STUDIA INFORMATICA
Volume 38
2017
Number 1B (130)
Paweł ZAJĄC, Mikhail MATALYTSKI
Institute of Mathematics, Czestochowa University of Technology
FINDING OF EXPECTED VOLUMES OF REQUESTS IN SYSTEMS
OF THE QUEUEING NETWORK WITH A LIMITED AMOUNT OF
SPACES OF EXPECTATIONS
Summary. We present a method of finding the expected volume of requests in
open HM-network with homogeneous requests, bypass of nodes the network service
systems. Were considered a case where the changes in volumes associated with
transitions between states of the network are deterministic functions dependent states
of network and time, and service systems are single line. Assumed that the probability
of state network systems, the parameters of entrance flow of messages and service
depend on time.
Keywords: queueing networks, volume of requests, capacity of claims,
HM-networks, wireless access point
ZNAJDOWANIE
OCZEKIWANYCH
OBJĘTOŚCI
ZGŁOSZEŃ
W SYSTEMACH SIECI KOLEJKOWYCH Z OGRANICZONĄ LICZBĄ
MIEJSC OCZEKIWANIA
Streszczenie. Opisano metodę znalezienia oczekiwanej objętości zgłoszeń
w otwartej HM-sieci z jednorodnymi zgłoszeniami i obejściami węzłów sieci
systemów obsługi. Rozpatrywano przypadek, gdy zmiany objętości związanych
z przejściami między stanami sieci są deterministycznymi funkcjami, zależnymi od
stanów sieci i czasu, a systemy obsługi są jednoliniowe. Zakłada się, że
prawdopodobieństwo stanów systemów sieci, parametry strumienia wejściowego
zgłoszeń i obsługi zależą od czasu.
Słowa kluczowe: sieci kolejkowe, objętość zgłoszeń, HM-sieci, bezprzewodowy
punkt dostępu
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P. Zając, M. Matalytski
1. Model of action wireless network with a limited amount of
simultaneous connections
In the IS the total amount of memory volume is bounded by some value, which is usually
called memory volume [1]. In the IS designing the main task is the determination of the
memory volume so as to take into account the conditions that limit the proportion of the lost
information. One of the methods for solving problems of IS design is the use of НМ-queueing
networks [2]. Further under the IS we will understand systems, converting objects which is
the information coming in portions in the form of messages [1]. HM-networks can be used to
determine the volume of a buffer storage of systems, which are representing processing IS
nodes and transferring messages. Note that considered problem is one of the main, for
example, in the design of communication centers or hubs in data communication networks.
Neglect time dependence of messages processing of their volumes can lead to serious
errors in finding the buffer memory in IS and in calculating of the probability of messages
loss. The solution in the general case the above problems can be based on the use of HMnetworks with revenues. In such networks, the request during the transition from one
queueing system (QS) to another brings some revenue last (which is equal to the volume of
this request), and revenue (volume) of the first QS is reduced by this amount.
QS with revenues in the stationary regime has been introduced in the consideration in [3],
and networks - in [4]. A survey results obtained by queueing systems and queueing networks
(QN) in the stationary regime contained in [5]. It is dedicated to finding the mean revenues in
the network systems which depend only on their states and do not depend on time, and
solving the problem of finding the optimal request service intensities in QS by the method of
dynamic programming. QN with revenues in the non-stationary regime has been studied in [6,
7]. Revenues from transitions between network states were depended on the states and their
time or were random variables with the given moments of the first and second orders. In a
survey article [7] the results of researching, optimization and selection of the optimal
strategies in Markov networks with revenues, various applications are described them as a
probabilistic models for forecasting expected revenues in the information and
telecommunication systems and networks, when, for example, requests service on the server
generates revenue for a servicer, as well as insurance companies, logistics transportation
systems, industrial systems and other facilities. For the first time application of the HMnetworks for estimating the memory volume in the IS has been described in [8].
Consider the action model a wireless network. Let us assume that the network is a set of
wireless access points S i , each of which gives the user the ability to connect to the network
Finding of Expected Volumes of Requests in Systems of the Queueing Network…
91
of information by the next available port and use its resources, i  1, n . Each access point can
simultaneously connect multiple users to a network or execute queries users (i.e. for users to
make bandwidth sharing for mi smaller teams). All other requests users do not receive service,
create maximum queue requests Li and are waiting for a reply access point Li   , i  1, n .
