QUEUEING SYSTEMS WITH DELAY

Context. In the queuing theory of a research of the G/G/1 systems are relevant because it is impossible to receive decisions for the average waiting time in queue in a final form in case of arbitrary laws of distributions of an input flow and service time. Therefore, the study of such systems for particular cases of input distributions is important. The problem of deriving solutions for the average waiting time in a queue in closed form for systems with distributions shifted to the right from the zero point is considered. Objective. Getting solutions for the main characteristics of the systems – the average waiting time of requirements in the queue for queuing systems (QS) of type G/G/1 with shifted input distributions. Methods. To solve this problem, we used the classical method of spectral decomposition of the solution of the Lindley integral equation. This method allows to obtaining a solution for the average waiting time for two systems under consideration in a closed form. The method of spectral decomposition of the solution of the Lindley integral equation plays an important role in the theory of systems G/G/1. For the practical application of the results obtained, the well-known method of moments of probability theory is used. Results. For the first time, spectral expansions are obtained for the solution of the Lindley integral equation for systems with delay, which are used to derive formulas for the average waiting time in a queue in closed form. Conclusions. It is shown that in systems with delay, the average waiting time is less than in in the usual systems. The obtained formula for the average waiting time expands and complements the well-known queuing theory incomplete formula for the average waiting time for G/G/1 systems. This approach allows us to calculate the average latency for these systems in mathematical packages for a wide range of traffic parameters. In addition to the average waiting time, such an approach makes it possible to determine also moments of higher orders of waiting time. Given the fact that the packet delay variation (jitter) in telecommunications is defined as the spread of the waiting time from its average value, the jitter can be determined through the variance of the waiting time.


LIE is a Lindley integral equation;
QS is a queuing system; PDF is a probability distribution function. G is a arbitrary distribution law; H 2 is a hyperexponential distribution of the second order; 2 H − is a shifted hyperexponential distribution of the second order; 2 HE is a hypererlangian distribution of the second order; 2 HE − is a shifted hypererlangian distribution of the second order; M is a exponential distribution law; M − is a shifted exponential distribution law; W is a average waiting time in the queue; * ( ) W s is a Laplace transform of waiting time density function; λ is a Erlang distribution parameter for input flow; 1 2 , λ λ is a parameters of the hyperexponential (hyperelangian) distribution law of the input flow; μ is a Erlang distribution parameter for of service time; 1 2 , μ μ is a parameters of the hyperexponential (hyperelangian) distribution law of service time; ρ is a system load factor;

INTRODUCTION
In the study of G/G/1 systems, an important role is played by the method of spectral decomposition of the solution of the Lindley integral equation (LIE) and most of the results in the queueing theory are obtained using this method. The most accessible this method with specific examples is described in the classic queueing theory [1].
This article is devoted to the analysis of QS with delay, i.e. systems defined by a pair of input distributions shifted to the right of the zero point. In [2] for the first time the results on the study of the M/M/1 system with delay with exponential input distributions shifted to the left from the zero point, obtained by the classical method of spectral decomposition, are presented. Hereinafter, the superscript "-" will denote the operation of shifting the distribution law.
In [2], it is shown that the average waiting time of a queue in the M /M /1 HE − are of type G/G/1. In the queueing theory, the studies of G/G/ systems are relevant due to the fact that they are actively used in modern teletraffic theory, moreover, it is impossible to obtain solutions for such systems in the final form for the general case.
The object of study is the queueing systems type G/G/1.
The subject of study is the main characteristics of the systems -the average waiting time of requirements in the queue.
The purpose of the work is obtaining a solution for the average waiting time of requirements in the queue in closed form for the above-mentioned systems.

PROBLEM STATEMENT
The paper poses the problem of finding the solution of the average waiting time of claims in a queue in the queueing systems, formed by four distribution laws shifted to the right from the zero point: M − , 2 E − , 2 H − , 2 HE − . These four laws of distributions form 4x4=16 different QS G/G/1. When using the method of spectral decomposition of an LIE solution to determine the average waiting time, we will follow the approach and symbolism of the author of the classical queuing theory [1]. To solve the problem, it is necessary to find the law of waiting time distribution in the system through the spectral decomposition of the  To solve the problem, it is necessary first to construct for these systems spectral decompositions of the form (2) in each case.

REVIEW OF THE LITERATURE
The method of spectral decomposition of the solution of the Lindley integral equation was first presented in detail in the classic queueing theory [1], and was subsequently used in many papers, including [6,7]. A different approach to solving Lindley's equation has been used in [8]. That work used factorization instead of the term "spectral decomposition" and instead of the functions ( ) is the characteristic function of a random variable ξ with an arbitrary distribution function C(t), and z is any number from the interval (−1, 1). This approach for obtaining end results for systems under consideration is less convenient than the approach described and illustrated with numerous examples in [1].
The practical application of the method of spectral decomposition of an LIE solution for the study of systems with hyperexponential and exponential input distributions shown in [10].
In [2], for the first time, the results of an analysis of In [9] presents the results of the approach of queues to the Internet and mobile services as queues with a delay in time. It is shown that if information is delayed long enough, a Hopf bifurcation can occur, which can cause unwanted fluctuations in the queues. However, it is not known how large the fluctuations are when the Hopf bifurcation occurs. This is the first publication in the English-language journals about queues with a delay. Approximate methods with respect to the laws of distributions are described in detail in [13][14][15], and similar studies in queuing theory have recently been carried out in [16][17][18].

