Journal of Applied Mathematics and Computational Mechanics 2013, 12(1), 85-92
FINDING OF THE EXPECTED INCOME OF THE CLOSED
QUEUEING STRUCTURE AND ITS APPLICATION
IN TRANSPORT LOGISTICS
Mikhail Matalytski 1, Olga Kiturko 2
1
Institute of Mathematics, Czestochowa University of Technology
Czestochowa, Poland
2
Faculty of Mathematics and Computer Science, Grodno State University
Grodno, Belarus
1
[email protected], 2 [email protected]
Abstract. Research of the closed exponential queueing structure with one-type messages
and the income is conducted. The differential equation in partial derivatives for an income
distribution density is received. The ordinary differential equation for its expected income
is constructed at particular starting conditions. The offered method of its decision in a case
when the intensity of the service of messages, the number of messages in networks, the
number of lines of the service in systems, the matrix of probabilities of transitions of
messages and the income from transitions between conditions of the closed queueing structure (CQS) depend on time is described. An example when change of parameters has
seasonal nature is reviewed. Results of this article can be applied at the prediction of the
income of the logistic transport system (LTS).
Introduction
In LTS in practice, the total number of the vehicles moving between various
objects and number of loading and unloading crews (workers or, for example,
operating loading and unloading tracks at stations) depend on time. Therefore the
following CQS can be considered as a LTS model.
Let's consider the closed queueing network consisting of n + 1 queueing
systems S 0 , S1 ,…, S n , the total number of one-type messages in which in an instant
t makes K (t ) . Usually the system S 0 is understood as environment, and as systems S1 , S 2 , …, S n - concrete queueing systems of network in which the service of
messages is made. The queueing networks closed on structure but having not constant total number of served messages, depending on time, are investigated for the
first time in special cases in [1] and called the closed queueing structures (CQS).
Let mi (t ) - number of service lines in system S i , i = 1, n , we will put m0 (t ) = K (t ) ,
and times of messages service of each of service lines are distributed under the
exponential law with intensity µ i (t ) , also time-dependent, i , j = 0, n . Messages
86
M. Matalytski, O. Kiturko
for the service get out according to discipline of FIFO. The message, whose service
in system S i ended, with probability pij (t ) passes to system S j , i , j = 0, n . The
matrix of transition probabilities P (t ) = pij (t ) , i , j = 0, n , is a matrix of transition
probabilities of a nonreducible Markov chain in each moment 0 ≤ pij (t ) ≤ 1 ,
n
∑p
ij
(t ) = 1. The primal problem of research of the given CQS consists in the
j =0
asymptotic analysis of the Markov process describing its behavior at a large
number of messages.
The condition of structure in a moment t is described by a vector
k (t ) = ( k , t ) = ( k1 , k 2 ,..., k n , t ) = ( k1 (t ), k 2 (t ),..., k n (t ))
where k i (t ) - number of messages in system S i in a moment t , t ∈ [0, T ] , i = 0, n ,
which forms the n-dimensional Markov process with the continuous time and
a finite number of conditions. It is apparent that the number of messages in system
n
S 0 is equal to k 0 (t ) = K (t ) − ∑ k i (t ) .
i =1
Believing that transitions of messages between systems bring in to CQS the
particular income, we will set the task of prediction of its total expected income.
Let V (k , t ) - the complete expected income which will receive CQS in time t if in
n
an initial moment it is in a condition (k , t ). It is apparent that V ( k , t ) = ∑ Vi ( k , t ) ,
i =0
where Vi ( k , t ) - the expected income which is gained by system S i in time t if in
an initial moment the CQS is in a condition k . In article [2], the concept of distribution density of the expected income of CQS v( x, t ) , x = ( x1 , x2 , ..., xn ) is injected,
and the following statement is proved.
Theorem. Income distribution density v( x, t ) under a condition that it is differentiated on t and is twice sectionally continuous differentiated on xi , i = 1, n , satis1
fies the terms of order of smallness ε 2 (t ) = 2
to the following differential
K (t )
equation in partial derivatives:
n
дv( x, t )
дv( x, t ) ε (t ) n
д2 v( x, t )
= −∑ Ai ( x, t )
−
B
(
x
,
t
)
−
∑ ij
дt
дxi
2 i , j =1
дxi дx j
i =1
n
− nK (t )ε ′(t )v( x, t ) + r (t ) + K (t ) ∑ µ j (t ) p ji (t ) min(l j (t ), x j )rji (t )
i, j =0
(1)
Finding of the expected income of the closed queueing structure and its application in transport logistics
87
where
n
n