From a technical point of view, the value of Li is determined taking into account the technical
characteristics of a wireless access point (WAP). For example, the greater the number of
divided bands of the original panel is more customers may be combined, and the slower the
speed of the bands assigned to them. The main limitation is the number of possible IP
addresses that are distributed to connect.
We also assume that in the absence free places in the queue, to the access point comes
the loss of user requests and redirecting this question to the next WAP. From a practical point
of view, redirecting associated with direct customer flow (ie, the device working with
computer network), or search for «distant» WAPs do not go beyond the radius of visibility of
the device.
At request, we assume a data packet, the sending by the source (ie. client device) and
tending to the recipient (by WAP). In computer networks, the package is designed in a way, a
block of data sent over the network in batch mode. Computer lines, which does not support
packet-mode such as traditional telecommunication point-to-point transfer data simply as a
sequence of bytes, characters, or bits individually. The package consists of two types of data:
the control information and user data (also called a payload). The control information
contains information necessary for the provision of user data: source and destination
addresses, error detection codes (such as checksums) and information about the order.
Typically, the control information contained in the packet header and tail, and the user data
disposed therebetween.
Various communication protocols use different conventions for separating elements and
formatting data. Protocol packets "Synchronous binary" formats in 8-bit bytes, and for the
separation of the used special characters. In other protocols, such as Ethernet header and the
beginning of data elements, their location in relation to the package they are registered. Some
protocols format the information at the level of bits and not bytes. In this case, it is assumed
that each data packet sent by a user will be deterministic or random length (volume).
Many networks cannot guarantee delivery, no duplicates packages, and order delivery,
such as the UDP protocol on the Internet. However, this can be done on top of a transport
packet (for one level of the OSI model), which can provide such protection. Packet header
identifies the type of data packets, the package, the total number of packets and IP address of
the source and destination. In our case, it should be assumed that the request sent by the user
92
P. Zając, M. Matalytski
will always be delivered to the recipient (AP), which will not be taken into account in case of
packet loss between the source and the destination.
Estimation of the total volume of data (data packets) for each wireless access point
(WAP) at the given point in time is an important task when designing a wireless network,
because it allows to locate the highly loaded AP and distribute the load evenly over them.
Therefore, you must determine the average total volume of data (packets of users)
received by the access points to the network information (for example, Internet), taking into
account the limited number of «simultaneous» connections on this point. This problem can be
solved by using HM-network storage service. According to the queueing network we mean a
collection of interconnected queueing systems Si with a limited buffer size Li to hold the
queue (queues) package requests. By application we mean the user's request to the WAP,
which is a data packet.
2. Analysis of queueing networks with homogeneous requests, bypass
of nodes and time depending parameters of entrance flow and
service time
Consider the open exponential queueing network comprising n queueing systems. With
some probability requests from have a chance to join the queue, or instantly pass in the matrix
of transition probabilities to the other queueing system or leave the network. The probability
of attachment to the queueing systems depend on the state and number of queueing system
from which the notification is sent.
The message is sent to the network at a given time interval
[t , t   t ]
with probability
 (t )t  o( t ) and is supported in queueing system S i in this interval with a probability of
 i (t )t  o(t ) .
Let
pij
– the probability of transition of requests after servicing from the the system Si to
S j , i, j  0, n , system S 0 we also understand the external environment. Now consider the case
where the stream parameters and service depend on the time. Notification is sent from the
external environment in the i -th SOM with the probability of p0 i ,
n
p
0i
 1 . A request to
i 1
the queuing system from outside or from the second system at time t with probability f (i) (k,t) ,
when the network is in state of k, t  , connects to the queue, and the probability of
1  f (i ) (k , t ) not join the requests queue, counts on service (i.e. the time service with a
probability of 1 is equal to zero). If the request ended service in the i-th queueing system, then
Finding of Expected Volumes of Requests in Systems of the Queueing Network…
93
with the probability of pij is immediately sent to the j-th system and with the probability pi 0
leaves the queueing network,
n
p
ij
 1 , i  1, n .
j 0
State of the network is described vector k (t )  (k , t )  (k1 (t ),..., k n (t ))  (k1 ,....k n , t ) ,
where k i (t )  the number of requests in the system S i (in a queue and handling), i  1, n .
