Journal of Applied Mathematics and Physics, 2014, 2, 138-149
Published Online April 2014 in SciRes. http://www.scirp.org/journal/jamp
http://dx.doi.org/10.4236/jamp.2014.25018
How to cite this paper: Alafif, H.A. (2014) Random Integral Equation of the Volterra Type with Applications. Journal of
Applied Mathematics and Physics, 2, 138-149. http://dx.doi.org/10.4236/jamp.2014.25018
Random Integral Equation of the Volterra
Type with Applications
Hamdin Ahmed Alafif
Khartoum University, Faculty of Education, Omdurman, Sudan
Email: hamdin@163.com
Received December 2013
Abstract
In this paper we shall present some of the most general results which have been obtained to date
concerning random integral equations of the Volterra type. Some results of Tsokos [4] are given
for the random integral equation
( )
;
t
xtwhtwktw fxwd
0
(;)( )(,;),(;)= +
τ ττ τ
(1.1)
where
t0
and 1) w is appoint of
; 2)
htw(; )
is the stochastic free term or free random variable
defined for
t0
and
w∈Ω
; 3)
xtw(; )
is the unknown random variable for each
t0
4) the
stochastic kernel
kt w(, ;)
τ
is defined for
and
w∈Ω
.
Keywords
Volterra Integral, Banach Space, Asymptotic Stability, Poincar e-Lyapunov,
Telephone Traffic Theory
1. Introduction
In Section 2, we present some results of [4] for the random integral equation in Section 2, investigating the exis-
tence and uniqueness of random solution of Equation (1.1). The asymptotic behavior of the random solution and
its stability properties are also considered. In Section 3 some applications of Equation (1.1) will be presented in
the areas of telephone traffic theory and a generalization of the classical poincare’-Lyapunov theorem [3]
Remark Below are some spaces which we use in our study
a)
( )
2
,(, ,)CC LP
+
= Ω
denote the space of all continuous and bounded functions on
+
with values in
2(, ,)LP
.
b)
( )
2
,(, ,)
gg
CC LP
+
= Ω
the space of all continuous functions from
+
into
2(, ,)LP
such that
{ }
1
22
(;)( )()x twdPwzgtt
+
≤∈
where Z is a positive number and g(t )is a positive continuous function defined on
+
(c)
( )
2
,(, ,)
CC
CC LP
+
= Ω
is the space of all continuous functions from
+
into
2(, ,)LP
with
H. A. Alafif
139
the topology of uniform convergence on the intervals [0, T] for T > 0.
Where
(, ,)P
a probability measure space.
2. The Random Integral Equation
( )
0
(;) (;)(,;),(,)
t
xtwhtwktwfxwd
τ ττ τ
= +
This equation seems to be more general than any random Volterra integral equation which has been studied to
date. The generality consists primarily in the choice of the stochastic kernel. In this Section we shall investigate
the existence of a random solution and its uniqueness and asymptotic behavior and shall consider a number of
special cases as corollaries of the main theorems. Finally, the stability properties of the random solution will be
investigated.
To accomplish our objectives here we employ certain aspects of the methods of “admissibility theory” which
has been utilized quite recently in the theory of deterministic integral equations by Corduneanu [4].
3. Existence and Uniqueness of a Random Solution
Let B and D be a pair of Banach spaces and T a linear operator. With respect to the study of this Section, we
state the following Lemma which will be used in the main theorems.
Lemma 3.1 Let T be a continuous operator from
()
2
,(, ,)
C
CL P
+
into itself. If B and D are Banach
spaces stronger than
C
C
and the pair (B, D) is admissible with respect to T, then T is a continuous operator
from B to D.
Proof
First we will prove that the operator T is closed form B to D. Let us consider the sequence
(; )
n
x twB
such
that
(; )(,)
B
n
x twxtw →
as
n→∞
. Let us assume
( )
(; )(; )
D
n
Tx twytw→
as
n→∞
. Now we must
show that
( )
(; )(; )Tx twytw=
. Since
(, )(, )
n
x twxtw
in B,
(, )(,)
n
x twxtw
in C
C
. But since T:
cc
CC
is continuous we have
( )
( )
(; )(; )
n
Txt wTxt w
in C
C
. On the other hand
( )
(; )(; )
n
Tx twytw
in D, which implies that
( )
(; )(; )
n
Tx twytw
in C
C
. Hence
()
(; )(;)
Txt wytw
because the limit is
unique in
C
C
. Therefore the operator T is closed. Then by the closed-graph theorem it follows that T is conti-
nuous operator from B to D.
Remark since T is closed and continuous linear operator, it is also bounded (Yosida [1], p. 10). Then it fol-
lows that we can find a constant
0k>
such that
( )
(; )(; )
B
D
Txtwkxtw
With respect to our aim here, we state and prove the following theorems.
Theorem 3.2 Let us consider (1.1) under the following conditions:
1) B and D are Banach spaces stronger than
( )
2
,(, ,)
C
CL P
+
such that (B, D) is admissible with respect
to the operator
( )
0
(;) (,;)(;)
t
Txtwktwxw d
τ ττ
=
2)
( )
(; ),(; )xtwftxtw
is a continuous operator on
{ }
(; ):(,),(; )
D
SxtwxtwD xtw
ρ
= ∈≤
with values in B, also satisfying
() ()
,(;),(;)(; )(;)
D
B
ftxtwftytwxtw ytw
λ
− ≤−
with
(; )xtw
,
(; )ytw S
and
λ
a constant.
3)
(; )
ht wD
Then there exists a unique random solution of the random integral Equation (1.1), provided that
( )
1,(;)(, 0)1
DB
kht wkftk
λ ρλ
<+ ≤−
,
where k is the norm of the operator T.
H. A. Alafif
140
Proof
Let us define an operator U on S into D as follows
( )()
0
(;) (;)(,,),(;)
t
Ux twhtwktwfxwd
τ ττ τ
= +
(3.1)
Now we must show that U is a contracting operator and
()US S
. Consider a function
(; )ytw
in S. We
can write
( )()
0
(;) (;)(,;) ,(;)
t
Uy twhtwktwfyw d
τττ τ
=+
(3.2)
Subtracting Equation (3.2) from Equation (3.1) we have
( )( )()()
0
(;)(,)(,;) ,(;),(;)
t
UxtwUytwktwfxwfywd
τττττ τ
−= −


