Investigation of some probabilistic characteristics of one class of semi-Markov wandering with delaying screens

ISSN 0146�4116, Automatic Control and Computer Sciences, 2014, Vol. 48, No. 2, pp. 109–119. © Allerton Press, Inc., 2014.Original Russian Text © T.I. Nasirova, B.G. Shamilova, 2014, published in Avtomatika i Vychislitel’naya Tekhnika, 2014, No. 2, pp. 64–75.

109

1. INTRODUCTION

Formulation of the Problem

Let the sequence of independent identically distributed random values ξi, ηi, be specified at the probabilistic space .

Let us consider that > 0, and the numbers a, b, and z, where a > b, and are given.

We will construct the following random process:

which will be said to be an uneven process of semi�Markov wandering.The following process is taken to be the one of semi�Markov wandering with delaying screens “b” and

“a”:

where

Note that most important problems from the storage control theory are described and solved with usingthe semi�Markov wandering processes ([4], ([5]).

As known from the condition [3], the absence of gratying and degeneracy provide the ergodicfactor of the process X(t). The investigations of the ergodic distribution for the processes of semi�Markovwandering have an important value in the random process theory [1–3, 6].

The purpose of the present article is to find the Laplace transform by time and the Laplace–Stieltjestransform by phase of the conditional and unconditional distributions of the semi�Markov wandering pro�

{ }, { },i iξ η 1,i = ∞ 1,i = ∞

F( , , )Ω Ρ

iξ 1 0E η > 0,b > 0 ,z a b< ≤ −

1 0

1

1 1 1 1

( ) , if , 0, 0 ,k k k

i i i

i i i

X t b z t k+

= = =

⎛ ⎞= + + η ξ ≤ < ξ ≥ =⎜ ⎟⎜ ⎟

⎝ ⎠∑ ∑ ∑ ∑

if1 0

1 1 1

( ) , , 0, 0 ,k k

k i i

i i

X t t k+

= =

⎛ ⎞= ζ ξ ≤ < ξ ≥ =⎜ ⎟⎜ ⎟

⎝ ⎠∑ ∑ ∑

{ }{ }1

0

min ,max , , 1,

.k k ka b k

b z−

ζ = ζ + η ≥

ζ = +

1 0Eη >

Investigation of Some Probabilistic Characteristics of One Class of Semi�Markov Wandering with Delaying Screens

T. I. Nasirova and B. G. ShamilovaBaku State University, ul. Z. Khalilova 23, Baku, AZ 1148 Azerbaijan

e�mail: [email protected], [email protected] September 2, 2013; in final form, December 18, 2013

Abstract—The process of semi�Markov wandering with delaying screens “b” and “a” (a > b > 0) is con�structed by the given sequence of independent and identically distributed random vectors

The integral equation for the Laplace transform by time and the Laplace–Stieltjes trans�form by the phase of its conditional distribution is derived. If the wandering occurs by a complicatedLaplace distribution, the ergodic distribution of the process and its moments are found. Then, the integral equation for the generating function of the conditional distribution of the numberof process steps at which it firstly reaches the level a is derived. When the wandering occurs by the sim�ple Laplace distribution, its generating functions and moments are found.

Keywords: process of semi�Markov wandering with delaying screen, ergodic distribution, Laplacetransform, generating function

DOI: 10.3103/S0146411614020059

( , ), 1.i i iξ η ≥

110

AUTOMATIC CONTROL AND COMPUTER SCIENCES Vol. 48 No. 2 2014

NASIROVA, SHAMILOVA

cess with delaying screens and its Laplace–Stieltjes transform of the ergodic distribution when the randomvalue has the Laplace transform. Moreover, the present authors have obtained the evident form of thegenerating function of the number of steps at which the process first reached the screen “a.” The two firstmoments were also found.

2. LAPLACE–STIELTIES TRANSFORM

This section discusses the Laplace–Stieltjes transform of the ergodic process of semi�Markov wander�ing X(t).

We introduce the following denotations:

is the conditional distribution of the process X(t);

is the Laplace transform by the time of the conditional dis�

tribution of the process X(t);

is the Laplace transform of the distribution of the random value ξ1;

is the Laplace transform by time and the Laplace–Stieltjes trans�

form by the phase of the conditional distribution of the process X(t).Theorem 1. The following correlation is true:

Proof. According to the probability formula,

(1)

Let us denote that

Then,

.

