We offer the model of applications parallel service in queuing system (QS) which consists of two units of service with an unlimited number of servers. A recurrent stream of binary applications enters to an input to the system. We recorded the approximate (asymptotic) equality of the first and second orders for the respective blocks of the system.
The research is made under supporting of ADAP "Development of the higher education scientific potential" (2009-2010 years) of the Federal education agency of RF concerning with the project "Investigation methods formulation of non-Markov queues systems and their appliance in complex economic systems and computer transmission network"

Moiseeva, S.

Sinyakova, I.

English

Электронная библиотека БГУ

INVESTIGATION OF QUEUING
SYSTEM GI
(2)
jM
2
j1
I. Sinyakova
1
, S. Moiseeva
2
1
The branh of Kemerovo state university in Anzhero-Sudzhensk,
2
Tomsk state university
1
Anzhero-Sudzhensk, Russia
2
Tomsk, Russia
1
Irinka_asfmail.ru
We oer the model of appliations parallel servie in queuing system (QS) whih
onsists of two units of servie with an unlimited number of servers. A reurrent stream
of binary appliations enters to an input to the system. We reorded the approximate
(asymptoti) equality of the rst and seond orders for the respetive bloks of the
system.
The researh is made under supporting of ADAP "Development of the higher ed-
uation sienti potential" (2009-2010 years) of the Federal eduation ageny of RF
onerning with the projet "Investigation methods formulation of non-Markov queues
systems and their appliane in omplex eonomi systems and omputer transmission
network".
Keywords: parallel servie, systems with an unlimited number of servers, a reurrent
stream, the method of asymptoti analysis.
1. INTRODUCTION
Currently, attention to the queuing theory is largely stimulated by the need to apply
the results of this theory to important pratial problems arising in onnetion with
the rapid development of ommuniation systems, the emergene of information and
omputing systems, appearane and omplexity of various tehnologial systems, the
reation of automated ontrol systems.
At the present stage of development of queuing theory, one of the urgent tenden-
ies is the queuing systems investigation with bath arrival of appliations and parallel
servie [1, 2℄. Range of appliation of the this QS is quite extensive, for example, model-
ing of modern information and omputer systems requires take into aount the paket
tra, as well as one of the basi priniples for the design of modern omputer net-
works - parallel information proessing [3, 4℄. Therefore there is a need to develop new
mathematial models of queuing systems, suh as systems with extraordinary inoming
ow and the various servie options, inluding two or more units of servie.
219
2. PROBLEM STATEMENT
We onsider a QS GI
(2)
jM
2
j1 with two servie bloks, eah of whih ontains an
unlimited number of servers.
Fig. 1. QS with parallel servies of multiple appliations GI
(2)
jM
2
j1
A reurrent ow [5℄ of dual appliations speied distribution funtion A(x) - length
of intervals between the moments of events in the onerned stream enters to the input
of the system. Servie intervals of various appliations are stohastially independent
and identially distributed in eah blok and have an exponential distribution with 
1
and 
2
parameters, respetively. Reeived appliations takes any of the free servers and
after ompletion of the servie the appliation leaves the system.
3. MATHEMATICAL MODEL
We dene i
k
(t) - number of requests in the k-th blok of servie. Beause the
inoming ow is non-Poisson, the two-dimensional proess {i
1
(t),i
2
(t)} is non-Markov,
and therefore we onsider an additional omponent of z(t), equal to the length of the
interval from t to themoment of the next event in the reurrent input stream.
For the onerned system a three-dimensional random proess {z(t),i
1
(t),i
2
(t)} is
Markov, so its probability distribution
P (z; i
1
; i
2
; t) = Pfz (t) < z; i
1
(t) = i
1
; i
2
(t) = i
2
g
We an write equality
P (z  t; i
1
; i
2
; t+t) = fP (z; i
1
; i
2
; t)  P (t; i
1
; i
2
; t)g (1  i
1