Since
()US D
and D is a Banach space then
( )( )
(,)(, )UxtwUytwD−∈
By assumptions 1) and 2),
() ()
,(; ),(; )f xwf ywB
ττ ττ
−∈


. From Lemma 3.1 we have seen that T is con-
tinuous operator from the Banach space B into D which implies that we can find a constert
0k>
such that
()
(; )(; )B
D
Txtwkxtw
That is
( )( )
(;)(;)(,(; )(,(; )
B
D
UxtwUytwkft xt wftyt w−≤ −
Now, applying Lipschitz’s condition given in (ii) we have
( )( )
(;)(;)(; )(; )
D
D
UxtwUytwkxt wyt w
λ
− ≤−
Using the condition that
1
k
λ
<
, the operator U is a contracting operator.
It now remains to be show that
()US S
. For every function
(; )xtw S
we have
( )()
0
(;) (;)(,;) ,(;)
t
Ux twhtwktwfxw
τ ττ
= +
(3.3)
Applying condition 3) and Lemma(3.1) we can write expression(3.3) as follows
( )( )
(; )(;),(;)
D
DB
Uxtwhtwkftxtwd
τ
≤+
(3.4)
In (3.4),
( )
, (;)B
ftxt w
can be written as
( )()( )
, (;), (;)(,0)(,0),(;)(,0)(,0)
B
B BB
ftxtwftxtwftftftxtwftft= −+≤ −+
Using Lipschitz’s condition we have
( )
, (;)(,)0(,0)
DB
B
ftxt wxt wft
λ
≤ −+
We can now write expression (3.4) as follows
()(; )(;)(; )(,0)
DDD B
Uxtwhtw kxtwkft
λ
≤+ +
(3.5)
Since
(; )xtw S
and
(; )
D
xtw
ρ
, (3.5) can be written as
()(;)(; )(,0)
DD B
Uxtwhtwkkft
λρ
≤ ++
(3.6)
Applying the condition of the theorem that
(;)(, 0)(1)
DB
htwkf tk
ρλ
+ ≤−
(3.6) becomes
()( ;)(1)
D
Uxt wkk
ρ λλρ
≤−+
or
()( ;)D
Uxt w
ρ
Which implies that
()(;)Uxt wS
for all random variables
(; )xtw S
or
()US S
. Therefore, since U is
a contracting operator and
()US S
applying Banach’s fixed-point theorem.
There exists a unique random solution
(; )xtw S
such that
H. A. Alafif
141
( )
0
( )(;)(;)(,;),(;)(;)
t
Ux twhtwktwfxwdxtw
τ ττ τ
=+=
4. Some Special Cases
Now we shall derive some particular cases of theorem 3.2 choosing in a convenient manner the spaces B and D.
Consider that a
g
C
space is a space of all continuous functions from
2(, ,)LP
+→Ω
such that
{ }
1
22
(; )(; )()()xtwxtw dPwzgt
= ≤
where
t
+
, Z is a number grater than zero, and g(t) is a continuous function greater than zero. Also
( )
2
,(, ,)
C
CL P
+
is the space of all continuous function from
+
into
2(, ,)LP
with the topology of
uniform convergence on the interval [0, T] for any
0T>
and the norm of the stochastic kernel of the integral
equation can be defined as follows:
(,)(, ;)sup(, ;)ktktwPessktw
ττ τ
==−
With respect to
w∈Ω
. That is
00
( ,;)infsup( ,;)ktwkt w
ττ
Ω−Ω