Upon applying the Laplace transform by t to both parts of (1), we obtain the following integral equa�tion:

Thus, the integral equation for is found:

(2)

1η

{ }( , ) ( ) (0)R t x b z X t x X b z+ = Ρ < = +

( )0

, ( , ) ,t

tR x b z e R t x b z dt

∞−θ

=

θ + = +∫� 0θ >

( ) { }10

t

te d t

∞−θ

=

ϕ θ = Ρ ξ <∫( ), ( , )

xa

xx b

R b z e d R x b z−α

=

θ α + = θ +∫�

� �

( )( ) ( ) ( ) { } ( )

( ) ( ) { } ( ) { } ( )

1

1 1

1, ,

, , .

b z

a

y

y b

R b z e z R b

R y d y b z a b z R a

−α +

=

− ϕ θθ α + = + ϕ θ Ρ η < − θ α

θ

+ ϕ θ θ α Ρ η < − − + ϕ θ Ρ η > − − θ α∫

� �

� �

� �

� �

{ }

{ } { }

1

1 1

0

( , ) ( ) ; (0)

( , ) min( ,max( , )) .

a t

y b s

R t x b z X t x t X b z

ds R t s x y a b b z dy

= =

+ = Ρ < ξ > = +

+ Ρ ξ ∈ − Ρ + + η ∈∫ ∫

( )0, 0,

1, 0.

xx

x

<⎧ε = ⎨

>⎩

{ } { } { }1 1 1( ) ; (0) ; ( )X t x t X b z b z x t x b z P tΡ < ξ > = + = Ρ + < ξ > = ε − − ξ >

( ) { }

{ } { }

1

0 0

1 1

0 0

( , )

min( , max( , )) ( , ) .

t t

t t

a t

t

y b t s

e R t x b z dt x b z e t dt

a b b z dy e ds R t s x y dt

∞ ∞

−θ −θ

= =

∞

−θ

= = =

+ = ε − − Ρ ξ >

+ Ρ + + η ∈ Ρ ξ ∈ −

∫ ∫

∫ ∫ ∫

( ),R x b zθ +�

( )( )

( ) ( ) { }11

, ( , ) min( ,max( , )) .

a

y b

R x b z x b z R x y d a b b z y

=

− ϕ θθ + = ε − − − ϕ θ θ Ρ + + η >

θ ∫� �


INVESTIGATION OF SOME PROBABILISTIC CHARACTERISTICS OF ONE CLASS 111

With taking the properties of the minimum and maximum of the two functions in Eq (2), we can writethat

(3)

Upon applying the Laplace transform by x to both parts of (2), we obtain the statement of Theorem 1.

Consequence 1. Let where has an Erlang distribution of the second order with the

parameter while is the Erlang distribution of the first order with the parameter µ, i.e.,

Then, we find from the statement of Theorem 1 that

(4)

From (4), we obtain the ordinary differential equation with constant coefficients with the right part

with the characteristic equation

and the general solution

(5)

Now, we should find The following boundary conditions result from (4):

( )( )

( ) ( ) ( ) { }

( ) ( ) { } ( ) ( ) { }

1

1 1

1, ,

, , .

a

y b

R x b z x b z R x b z

R x y d y b z R x a a b z

=

− ϕ θθ + = ε − − + ϕ θ θ Ρ η < −

θ

+ ϕ θ θ Ρ η < − − + ϕ θ θ Ρ η > − −∫

� �

� �

1 1 1 ,+ −

η = η − η 1+

η

,λ 1−

η

{ }( )

2

2

1

, 0, 0,

21 , 0, 0.

x

x

e x

x

x e x

μ

−λ

⎧ λ < μ >⎪ λ + μ⎪Ρ η < = ⎨

⎡ ⎤μ λ + μ⎪ − + λ > λ >⎢ ⎥⎪ λ + μ λ + μ⎩ ⎣ ⎦

( )( ) ( ) ( )

( )( )

( )

( )

( )( )

( )

( )

( )( ) ( )

( ) ( )( ) ( )

( )

( )

( )( ) ( )

( )( ) ( )

2 2

2 2

2

2 2

2

1, , ,

2, ,

, , .

b z

b z b zz y

y b

a b z a b z

a

b z b zy y

y b z y

R b z e e R b e R y e dy

a b ze R a e R a

e e R y dy e y b z e R y dy

+

−α + −μ +−μ μ

=

−λ − − −λ − −

λ + λ +−λ −λ

= +

− ϕ θ λ ϕ θ λ μϕ θθ α + = + θ α + θ α

θ λ + μ λ + μ

μ λ + μ λμ − −+ ϕ θ θ α + ϕ θ θ α

λ + μλ + μ

λ μϕ θ λ μ+ θ α + ϕ θ − − θ α

λ + μλ + μ

∫

∫

� � �

� � �

� �

� �

� �

� �

a

b z= +

∫

( ) ( ) ( ) ( ) ( )