1
t) (1  i
2

2
t)+
+P (t; i
1
  1; i
2
  1; t)A(z) + P (z; i
1
; i
2
+ 1; t) (i
2
+ 1)
2
t+
220
+P (z; i
1
+ 1; i
2
; t) (i
1
+ 1)
1
t+ o(t);
from whih we an readily obtain a system of dierential equations of Kolmogorov [6℄
P (z; i
1
; i
2
; t)
t
=
P (z; i
1
; i
2
; t)
z
 
P (0; i
1
; i
2
; t)
z
 P (z; i
1
; i
2
; t)i
1

1
 P (z; i
1
; i
2
; t)i
2

2
+
+P (z; i
1
+1; i
2
; t) (i
1
+ 1)
1
+P (z; i
1
; i
2
+1; t) (i
2
+ 1)
2
+
P (0; i
1
  1; i
2
  1; t)
z
A(z):
For the stationary probability distribution this system an be rewritten as
P (z; i
1
; i
2
)
z
 
P (0; i
1
; i
2
)
z
  P (z; i
1
; i
2
)i
1

1
  P (z; i
1
; i
2
)i
2

2
+
+P (z; i
1
+ 1; i
2
) (i
1
+ 1)
1
+ P (z; i
1
; i
2
+ 1) (i
2
+ 1)
2
+
P (0; i
1
  1; i
2
  1)
z
A(z) = 0:
Dening H(z; u
1
; u
2
) =
1
P
i
1
=0
1
P
i
2
=0
e
ju
1
i
1
e
ju
2
i
2
P (z; i
1
; i
2
), where j =
p
 1 imaginary unit
and taking into aount that
H(z; u
1
; u
2
)
u
1
= j
1
X
i
1
=0
1
X
i
2
=0
i
1
e
ju
1
i
1
e
ju
2
i
2
P (z; i
1
; i
2
);
H(z; u
1
; u
2
)
u
2
= j
1
X
i
1
=0
1
X
i
2
=0
i
2
e
ju
1
i
1
e
ju
2
i
2
P (z; i
1
; i
2
);
for the funtions H(z,u
1
,u
2
) we obtain the basi equation for investigating the system
GI
(2)
jM
2
j1
H (z; u
1
; u
2
)
z
+
H (0; u
1
; u
2
)
z

e
j(u
1
+u
2
)
A (z)  1
	
  j
1
 
e
 ju
1
  1

H (z; u
1
; u
2
)
u
1
 
  j
2
 
e
 ju
2
  1

H (z; u
1
; u
2
)
u
2
= 0: (1)
We dene additional onditions as
H(z; 0; 0) = R(z);
where R(z)  stationary distribution of the z(t). Solution H(z,u
1
,u
2
) denes the hara-
teristi funtion of the number of servers employed in the stationary state in eah blok
of the system GI
(2)
jM
2
j1 as equalities
Me
ju
1
i
1
(t)
= H(1; u
1
; 0);
Me
ju
2
i
2
(t)
= H(1; 0; u
2
):
221
4. THE METHOD OF ASYMPTOTIC ANALYSIS
4.1. Asymptoti form of the rst order. For a omplete analysis of the studied
queuing systems apply the method of asymptoti analysis [5℄.
We onsider the basi equation for the harateristi funtion (1):
H (z; u
1
; u
2
)
z
+
H (0; u
1
; u
2
)
z
 
e
j(u
1
+u
2
)
A (z)  1

  j
1
 
e
 ju
1
  1

H (z; u
1
; u
2
)
u
1
 
 j
2
 
e
 ju
2
  1

H (z; u
1
; u
2
)
u
2
= 0;
whih will be solved in the asymptoti onditions of the growing servie-time, believing
that 
1
,
2
!0. We dene 
1
= ", 
2
= "q and do hanges in equation (1)
u
1
= "w
1
; u
2
= "w
2
; H(z; u
1
; u
2
) = F
1
(z; w
1
; w
2
; "); (2)
In the result we get the equation for F
1
(z,w
1
,w
2
,") in the following form:
F
1
(z; w
1
; w
2
; ")
z
+
F
1
(0; w
1
; w
2
; ")
z
 