=

With
0
() 0.PΩ=
Theorem 4.1 Let us consider the random integral Equation (1.1) under the following conditions
1) there exist a number
0A>
and a continuous function
() 0gt >
such that
0
(,;)( ),
t
ktw gAt
ττ
+
≤∈
2)
(, )ftx
is a continuous vector-valued function for
t
+
,
(; )xtw
ρ
such that
() ()
(,0),,(; ),(; )()(; )(; )
g
ftCftxtwf tytwgtxtwytw
λ
∈ −≤−
3)
(, )htw
is a continuous bounded function on +
whose values are in
2(, ,)LP
. Then there exists a
unique random solution
(; )xtw C
of the random integral Equation (1.1) such that
{}
1
22
0
(; )sup(; )()
Ct
xtwxtw dPw
ρ
= ≤
for
t
+
as long as
(; )htw
,
λ
and
( ,0)g
C
ft
are small enough.
Proof We must show that under condition (i) of the theorem the pair of Banach spaces
( ,)
g
CC
is admissible.
That is
( ,)
g
CC
is admissible with respect to the integral operator
( )
0
(;) (,;)(,)
t
Txtwktwxw d
τ ττ
=
for
(; )
g
xtw C
we have
( )
0
(;) (,;)(;)
t
Txtwktw xwd
τ ττ
=
or
( )
00
(;)(,;)(;)(,;)(;)/() ()
tt
Txtwkt wxwdkt wxw ggd
ττττττ ττ
≤ ≤

∫∫
(4.1)
where
(,; )sup(,; )
w
ktwPessktw
ττ
= −
is a function only of
(, )t
τ
. Using the definition of the norm in
g
C
, that is
{ }
1
22
0
1
sup(; )()(; )
()
g
tC
xtw dPwxtw
gt