( )[ ] ( ) ( ) ( )( ) ( )22

''' ''', 2 , 2 ,

11 , ,

z z z

b z

R b z R b z R b z

R b z e−α +

θ α + − λ − μ θ α + − λ λ − μ θ α +

− ϕ θ+ λ μ − ϕ θ θ α + = − λ + α α − μ

θ

� � �

� � �

�

�

( ) ( )[ ]3 2 2( ) (2 ) ( ) 2 ( ) 1 0k k kθ − λ − μ θ − λ λ − μ θ + λ μ − ϕ θ =

( ) ( )( ) ( ) ( )

( )[ ]

( )3 2

( )( )

3

1

1

1, , .i b zk b z

i

ii

i

R b z c e e

k

−α +θ +

=

=

− ϕ θ λ + α α − μ

θ α + = θ α +

θ

α + θ

∑∏

�

�

( ), , 1,2,3.ic iθ α =

112


NASIROVA, SHAMILOVA

(6)

On the other hand, from (4), at z = 0 we have

By substituting the expressions and on the left and formula (5) insteadof the desired function under the integrals into (6), we obtain the system of algebraic equations in relationto

( )( )

( )( )

( )( )

( )( )

( )

( )( )

( )

( )( )

( )

2

2

2 2

2

2

1 2, , ,

,

, , ,

a bb

a

b y b

y b

a a

y b y

y b y b

R b e R b a b e R a

e e R y dy e

bye R y dy e e R y dy

−λ −−α

λ −λ λ

=

−λ λ −λ

= =

− ϕ θ μϕ θ ⎡ ⎤λ μ λ + μθ α = + θ α + + λ − θ α⎢ ⎥θ λ + μ λ + μλ + μ ⎣ ⎦

λ μϕ θ λ μϕ θ+ θ α +

λ + μλ + μ

λ μ ϕ θ× θ α − θ α

λ + μ

∫

∫ ∫

� � �

� � �

�

�

� �

� �

( )( ) ( )

( )( )

( )( )

( )( )

( )

( )( )

( )

( )( )

( )

2 2

2

2 2 3

2

3

1 1' , , ,

,

, , ,

a bbz

a

b y b

y b

a a

y b y

y b y b

R b e R b a b e R a

e e R y dy e

bye R y dy e e R y dy

−λ −−α

λ −λ λ

=

−λ λ −λ

= =

− ϕ θ λ μϕ θ λ μϕ θ ⎡ ⎤θ α = − α − θ α + + − θ α⎢ ⎥θ λ + μ λ + μλ + μ ⎣ ⎦

λ μ ϕ θ λ μϕ θ− θ α +

λ + μλ + μ

λ μ ϕ θ× θ α − θ α

λ + μ

∫

∫ ∫

� � �

� � �

�

�

� �

� �

( )( ) ( )

( )( )

( )( )

( )( )

( ) ( )

( )( )

( )( )

( )( )

2 2 22

2

3 2

2

4 4

1'' , , ,

2,

, , .

a bbz

a

b y

y b

a a

b y b y

y b y b

R b e R b a b e R a

e e R y dy

be ye R y dy e e R y dy

−λ −−α

λ −λ

=

λ −λ λ −λ

= =

− ϕ θ λ μ ϕ θ λ μϕ θ ⎡ ⎤μθ α = α + θ α − − λ − θ α⎢ ⎥θ λ + μ λ + μλ + μ ⎣ ⎦

λ μ λ + μ ϕ θ− θ α

λ + μ

λ μϕ θ λ μ ϕ θ+ θ α − θ α

λ + μ λ + μ

∫

∫ ∫

� � �

� � �

�

�

� �

� �

( )( ) ( ) ( )

[ ]

θ −α

=

=

− ϕ θ λ + α α − μ

θ α = θ α +

θ

α + θ

∑∏

�

�

3 2( )

3

1

1

1, ( , ) ,

( )

ik b bi

ii

i

R b c e e

k

( )( ) ( ) ( )

[ ]

θ −α

=

=

− ϕ θ α λ + α α − μ

θ α = θ α θ −

θ

α + θ

∑∏

�

�

3 2( )

3

1

1

1' , ( , ) ( ) ,

( )

ik b bz i i

ii

i

R b c k e e

k

( )( ) ( ) ( )

[ ]

θ −α

=

=

− ϕ θ α λ + α α − μ

θ α = θ α θ +

θ

α + θ

∑∏

�

�

3 222 ( )

3

1

1

1'' , ( , ) ( ) .