e
j(w
1
+w
2
)
A (z)  1

 
  j
 
e
 j"w
1
  1

F
1
(z; w
1
; w
2
; ")
w
1
  j
 
e
 j"w
2
  1

F
1
(z; w
1
; w
2
; ")
w
2
= 0: (3)
We prove the following statement:
Theorem 1. The limit (when "!0) solution of equation (2) has the form:
F
1
(z; w
1
; w
2
) = R(z)e
fj(w
1
+w
2
)g
;
where R(z) - stationary probability distribution of random proess values{z(t)}, and the
parameter  is given by:
R(0)
z
= 
Proof. The proof 1. In the equation (3) limiting transition is exeuted at " ! 0, we
will reeive that F
1
(z,w
1
,w
2
) the equation deision is
F
1
(z; w
1
; w
2
)
z
+
F
1
(0; w
1
; w
2
)
z
fA(z)  1g = 0;
whih denes a vetor funtion R(z), therefore
F
1
(z; w
1
; w
2
) = R(z)
1
(w
1
; w
2
) (4)
We nd the salar funtion 
1
(w
1
,w
2
) as follows. In the equation (3) limiting transition
is performed at z !1, we will reeive an equality
F
1
(0; w
1
; w
2
; ")
z
 
e
j(w
1
+w
2
)"
  1

  j
 
e
 j"w
1
  1

F
1
(1; w
1
; w
2
; ")
w
1
 
 j
 
e
 j"w
2
  1

F
1
(1; w
1
; w
2
; ")
w
2
= 0;
222
dividing the left and right parts of the equation by " and exeuting limiting transition
at "! 0, we will reeive the equality
F
1
(0; w
1
; w
2
)
z
j (w
1
+ w
2
) + j
2
w
1
F
1
(1; w
1
; w
2
)
w
1
+ j
2
w
2
F
1
(1; w
1
; w
2
)
w
2
= 0;
in whih we substitute the expression (4) and write down the dierential equation in
partial derivatives

1
(w
1
; w
2
)
w
1
+ j
2
w
2

1
(w
1
; w
2
)
w
2
= j (w
1
+ w
2
)
R(0)
z

1
(w
1
; w
2
) =
= j (w
1
+ w
2
)
1
(w
1
; w
2
) ;
whose solution is satisfying the initial ondition 
1
(0,0)=1 and it has the form

1
(w
1
; w
2
) = e
fj(w
1
+w
2
)g
:
We get F
1
(z; w
1
; w
2
) = R(z)e
fj(w
1
+w
2
)g
.
The theorem is proved.
Using (2) and (4) we an write the approximate (asymptoti) equality
H(z; u
1
; u
2
) = R(z)e
f
j
(
u
1
"
+
u
2
"
)
g
= R(z)e
n
j

u
1

1
+
u
2

2
q
o
from whih we an obtain following expression for the harateristi funtion of proess
{i
1
(t),i
2
(t)} in the stationary state,
Me
ju
1
i
1
(t)
= H(1; u
1
; 0) = e
n
ju
1


1
o
; Me
ju
2
i
2
(t)
= H(1; 0; u
2
) = e
n
ju
2


2
q
o
:
The reeived equations is alled the rst order asymptoti form for the servie bloks
of systems servie GI
(2)
jM
2
j1.
For a more detailed investigation, we onsider the seond order asymptoti form.
4.2. Asymptoti form of the seond order. In equation (1) we make the
substitution
H(z; u
1
; u
2
) = H
2
(z; u
1
; u
2
)e
n
j

u
1

1
+
u
2

2
q
o
;
whih later beomes
H
2
(z; u
1
; u
2
)
z
+
H
2
(0; u
1
; u
2
)
z
 
e
fj(u
1
+u
2
)g
A (z)  1

+
+j
1
 
1  e
 ju
1

H
2
(z; u
1
; u
2
)
u
1
+ j
2
q
 
1  e
 ju
2

H
2
(z; u
1
; u
2
)
u
2
 
 
 
1  e
 ju
1

H
2
(z; u
1
; u
2
)  
 