=





For
() 0gt >
we can write (4.1) as follows
H. A. Alafif
142
( )
00
(;)(,;)(;)()(;)(,;)()(;)
gg g
tt
CC C
Txtwktwx wg dxtwktwgdAxw
τττττ ττ τ
≤≤≤
∫∫
Therefore
()
(; )Txt w
is bounded and hence
( )
(; )Txt wC
for all
( )
(; )Txtw
, hence
g
TC C
, which
implies that the pair of Banach spaces
( )
,
g
CC
is admissible with respect to the integral operator as defined
here.
The remainder of the proof is analogous to that of theorem 3.2 and is omitted.
For the special case where
() 1gt =
we state and prove the following corollary 4.2 let us assume that the
random integral Equation (1.1) satisfies the following conditions:
1)
0
(, ;)
t
kt wdA
ττ
,
t
+
, where A is a constant greater than zero.
2)
(, )ftx
is a continuous function from
+
into
uniformly in x such that
(, )(,)ftxftyx y
λ
− ≤−
3)
(; )htw
is a continuous bounded function from +
into
2(, ,)LP
. Then there exists a unique
bounded random solution on
+
of the random integral Equation (1.1) if
λ
is small enough.
Proof We must show that under condition (i) of the corollary the pair of Banach spaces
( )
,CC
is admissible.
For a function
(; )xtw C
we have
( )
0
(;) (,;)(;)
t
Txtwktwxw d
τ ττ
=
or
( )
0
(;)(,;)(;)
t
Txtwk twxwd
τ ττ
(4.2)
Applying the definition of the norm as used in theorem 4.1, in equality (4.2) can be written as follows
()
0
(;) (;)(,;)(;),
t
Txtwxtwktw dAxtwt
ττ
+
≤ ≤∈
Therefore
()( ;)Txt wC
for every random variable
(; )xtw
or
TC C
, which implies that
(,)CC
is
admissible. The remainder of the proof follows from theorem 3.2
The following two corollaries are particular cases of theorem 4.1.
Corollary 4.3 Assume that the random integral equation (1.1) satisfies the following conditions:
1)
1
(, ;)kt w
τ
≤Λ
, for
0t
τ
≤≤<∞
and
0
()g t dt
<∞
2) Same condition as in theorem 4.1, condition 2).
3) Same condition as in theorem 4.1, condition 3).
Then there exists a uniqe random solution of Equation (1.1) bounded on
+
if
(; )htw
,
λ
and
( ,0)ft
are sufficiently small.
Proof It is only necessary to show that the pair of Banach spaces
( )
,
g
CC
is admissible with respect to the
integral operator
( )
0
(;) (,;)(;)
t
Txtwktwxw d
τ ττ
=
(4.3)
Along with condition (1) of the corollary. For a function
(; )g
xtwC
expression (4.3) implies that
( )
0
(;)(,;)(;)
t
Txtwk twxwd
τ ττ
Applying hypothesis (i) of the corollary, we have
( )
11
00
(; )
(;) (;)()
()
tt
xtw
Txt wxwdgd
g
τ τττ
τ

≤Λ≤Λ 

∫∫
(4.5)
Utilizing the definition of norm as applied in theorem 4.5 to in equality (4.5) we have
( )
10
(; )(; )()
g
t
C
Tx twxtwgd
ττ
≤Λ
(4.6)
H. A. Alafif
143
but applying condition (1) of the corollary (4.6) is written as
( )
(; )TxtwM
for all
0t
Therefore the function
(; )
g
xtwC
implies that
()( ;)TxtwC
or
g
TC C
. Hence the pair
( ,)
g
CC
is
admissible, and, since condition (2) and (3) are the same as in theorem 4.1 the proof is complete.
Corollary 4.4 Let us consider the random integral Equation (1.1) under the following conditions
1)
()
2
(, ;)t
kt we
ατ
τ
−−
≤Λ
, for
0t
τ
≤≤< +∞
, and
{ }
1
sup( )
t
tt
gd
ττ
+
+
<∞
where
2
Λ
and
α
are positive numbers.
2) same as condition (2) of theorem 4.1.
3) same as condition (3) of theorem 4.1.
Then there exists a unique random solution of the random integral Equation (1.1) bounded on
+
if
(; )htw
,
λ
and
( ,0)ft
are small enough.
5. Asymptotic Stability of the Random Solution
With respect to the asymptotic behavior of the random solution of the stochastic integral Equation (1.1), we state
and prove the following theorem, the objective of which is to investigate the possibility of the random solution
being asymptotically exponentially stable.
Theorem 5.1 Let us consider the stochastic integral Equation (1.1) under the following conditions
1)
()
2
(, ;)t
kt we
ατ
τ
−−
≤Λ
, for
0t
τ
≤≤< +∞
,
20Λ>
, and
0
α
>
2)
(, )ftx
is a continuous function from
+
×
into
such that
( ,0)0ft =
and
(, )(,)ftxftyx y
λ
− ≤−
3)
(; )t
htw e
β
ρ
where
ρ
and
β
are positive numbers such that
0
βα
<<
.
Then there exists a unique random solution of Equation (2.1.1) such that
{ }
1
22
(;)()
t
x twdPwet
β
ρ
+
≤∈
as long as
λ
is small enough.
proof We must show that the pair of Banach spaces
( )
,
gg
CC
with
() t
gte
β
=
, is admissible under condi-
tions (1) and (2) of the theorem.
That is
( )
,
gg
CC
is admissible with respect to the operator defined by
( )
0
(;) (,;)(;)
t
Txtwktwxw d
τ ττ
=
(5.1)
The norm of the expression (5.1) can be written as
( )
0
(;)(,;)(;)
t
Txtwk twxwd
τ ττ
(5.2)
Applying condition (1) of the theorem, we have
()
20
()(; )(; )
tt
Tx twextwd
ατ
τ
−−
≤Λ
(5.3)
Hence
()
20
(; )
()( ;)()
()
ttxw
Txt wegd
g
ατ τττ
τ
−−