( )

ik b bz i i

ii

i

R b c k e e

k

( ) ( )', , , ,zR b R bθ α θ α� �

� � ( )'' ,zR bθ α�

�

( ) :, , 1,2,3ic iθ α =



Clearly, the determinant of the system is not zero. Thus, using the Kramer formula, we find that

and

( )[ ] ( )( )

( ) ( )[ ]

( ) ( ) ( )[ ]( )[ ]( )[ ]

( ) ( )

[ ]

( ) ( )

( ) ( ) ( )[ ]( )

( ) ( )[ ] ( ) ( )[ ]( ){ } ( )[ ]

( ) ( ) ( )

32

1

22

3

1

3

1

2

{

2 } ( , )

1{

( )

2 } ,

2 ( , )

1

(

i

i

i

k bi i i i

i

b k ai i i

b

i

i

b a

b k ai i i i i

i

i

k k e k k

k k a b e c

e

k

a b e

k k k k a b e c

k

θ

=

λ − λ− θ

−α

=

λ − λ+α

λ − λ− θ

=

λ − θ θ + θ μ + θ

× λ + μ − θ + λ + μ λ − θ − θ α

− ϕ θ λ αμϕ θ= − λ + α + λ − μ

θα + θ

× λ + μ + α + λ + μ λ + α −

θ μ + θ λ − θ + λ λ − θ − θ α

− ϕ θ λ αμϕ θ α − μ=

θα + θ

∑

∏

∑

[ ]

( ) ( )[ ][ ]

( ) ( )[ ] ( )[ ]( ){ } ( )[ ]

( ) ( ) ( )

[ ]

( ) ( )[ ][ ]

3

1

3

1

2

3

1

2 ,

)

1 ( , )

11 .

( )

i

b a

i

b k ai i i i

i

b a

i

i

a b e

k k k a b e c

a b e

k

λ − λ+α

=

λ − λ− θ

=

λ − λ+α

=

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪λ + α + λ + α − ⎪⎪⎪⎪⎪

θ μ + θ + λ − θ − θ α ⎪⎪⎪− ϕ θ λ αμϕ θ α − μ ⎪= + λ + α −⎪θ

α + θ ⎪⎪⎭

∏

∑

∏

( ) ( )

[ ] ( )

( ) [ ] [ ][ ]

( )[ ][ ][ ][ ]

( )[ ][ ][ ][ ]

3 2

3 2

2 3

2 2 ( ) ( )21 3 2

1 3

1

1

2 ( ) ( )2 2 3

2 ( ) ( )3 2 2

1 ( ) ( ) ( )( , )

( )

( ) ( ) ( )

( ) ( ) ( ),

b k k a

i

i

k b k a

k b k a

k k k ec

lk k

k k k e

l

k k k e

l

−α + θ + θ

=

θ − α− θ

θ − α− θ

⎡− ϕ θ λ μαϕ θ − λ + α λ − θ θ − θθ α = ⎢

θ ⎣α + θ θ

α − μ μ + θ α + θ λ − θ+

⎤α − μ μ + θ α + θ λ − θ− ⎥

⎦

∏

( ) ( )

[ ] ( )

( ) [ ] [ ][ ]

( )[ ][ ][ ][ ]

( )[ ][ ][ ][ ]

1 3

3 1

1 3

2 2 ( ) ( )22 3 1

2 3

2

1

2 ( ) ( )1 1 3

2 ( ) ( )3 3 1

1 ( ) ( ) ( )( , )

( )

( ) ( ) ( )

( ) ( ) ( ),

b k k a

i

i

k b k a

k b k a

k k k ec

lk k

k k k e

l

k k k e

l

−α + θ + θ

=

θ − α− θ

θ − α− θ

⎡− ϕ θ λ μαϕ θ λ + α λ − θ θ − θθ α = × ⎢

θ ⎣α + θ θ

α − μ μ + θ α + θ λ − θ−

⎤α − μ μ + θ α + θ λ − θ+ ⎥

⎦

∏

114


NASIROVA, SHAMILOVA

where

Putting the values of into (5), we obtain the expression for

Thus, we found the Laplace transform by time and the Laplace–Stieltjes transform by the phase of theconditional distribution of the process

Using the formula of the total probability, we can determine the unconditional distribution from theconditional distribution of the process; i.e.,

where has an Erlang distribution of the second order.

