About one method of finding expected incomes in HM-queueing network with positive customers and signals
Mikhail Matalytski
Journal of Applied Mathematics and Computational Mechanics |
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ABOUT ONE METHOD OF FINDING EXPECTED INCOMES IN HM-QUEUEING NETWORK WITH POSITIVE CUSTOMERS AND SIGNALS
Mikhail Matalytski
Institute of Mathematics, Czestochowa
University of Technology
Częstochowa, Poland
m.matalytski@gmail.com
Abstract. In the paper an open Markov HM(Howard-Matalytski)-Queueing Network (QN) with incomes, positive customers and signals (G(Gelenbe)-QN with signals) is investigated. The case is researched, when incomes from the transitions between the states of the network are random variables (RV) with given mean values. In the main part of the paper a description is given of G-network with signals and incomes, all kinds of transition probabilities and incomes from the transitions between the states of the network. The method of finding expected incomes of the researched network was proposed, which is based on using of found approximate and exact expressions for the mean values of random incomes. The variances of incomes of queueing systems (QS) was also found. A calculation example, which illustrates the differences of expected incomes of HM-networks with negative customers and QN without them and also with signals, has been given. The practical significance of these results consist of that they can be used at forecasting incomes in computer systems and networks (CSN) taking into account virus penetration into it and also at load control in such networks.
Keywords: HM-queueing network, positive and negative customers, signals, expected incomes, transient regime
1. Introduction
For the first time Markov nets with positive customers and signals were introduced and investigated at the non-stationary behavior by E. Gelenbe, see [1]. The action of the signal consists of instantaneous movement of positive customers of this system to some other network system. The signal may work as a trigger, which doesn’t destroy the customers, but only moves them instantly with a given probability of a given system to another network system.
In developing models of computer viruses we can use negative customers. And for load control in the network can be inputted signals (triggers). When viruses penetrate into computers in the information system, it suffers costs or losses due to the loss of information or CSN distortion. The accounting of the losses in CSN can be realized with the help of a Markov QN model with incomes (HM-networks), positive and negative customers and signals.
In this paper, an open Markov HM-network with positive and negative customers, signals, when the incomes from the transitions between the states of the network are random variables (RV) with time-dependent customer servicing in the systems has been carried out. The expressions for the variances of incomes of queueing systems (QS) was also obtained.
A description of the network is given in [2, 3].
Signal, coming in an empty
system (in which there are no positive
customers), does not have any impact
on the network and immediately disappeared from it.
Otherwise, if the system
is not empty, when it receives a signal, the following events may
occur: incoming signal instantly moves the positive customer from the system
into the system
with probability
in this case, signal is referred
to as a trigger; or with
probability
signal is triggered by a
negative customer and destroys in QS
positive customer. The state
of the network meaning the vector
where
- the number of customers
at the moment
of time
at the system
,
.
2. Description of incomes from the transitions between the states of the network
Let -
time of customers service in the system
with
the distribution function (DF)
,
. Consider the dynamics of income
changes of
a network system
,
. Let at the initial moment of time the income of
this QS be equal to
. We are interested
in income
at time t. The income of its QS
at moment time
can be represented in the
form
, where
-
income changes of the system
at the time interval
,
.
To find the income of the system we write the conditional probabilities
of the events that may occur during
. The following cases
are possible:
1) with
probability to the system
from the external environment
a positive customer will arrive, which will bring an income in the amount of
, where
- RV with expectation (E) of
which equals
,
;
2) with probability in the QS
from the external environment
a signal will arrive, which will bring an income (loss) in the amount of signal
is triggered by a negative customer and destroys in QS
positive customer, which will bring a
loss in the amount of
,
where
- RV with E
,
;
3) the incoming
signal instantly moves the positive customer from the system into the system
probability
of this event equals
,
by this transition the income of
is reduced by the amount
, and income of
is
increased by this amount, where,
- RV with E
,
;
4) with probability a positive customer will depart from
the network to the external environment, while the total amount of income of QS
is reduced by an amount which is equal
to
, where
- RV with E
,
;
5) with probability a customer from the
system
transit to the system
as a signal, if in it there were no
customers, and the income of
is reduced by the
amount
, where
- RV with E
,
;
6) a customer from
the QS transit to the system
with probability
,
by such a transition the income of
system
is reduced by the amount
and the income of system
is
increased by this amount,
,
,
7) with probability positive customer transit from the
system
to the system
, wherein the income of the QS
will increase
by the value of
, and the income of
is reduced by this amount,
,
,
8) after
finishing servicing of a positive customer in QS it
is sent to
as
a signal, which is triggered by a negative customer and destroyed in QS
positive customer; the probability of
this event equals
; wherein the income of the QS
will reduce by value
, and the income of
is reduced also by the amount,
,
,
9) after finishing
servicing of positive customer in QS , it is sent to
as
a signal, which instantly moves the
positive customer from the system
into the system
the probability of this
event equals
,
,
,
by such a
transition the income of system
and
reduced
by the amount
, and income of system
is increased by
this amount respectively, where
- RV with
the E
,
,
10) with probability on time interval
network state will not change;
11) for every small time interval system
because of customers’ presence
in it increases its income by the amount of
,
where
- RV with the E
,
.