Let  i k , t  - conditional probability that the request comes from outside and to the i-th
queueing system at time t, when the network is in state k, t  cannot be handled by queueing
system;  ij k , t  - conditional probability that request by introducing point i-th system the
outside, at time t, when the network is in state k, t  , for the first time will receive service in
the j-th system;  i k , t  - conditional probability that requests serviced at time t in the i-th
queueing system, when the network is to state k, t  , will not be supported further in either a
single system;  ij k , t  - conditional probability that the request serviced in the i-th system at
time t, when the network is to state k, t  , for the first time will be supported in the j-th
queueing system, i, j  1, n .
The formula for total probability we get:


n

j 1




 i k , t   1  f i  (k , t )  p i 0   p ij  j k , t  , i  1, n ,
n
 ij k , t   f ( i ) (k , t ) ij  (1  f ( i ) (k , t )) pil lj k , t  , i, j  1, n .
l 1
I the function f
(i )
(k , t )  f
(i )
(k i , t ) depends only on the number of requests is
n
 i (k , t )  (1  f (i ) (k i , t ))( p i 0   p ij  j (k , t )), i  1, n, ,
(1)
j 1
n
 ij (k , t )  f (i ) (k i , t ) ij  (1  f (i ) (k , t )) pil lj (k , t ), i, j  1, n .
(2)
l 1
Furthermore,
n
n
 ij (k , t )  pi 0   pij j (k  I i , t ),  ij (k , t )  pi 0   pil lj (k  I i , t ), i, j  1, n .
j 1
l 1
(3)
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P. Zając, M. Matalytski
We also have equality
n
 i k , t    ij k , t   1, i, j  1, n
(4)
j 1
n
 i k , t   1    ij k , t , i, j  1, n.
(5)
j 1
From (1) and (3) we find


 ij k , t   f (i ) (k , t ) ij  1  f (i ) (k , t )  ij k  I i , t , i, j  1, n.
(6)
3. The system of difference-differential equations for the expected
volume of requests in the HM-systems network
Let vi k , t  - expected volume of requests, which accumulate in the system S i at time t ,
when, during the initial network is to state k , and assume that the function is differentiable
with respect to t; ri (k ) - increasing the volume of requests in the system S i per unit time
when the network is to state k ; r0 i ( k  I i , t ) – volume of requests, which increases the total
volume of requests in the system S i , when the network passes from the state of
(k , t ) in to
state k  I i , t  t  in time t ;  Ri 0 k  I i , t  – reducing the volume size of the system, if


the network causes a transition from state (k , t ) in to state k  I i , t  t  ; rij k  I i  I j , t –
the volume of requests of the system S i (reducing the volume of requests of the system S j ),
to which increases the total volume of requests, the network changes its state from (k , t ) to
k  I
i
 I j , t  t  in time t , i, j  1, n . Note that we consider now the case when the
value of r0 i , Ri 0 , rij are deterministic functions dependent states of network and time.
Assume that the network is in a state
the states
k  I  , k  I , k  I
i
i
i
(k , t ) . At time t it may be in a state k or go to
 I j , i, j  1, n . If the network is still in the state
(k , t  t ) , volume of requests in the system S i is ri ( k ) t plus volume vi (k , t ) , to which
increases its volume in the remaining t time units. The probability of this happening is
n
1    t  p0i 1  i k , t   i t  1   ii k , t t  o(t ) . If the network passes to the state
i 1
of k  I i , t  t  with probability  t  p0 i ij k  I i , t t  o(t ) , then the total volume of
Finding of Expected Volumes of Requests in Systems of the Queueing Network…
95
requests in the system S i is r0 i k  I i , t   vi ( k  I i , t ), and if the state k  I i , t  t  with
probability  i (k , t ) i k  I i , t u(ki , (t ))t  o(t ) , then the total volume of requests of this
system will be  Ri 0 k  I i , t   vi k  I i , t , i  1, n .
