Stationary Characteristics of the Single-Server Queue System with Losses and Immediate Service Quality Control

Semi-Markovian model of operation of a single-server queue system with losses and immediate service quality control has been built. In case of unsatisfactory request service quality, its re-servicing is carried out. Re-servicing is executed till it is regarded satisfactory. Time between request income, and request service time are assumed to be random values with distribution functions of general kind. An explicit form of the system stationary characteristics has been defined.


Introduction
A large number of works [1][2][3][4] are dedicated to the queue systems with losses.In most of them the incoming flux of requests is supposed to be Poisson one or request service time is considered to have exponential distribution.This admission allows efficient modeling of the system operation by means of Markovian processes.But if the incoming flux of requests makes a renewal process and request service time distribution is of general kind, essential difficulties arise when defining system stationary characteristics in explicit form.To overcome them an apparatus of semi-Markovian processes with a common phase field of states is used in the works [5,6].Thus, in [6] stationary characteristics of the single-server queue system 1 0 GI G with losses were found.In the present article analogical characteristics for the same system have been defined under the assumption that the server admits both satisfactory and unsatisfactory request service.In case of the latter one the re-servicing begins immediately.It is repeated until the service is regarded satisfactory.
In the second chapter of the article the system operation is described, mathematical problem definition and research purpose are stated.In the third section semi-Markovian model of system operation is built, and stationary distribution of embedded Markovian chain is given.In the fourth section system stationary characteristics are defined.These are: final probabilities that the server is free, is in state of service or re-servicing, and mean dwelling times in these states.Besides, formulas of the system stationary characteristics for some subcases are given here.One of them is the exponential distribution of time between request income and request service time.

The Problem Definition
Let us investigate the queue system (QS) 1 0 GI G with losses and a single server.Request service time is a random value (RV)  with an absolutely continuous dis-    respectively.It is necessary to define the following stationary characteristics of the system: the final probabilities that the queue system is in a standby state; that the system is busy with the primary service or re-servicing; mean time periods of system's dwelling in the above-mentioned states.

Semi-Markovian Model Building
In order to build the model of the system operation an apparatus of semi-Markovian processes with a discretecontinuous phase field of states [6,7]  Let us write out the codes of the states: 21 -service of the request that has come begins; 10x -request service has been successfully completed, the server passes into a standby state that lasts time x (till the next request income); 21x -the incoming request has been lost, the server is busy with the primary service that will last time x; 22x -the incoming request has been lost, the server is busy with re-servicing that will last time x; 12x -request service has been completed, and its re-servicing has begun, time x is left till the next request income.
In Figure 1 time diagram of the system operation is shown, and in Figure 2 there is the system transition graph.
System dwelling times in the states are defined by the formulas: , , , Now we can proceed to EMC stationary distribution definition, the system of integral equations for its definition is the following one: x g y x y y g y x f y y x g y x y y y x y y x q y x y y q f y x g y y q g y x y y y x p y x y y p f y x g y y p g y x y y y Let us introduce the following integral operators:  Then the system of Equations ( 1) can be rewritten in such a way-Equation ( 2).
We shall exclude   Here With regard to the introduced operators the Equation (3) will have the form Let us write down kernels of integral operators K  , f K in explicit forms and single out their probability sense q y x q t y v t x t y x k x y q t y v t x t y x  As it is not difficult to ensure that the operator K  is the operator of contraction in the space of summable functions.That is why the solution of Equation ( 4) can be found by means of successive approximations.This solution with regard to the identity where is the density of RV that is time period between the end of primary request service and the moment of the next request income.The is the density of random time period between the end of the   request re-servicing and the moment of the next request income.Here is the density of the direct residual time for the renewal process generated by the functions Let us note that the density of renewal function complies with the ratio The rest of EMC stationary distributions are defined by the formulas x y l y y q y x y s l s s The stationary probability 21  of EMC's dwelling in the state 21 is found with the help of normalization requirement:

Definition of System Stationary Characteristics
Mean values of system dwelling times in the states are defined by the formulas Let us consider the following disjoint subsets of system states: . Dwelling in the state 0 E means that the server is free and the system is in a standby mode.Dwelling in the state 1 E or 2 E signifies that the server is busy with the primary request service or with re-servicing respectively.
Let us introduce SMP   t  transition probabilities: .
As it is known [9], under the condition of the unique stationary distribution existence the following ratios take place where   m e is mean SMP   t  dwelling time in the state e E  .
With the help of Formulas ( 5)-( 7) integrals contained in the Formula (8) are converted into:

G t t x G t t h y x f y y M
m e e q l x x t t dx x y l y y G t t q x G t t h y y x y s l s s M p Consequently, the final probability that the server is free is defined by the formula and the final probabilities p 1 and p 2 that the server is busy either with the primary service or with the re-servicing are: It is necessary to note that the ratio determines an average value of requests lost per time unit of a complete request service.
Let us define mean stationary dwelling times   i T E in the extracted subsets , 02 i E i  . .According to [7] we have ) One can define the values of expressions in denominators of Formulas (12).To do it let us take integrals of both parts of the system (1) equations.The result is Consequently, mean system dwelling times in the extracted subsets are defined by the formulas .
It is necessary to note that in case if with the probability equal to 1 satisfactory request service is carried out the system characteristics defined by the Formulas ( 9), (10) and ( 13) coincide with ones found in [6].
To illustrate some subcases of the results gained let us write down QS 1 0, 1 0, The system The system The Formulas ( 9)-( 11) and (13) for defining system stationary characteristics convert into The system 1 0 GI M stationary characteristics.For this system the incoming flux of requests is generated by RV  with density   g t of a general kind, and Let us find the functionals contained in the formulas for defining stationary characteristics.We have This solution obeys the condition Consequently, system stationary characteristics are defined by the following ratios

Conclusions
In the present work semi-Markovian model of the operation of the single-server queue system 1 0 GI G with losses in which unsatisfactory request service quality is admitted has been built.Service quality control is carried out immediately, and in case of unsatisfactory service quality re-servicing is executed until the service is regarded as satisfactory.With the help of this model stationary characteristics in explicit form have been defined.These are: the final probabilities of system's dwelling in a standby state, in the states of primary service and re-servicing, and, besides, mean stationary dwelling times in these states.These characteristics depend on mean time periods between requests' income, mean service time, average value of requests lost per time unit of a complete request service and the probability of unsatisfactory service.In case of satisfactory service only the characteristics found coincide with the formerly defined ones.

Figure 1 .
Figure 1.Time diagram of the system operation.
and second equations of the system (2) respectively, and then substitute it in the third equation.The result is the request is sent to re-servicing procedure.This process lasts until the service is considered sufficient.It is assumed that RV , f t .Time period between requests' income is a RV  with an absolutely continuous DF     t  .After re-servicing with the probability p the service is regarded satisfactory and with the probability q is used.Let us describe the system operation with the help of semi-Marko-