≤Λ 

(5.4)
Using the definition of the norm on the
g
C
space, inequality (5.4) can be written as
( )()
() ()
00
() 1
()( ;)()
11()
()
tt
tt
t ttt
TxtwMegdMeed
MeeMe e
α τατβτ
ααββ α
ττ τ
αβ
αβ
−−−− −
−−−− −
≤=

=−=−−


∫∫
(5.5)
H. A. Alafif
144
Since
0
βα
<<
, we can majorize inequality (5.5) as follows
( )
( )
1
1
()(;)(),
tt t
Txt wMeeMet
βα β
αβ αβ
−− −−
+
≤−−< −∈
Which implies that
()( ;)g
Txt wC
for a function
(; )
g
xtw C
. Therefore the pair of Banach spaces
( )
,
gg
CC
is admissible with respect to the operator T where
()
t
gte
β
=
. Condition (3) of the theorem means
that
(; ).
g
htw C
Applying condition (2), we have
() ()
,(; ), (; )(; )(; )
g
g
C
C
f txtwf tytwxtwytw
λ
− ≤−
Hence, all the conditions of theorem 4.1 have been satisfied, which implies that there exists a unique random
solution of the integral Equation (1.1) such that
{ }
1
22
(;)( )
t
x twdPwe
β
ρ
Remark It is now clear that under these conditions there exists a random solution of the random integral
Equation (1.1) which is exponentially asymptotically stable that is
{ }
1
22
lim( ;)()0
tx twdPw
→∞ =
6. Some Applications of the Equation
( )
0
(;) (;)(,;),(;)
t
xtwhtwktwfxwd
τττ τ
=+
In this Section we shall present some applications of the results of the previous Section. We shall first
consider a generalization of the classical stability theorem of Poincaré and Lyapunov. We shall then study a
stochastic integral equation arising in the theory of telephone traffic, arelated study of which was done by Fortet
[1].
6.1. Generalization of Poincaré-Lyapunov Stability Theorem
As an example to illustrate our results, we shall generalized the classical stability theorem of Poincaré and
Lyapunov(Tsokos [3]). That is consider the following random differential system
( )
(; )()(; ),(;)0
xtwAwxtwf txtwt=+≥
(6.1)
where 1)
(; )xtw
is the unknown
1n×
random vector; 2)
()Aw
is an
nn×
matrix whose elements are
measurable functions; and 3)
(, )ftx
is, for
t
+
and
x
, an
1
n×
vector-valued function.
Now we shall reduce the random differential system (6.1) to astochastic integral equation which will be a
special case of the stochastic integral Equation (1.1). Myltiplying the random system (6.1) by
()Awt
e
, we have
( )
()() ()
(; )()(; )(,(; )
AwtAwt Awt
e xtwAwe xtweftxtw
− −−
−=
But
( )
{ }
( )
() ()()
(; )(;)()(; )
Awt AwtAwt
dd
e xtwextwAwe xtw
dt dt
−− −
= −
Therefore
( )
{ }
( )
() ()
(; ),(; )
Awt Awt
dextwef txtw
dt
−−
=
(6.2)
Integrating both sides of Equation (6.2) from
0
t
to t we have
( )
0
0
()
() ()
0
(;) (;)(,(;)
t
Awt
Awt Aw
t
extwext wefxw d
τ
ττ τ
−−
−=
(6.3)
Multiplying Equation (6.2) by
()Awt
e
and letting
0
t
= 0, it reduces to
( )
()()( )
00
(;) ()(,(;)
t
AwtAw t
xtwex wefxwd
τ
τττ
= +
(6.4)
H. A. Alafif
145
where
0
( )(0;)xwx w=
. Hence; if we let
() 0
(;)()
Awt
htwex w=
and
( )()
(, ;),0
Awt
kt w et
τ
ττ
=≤≤ <∞
Equation (6.4) can be written as
( )
0
(;) (;)(,;),(;)
t
xtwhtwktwfxw d
τττ τ
= +
Hence the stochastic differential system (6.1) reduces to the stochastic integral Equation (6.4), which is a spe-
cial form of Equation (1.1).
Now we state the following theorem.
Theorem (6.1) Let us assume that the following conditions hold with respect to the stochastic integral Equa-
tion (6.4)
1) The matrix A(w) is stochastically stable, that is there exists an
0
α
>
such that
{ }
; Re(),1, 2,,1
k
Pww kn
α
Ψ<−==
where
()
kwΨ
,
1, 2,,kn=
are the characteristic roots of the matrix.
2)
(, )ftx
is a continuous function from
nn
+
×→ 
such that
(, )(,)ftxftyxy
λ
− ≤−
With
( ,0)0ft =
and
λ
sufficiently small.
Then there exists a unique random solution of the stochastic integral Equation (6.4) such that
{ }
1
22
lim( ;)()0
t
x twdPw
→∞
=
Proof To prove this result we want to prove that the pair of Banach spaces
( )
,
gg
CC
is admissible under
conditions 1) and 2) with
()
t
gte
β
=
, and then apply theorem 4.1.
Recall that the norm in the space
( )
2
,(, ,)
g
CL P
+
is defined by
{ }
1
22
1
(; )sup(; )()
()
g
CtR
xtwxtw dPw
gt