Similarly, if the network goes from (k , t ) in to a state
k  I
i
 I j , t  t  with
probability  j ( k j , t )  ji k  I i  I j , t  u ( k j , (t ))t  o( t ) , then it will increase the total
volume of requests in the system S i by the amount rij k  I i  I j , t  plus the volume of the
remaining time, if the initial state of the network was k  I i  I j . The above description has
been collected in Table 1. Then, using the formula of the total probability for the expected
volume of requests in the system, you can get a system of difference-differential equations
(DDE):
n
dvi (k , t )
 ri k     t  p0i 1  i k , t   i t  1   ii k , t  vi (k , t ) 
dt
i 1
n
   t  p0 j ij k  I j , t vi (k  I j , t )  j (t ) j k  I j , t u k j (t ) vi (k  I j , t ) 
j 1
n
   j (t )  ji k  I i  I j , t u k j (t ) vi ( k  I i  I j , t ) 
j 1
j i
 i (t )  ij k  I i  I j , t u ki (t ) vi (k  I i  I j , t ) 
(7)
n
   j (t )  ji k  I i  I j , t u k j (t ) rij ( k  I i  I j , t ) 
j 1
j i
 i (t )  ij k  I i  I j , t u ki (t ) rji (k  I i  I j , t ) 
n


s
(t )  sc k  I c  I s , t u k s (t ) vi ( k  I c  I s , t ) 
c , s 1
c ,s i
  t  p0i ij k  I i , t r0i (k  I i , t )   i (t ) i k  I i , t u k i (t ) Ri 0 k  I i , t .
Expressions for conditional probabilities
i k , t  , ij k , t  ,  i k , t  , ij k , t  , i, j  1, n ,
originate from (1) – (5).
For a closed networks system of equations (7) can be simplified. In this case, we can
reduce it to a finite number of linear non-uniform ordinary differential equations (ODEs) with
variable coefficients, which in matrix form can be written as
dVi (t )
 Qi (t )  At Vi (t ),
dt
(8)
96
P. Zając, M. Matalytski
where Vi (t )  vi (1, t ), vi (2, t ), ..., vi ( L, t )  – vector of unknown volume of requests of
T
the system S i ,
L is the number of labeled states in the network.
Table 1
Possible transitions between states of the network, their probabilities and expected volume of
requests in the system S i
expected volume of
possible transitions between
transition probabilities
requests in the system Si
states of the network
n
( k , t )  ( k , t  t )
1    t  p0i 1   i k , t  
i 1
 i t  1  ii k , t t  o(t )
( k , t )  ( k  I j , t  t ) ,
ji
 t  p0 j ij k  I j , t t  o( t )
( k , t )  ( k  I j , t  t ) ,
ji
 j (t ) j k  I j , t u k j (t ) t 
(k , t )  (k  I c  I s , t  t ) ,
c, s  i
 s (t )  sc k  I c  I s , t  
(k , t )  (k  I i , t  t )
(k , t )  (k  I i , t  t )
( k , t )  ( k  I i  I j , t  t ) ,
ji
( k , t )  ( k  I i  I j , t  t ) ,
ji
 o ( t )
ri (k )t  vi (k , t )
ri ( k )t  vi (k  I j , t )
ri ( k ) t  vi ( k  I j , t )
ri (k )t  vi (k  I c  I s , t )
 u k s (t ) t  o(t )
 t  p0 i ij k  I i , t t 
r0i (k  I i , t ) 
 o(t )
 vi (k  I i , t )
i (t ) i k  I i , t uki (t ) t 
 Ri 0 k  I i , t  
 o(t )
 vi (k  I i , t )
 j (t )  ji k  I i  I j , t 
rij (k  I i  I j , t ) 
 vi ( k  I i  I j , t )
 u k j (t ) t  o(t )
 i (t )  ij k  I i  I j , t 
 u k i (t ) t  o ( t )
 rji ( k  I i  I j , t ) 
 vi ( k  I i  I j , t )
The solution of (7) provided in the form of a matrix can also be prepared by the method of
Laplace transformation. Let U i (s) , Gi (s ) , i (s) – Laplace transformation vectors the
vector function properly: vi ( j, t ) , Qi (t ) , Ai (t ) , i  1, L .
Next Ui s   Vi 0  Gi s   fi i s ,Ui s  . Solving this equation with respect to U i (s) ,
we get:
U i ( s)  Fi (Gi ( s),  i ( s)), i  1, n.
Finding of Expected Volumes of Requests in Systems of the Queueing Network…
97
Taking the inverse Laplace transform on both sides of the equation (8), we can find
functions vi ( j, t ) , i  1, L .