1  e
 ju
2

H
2
(z; u
1
; u
2
) = 0:
We dene 
1
= "
2
, 
2
= "
2
q in the last equation and make substitutions
u
1
= "w
1
; u
2
= "w
2
; H
2
(z; u
1
; u
2
) = F
2
(z; w
1
; w
2
; ");
223
we get the result
F
2
(z; w
1
; w
2
; ")
z
+
F
2
(0; w
1
; w
2
; ")
z
 
e
j"(w
1
+w
2
)
A (z)  1

+
+j"
 
1  e
 j"w
1

F
2
(z; w
1
; w
2
; ")
w
1
+ j"
 
1  e
 j"w
2

F
2
(z; w
1
; w
2
; ")
w
2
 
  
 
1  e
 j"w
1

F
2
(z; w
1
; w
2
; ")  
 
1  e
 j"w
2

F
2
(z; w
1
; w
2
; ") = 0: (5)
The following theorem is proved similarly theorem for rst order asymptoti form.
Theorem 2. The limit (when "!0) solution of equation (5) has the form
F
2
(z; w
1
; w
2
) = R(z)e

(j(w
1
+w
2
))
2
2

2

;
where R(z) - stationary probability distribution of the random proess values {z(t)}, the
parameter  is given by:
R(0)
z
= , the value 
2
is determined as 
2
= +
f
2
(0)
z
.
The vetor funtion f
2
(z) satises the ondition f
2
(1)=0 and it is a solution of the
equation
f
2
(z)
z
+
f
2
(0)
z
(A(z)  1) +
R(0)
z
A(z)  R(z) = 0:
Also the asymptoti (approximate) equality an be written
H
2
(z; u
1
; u
2
) = F
2
(z; w
1
; w
2
")  F
2
(z; w
1
; w
2
) =
= R(z)e

j
2


2
(w
1
+w
2
)
2
2
 
w
1
w
2
2

=
= R(z)e

j
2

2
2

u
1
p

1
+
u
2
p

2
q

2
 j
2
u
1
u
2
2
p

1

2
q

;
from whih we an obtain following expression for the harateristi funtion H(z,u
1
,u
2
)
H(z; u
1
; u
2
) = R(z)e

j

u
1

1
+
u
2

2
q

+
j
2

2
2

u
1
p

1
+
u
2
p

2
q

2
 j
2
u
1
u
2
2
p

1

2
q

;
Then the marginal harateristi funtions have the form
Me
ju
1
i
1
(t)
= H(1; u
1
; 0) = e

j
u
1

1
+
(ju
1
)
2
2
1

2

;
Me
ju
2
i
2
(t)
= H(1; 0; u
2
)E = e

j
u
2

2
q
+
(ju
2
)
2
2
2
q

2

:
The reeived equations is alled seond order asymptoti form for servie bloks of the
system GI
(2)
jM
2
j1.
5. CONCLUSION
So we have onstruted a model of multiple parallel servie requests of reurrent
ow. We investigate two-dimensional stohasti proess haraterizing the number of
engaged servers in the rst and the seond servie blok. We have found expressions for
the harateristi funtions at the asymptoti onditions of the growing laims servie-
time.
224
REFERENCES
1. Tatashev A. G. Queuing system group with group arrival and inverse disipline //
Cybernetis and of systems analyses 1995.  6. P. 163165.
2. Chehelnitsky A. A., Kuherenko O. B. Stationary harateristi of queuing system
with parallel servie // Siene journal // Minsk, 2009. P. 262268.
3. Endrus G. R. Basis of multithreaded, parallel and distributed software tehnology:
Transl. from English. //M.: Publishing house Williams 2003. P. 512.
4. Toporkov V. V. Models of grid omputing. // M.: FIZMATLIT, 2004. P. 320.
5. Nazarov A. A., Moiseeva S. P. Method of asymptoti analysis in theory of queuing
systems. // Tomsk: NTL, 2006. P. 112.
6. Kleinrok L. Queuing systems. Transl. from English Glushko. // Ì. : Mehanial
engeneering, 1979. P. 432.
225


Investigation Of Queuing System Gl(2)|M2|∞

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Investigation Of Queuing System Gl(2)|M2|∞

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