=

and for any function
( )
2
(; ),( ,, )
g
xtw CLP
+
∈Ω
, let us define the following integral operator
0
( )(;)(,;)(;)
t
Txt wk tw xwd
τ ττ
=
(6.5)
Since
( )()
(, ;),0
Awt
kt w et
τ
ττ
=≤≤ <∞
Equation (6.5) becomes
( )()
0
( )(;)(;)
tAw t
Txt wexw d
τ
ττ
=
or
( )()
0
( )(;)(;)
tAw t
Txt wexwd
τ
ττ
(6.6)
It has been shown by Morozan [2] [3] that there exists a subset D of
such that
()1PD=
and
()() ()Aw tt
e ke
τ ατ
− −−
(6.7)
for
,0w Dk∈>
and
α
as defined previously now, putting (6.7) into in equality (6.6) we have
( )( )
()() ()
00
(; )
( )(;)(;)
tt
t tt
xw
Txt wkexwdkeegd
g
ατατ ατ
τ
τ τττ
τ
−−−−−−

≤≤


∫∫
(6.8)
Since
() ,0
t
gte
β
βα
= <<
, inequality (6.8) becomes
()
( )
( )
()
0
1
()
0
1
( )(;)(;)
(; )(; ),0
gg
tt
t
t tt
CC
Txt wkexwed
e
k xtweedk xtweet
α τβτ
βτ
ατα ββα
ττ
τ αβ
−− −
− −−−
≤≤− −≥
(6.9)
H. A. Alafif
146
Inequality (6.9) can be majorized as follows
( )
1
( )(;)(;)
g
t
C
Txt wk xwe
β
τ αβ
≤−
(6.10)
Because
0
βα
<<
. Dividing inequality (6.10) by
t
e
β
, we have
( )
1
()(; )(; )
gg
CC
Tx twkxtw
αβ
≤−
Hence for
( )()
2
;,(, ,)
g
xtw CLP
+
∈Ω
we have
()()
22
, (,,),(,,)
gg
TCL PCL P
++
Ω⊂ Ω
and the
pair of Banach spaces
( )
,
gg
CC
is admissible.
The rest of the proof is due to theorem 4.1.
6.2. A Problem in Telephone Traffic Theory
In this subsection we shall examine a stochastic integral equation arising in the study of telephone traffic. We
shall describe the problem in detail and then apply corollary 4.3 to show existence of a unique random solution
Consider a telephone exchange and suppose that calls arrive at the exchange at time instants
12
,, ,,
n
tt t
,
where
12
0
n
tt t< <<< <∞
.
These arrival times must be considered as random instants, so we denote the distribution function by A(t) on
the time axis. For a call arriving at time t let the random variable
(; )Htw
denote the holding time that is the
length of time that a “conversation” is held for a call arriving at the exchange time t.
The
12
(; ),(; ),H tw Htw
are considered as being mutually independent for different times
12
,,tt
and as
being independent of the state of the exchange, where the state of the exchange is the number of busy channels.
The number m of trunks or channels of the exchange is assumed to be finite and large, so that we approximate
a continuous process. It is also assumed that any channel not being used may be utilized by an incoming call and
that the holding time for a channel beings at the time instant that the call arrives at the exchange.
A conversation (or connection) is realized if a channel is not busy at the time a call arrives. If all channels are
busy at the time t that a call arrives, then either the call is lost or a queueing problem develops. Only the first
case will be considered here. Various problems have been studied in this situation. For example, the probability
()
k
Pt
that at time t, k of the m channels or busy has been examined in detail (Fortet [1]). We are concerned with
the total number of “conversations” held (the number of busy channels) at time t which for each t is a random
variable and may be described by as to chastic integral equation.
Let