For
a
closed
network,
ie.
if
 (t )  0, p0i  pi 0  0, r0i (k i , t )  Ri 0 (k i , t )  0,
n
 k (t )  K , i  1, n , then we have from (7)
i
i 1
dvi (k , t )
 ri (k ) 
dt
n
n
  (t )(1  
i
ii ( k i , t ))v i ( k , t )
i 1


 j (k  I j , t )u (k j (t ))vi (k  I j , t ) 
j (t )
j 1
n
   j (t )  ji (k  I i  I j , t )u (k j (t ))vi (k  I i  I j , t ) 
j 1
j i
  i (t )  ij ( k  I i  I j , t )u ( ki (t ))vi ( k  I i  I j , t ) 
n
  [  j (t )  ji ( k  I i  I j , t )u ( k j (t )) rij ( k  I i  I j , t ) 
j 1
j i
  i (t )  ij ( k  I i  I j )u ( k i (t ))r ji ( k  I i  I j , t )] 
(9)
n
   s (t )  sc ( k  I c  I s , t )u (k s (t ))vi (k  I c  I s , t ), i  1, n.
c , s 1
c ,si
4. Solution of DDE for the closed network of queues with a limited
amount of spaces of expectations of homogeneous requests in
queueing systems
Consider the closed queueing network with a limited length of the request for the system
S i , equal Li , i  1, n . It is assumed that the notification coming in the system S i to service,
takes place in the queue, i  1, n . If on arrival request into the system Li
the number of
requests is less than S i , then the incoming request takes place in the queue, otherwise it is
immediately operated and goes according to the matrix switch P || p ij || nn by other system:
1
f k(i i ) (t )  
0
if
0  k i (t )  Li ,
if
k i (t )  Li .
(10)
For the closed network  ij (k , t )  conditional probability that the request occupy the
i  th system in to a time t , where network is in the state (k , t ) immediately after the time of
its transfer to the system (Ie, without the transfer of the last report), then will operate in j  th
n
system (  k i (t )  K  1, i, j  1, n );  ij (k , t )  conditional probability that the request handled
i 1
98
P. Zając, M. Matalytski
in i  th system at the time of t , where network is in the state of
(k , t ) immediately before
n
the end of its operation, will first serve in j  th queueing system,
 k (k )  K , i, j  1, n.
i
i 1
According to the formula for the probability of complete:
n
n
i 1
i 1
 ij (k , t )  f ( i ) (k , t ) ij  (1  f ( i ) (k , t ) pil lj (k , t )),  ki (t )  K  1, j  1, n
,
(11)
n
n
l 1
i 1
(12)
 k (t )  K  1,
(13)
 ij (k , t )   pil lj (k  I i , t ),  ki (t )  K , j  1, n,
So
n

n
ij
(k , t )  1,
j 1
i
i 1
n
n
  ij (k , t )  1,  ki (t )  K .
j 1
(14)
i 1
n
In a closed network must be: if Li  K , i  1, n , then  Li  K , the number of requests
i 1
in the network should not be larger than the maximum number of requests that can be
accepted by all queueing systems [Gordon-Newell].
Example
1.
Consider
a
n  2, mi  1, i  1,2, L1  1, L2  2, K  3 .
Let
closed
network
f ( i ) (k , t )  f ( i ) (k i , t ), i  1, n ,
parameters
that
is
probability of join the queue requests in queueing system depends only on the number of
requests in system. The number of states is the network is L=3. Relationship (3,0) system
network is unavailable because the total number of requests is not exceed L1  m1  2 . We
number them in order of {(0,3), (1,2), ( 2,1)} . The graphic interpretation of queueing network
shown in Fig. 1.
p1,2  1
S2
S1
p 2 ,1  1
Fig. 1. Graphical representation of the network
Rys. 1. Graficzna reprezentacja sieci
From the relationship (11) where k1 (t )  k 2 (t )  K  1 , we receive the system
Finding of Expected Volumes of Requests in Systems of the Queueing Network…
(15)
(16)
(17)
(18)
  11 ( k , t )  f (1) (k1 , t )  (1  f (1) ( k1 , t )) 21 (k , t ),

(1)
  12 (k , t )  (1  f ( k1 , t )) 22 ( k , t )

(2)
  21 ( k , t )  (1  f (k 2 , t )) 21 ( k , t )
 ( k , t )  f ( 2 ) ( k , t )  (1  f ( 2 )( k , t )) ( k , t ) .
