The method of a finding of mean characteristics of an open exponential queueing network (QN) and solution of the problem of coasts optimization in such network by means of parallel calculations is considered. The software for solution of the formulated problems, developed with usage of technology of programming MPI and libraries PETSc is described

Koluzaeva, E.

Pankov, A.

English

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ANALYSIS AND OPTIMIZATION OF
EXPONENTIAL QUEUEING
NETWORKS BY MEANS OF
PARALLEL CALCULATIONS
A. Pankov, E. Koluzaeva
Grodno State University
Grodno, Belarus
koluzaevagmail.om
The method of a nding of mean harateristis of an open exponential queueing
network (QN) and solution of the problem of oasts optimization in suh network by
means of parallel alulations is onsidered. The software for solution of the formulated
problems, developed with usage of tehnology of programmingMPI and libraries PETS
is desribed.
Keywords:queueing network, parallel alulations.
1. INTRODUCTION
Now one of the most ompliated problems in the world is onsidered the solution
of tasks whih need big alulation resoures. During researh of the time-probability
harateristis of the QN with usage of lassial methods [1℄ it is often neessary to solve
systems of great number of dierene-dierential equations whih they satisfy. The
number of equations in these systems orresponds to number of the network states and
grows exponentially with adding in the network at least one message or new queueing
system (QS) (for losed networks). For open QN the situation espeially worse ause the
number of states of suh networks is ountable. Therefore evaluation of harateristis of
QN by regular omputer mahines an vary from several hours (for numerial methods)
to several days and more (for analytial methods). The next step in development of
QN researh methods is usage of high performane omputing (HPC) for alulations.
However, for parallel omputing one needs to develop speial software. That is time
and eort onsuming proess.
The problem of nding of the exponential QN harateristis as well as solution of
the network ost optimization problem using parallel omputations are onsidered in
the paper.
2. FINDING OF MEAN CHARACTERISTICS OF
JACKSON OPEN NETWORK
Let's onsider an open QN onsisting of n QS S
1
; S
2
; :::; S
n
. For uniation we
enter QS S
0
, orresponding to an external environment. Eah system has FIFO servie
196
disipline. An arrival ow of messages in the network is Poisson with rate . Servie
times of messages in various QS are independent and don't depend on ows arriving
in the systems; thus the servie time of the messages in i-th QS has an exponential
distribution with parameter 
i
(l), where l  number of messages in QS. The message
servied by i-th QS, passes into j-th QS with probability p
ij
, and leaves the network
with probability p
i0
,
n
P
j=0
p
ij
= 1, i = 1; n.
From the Jakson expansion theorem [2℄ follows that nal stationary probabilities
of Jason's network states look like
P (k) =
n
Y
i=1
P
i
(k
i
); (1)
where P
i
(k
i
), P
i
(0) are dened by following:
P
i
(k
i
) =
e
i

i
(k
i
)
P
i
(k
i
  1) =
e
i

i
(k
i
)
e
i

i
(k
i
  1)
P
i
(k
i
  2) = ::: =
k
i
Y
l=1
e
i

i
(l)
P
i
(0); (2)
P
i
(0) =
"
1 +
1
X
k
i
=1
k
i
Y
l=1
e
i

i
(l)
#
 1
; i = 1; n: (3)
Here values e
i
, i = 1; n, are roots of linear equations system:
e
j
= p
0j
+
n
X
i=1
e
i
p
ij
; j = 1; n: (4)
Thus, in steady state mode the number of messages in the network systems at eah
xed time instant is independent random variables.
In ase when systems of the Jakson open network are multilinear, i.e. QS S
i
onsists
of m
i
idential servie lines, and servie rate of messages equals 
i
, i = 1; n, for eah
line, steady probabilities of the network states also look like (1), thus
P
i
(k
i
) =
8
<
:

k
i
i
k
i
!
P
i
(0); k
i
 m
i
;

k
i
i
m
i
!m
k
i
 m
i
i
P
i
(0); k
i
> m
i
;
P
i
(0) =
"
m
i
X
j=0

j
i
j!
+

m
i
+1
i
m
i
!(m
i
  
i
)
#
 1
;
here 
i
=
e
i


i
, i = 1; n.
Knowing values of network states probabilities, it is possible to alulate various
probability harateristis of the QN. For example, average number of messages N
i0
in
197
system S
i
queue, average number of messages N
i
in system S
i
(in queue and on servie)
and average time of stay 
i
in system S
i
are nding by
N
i0
= P
i
(m
i
)

i
(m
i
  
i
)
2
; i = 1; n;
N
i
= N
i0
+
m
i
 1
X
k
i
=1
k
i
P
i
(k
i
) +
P
i
(m
i
)
1 

i
m
i
; i = 1; n;