Taking (5) into account, we obtain the expression for

(7)

where

Since the process X(t) is ergodic, the Tauber theorem can be applied to (7), or

(8)

( ) ( )

[ ] ( )

( ) [ ] [ ][ ]

( )[ ][ ][ ][ ]

( )[ ][ ][ ][ ]

1 2

1 2

2 1

2 2 ( ) ( )21 2 1

3 3

3

1

2 ( ) ( )2 2 1

2 ( ) ( )1 1 2

1 ( ) ( ) ( )( , )

( )

( ) ( ) ( )

( ) ( ) ( ),

b k k a

i

i

k b k a

k b k a

k k k ec

lk k

k k k e

l

k k k e

l

−α + θ + θ

=

θ − α− θ

θ − α− θ

⎡− ϕ θ λ μαϕ θ − λ + α λ − θ θ − θθ α = ⎢

θ ⎣α + θ θ

α − μ μ + θ α + θ λ − θ+ −

⎤α − μ μ + θ α + θ λ − θ− ⎥

⎦

∏

( )[ ] [ ]( ) ( ) ( )[ ]

( )[ ] ( ) ( )[ ]( ) ( )[ ] ( )

( )[ ] ( ) ( )[ ]( ) ( )[ ] ( )

θ + θ + θ

θ + θ + θ

θ + θ + θ

= λ − θ θ − θ

− λ − θ θ − θ

+ λ − θ θ − θ

1 2 3

3 1 2

2 1 3

41 3 2

42 3 1

43 2 1

( ) ( )

.

k b k k a

k k a k b

k k a k b

l k k k e

k k k e

k k k e

( ), , 1,2,3ic iθ α = ( ), .R b zθ α +�

�

( ).X t

( ){ }1

0

( , ) ( , ) min , ,

a b

z

R R b z d a b b z

−

+

=

θ α = θ α + Ρ + η < +∫� �

� �

1+

η

( ) :,R θ α�

�

( )[ ]

[ ]

[ ]

[ ]

[ ]

( ) ( )

( ) ( ) ( )

− λ− θ +λθ

=

− λ− θ +λ − λ+α +λ − λ+α +λ−λ

⎡ λ θ − θλθ α = θ α −⎢λ − θ λ − θ⎣

⎡ ⎤λα −λ θ λ λα + α− + θ α + +⎢ ⎥λ − θ λ + αλ + α λ + α⎣ ⎦

∑�

�

3 22( )( )

2 2

1

2 2( )

2 2

2 ( ) ( ), ( , )

( ) ( )

( ) 2( , ) ,( )

ii

i

k a bk b i ii

i ii

k a b a b a bbi

i

k kR c e e

k k

a bke c e e e

k

( )( ) ( ) ( )

( )[ ]

( )2

3

1

1, .b z

i

i

c e

k

−α +

=

− ϕ θ λ + α α − μ

θ α =

θ

α + θ∏

( )0

lim ( , )R Rθ→

α = θ θ α�

� �

( )( )

[ ]( ) ( )[ ] ( )[ ] ( )

( ) ( )( )[ ] ( )

( ) ( )

( ) ( ) ( )[ ]( ) ( )[ ]

( )[ ] ( )[ ]

( ) ( )[ ] ( )[ ] ( )[ ]( ) ( )[ ]

( )[ ] ( )[ ]

( ) ( )[ ] ( )[ ] ( )[ ]( ) ( )[ ]

( )[ ]

2 3 3 2 2 3

2 3

3 2

2 3

4 40 0 0 0 0 043 2 2 3 3 2

2 0 023 2

2 3

2 0 02 2 3

2 3

2 0 03 3 2

2 3

2

(0) (0) 0 0 0 0

0 0

0 0

0 0 0

0 0

0 0 0

0

k k a k a k b k a k b

b k k a

k b k a

k b k a

Rk k e k k e k k e

k k e

k k

k k k e

k k

k k k e

k k

+ + +

−α + +

− α−

− α−

λ λ − μα =

⎡ ⎤λ − − λ − + λ −⎣ ⎦

⎧λ λ + α −⎪× ⎨

α + α +⎪⎩

α − μ μ + α + λ −−

α + α +

α − μ μ + α + λ −+

α + α +

�

( )[ ].

0

⎫⎪⎬⎪⎭



Let us denote the random value by X. Its distribution coincides with the ergodic distribution of the pro�cess X(t); i.e.,

Let us find EX and DX.

Since is the Laplace–Stieltjes distribution of the random value X, we define their first moments

by using the known formulas and

The mathematical expectation and the dispersion of the ergodic distribution of the process arefound from (8):

where

3. GENERATING FUNCTION OF THE STEP NUMBER DISTRIBUTION

The authors in this section studied the generating function of the distribution of the number of steps ofthe process at which it firstly gains the screen “a” (a > 0).