2
2
12
 22
After substituting (17) to (15) we get
 12 ( k , t )  f (1) ( k1 , t )  (1  f
(1)
( k1 , t ))(1  f
( 2)
( k 2 , t )) 11 ( k , t ),
so
 11 ( k , t )[1  (1  f (1) ( k1 , t ))(1  f ( 2 ) ( k 2 , t ))]  f (1) ( k1 , t ),
therefore it follows that
 11 (k , t ) 
f
1  (1  f
(1)
(1)
( k1 , t )
( 2)
(k1 , t ))(1  f
(k 2 , t ))
,
and from equation (17) results
 21 (k , t ) 
(1  f ( 2 ) (k 2 , t )) f
1  (1  f
(1)
(1)
( k1 , t )
.
(k1 , t ))(1  f
( 2)
( k 2 , t ))
(1  f
(1)
(k1 , t )) f
Then, from (13):
 12 (k , t )  1   11 (k , t ) 
 22 (k , t )  1   21 (k , t ) 
1  (1  f
(1)
(2)
(k 2 , t )
(k1 , t ))(1  f
(2)
(k 2 , t ))
,
f ( 2) ( k 2 , t )
.
1  (1  f (1) (k1 , t ))(1  f ( 2) (k 2 , t ))
For our queueing network conditional probabilities take the form of
 11 (0,2, t ) 
 11 (1,1, t ) 
f (1) (0, t )
1  (1  f (1) (0, t ))(1  f
f (1) (1, t )
1  (1  f (1) (1, t ))(1  f
( 2)
( 2)
(2, t ))
(1, t ))
 1,
 0,
 11 (2,0, t ) 
f (1) (2, t )
1  (1  f (1) (2, t ))(1  f
 21 (0,2, t ) 
(1  f (1) (2, t )) f ( 2 ) (0, t )
 1,
1  (1  f (1) (0, t ))(1  f ( 2 ) (2, t ))
 21 (1,1, t ) 
( 2)
(0, t ))
 0,
(1  f (1) (1, t )) f ( 2) (1, t )
 0,
1  (1  f (1) (1, t ))(1  f ( 2 ) (1, t ))
 21 (2,0, t ) 
99
(1  f ( 2) (0, t )) f (1) (2, t )
0.
1  (1  f (1) (2, t ))(1  f ( 2 ) (0, t ))
100
P. Zając, M. Matalytski
If k1 (t )  k 2 (t )  K , then with (12):
11 (k , t )   21 (k  I1 , t ) 
(1  f ( 2 ) (k 2 , t )) f (1) (k1  1, t )
,
1  (1  f (1) (k1  1, t ))(1  f ( 2) (k 2 , t ))
12 (k , t )  1  11 (k , t ) ,  22 (k , t )  1   21 (k , t ) ,
 21 (k , t )   11 (k  I 2 , t ) 
f (1) (k1 , t )
.
1  (1  f (1) (k1 , t ))(1  f ( 2 ) (k 2  1, t ))
conditional probabilities  ij ( k , t ), i, j  1, n , have the form:
11 (0,3, t )  0 , 12 (0,3, t )  1
11 (1,2, t ) 
(1  f ( 2) (2, t )) f (1) (0, t )
(1  0) 1

 1,
(1)
( 2)
1  (1  f (0, t ))(1  f (2, t )) 1  (1  1)  (1  0)
(1  f ( 2 ) (1, t )) f (1) (1, t )
11 (2,1, t ) 
 0,
1  (1  f (1) (1, t ))(1  f ( 2 ) (1, t ))
(1  f ( 2 ) (2, t )) f (1) (0, t )
12 (1,2, t )  1 
 0,
1  (1  f (1) (0, t ))(1  f ( 2 ) (2, t ))
12 (2,1, t )  1 
(1  f ( 2 ) (1, t )) f (1) (1, t )
 1,
1  (1  f (1) (1, t ))(1  f ( 2 ) (1, t ))
 21 (0,3, t ) 
f (1) (0, t )
 1,
1  (1  f (1) (0, t ))(1  f ( 2 ) (2, t ))
 21 (1,2, t ) 
f (1) (1, t )
 0,
1  (1  f (1) (1, t ))(1  f ( 2 ) (1, t ))
f (1) ( 2, t )
 21 (2,1, t ) 
 0,
1  (1  f (1) (2, t ))(1  f ( 2 ) (0, t ))
f (1) (0, t )
 22 (0,3, t )  1 
 0,
1  (1  f (1) (0, t ))(1  f ( 2 ) (2, t ))
f (1) (1, t )
 22 (1,2, t )  1 
 1,
1  (1  f (1) (1, t ))(1  f ( 2 ) (1, t ))
 22 (2,1, t )  1 
f (1) (2, t )
 1.