i
=
N
i

i

i
; i = 1; n;
where
m
i
 1
P
k
i
=1
k
i
P
i
(k
i
) = 0 when m
i
= 1.
Proeeding from theoretial results follows that alulation of state probability using
(1) an be easy parallelized, for example, eah fator P
i
(k
i
), i = 1; n, an be omputed
by separate proessor irrespetively of other fators. Probability omputation proess
P
i
(0), i = 1; n, an also be parallelized, for example, eah item of the innite sum in
(3) an be evaluated irrespetive of other items on dierent proessors of a omputing
luster.
Solution of linear equation system by omputing lusters is well studied problem.
The matrix of the equation system (4) is rareed sine the matrix of probabilities of
message transitions between QS is suh. So the problem of values e
i
, i = 1; n, nding is
redued to usage of one of well studied methods of the solution of the equation systems
multisequening [4℄.
3. COSTS OPTIMIZATION IN OPEN JACKSON
NETWORK
Let's onsider the problem of osts minimization of the QN by the number of servie
lines in the queueing systems. We introdue next designations: d
i
 ost of one message
in i-th QS and E
i
 ost of one servie line in i-th QS, i = 1; n. So the network
optimization problem is
8
<
:
W (m) = W (m
1
; m
2
; :::; m
n
) =
n
P
i=1
(d
i
N
i
(m
i
) + E
i
m
i
)! min
m
1
;m
2
; :::;m
n
;
m
i
 a
i
; i = 1; n;
(5)
where a
i
 given numbers in whih limits the number of servie lines in i-th QS an
hange, i = 1; n.
The problem (5) is integer programming problem, it ould be solved by exhaustive
searh mathod. However even for the networks of rather small dimension with number
of the systems in the network equal, for example 20, and number of servie lines in
systems equal to 10, the number of suh searhes will be 10
20
and the solution of the
problem (5) on one proessor ould oupy some days.
198
The MPI tehnology was applied for parallel omputing to solve the optimization
problem (5). Software based on programming library PETS [3℄ was developed. Open
soure library PETS, distributing by GPL liense, was developed taking into aount
usage in the big salable appliations. PETS inludes a huge set of subroutines for
parallel solving of linear and non-linear equations that are easily integrated into appli-
ations written in programming languages C, C++, Fortran and Python. Also PETS
provides set of mehanisms used for development of parallel programs, for instane re-
ation of parallel matrixes and vetors, allowing to lead a data interhange and to arry
out various alulations.
The riterion in optimization problem (5) an be presented in a matrix form:
W
 
m
(l)

L
=M
 
m
(l)

L2n
 E
2n
! min
m
(l)
; l=1; L
;
where E
2n
= (d
1
; :::; d
n
; E
1
; :::; E
n
)
T
,
M
 
m
(l)

L2n
=











m
(1)
1
::: m
(1)
n
N
1
 
m
(1)

::: N
n
 
m
(1)

m
(2)
1
::: m
(2)
n
N
1
 
m
(2)

::: N
n
 
m
(2)

:::
m
(L)
1
::: m
(L)
n
N
1
 
m
(L)

N
n
 
m
(L)












;
where m
(l)
=

m
(l)
1
; m
(l)
2
; :::; m
(l)
n

, m
(l)
i
 number of servie lines in i-th system in l-th
searh, L =
n
Q
i=1
a
i
 number of all possible searhes, l = 1; L, i = 1; n.
The proess of problem (5) solving with library PETS is redued to reation of
parallelized matrix M
 
m
(l)

L2n
arranged line by line on all proessors, and further
multipliation of it by parallelized vetor E
2n
. The resulting vetor W
 
m
(l)

L
will
be also parallelized. The minimum of ost riterion is a minimum element of vetor
W
 
m
(l)

L
with number l

, and the solution of problem (5) is the rst n elements of a
line with number l

in matrixM
 
m
(l)

L2n
.
4. PROGRAM IMPLEMENTATION
From implementation point of view, the developed software is a bunh of several
utilities, eah one is dediated to one task: utility of intermediate and auxiliary matrixes
reation, utility for intermediate alulations obtaining (for omputing solution of set of
linear algebrai equations on a luster), utility for alulation of QN's harateristis on
a luster, optimization problem utility and also set of utilities for vetors and matrixes
visualization.
Speied partition into set independent subroutines allowed to make proess of
harateristis nding as transparent as possible for end user. Cluster resoures are used
more eetively sine temporary results are not omputing several times and ould be
taken from some intermediate storage. The only user interfae with developed utilities
is ommand line, all parameters are speied by ommand line arguments.
199
The given set of developed utilities ould be used not only on a superomputer but
on regular omputers driven by UNIX or Linux operating system with PETS installed,
for example when dimension of task being solving is rather insigniant.
5. THE EXAMPLE
Let's onsider queueing network onsisting of n = 5 QS, the number of servie lines in
eah system an vary from 1 to 10, so the number of possible searhes equal L = 160807.
Probabilities of messages transition between systems are equal p
ii+1
= 1; i = 0; n  1,
p
n0
= 1, all remaining probabilities in transition matrix are equal to zero. The vetor
of ost oeients is E
2n
= (22, 31, 11, 21, 13, 15, 21, 43, 54, 32)
T
.
Solution of optimization problem (5) took about 53 mins on regular AMD64 Linux-
based mahine in single-proessor mode (1 ore was used). It took about 17 mins on
the same mahine with 2 ores dediated. Obtained results are: W
 
m
(l

)

= 198:275,
m
(l

)
= (2; 1; 2; 2; 3), N
 
m
(l

)

= (0:07; 0:953; 0:052; 0:126; 0:009).
REFERENCES
1. Ivnitsky V.A. Theory of queueing networks, Fizmatlit, Mosow, 2004.
2. Matalytsky M.A., Tikhonenko O.M. and Pankov A.V. Queuing theory and its ap-
pliation, GrSU, Grodno, 2008.
3. http://www.ms.anl.gov/pets/pets-as/
4. Toporkov V.V. Models of the distributed alulations, Fizmatlit, Mosow, 2004.
200


Analysis And Optimization Of Exponential Queueing Networks By Means Of Parallel Calculations

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Analysis And Optimization Of Exponential Queueing Networks By Means Of Parallel Calculations

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