3. Finding the expected incomes of the network systems
Income changes of the QS on interval
can
be written as:
![]() | (1) |
Let us find the expression for the expected
income of the system in time
suppose, all the network systems
operate under heavy-traffic regime, i.e.
Taking into account (1) for the E
or income changes we can write:
![]() |
Then, similarly as in [3], we obtain
![]() | (2) |
4. Finding variances of the incomes of the network systems
As in
[4], system income can be presented in form
, where
- count of partitions of the interval
by equal parts,
;
- income changes i-th
QS on l-th time interval,
,
. To calculate the variance of the
system income in the network, we introduce the following designations:
![]() |
Let us consider the square of the difference
![]() | (3) |
Let us find the expectations of summands in
the right side of the last equality. For
this we write support equalities considering that RV and functions of RV
,
,
,
,
,
,
,
,
pairwise independent from
,
. Then
![]() | (4) |
![]() | (5) |
![]() | (6) |
![]() | (7) |
![]() | (8) |
![]() | (9) |
![]() | (10) |
![]() | (11) |
Considering (4)-(11), we have:
![]() | (12) |
Insofar
as the values and
are independent at
using (12)
one can find
![]() | (13) |
Then, further, passing to the limit from (3),
(12), (13) and also, that
obtain
![]() | (14) |
Now let us find an expression for , using (2):
![]() | (15) |
In this way, the variance of income of i-th QS, considering (14), (15), can be written in the form
![]() | (16) |
5. Numerical example
As is
known (see [5]), relation for the expected income of the system in the case,
where there are no negative customers in the network and all systems operate in
a heavy-traffic regime, has the form:
![]() | (17) |
where: - input rate
of customers;
- probability of a
customer arriving to the system
,
,
;
- probability that the customer
which finished servicing in i-th QS move to j-th QS,
,
;
- probability of the
departure of a customer,
. Relation for the expected income of
the system
with negative customers but without
considering input signals, can be written as [6]:
![]() | (18) |
Let n = 10; input rates of
positive customers and signals and
respectively equal
,
,
. Service rates of customers
equal
,
,
,
,
,
,
,
. Let
us transition probabilities of positive customers
respectively
equal:
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, other equal zero. Probabilities, that
were serviced in
, move to
as
negative, equal:
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
other equal zero. Departure probabilities equal
,
. Probabilities of a signal
arriving, which are instantly moves the positive customer from the system
to the system
respectively equal:
,
,
,
,
,
,
,
,
,
,
,
,
,
,
other equal zero. Probabilities that signal is triggered as a negative customer
and destroys
in QS positive customer equal:
,
,
,
,
,
,
,
,
,
.
Let us set values for the required
expectations: ,
,
,
,
,
,
,
;
,
,
,
,
,
,
,
,
;
,
,
,
,
,
,
,
,
,
;
,
,
,
,
,
,
,
;
,
,
,
,
,
,
,
,
,
,
,
,
,
;
,
,
,
,
,
,
,
,
,
,
other equal zero.
Expectations
for the random system of incomes has been calculated. Their values have the form: ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
;
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
.
Let us suppose that income at the initial time equals
.
Consider the length of the time interval of 10 hours,
,
. Then using formulas (2), (17), (18)
analytical expressions have been found for the expected system
incomes of the networks.
In Figure 1, income changes is shown of the QS for HM-network with
negative customers and without them, and also with
signals. One can see that
the negative customers reduce
expected income of the system
Signals inputting also influence the income changes, reducing it.
Fig. 1. Income changes of the S2 QS, solid line - a case without negative customers, dashed - a case with negative customers, dotted - taking into account the signals
6. Conclusions
In this paper a method was proposed for finding expected incomes in HM-network systems with positive and negative customers and also with signals. Incomes from the transitions between the states of the network are RV with given mean values. This method is based on the using of found approximate and exact expressions for the mean values of the random incomes. An example was calculated. The expres- sions for the variances of incomes of QS was obtained. The obtained results can be used in modeling income changes in various CSN, the virus penetration into it, and also to load control in CSN.
References
[1] Gelenbe E., G-networks with triggered customer movement, Journal of Applied Probability 1993, 30, 742-748.
[2] Matalytski M., Naumenko V., Investigation of G-network with signals at transient behavior, Journal of Applied Mathematics and Computational Mechanics 2014, 13(1), 75-86.
[3] Naumenko V., Matalytski M., Finding the expected incomes in the Markov G-network with signals, Vestnik of GrSU. A Series of Mathematics, Physics, Computer Science, Computer Facilities and Management 2014, 2, 134-143.
[4] Naumenko V., Matalytski M., Investigation and application of G-networks with incomes and signals with random time activation Vestnik of GrSU. A Series of Mathematics, Physics, Computer Science, Computer Facilities and Management 2014, 3, 142-152.
[5] Matalytski M., On some results in analysis and optimization of Markov networks with incomes and their application, Automation and Remote Control 2009, 70(10), 1683-1697.
[6] Naumenko V., Matalytski M., Analysis of Markov net with incomes, positive and negative customers, Informatics 2014, 1, 5-14.