Let us denote the number of steps of the process at which it firstly gains the screen “a” by i.e.,

We introduce the following denotations:

{ } { }lim ( ) .t

X t x X x→∞

Ρ < = Ρ <

( )R α�

( )' 0EX R= −� ( ) ( )

2'' 0 ' 0 .DX R R= − ⎡ ⎤⎣ ⎦

� �

( )X t

( )

2,

2rEX

λ − μ= −λ λ − μ υ

( )

( )

( ) ( ) ( )( )( )

( ) ( ) ( )( )( )

2

2

22 22 2 2

2 22

2 2 42 2 2 2 2

2 2 42 2 2 2 2

16 4 12

2

2 2 4 2 2 3 4 2 4

2 2 4 2 2 3 4 2 4,

a b

a b

brDX

a a e

a a e

−− λ−μ− λμ+μ

−− λ−μ+ λμ+μ

λ λμ + μ − λλ + μ= − +

υυλ λ − μ

⎡ ⎤μ λ + μ + λμ + μ − λ − μ + λμ + μ − μ λ − μ + λμ + μ⎣ ⎦+

υ

⎡ ⎤μ λ + μ − λμ + μ − λ − μ − λμ + μ − μ λ − μ − λμ + μ⎣ ⎦−

υ

( ) ( )( )( )

( ) ( )( )( )

2

2

2 2 44 2 2 2 2 2

2 2 42 2 2 2

8 4 2 4 2 4

2 4 2 4 ,

a b

a b

e

e



υ = λ λμ + μ + μ λ + μ − λμ + μ λ − μ − λμ + μ

− μ λ + μ + λμ + μ λ − μ + λμ + μ

( ) ( )( )

( )( )

( ) ( )( )( )

2

2

23 2 2

2 42 2 2

2 2 42 2 2 2

8 4 2 2 4

2 3 4 2 4

2 4 2 3 4 2 4 .

a b

a b

r b

a e

a e



= λ λμ + μ − λ + μ λ + μ + λμ + μ

⎡ ⎤× λ − μ + λμ + μ + μ λ − μ + λμ + μ⎣ ⎦

⎡ ⎤− μ λ + μ − λμ + μ λ − μ − λμ + μ + μ λ − μ − λμ + μ⎣ ⎦

( ),X t

( ),X t 1;aν

{ }1 min : .akk aν = ζ =

( )

( ) ( )( )1

1

, 1,

0 .

a

a

u Eu u

u z E u X z

ν

ν

ψ = ≤

ψ = =

116


NASIROVA, SHAMILOVA

Theorem 2. The following expression is true:

Proof. At

(9)

Denote that

Clearly,

Then, (9) takes the form

(10)

Consequence 2. If has a Laplace distribution, i.e.,

we then obtain from (10) that

(11)

The differential equation follows from the integral one:

which has the solution

(12)

where are the roots of the characteristic equation

( ) { } ( ) { }

( ) { }

1 1

1 .

a

y

y b

u b z uP a b z u u b P z

u u y d P y b z

=

ψ + = η > − − + ψ η < −

+ ψ η < − −∫

2,k ≥

( ){ } ( ){ }

( ){ }

1 1

1 1

0 1 0

; max ; .

a

a a

y b

y

P k X b z P k X y

d P b z a b b z y=

ν = = + = ν = − =

× + + η < + + η <

∫

( ) ( ){ }1

1

0 , 0 1.k a

k

u b z u P k X b z u∞

=

ψ + = ν = = + < ≤∑

( ){ } { } { }1 1 11 0 .aP X b z P b z a P a b zν = = + = + + η > = η > − −

( ) { } ( ) { }

( ) { }

1 1

1 .

a

y

y b

u b z uP a b z u u b P z

u u y d P y b z

=

ψ + = η > − − + ψ η < −

+ ψ η < − −∫

1η

{ }1

, 0, 0,

1 , 0, 0,

x

x

e x

xe x

μ

−λ

λ⎧ < μ >⎪λ + μ⎪

Ρ η < = ⎨ μ⎪ − > λ >⎪ λ + μ⎩

( )( )

( )( )

( )( )

( ) .

a b z b zz

b z a

b zy y

y b y b z

u uuu b z e e u b e

uu y e dy e u y e dy

−λ − − −μ +−μ

+

λ +μ −λ

= = +

μ λμλψ + = + ψ +

λ + μ λ + μ λ + μ

λμ× ψ + ψ

λ + μ∫ ∫

( ) ( ) ( ) ( ) ( )'' ' 1 0,z zu b z u b z u u b zψ + − λ − μ ψ + − λμ − ψ + =

( ) ( )( )( )