(; )xtw
be the total number of conversations held at time t. That is
(; )xtw
is a random variable for
each
t
+
and
(0; )0xw=
. Let
(; )Jtw
be a random function with value one if a call arising at time
0
t>
is not lost and value zero if the call is lost
Let
[ ]
[ ]
10,( ;);
(, ;)00,( ;);
if tHw
kt wiftHw
ττ
τττ
−∈
=−∉
Such that
(, ;)kt w
τ
is equal to one if a conversation from a call arising at time
τ
is still being held at time
t
τ
and is equal to zero otherwise. Thus we may write
0
(; )(;)(,; )()
t
xwJw ktwdA
τ τττ
=
(6.11)
Equation (6.11) is interpreted as the total number of telephone calls arising at times
,0 t
ττ
≤≤
that were not
lost such that the conversation is still being held at time t.
Suppose that
()Vk
is any function such that
10,1, ,1,
() 0 otherwise,
if km
Vk = −
=
Clearly,
(; )xtwm
for all
t+
and
w∈Ω
. Hence we may write
[ ]
1( ;)0,1,,1,
(; )0otherwise,
ifxtwm
V xtw=−
=
H. A. Alafif
147
Which means that
[]
(; )V xtw
has values one if a call arising at time t is not lost and values zero otherwise.
Then Equation (6.11) may be written as the nonlinear stochastic integral equation
[ ]
0
(;)(,;) (;)()
t
xtwktwvxwdA
τττ
=
(6.12)
Which Bharucha-Reid [7] refers to as the Fortet integral equation.
Suppose that the distribution A(t) of arrival times has a density function a(t). Then Equation (6.12) reduces to
a stochastic integral equation of the voltera type
[ ]
0
(;)(,;)(;) ()()
t
xt wktwvxwad
ττττ
=
If we let
( )
[]
( )( ;)0,1,,1
, (;)(;)()0 otherwise
aifxwm
fxwvxw a
ττ
τττ τ
= −
=
Then we obtain a stochastic volterra integral equation of the form of (1.1) with the stochastic free term iden-
tically zero
( )
0
(;)(,;),(;)()
t
xtwktwfxwd
τ τττ
=
(6.13)
(, ;)kt w
τ
is the stochastic kernel defined for
0t
τ
≤≤<∞
and taking only the value one or zero.
Before showing that (6.13) possesses a unique random solution we observe that the above description applies
to many systems. If we replace the word “telephone exchange” with “serving mechanism” and the words
“channel”, “call”, and “conversation” with the words “server”, “customer”, and “service”, respectively, then we
are dealing with a general system in which “customers” are being “served” by
m<∞
“servers”. If we assume
that a customer does not wait when he finds all m servers busy so that no queue develops, then the random solu-
tion of the stochastic integral equation (6.13) gives the total number of “services” being performed at time t. Al-
so the functions in (6.13) may be any functions describe the physical situation. For example the stochastic kernel
may be of the form
()
()
(, ;)()
t
Xw
kt wIte
τ
ττ
−−
= −
where
( )
()Xw
I
is the indicator function of a random set
()Xw
, which means that solutions at earlier times
t
τ
have a decaying effect on the system.
We now show that the stochastic integral Equation (6.13) satisfies the conditions of corollary 4.3. We first
show that
()
2
, (;)(,,
f txtwL
∈Ω
and
(, ;)(,,))kt w LPP
τ
∈Ω
Let
0t
be fixed. Since
()at
is a density function, it may be assumed to be bounded for all t except on a
set of measure zero. Hence for some
0M>
and all t,
0 ()
at≤<∞
we have
( )
222