1  (1  f (1) (2, t ))(1  f ( 2 ) (0, t ))
From the definition of conditional probability  ij ( k , t ) we assume further that
12 (0,3, t )   21 (3,0, t )  12 (3,0, t )  11 (3,0, t )   22 (3,0, t )  0 ,  ii (k , t )  0 , i  1,2 .
Finding of Expected Volumes of Requests in Systems of the Queueing Network…
101
As the volumes of requests ri (k )  4 bytes used by the amount equal to the volume of
data that represent the statistical information about the specific operating range of user (Ie.
incoming packet data user), as well as route information packet network information (eg
address of the recipient), i  1,2 . From a practical point of view, IPv4 uses 32-bit (4-byte)
address limiting, as you know, address space 2 32 possible unique addresses.
Let rij ( k , t ) – the volume of user data packets already received from one WAP and
transmitted from the sender to the recipient i, j  1,2 , i  j . Typically the volume varies in the
range of 20 bytes (in 5  32  160 bits) to 65535 bytes depends on the volume and the user data
generated in the data packet. In this case the volumes rij ( k , t ) are linear for t independent of the
state of k functions 20 t  a bytes. Let the intensity of use requests in the queueing network
system will be  i (t )  cos(at )  1 , i  1,2 . The initial conditions are
v1 (0,3,0)  0, v1 (1,2,0)  20, v1 (2,1,0)  40, v2 (0,3,0)  60, v2 (1,2,0)  40,
v2 (2,1,0)  20. Then, equations (9) has the form:
dv1 (0,3, t )
 2(cos(at )  1)v1 (0,3, t )  4 ,
dt
(19)
dv1 (1,2, t )
 2(cos(at )  1)v1 (1,2, t )  4 ,
dt
dv1 (2,1, t )
 2(cos(at )  1)v1 (2,1, t )  4 ,
dt
(20)
(21)
dv2 (0,3, t )
 2(cos(at )  1)v2 (0,3, t )  4 ,
dt
(22)
dv2 (1,2, t )
 2(cos t  1)v2 (1,2, t )  2(cos t  1)v2 (2,1, t ) 
dt
 2(cos t  1)v2 (0,3, t )  (cos t  1)r21 (t )  (cos t  1)r12 (0,3, t )  4,
(23)
,
dv2 (2,1, t )
 2(cos t  1)v2 (2,1, t )  4.
dt
(24)
As you know, the general solution of a linear equation
dv(k , t )
  p(t )v( k , t )  q (t ), v(k ,0)  v0 (k ) ,
dt
has the form:
t

 p ( x ) dx
v( k , t )  e
0
x
t
[v0 ( k )   q ( x)e
0

 p ( ) d
0
dx ] .
(25)
102
P. Zając, M. Matalytski
Therefore, the solution of equations (19) - (22) will have the form:
t
v1 (0,3, t )  4e 2 ( t sin t )  e 2 ( x sin x ) dx ,
(26)
0
t
v1 (1,2, t )  4e
2 ( t sin t )
(5   e 2 ( x sin x ) dx)  v2 (2,1, t ) ,
(27)
0
t
v1 ( 2,1, t )  4e
2 ( t sin t )
(10   e 2 ( x sin x ) dx )
(28)
0
t
v1 (0,3, t )  4e
2 ( t sin t )
(15   e 2 ( x sin x ) dx)
(29)
0
Substitute (27), (29), (23) and find a solution to this equation:
t
2
v2 (1,2, t )  e (t  sin t )(40   e
0
t
t
2 ( x sin x )
( 4  8e
2 ( t sin t )
(1  cos x)(5   e 2 ( l sin l ) dl ) 
0
x
 8e 2 ( xsin x ) (1  cos x )(5   e 2 ( l sin l ) dl )  8e 2 ( xsin x ) (1  cos x)(15   e 2 ( z sin z ) dz ))dx).
