! ×
i =1 
 i i i

n
(∫ α (t )(cos(ω t ) + 1)dt )
qi
i
i
ri


 n
 ∏ ∫ βij (t )(cos(ω i t ) + 1)dt  zixi + li − qi − ri + R 

 j =1

×
(17)
Suppose we have found, for example, the probability of state It is the coefficient
of z1 z 2 ⋅... ⋅ z n in the expansion of Ψn ( z , t ) in multiple series (17), so that when the
degree of z i must satisfy the relation xi + li − qi − ri + R = 1 , i = 1, n , it follows that
M. Matalytski, V. Naumenko
102
qi = xi + li + ∑ r j − 1 , i = 1, n , qi + ri = xi + li + ∑ r j − 1 , i = 1, n ,
j =1
j ≠i
j =1
li + qi + ri = xi + 2li + ∑ r j − 1 , i = 1, n ,
j =1
n
n
i =1
i =1
∑ (li + qi + r ) = ∑ (xi + 2li ) + n(R − 1) .
Therefore, from (17) follows that
∞
∞
∞
∞
P (1,1,...,1, t ) = a0 (t ) ∑ ... ∑ ∑ ... ∑
l1 = 0

 n
×  ∏ ∫ψ ij (t )tdt 

 j =1
li
ln = 0 r1 = 0
rn = 0

n

∑ li
=
1
i

n
p0lii
λ 2
x + l + R −1

×
 t  µi i i
∏

 2 
i =1
 l ! x + l + r − 1!r !
j
 i
 i  i i ∑
j =1
≠
j
i

 
(∫ α (t )(cos(ω t ) + 1)dt )
xi + li + ∑ r j −1
i
i
j =1
j ≠i

 n
 ∏ ∫ β ij (t )(cos(ω i t ) + 1)dt 

 j =1
ri

.

We assume that n = 10 and the initial time the network is in a state
(2,2,2,2,2,2,2,2,2,2,0) . We find the probability of the state P(1,1,1,1,1,1,1,1,1,1, t) ,
using the formula (17). Let the intensities λ (t ) = λ t , µ i (t ) = µ i [cos(ω i t ) + 1] , transition probabilities of messages are equal: p0i = 0.1, i = 1,10 , p10i = 0.1, i = 0, 9 ,
pi 0 = 0.5, i = 1,9 , pi10 = 0.5, i = 1,9 , pii = 0, i = 0,10 . In addition, the intensity of
the service messages in the systems are: µ i = 20.8, i = 1,4 , µ i = 10.22, i = 4,9 ,
µ10 = 20.5 . The expression for the time-dependent probability of the state in the
systems of the network obtained by on a computer using a mathematical calculation package Mathematica. Figure 1 shows a chart of this probability depending on
the time t.
P(1,1,1,1,1,1,1,1,1,1,t)
1.00
0.99
0.98
0.97
0.96
0
0.
0.
0.
0.
Fig. 1. The chart of probability of the state P(1,1,1,1,1,1,1,1,1,1, t )
1 t
103
Analysis of networks with time-dependent transition probabilities and messages bypass …
6. Example 2
We will consider the network described in Example 1. Let λ (t ) = λ t ,
µ i (t ) = µ i [cos(ω i t ) + 1] , transition probabilities of messages are equal:
p0i = 0.1, i = 1,10 ,
p10i = 0.1, i = 0, 9 ,
pi 0 = 0.5, i = 1,9 ,
pi10 = 0.5, i = 1,9 ,
pii = 0, i = 0,10 , intensity of service messages in the systems are: µ i = 20.8, i = 1,4 ,
µ i = 10.22, i = 4,9 , µ10 = 20.5 . The average number of messages in the systems
network (in the queue and service) at the initial time t = 0 equally N i (0 ) = 0 ,
i = 1,10 . Equation (15) to find the average number of messages in the systems network when the network parameters and the conditional probabilities depend on the
time takes the form:
 n λ
N c (t ) = exp − ∑  t 2 − ∫ ϕ i (t )tdt p0 i +
 i =1  2
∞
∞
 sin(ωi t )
   ∞ ∞
+ µi 
+ t − ∫ βii (t ) ( cos(ωi t ) + 1) dt   × ∑ ... ∑ k c ∑ ... ∑
 ωi
   k1 =1 kn =1 r1 = 0 rn =0
(
)
xn − rn + R −1
x1 − r1 + R −1
∑
...
q1 = 0
∑
×
qn = 0
ki − xi + qi + ri − R ri

λki − xi + qi + ri − R p0lii
 n

×∏
×
 ∏ ∫ψ ij (t )tdt 
i =1  (k i − xi + qi + ri − R )!qi !ri !  j =1


ri
n
qi 
 
× (∫ α i (t )(cos(ω i t ) + 1)dt )  ∏ ∫ β ij (t )(cos(ω i t ) + 1)dt   .
 j =1
 
n
This example is designed on a computer using a mathematical calculation
package Mathematica. Here are some of the values of the average number of
applications in the systems of the network (in the queue and service) at time t = 0.2 ,
found using the program: N 1 (t ) = 0.547 , N 2 (t ) = 0.323 , N 3 (t ) = 0.429 ,
N 4 (t ) = 0.522 , N 7 (t ) = 0.742 , N 10 (t ) = 0.654 . Figure 2 shows a chart of the
average number of messages in the QS S3 depending on time t.
N3(t)
0.578
0.511
0.489
0.429
0
0.2
0.4
0.6
0.8
1 t
Fig. 2. The chart of changes of the average number of messages N 3 (t ) in QS S3
104
M. Matalytski, V. Naumenko
References
[1] Ivnitsky V., Theory of Queuing Network. Monograph, Fizmatlit, Moscow 2004 (in Russian).
[2] Malinkovsky Y., Queuing Networks with Bypass of Nodes. Automation and Remote Control
1991, 2, 102-110 (in Russian).
[3] Matalytski M., Queuing theory and its applications, Monograph., Grodno 2011.
[4] Matalytski M., Naumenko V., Analysis of queueing network with messages bypass of systems in
transient behavior, Scientific Research of the Institute of Mathematics and Computer Science
2012, 11(2).
[5] Koluzaeva E., Finding the probabilities of states of models of information and computer networks
with time-dependent parameters of flow and maintenance of operating under high load Vestnik
GrSU. Ser. 2.2008, 3, 22-29 (in Russian).
[6] Statkevich S., Investigation of queuing networks with unreliable systems in the transient behavior,
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2012, 1, 112-125 (in Russian).