# 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 *S*_{2 }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.