Ai ( x, t ) = ∑ µ j (t ) p *ji (t ) min(l j (t ), x j ) + µ 0 (t ) p0 i (t )1 − ∑ xi  ,
j =1
i =1


m j (t )
 p (t ) − 1, j = i,
p *ji (t ) =  ji
l j (t ) =
, j = 1, n ,
p
(
t
),
j
≠
i
,
K (t )
 ji
(2)
Bij ( x, t ) - sectionally continuous function concerning x ,
n
Bii ( x, t ) = ∑ µ j (t )q*ji (t ) min(l j (t ), x j ) , Bij ( x, t ) = µ i (t ) pij (t ) min(li (t ), xi ) , (3)
j =0
− 1 − p ji (t ), j = i,
q *ji (t ) = 
− p ji (t ), j ≠ i.
1. Differential equation for the average income of structure
Considering (3), expression
д 2 v ( x, t )
ε(t ) n
Bij ( x, t )
∑
2 i , j =1
дxi дx j
can be referred to
O(ε 2 (t )) . Therefore we will consider the following equation:
n
дv(k , t )
дv( x, t )
= −∑ Ai ( x, t )
− nK (t )ε′(t )v( x, t ) +
дt
дxi
i =1
n
+ K (t ) ∑ µ j (t ) p ji (t ) min(l j (t ), x j )r ji (t ) + r (t ) .
i , j =0
Having integrated both parts of this equation on x = ( x1 , x 2 ,..., x n ) in area
n


G =  x = ( x1 , x2 ,..., xn ) : xi ≥ 0, ∑ xi ≤ 1 and having divided both members of
i =1


equation into the volume of area G , equal to m(G ) , we receive:
1
1 n
дv( x, t )
дv( x, t )
...
... ∫ Ai ( x, t )
dx
=
−
dx −
∑
∫∫
∫
∫∫
m(G ) G
дt
m(G ) i =1 G
дxi
−
nK (t )ε′(t )
K (t ) n
...
v
(
x
,
t
)
dx
+
∑ µ j (t ) p ji (t )rji (t ) ∫∫... ∫ min(l j (t ), x j )dx +
m(G ) ∫∫G ∫
m(G ) i =0 ,
G
j =1
+
n


K (t )
r (t )

µ
(
t
)
p
(
t
)
r
(
t
)
...
1
−
x j  dx +
... dx .
∑
∑
0
0i
0i
∫∫
∫

m(G ) i = 0
m
(G ) ∫∫G ∫
j =1
G


n
(4)
88
M. Matalytski, O. Kiturko
Let's consider that in the left part of this equality the change of order of an integration and derivation (we assume that in closed area G function v( x, t ) is the
continuous) is admissible:
1
дv( x, t )
1 д
d
... ∫
dx =
... ∫ v( x, t )dx = vG (t ) ,
∫∫
∫∫
m(G ) G
дt
m(G ) дt G
dt
where vG (t ) - an average on x value of the income on condition of change of
a reference state ( x, t ) in area G .
Let's consider integrals in a right member (4). It is apparent that
nK (t )ε′(t )
... v( x, t )dx = nK (t )ε′(t )vG (t ) . During calculation of the remaining
m(G ) ∫∫G ∫
integrals we use integration by parts, and also we will assume that boundary conditions [3] are satisfied:
Ai ( x, t )v( x, t ) x∈Γ (G ) = 0 , i = 1, n ,
where
area
border
G,
x =1− x1 − x2 −...− xn −1
An ( x, t )v( x, t ) xn =0
i.e.
n
xn −1 =1− x1 − x2 −...− xn − 2 − xn
An −1 ( x, t )v( x, t ) x
n −1 = 0
x1 =1− x2 − x3 −...− xn
= 0 , …, A1 ( x, t )v( x, t ) x =0
1
= 0,
= 0 , which means
that the flow of the income through the border of area G is not assumed or that in
boundary points of area G reflecting screens are applied. Then, considering that
дАi ( x, t )
does not depend on x j , j = 1, n , we receive
дxi
дA ( x, t )
1
дv( x, t )
... ∫ Ai ( x, t )
dx = − i
vG (t ) , i = 1, n .
∫∫
m(G ) G
дxi
дxi
Therefore we come to the following differential equation
 n дA ( x, t )

d
vG (t ) = ∑ i
− nK (t )ε′(t )  vG (t ) +
dt
 i =1 дxi

+
+
K (t ) n n
∑∑ µ j (t ) p ji (t )rji (t ) ∫∫... ∫ min(l j (t ), x j )dx +
m(G ) j =1 i = 0
G
K (t )
m (G )
n
∑ µ (t ) p
0
i =0
0i
(t )r0i (t )