2

1

,ik u b zi

i

u b z s u e +

=

ψ + =∑

( ), 1,2,ik u i =

( ) ( ) ( ) ( )2 1 0.k u k u u− λ − μ − λμ − =



We find the following boundary conditions from (12) at z = a – b:

Taking the general form of the differential equation, we obtain the following system of nonuniformalgebraic equations from the previous system:

According to the Kramer theorem,

(13)

Putting into (12), we obtain

Let us find and According to the known property of the generating function,

and

From the characteristic equation, it follows that

From (13) we find

( )( )

( ) ( )

( ) ( ) ( ) ( )'2

,

.

a

a b a y

y b

a

a b a y

y b

u uuu a e u b e u y e dy

u u uu a e u b e u y e dy

−µ − −µ µ

=

−µ − −µ µ

=

⎫μ λμλ ⎪ψ = + ψ + ψλ + μ λ + μ λ + μ ⎪⎪

⎬⎪λμ λμ λμψ = − ψ − ψ ⎪λ + μ λ + μ λ + μ ⎪⎭

∫

∫

( )[ ]( )

( )[ ] ( )( ) ( ){ } ( )

( )[ ]( )

( )[ ] ( )( ) ( ){ } ( )

( )[ ]( )

( )[ ] ( )( ) ( ){ } ( )

( )[ ]( )

( )[ ] ( )( ) ( ){ } ( )

1 1

2 2

1 1

2 2

2 2 1 1

21 1 2 2

1 2 1 1

2 1 2 2

,

.

k u a a b k u b

k u a a b k u b

k u a a b k u b

k u a a b k u b

k u e k u k u e s u

k u e k u k u e s u u

k u e k u k u e s u

k u e k u k u e s u u

−µ − +

−µ − +

−µ − +

−µ − +

⎫μ λ − − μ +⎪⎪+ μ λ − − μ + = μ ⎪⎬⎪λ μ + + μ +⎪⎪+ λ μ + + μ + = λμ ⎭

( )( )[ ] ( )

( )

( )[ ] ( )( ) ( )

( )[ ] ( )( ) ( )

( )( )[ ] ( )

( )

( )[ ] ( )( ) ( )

( )[ ] ( )( ) ( )

2

1 2 2 1

1

1 2 2 1

2 21 2 2

2 2 1 1

1 12 2 2

2 2 1 1

,

.

k u b

k u a k u b k u a k u b

k u b


u k u k u es u

k u k u e k u k u e

u k u k u es u

k u k u e k u k u e

+ +

+ +

⎫μ λ −= ⎪

λ − − λ − ⎪⎬

μ λ − ⎪= − ⎪λ − − λ − ⎭

( ), 1,2,is u i =

( )( )[ ] ( )

( ) ( )( )

( )[ ] ( )( ) ( )

( )[ ] ( )( ) ( )

( )[ ] ( )( ) ( )( )

( )[ ] ( )( ) ( )

( )[ ] ( )( ) ( )

2 1

1 2 2 1

1 2

1 2 2 1

2 22 2

2 2 1 1

1 12 2

2 2 1 1

.

k u b k u b z


k u b k u b z


u k u k u eu b z

k u k u e k u k u e

u k u k u e

k u k u e k u k u e

+ +

+ +

+ +

+ +

⎫μ λ −ψ + = ⎪

λ − − λ − ⎪⎬⎪μ λ −

− ⎪⎭λ − − λ −

1aEν 1.aDν

'1 (1)aEν = ψ [ ]''(1) '(1) '(1)2

1 .aDν = ψ + ψ − ψ

(1)1' ,kλμ

= −λ − μ

(1)2' ,kλμ

=λ − μ

( )(1)

2 2

1 3

2'' ,kλ μ

= −λ − μ ( )

(1)2 2

2 3

2'' .kλ μ

=λ − μ

( )

( )(1)2

1 2' ,bs e− λ−μλ

= −λ − μ

( )( )

( )

( )

( )( )

( )

( )(1)3 4 2 2 35

21 3 4 4

2 2 4 22'' 2 ,b a b bs a b e e e− λ−μ − λ−μ − λ−μλ μ λ − λ μ − λ μλ

= − − +λ − μ μ λ − μ λ − μ

( )

( )( )(1)3

2 2' ,a bs a e λ−μ −μ λμ λ

= − − +λ − μ λ − μ μ λ − μ

118


NASIROVA, SHAMILOVA

Clearly,

Because has an Erlang distribution of the first order with the parameter λ, takes the form

Thus, we obtain the unconditional generating function for distributing the random value