,(;)()() ()ftxtwdPwa tdPwM
ΩΩ
≤≤ <∞
∫∫
by definition of
( )
,(;),f txtw
so that
()
2
, (;)(,,)f txtwLP∈Ω
for each
.t
+
By definition of
( ,;),kt w
τ
0,t
τ
≤≤ <∞
we obviously have that the P-measure of
{ }
:(,;)1 ,0w ktwt
ττ
>≤≤ <∞
is zero, that is a P-null set. Hence
(, ;)kt w
τ
is bounded P-a.e. and is in
(, ,)LP
. Also if
( )
,,(, )
nn
tt
ττ
as
n→∞
we have
{ }
:(,;)(,;)00
nn
Pw ktwktw
ττ
− >→
as
n→∞
Since
(, ;)kt w
τ
has values zero ore one only.
That
(; )xtw C
, a continuous bounded function for each w, is easily shown. Let
w∈Ω
and choose
t+
.
For
0
ε
>
and
0h>
H. A. Alafif
148
(;) (;)(,;)(,(;))()
th th
tt
xthwxtwktwfxwdad
ττ ττττ
++
+− =≤
∫∫
Thus for
0
ε
>
there is
0a
δ
>
such that when
()
th t
δ
+−<
()
th
t
ad
ττε
+
<
Therefore
(; )xtw
is continuous on
+
for each
w∈Ω
and bounded by m also
22
(;)( )x twdPwmt
+
≤ <∞∈
Which means that
2
(; )(,, )xtwLP∈Ω
for each t.
We note that
1
1Λ=
in corollary 4.3 since
(, ;)sup(, ;)1
w
kt wPesskt w
ττ
=−≤
Since
(; )0
htw
P. a.e, trivially, it belongs to C. Since a(t) is a density function,
( ,0)( )f tat=
is conti-
nuous, bounded, and nonnegative, and
()a tdt
−∞
<∞
Suppose g(t) is any function satisfying condition 1) of corollary 4.3.
Then we must have
{ }
0 ()0
(, )(,)0()(1);,0,1,,1
( )()andorand
g tif xy
ftxftygt mifxyxym
atgtm ifxmymxmym
λ
λ
λ
≤⋅ =
− =≤−≠∈−
≤ ≠==≠
If we choose
1m
λ
=
, then in order for condition 2) of corollary 4.3 to be satisfied, there must exist apposi-
tive continuous function g(t) on
+
such that
0()g tdt
<∞
and
() ()
at gt
t
+
This restriction is not too sever, since a(t) is a density function. We may take
(),0
qt
gt eq
= >
, for example.
Therefore, since
(, )0
C
htw =
,
( ,0)( )f tatM=≤
and
1m
λ
=
is small for large m there exist a unique
random solution of Equation (6.13) if
() ()at gt
, where g(t) satisfies the given conditions.
References
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[2] Tsokos, C.P. (1969) On Some Nonlinear Differential System with Random Parameters. IEEE Proceedings Ann.
Princeton Conf. on Information Sci. and Systems, 3rd, 228-234.
[3] Tsokos, C.P. (1969) On the Classical Stability Theorem of Poincare-Lyapunov. Proceedings of the Japan Academy, 45,
780-785. http://dx.doi.org/10.3792/pja/1195520593
[4] Tsokos, C.P. (1969) On the Stochastic Integral Equation of the Volterra Type. Mathematical Systems Theory, 3,
222-231. http://dx.doi.org/10.1007/BF01703921
[5] Fortet, R. (1956) Random Distributions with Application to Telephone Engineering. Proc. Berkeley Symp. Math. Sta-
tist and Probability, 3rd, 11, 81-88.
[6] Bharucha-Reid, A.T. (1972) Random Integral Equations. Academic Press, New York.
[7] Corduneanu, C. (1968) Admissibility with Respect to an Integral Operator and Applications. Math. Tech. Rep., Univ.
of Rhode Island.
[8] Tserpes, N.A. and Mukherjea, A. (1971) Invariant Measures on Semigroups with Closed Translations. Z. Wahr schein-
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[10] Adomian, G. (1970) Random Operator Equations in Mathematical Physics 1. Journal of Mathematical Physics, 11.
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http://dx.doi.org/10.1137/0118045