... 1 −

G
∫∫ ∫

n
(5)
r (t )
∑ x dx + m(G ) ∫∫ ... ∫ dx .
j
j =1
G
From (2) we see that the coefficients Ai ( x, t ) represent piecewise linear functions on x j , j = 1, n , that is, (5) is a differential equation with a piecewise constant
right member. Let's designate a set of indexes of components of a vector
Finding of the expected income of the closed queueing structure and its application in transport logistics
89
x = ( x1 , x2 ,..., xn ) through Ω = {1, 2, ..., n}. Let's break Ω into two non-overlapping
sets Ω0 (τ ) , Ω1 (τ ) so that Ω0 (τ ) = { j : l j (t ) < x j ≤ 1} , Ω1 (τ) = { j : 0 ≤ x j ≤ l j (t )},
τ - splitting number. At fixed t , the number of splittings of this kind is equal to 2 n ,
τ = 1,2 n . Each splitting will set in a set G not being crossed areas Gτ so that
n


Gτ =  x(t ) : li (t ) < xi ≤ 1, i ∈ Ω 0 (τ); 0 ≤ x j ≤ l j (t ), j ∈ Ω1 ( τ); ∑ xi ≤ 1 ,
i =1


2n
UG
τ = 1, 2, ..., 2 n ,
τ
=G.
τ=1
Now in each area splitting of a phase space we can write down an apparent look
(5) and at particular starting conditions find the average expected income for each
of the areas Gτ .
Let's set, for example, splitting: Ω0 (1) = {1, 2, ..., n} , Ω1 (1) = {Ο/} , τ = 1 , that
corresponds to existence of queues S1 , S 2 , …, S n . Then, solving the equation (5)
at starting conditions vG1 (0) = S , it is possible to determine the average expected
income
changing
of
a
reference
state
of
area
n


G1 =  x(t ) : li (t ) < xi ≤ 1, i = 1, n, ∑ xi ≤ 1 . The equation (5) thus looks like
i =1


n
d


vG (t ) = µ 0 (t )∑ p0i (t ) − nK (t )ε′(t ) vG (t ) +
dt
i =1


1
+
1
1  n n
∑∑ µ j (t ) p ji (t )m j (t )r ji (t ) +
m(G1 )  j =1 i =0
(6)
n
n

 
+ K (t )∑ µ 0 (t ) p0i (t )r0i (t ) ∫∫ ... ∫ 1 − ∑ x j dx  + r (t ) .
i =0
j =1
G

 
1
2. Example
The transport enterprise (TE, system S3 ) at the disposal of which a large number
of cars (messages) is available, sends the cars for realization of a number of particular transportations between various cities (environment, system S 0 ). After that
they come back to the TE base, before having passed in two points (systems S1 and
S 2 ) technical inspection which can also include car repairs. The number of
functioning lines of a service in points S1 and S 2 in a moment t are equal respec-
90
M. Matalytski, O. Kiturko
tively to m1 (t ) and m2 (t ) . In system S3 loading of cars before the flight in which
are engaged m3 (t ) loading crews (service lines) is carried out.
A similar situation can arise when cars come back from the environment and are
unloaded in two warehouses TE (systems S1 and S 2 ); in this case m1 (t ) and m2 (t )
- number of crews of unloading accordingly in the warehouse S1 and S 2 .
In both cases the model of functioning of transportations is CQS represented
in Figure 1.
It is apparent that in this case the matrix P(t ) looks like
P (t ) = pij (t )
4× 4
0

= 0
0
1
p01 (t )
0
0
0
p02 (t )
0
0
0
0
1  , p (t ) + p (t ) = 1 , ∀t ∈ [0, T ] .
02
1  01
0
S1
S3
S2
S0
Fig. 1. Model of cars’ movement
It is apparent that in this case the matrix looks like
P (t ) = pij (t )
4×4
0

= 0
0
1
p01 (t )
0
0
0
p02 (t )
0
0
0
0
1  , p (t ) + p (t ) = 1 , ∀t ∈ [0, T ] .
02
1  01
0
Let the change of the CQS parameters have a seasonal nature, for example, they
can have one value during the winter period and others in the summer. Let's establish that

 (1)
 T
 T
K1 , t ∈ 0, ,
µ j , t ∈ 0, ,



 2  µ (t ) = 
 2  j = 0,3 ,
K (t ) = 

j
T


K 2 , t ∈ , T ,
µ (j2 ) , t ∈  T , T ,


2 


2 




Finding of the expected income of the closed queueing structure and its application in transport logistics
91
 (1)
 (1)
 T
 T
m j , t ∈ 0, ,
p0i , t ∈ 0, ,