According to we find

Similarly,

4. CONCLUSIONS

The evident form of the Laplace transform by time and the Laplace–Stieltjes transform by the phaseof the conditional and unconditional distributions of the process of semi�Markov wandering with two

( ) ( ) ( ) ( )

( )

( )( ) ( )

( )( )

(1) 1

1

2 3 2 2 622

2 3 3 2 42

3 2 24

4 3

2 2 2''

2 2 3 2 3 2 .

k a b

k a b

s a a e

a b e

−

−

λμ λμ λ μ λ= − − + +λ − μ λ − μ λ − μ μ λ − μ

⎡ ⎤λ λ − μ − λμ λ− + −⎢ ⎥μ λ − μ λ − μ⎢ ⎥⎣ ⎦

( ) ( ) ( ){ } ( ) ( ){ }

( ) { } ( ) { }

1

0 0

1 1

0

0 min ,

.

a b a b

z z

a b

z

u u b z dP X b z u b z dP a b b z

u b z dP z u a P a b

− −

+

= =

−

+ +

=

ψ = ψ + < + = ψ + + η < +

= ψ + η < + ψ η > −

∫ ∫

∫

1+

η ( )uψ

( )( ) ( )

0

( ) .

a b

a b z

z

u e u a e u b z dz

−

−λ − −λ

=

ψ = ψ + λ ψ +∫

:1a

ν

[ ][ ]{ }

4

141

( ) 1 ( ) .( )

buu b k u ek u

−λλψ = + λ −

λ −

' 1(1) ,aEψ = ν

( )

( ) ( )( )

3 2

1 2

3

2

( )( )( )

.( )

a ba

a b a b

E a b e

e e

−μ −

−λ − − λ−μ

λμλ λν = + − −

λ − μ μ λ − μμ λ − μ

μ λ+ −λ − μ μ λ − μ

( )( )

( )

( ) ( ) ( ) ( )

( ) ( ) ( )( )

2 26 5 4 2 3 3 22

1 2 4 3 2

6 3 2 2 462 ( ) ( )

2 4 2 4 3

23 2 3 2

3 2 2

3 2 5

( ) ( ) ( )

2 4

( ) ( ) ( )

22 2( ) ( ) ( )

a ba

a b a b

a b a b

D a b e

a be e

a be e

− λ −

− λ−μ − λ−μ

−λ − − − λ+μ

λμ λ + μλ − λ μ + λ μ + λ μ μν = − − − −

μ λ − μ λ − μ λ − μ

⎡ ⎤λ − λ μ λ − μ λ −λ+ + −⎢ ⎥μ λ − μ μ λ − μ λ − μ⎢ ⎥⎣ ⎦

⎡ ⎤λμ −λ − λ μ + μ λ− + +⎢ ⎥λ − μ λ − μ λ − μ⎣ ⎦

+( ) ( ) ( ) ( ) ( )

3 2 2 2 2 2 3 42

2 3 2 2 2

2 3 2.

( ) ( ) ( )

a b a ba be e−μ − − μ −

⎡ ⎤λ λ + μ − λ μ λ − μ λ − λ− −⎢ ⎥μ λ − μ λ − μ μ λ − μ⎢ ⎥⎣ ⎦



delaying screens, as well as its Laplace–Stieltjes transform of the ergodic distribution, was found. The pro�posed method can be applied when the random value takes the form

where the random values have an exponential distribution.

REFERENCES1. Borovkov, A.A., Ergodichnost’ i ustoichivost' sluchainykh protsessov (Ergodicity and Stability of Accidental Pro�

cesses), Moscow: Editorial URSS; Novosibirsk: Inst. Matem., 1999.2. Borovkov, A.A., Teoriya veroyatnostei (Theory of Probabilities), Moscow: Nauka, 2003.3. Gikhman, I.I. and Skorokhod, A.V., Teoriya sluchainykh protsessov (Theory of Accidental Processes), Moscow:

Nauka, 1973.4. Nasirova, T.I., Protsessy polumarkovskogo bluzhdaniya, (Processes of Semi�Markov Error), Baku: Elm, 1984.5. Kharlamov, B.P., Nepreryvnye Polumarkovskoe protsessy (Continuous Semi�Markov Processes). Moscow:

Nauka, 2001.6. Shurenkov, V.M., Ergodic Theorems and Related Problems. Utrecht: VSP, 1998.

Translated by A. Evseeva

1η

,

1 1

,k m

k m i i

i i

+ −

= =

η = η − η∑ ∑

,i+

η 1, ,i k= , 1, ,i i m−

η =

Documents

Investigation of some probabilistic characteristics of one class of semi-Markov wandering with delaying screens