Semi-Markovian Model of Two-Line Queuing System with Losses

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.


Introduction
A large number of works, in particular [1]- [5], have been dedicated to the queuing systems (QS) with losses.Building of QS models and determining their characteristics are simplified, if it is assumed that the incoming flow of requests or their service times are exponentially distributed.The rejection of this assumption leads to a considerable complication of the models.In this paper, the model of two-line QS with losses was built on the assumption that the incoming flow of requests and their service times have distributions of general form.For building QS model and determining its stationary characteristics, the theory of semi-Markov processes with arbitrary phase state space [5]- [10] was used.time between their arrival is a random variable (RV) β with the distribution function (DF) ( ) { } G x P x β = ≤ .A received request, with equal probability, starts to be served by one of the available servers or gets lost, if no servers are available.The service time of request by the th i server-RV i α with DF . It is assumed that RV i α , β are independent, and have densities ( ) i f x , ( ) g x , finite mathematical expectations and variances.
To describe the QS operation, the semi-Markov process [5]- [7] ( ) with the following set of states is used: { } .
The meaning of state codes is the following: ( ) 10 01 : first (second) server started serving the received request, and second (first) server is available; ( ) We define the transition probabilities of the embedded Markov chain (EMC) { } for states ) ( )

Definition of the Stationary Distribution of the Embedded Markov Chain
We  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 ( ) ( ) ( ) . Then, due to the symmetry of states, we get that ( ) ( ) z z ϕ ϕ = .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: ( ) ( ) ( ) -is the density of the renewal function [11] ( ) g H t of the renewal process generated by -is the density of the direct residual time distribution [11] for the renewal process generated by RV β ; ( -is the density of the renewal function, renewal process generated by RV with the distribution density ( )

n n n n h t g t h g t x t x t x y t h y y x y x y x y x t t y t n x t x t x f x t h t t x y y f y t h t t
Using the method of successive approximations [12], we can show that the system (4) has the following solution: The constant 0 ρ is found by means of normalization requirement; its explicit form is not used when finding the QS stationary characteristics.The system of equations, which is almost identical to the system (3), and its solution method are covered in [13].

Definition of Stationary Characteristics of System
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: -all servers are available; { } -two servers are in service; We will introduce the transition probabilities of the semi-Markov processes ( )
We will show that the stationary probabilities of QS / / 2 / 0 GI G are defined by the following formulas:

P G t F t t M y f y y M F y y y C f y y V y z z y z y y V y z F z y z P F t G t t f y y x y V y x x y y F x y V y x x C
where

x t h t t x y y f y t h t t x t t F x y h y y h t t x y f y t y x y t y t
-is the renewal function [11]; ( ) , g V y x -is DF of the direct residual time [11]; -is the mathematical expectation of the direct residual time [11].The proof.As is known [5] [6], the following equalities are true: where ( ) m e -is the average sojourn time of SMP ( ) in state e E ∈ ; ( ) 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:

z g z t f t t z z t t y f y v y z t y z z t y f t y v y z t y G t F t t M y f y y M F y y y f y y V y z z y z y y V y z F z y z
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: x x -a received request was lost.We will find ( ) Therefore, the stationary probability of request loss equals: Important characteristics of the QS under consideration are average stationary sojourn times ( ) i T E of the system in the selected subsets of states , 0, To determine them we will use Formulas [5] [6]: Let us find the values of the expressions in the denominators of Formulas (14).
g E E z g z P e E e x g t f x t t z x x y f y y z y z x y f x y y z y 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:

Particular Cases of QS GI/G/2/0
Let us look at particular cases of QS GI/G/2/0.1) We find the stationary characteristics of QS / / 2 / 0 M M .In this case, Using Formulas ( 8), ( 13), (19), we obtain: 2) Let us examine QS M/G/2/0, ( ) e t g t . The direct substitution into the system (4) can show that the stationary distribution of EMC is determined by the formulas: (  Using Formulas (8), ( 13), (19), we obtain that the stationary characteristics of QS M/G/2/0 are written as: 2 0 2 Mα = Thus, in this case, as shown in [4], stationary probabilities , 0, 2 i P i = are invariant under the laws of distribution of service time.

x t f y x t h y y x f x t h t t x y y f y t h t t x
The semi-Markov model of QS / / 2 / 0 GI M is considered in [14].
In the paper [5], a similar approach to the building of QS model under consideration is used.To find the stationary distribution of EMC, a method based on the usage of taboo-probabilities is applied.
In monograph [13], the semi-Markov model of QS / / / 0 M G N  is considered and stationary characteristics are defined.
200 z z : first (second) server became available; second (first) server is available; 0 z > is the time un- til the arrival of the next request; (second) server started serving the received request; 0 x > is the time until the end of the request service by second (first) server; (second) server became available; 0 x > is the time until the end of the request service by second (first) server; 0 z > is the time until the arrival of the next request; 1 2311x x : the received request was lost; the times until the end of the request service by first (second) servers are respectively of the system is shown in Figure1.Let us define the sojourn times in states of the system.For instance, the sojourn time 211xθ , in the state 211x is determined by three factors: the time x left until the end of request service by the first server, the time 2 α of request service by the second server, and the time β between the request arrivals., where ∧ is the minimum sign.Similarly, the sojourn times in other states are determined as follows:

Figure 1 .
Figure 1.The time diagram of the system functioning.

Functions ( 5 )
in this case are as follows: will find the stationary distribution of EMC { }