_{1}

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In the present paper, to build model of two-line queuing system with losses GI/G/2/0, the approach introduced by V.S. Korolyuk and A.F. Turbin, is used. It is based on application of the theory of semi-Markov processes with arbitrary phase space of states. This approach allows us to omit some restrictions. The stationary characteristics of the system have been defined, assuming that the incoming flow of requests and their service times have distributions of general form. The particular cases of the system were considered. The used approach can be useful for modeling systems of various purposes.

A large number of works, in particular [

Two-line QS with losses GI/G/2/0 is being considered. It is assumed that the system receives requests, and the time between their arrival is a random variable (RV)

To describe the QS operation, the semi-Markov process [

The meaning of state codes is the following:

The time diagram of the system is shown in

Let us define the sojourn times in states of the system. For instance, the sojourn time

Therefore,

We define the transition probabilities of the embedded Markov chain (EMC)

We will find the stationary distribution of EMC

Introduce the notations:

Using (2), set up a system of integral equations to determine the stationary distribution:

The last equation in the system (3) is the normalization requirement.

Next, for the sake of simplicity, a homogenous case is considered, and a inhomogeneous case leads to lengthy transformations and results. Let

The system (3) is reduced to the following system of equations:

Let us introduce the following functions, which are used to record the stationary distribution of EMC:

Using the method of successive approximations [

The constant

The system of equations, which is almost identical to the system (3), and its solution method are covered in [

Let us turn to the determination of the stationary characteristics of the QS. Using Formulas (1), we will define the average sojourn times in states of the system:

We divide the set of states E into three following subsets:

We will introduce the transition probabilities of the semi-Markov processes

and

We will show that the stationary probabilities of QS

where

The proof. As is known [

where

Let us calculate the integrals entering into the right side of equalities (10). Using (6), (7), we get:

In the transformations, the following formula was used:

By substituting the determined expressions in Formulas (10), we get Formulas (8).

Let us define the stationary probability of request loss. We will consider the subset of states:

We will find

Therefore, the stationary probability of request loss equals:

Important characteristics of the QS under consideration are average stationary sojourn times

Let us find the values of the expressions in the denominators of Formulas (14).

The transformations used the following formula:

which results from the first equation of the system (4),

In the derivations of equalities (17), (18) Formula (16) was used in the same way.

Having placed the determined values of the denominators into Formulas (14), we obtain:

Let us look at particular cases of QS GI/G/2/0.

1) We find the stationary characteristics of QS

Using Formulas (8), (13), (19), we obtain:

2) Let us examine QS M/G/2/0,

The direct substitution into the system (4) can show that the stationary distribution of EMC is determined by the formulas:

Functions (5) in this case are as follows:

Consequently,

Using Formulas (8), (13), (19), we obtain that the stationary characteristics of QS M/G/2/0 are written as:

Thus, in this case, as shown in [

The semi-Markov model of QS

In the paper [

In monograph [

Using built semi-Markov model, limiting theorems and Markov renewal equations [

Yuriy E. Obzherin, (2016) Semi-Markovian Model of Two-Line Queuing System with Losses. Intelligent Information Management,08,17-26. doi: 10.4236/iim.2016.82003