0
0
Numerical solutions of differential equations is obtained using a mathematical package
Mathematica, using the operator NDSolve, which by default performs calculations using the
method of Runge-Kutta fourth order. Charts for volumes of requests for network systems
obtained are given in fig. 2 and 3.
Fig. 2. The volume of requests in the system S1 for T = [0.10]
Rys. 2. Liczba żądań w systemie S1 dla T = [0.10]
Finding of Expected Volumes of Requests in Systems of the Queueing Network…
103
Fig. 3. The volume of requests in the system S2 for T = [0.10]
Rys. 3. Liczba żądań w systemie S2 dla T = [0.10]
BIBLIOGRAPHY
1.
Tikhonenko O.: Metody probalistyczne analizy systemów informacyjnych. Akademicka
Oficyna Wydawnicza EXIT, Warszawa 2006.
2.
Matalytski М, Tikhonenko O., Koluzaeva E.: Systems and queueing networks: analysis
and application. GrSU, Grodno 2011 (in Russian).
3.
Crabill T.: Optimal Control of a Service Facility with Variable Exponential Service
Times and Constant Arrival Rate. Management Science, V. 18 (9), 1972, p. 560÷566.
4.
Foschini G.: On heavy traffic diffusion analysis and dynamic routing in packet switched
networks. Computer Performance. 10 (1977), p. 499÷514.
5.
Stidham S., Weber R.: A survey of Markov decision models for control of networks of
queue. Queueing Systems,. V.13 (3). 1993, p. 291÷314.
6.
Matalytski M., Pankov A.: Analysis of the stochastic model of the changing of incomes
in the open banking network. Computer Science, V. 3 (5), 2003, p. 19÷29.
7.
Matalytski M.: On some results in analysis and optimization of Markov networks with
incomes and their applications. Automation and Remote Control, V. 70 (10), 2009,
p. 1683÷1697.
8.
Matalytski M., Naumenko V.: Zastosowanie HM-sieci kolejkowych dla wyznaczenia
objętości pamięci systemów informacyjnych. Studia Informatica, V. 35 (3), 2014,
p. 63÷69.
104
P. Zając, M. Matalytski
Omówienie
W niniejszym artykule przedstawiono metodę znalezienia oczekiwanej objętości zgłoszeń
jednorodnych w otwartej HM-sieci z obejściami węzłów systemów obsługi. Rozpatrywano
przypadek, gdy zmiany objętości związanych z przejściami między stanami sieci są
deterministycznymi funkcjami, zależnymi od stanów sieci i czasu, natomiast systemy obsługi
są jednoliniowe, przy założeniu że prawdopodobieństwo stanów systemów sieci, parametry
strumienia wejściowego zgłoszeń i obsługi zależą od czasu.
Wyniki te można zastosować w celu znajdowania objętości pamięci w systemach
informacyjnych. Przedstawiono model działania bezprzewodowej sieci komputerowej
z ograniczoną liczbą równoczesnych połączeń. Dokonano analizy tej sieci z jednorodnymi
zgłoszeniami, obejściami węzłów obsługi i parametrami strumienia zgłoszeń, zależnymi od
czasu. Prawdopodobieństwa warunkowe dla obsługi zgłoszeń wyrażają się wzorami (1)-(6).
W tabeli 1 znajdują się możliwe przejścia między stanami sieci, ich prawdopodobieństwa
i oczekiwane objętości zgłoszeń systemów sieci.
Otrzymano
układ
równań różniczkowych dla oczekiwanej objętości zgłoszeń
jednorodnych w systemach sieci z ograniczoną liczbą miejsc oczekiwania w kolejkach
systemów i zaprezentowano przykład dla rozwiązania tego układu dla sieci, pokazanej na
rysunku 2, w przypadku gdy prawdopodobieństwa przyłączenia do kolejek zgłoszeń
w systemach obsługi zależą tylko od liczby zgłoszeń w nich. Wykresy oczekiwanych
objętości zgłoszeń dla systemów sieci znajdują się na rysunkach 3 i 4.
Addresses
Paweł ZAJĄC: Czestochowa University of Technology, Institute of Mathematics
al. Armii Krajowej 21, 42-201 Częstochowa, Poland, [email protected].
Mikhail MATALYTSKI: Czestochowa University of Technology, Institute of Mathematics
al. Armii Krajowej 21, 42-201 Częstochowa, Poland, [email protected].