 2  j = 1,3 , p (t ) = 
 2  i = 1,2 .
m j (t ) = 

0i
T
m (j2 ) , t ∈  , T  ,
 p0( 2i ) , t ∈  T , T ,


2 


2 




Also let the income from transitions between conditions of CQS also be step functions with two intervals of constancy:
 (1)
 (1)
 T
 T
r ji , t ∈ 0, ,
r , t ∈ 0, ,



 2  i, j = 0,3 , r (t ) = 
 2
r ji (t ) = 

T
r ji( 2) , t ∈  , T ,
r ( 2 ) , t ∈  T , T .


2 


2 




It is clear that ε ′(t ) = 0 .
3


Let's find area volume G1 =  x : li < xi ≤ 1, i ∈ {1, 2, 3}; 0 ≤ x0 ≤ l0 ; ∑ xi ≤ 1 . We
i =1


have
m(G1 ) = ∫∫∫ dx =
G1
1−l2
∫
1− x1
1− x1 − x2
∫
dx1
l1
dx2
l2
It is possible also to find integral
(1 − l1 − l2 ) 2 (1 − l1 − l2 − 3l3 )
.
6
∫ dx3 =
l3
∫∫∫(1 − x1 − x2 − x3 )dx :
G1
∫∫∫ (1 − x1 − x2 − x3 )dx =
1−l2
G1
∫
1− x1
∫
dx1
l1
=
1− x1− x2
dx2
l2
∫ (1 − x1 − x2 − x3 )dx3 =
l3
(
)
(1 − l1 − l2 ) 2 (1 − l1 − l2 ) 2 − 4(1 − l1 − l2 )l3 + 6l32
.
24
 T
Let's first find the expected income of CQS v1 (t ) on an interval 0,  . The equa 2
tion (6) in this case looks like:
d v1 (t )
1
= µ 0(1) v1 (t ) +
µ1(1) m1(1) r13(1) + µ 2(1) m2(1) r23(1) + µ 3(1) m3(1) r30(1) +
dt
m(G1 )
[
(
(1) (1)
(1) (1)
+ K1µ 0(1) p01
r01 + p02
r02
)∫∫∫ (1 − x − x
1
G1
2

− x3 )dx + r (1)  = µ 0(1) v1 (t ) + r1 .

92
M. Matalytski, O. Kiturko
And let v1 (0) = v0(1) . Its decision is
v1 (t ) = v0(1) e
µ 0(1)t
t
+ r1 ∫ e
µ 0(1) ( t −τ )
dτ = v0(1) e
µ 0(1)t
+
0
r1
µ 0(1)
 e µ0(1)t − 1 =





r  µ (1)t
r
 T
=  v0(1) + 1(1) e 0 − 1(1) , t ∈ 0,  .
µ0 
µ0
 2

T 
Let's now find the expected income v2 (t ) of CQS on interval  , T  . The
2 
equation (6) on this interval becomes
d v2 (t )
= µ 0( 2 ) v2 (t ) + r2 ,
dt
where r2 =
(7)
1
( 2) ( 2)
( 2) ( 2)
µ1( 2) m1( 2) r13( 2) + µ 2( 2) m2( 2) r23( 2) + µ 3( 2) m3( 2) r30( 2) + K 2 µ 0( 2) p01
r01 + p02
r02 ×
m(G1 )
[
(
)

× ∫∫∫ (1 − x1 − x2 − x3 ) dx + r ( 2)  , as starting conditions it is necessary to take

G1
T 
T 
v2   = v1   . The solution of the equation (7) is
2
 
2
(1) T
 (1)
r1  µ 0 2
r
r  µ ( 2 )t r
T 


v2 (t ) =  v0 + (1) e
− 1(1) + (22 )  e 0 − (22 ) , t ∈  , T  .
µ0 
µ0
µ 0 
µ0
2 

References
[1] Matalytski M., Rusilko T., Mathematical analysis of stochastic models for claim processing
in insurance companies, GrSU, Grodno 2007 (in Russian).
[2] Kiturko O., Matalytski M., Rusilko T., Asymptotic analysis of the total expected income of closed
queueing structure with the one-type messages and application, Wiestnik GrUP 2013, 1(2)
(in Russian).
[3] Tikhonov V., Mironov V., Markov processes, Sov. Radio, Moscow 1977 (in Russian).