MATERIALS AND METHODS
Consider the class of density functions f(t), which are Laplace-convertible, that is, for which there is a function Next, we use the delay theorem as a property of the Laplace transform: for any 0 0 t > , the equality will be satisfied where Re( ) 0 s > . The considered density functions M, E 2 , H 2 , HE 2 belong to this class.
Let us write the Laplace transforms of functions (4) and (5) Here, exponents with opposite signs of exponential functions are reset to zero, and thus the shift operation in the spectral decomposition is leveled. The right-hand side of expression (6) completely coincides with the spectral decomposition of the solution of an LIE for the classical system M/M/1. It will be similar for other systems, the spectral expansions for systems with delay and ordinary systems will coincide [2][3][4][5].
Assertion is proved. Corollary. The formulas for the average waiting time for all systems with shifted distributions will have exactly the same form as for the corresponding systems with ordinary distributions, but with changed parameters due to the time shift operation [2][3][4][5]. Consequently, the average waiting time for systems with lag actually depends on the magnitude of the shift parameter 0 0 t > . Further, taking into account conditions (1) and (2) Note that the function ( ) According to the method of spectral decomposition, the constant K is determined from the condition: where the parameters λ and µ are determined by expressions (7) and (8) by the method of moments: The constant K determines the probability that the demand entering the system finds it free.
In this case, the ratio λ/µ does not determine the load factor as in the case of the M/M/1 system and the parameters λ and µ are not the intensity of input flow and service time, respectively. The Laplace transform for the waiting time distribution function, following [1], has the form: Distributions (4) and (5) contain two parameters; to determine them by the moments method, the moment equations (7) and (8) must be supplemented with expressions for the second-order initial moments. As the second moments it is more convenient to use the coefficients of variation: Then, as input parameters for calculating the QS M /M /1 − − , we set the values 0 , , , , t c c λ λ μ μ τ τ , and the unknown parameters λ, μ and t 0 are determined from the system of moment equations (7), (8), (11) and (12). This system of equations is overdetermined and the input parameters will be bound by the condition where / μ λ ρ = τ τ is load factor.
Similar reasoning for other systems with a delay will lead to similar results. To do this, in Table 1 we give the numerical characteristics of the considered laws of distributions, which were used in [2][3][4][5].
The numerical characteristics of the shifted distributions (Table 1) clearly indicate a significant influence on them of the shift parameter t0. The numerical characteristics of the shifted distributions (Table 1) clearly indicate a significant influence on them of the shift parameter t 0 . Now it is necessary to determine the unknown parameters of these distributions. These parameters were also obtained in [2][3][4][5] and for the cases of density functions of the distribution of intervals of input flows a(t) are given in Table 2. Similar parameters for the service time distributions b(t) will take place by replacing λ with µ. Table 3 shows the Laplace transformations of the waiting time density functionы in the queues in the systems under consideration, the components of the spectral expansions of the LIE solution, as well as the expressions for the average waiting time in the corresponding systems.
A detailed description of the algorithms for calculating the average waiting time for the systems under consideration can be found in [2][3][4][5]. In this way, are the published results for eight of the sixteen systems.     The expressions for the average waiting time s s s s s s s        Tables 4-7

RESULTS
As one would expect, a decrease in the coefficients of variation λ c and μ c due to the introduction of the shift parameter 0 0 t > into the laws of the distributions of the input flow and service time, entails a decrease in the average waiting time in systems with a delay several times. The adequacy of the presented results is fully confirmed by the fact that when the shift parameter 0 t tends to zero, the average waiting time in the delayed system tends to its value in the usual system. The above calculation results are in good agreement with the results of work [11] in the range of parameters in which the systems under consideration are valid.

DISCUSSION
The operation of the shift in time on the one hand, leads to an increase in system load with a delay. (1 ) / (1 ) t t + μ +λ compared to a usual M/M/1 system.
The time shift operation, on the other hand, reduces the variation coefficients of the interval between receipts and of the service time of requirements. Since the average waiting time in the system G/G/1 is related to the coefficients of variation of the arrival and servicing intervals by a quadratic dependence, the average waiting time in the delayed system will be less than in the usual system under the same load factor. For example, for the M / M /1 − − system when loading ρ =0,9 and the shift parameter t 0 =0,9, the variation coefficient c λ of the interval between receipts decreases from 1 for a usual system to 0,19, the service time variation coefficient μ c decreases from 1 to 0,1, and the waiting time decreases from 9 units of time for a usual system to almost 0.,09 units of time for a delayed system (Table 4). In addition, the introduction of the shift parameter leads to a fairly wide range of variation in the coefficients of variation λ c and μ c , in contrast to usual systems, which are applicable only in the case of fixed values of the coefficients of variation. Therefore, systems with delay extend the range of their applicability in the modern theory of teletraffic.

CONCLUSIONS
The paper presents the spectral expansions of the solution of the Lindley integral equation for eight systems with delay, which are used to derive expressions for the average waiting time in the queue for these systems in closed form.
The scientific novelty of the results is that spectral expansions of the solution of the Lindley integral equation for the systems under consideration are obtained and with their help the calculated expressions for the average waiting time in the queue for systems with delay in closed form are derived. These expressions complements and expands the well-known incomplete formula for the average waiting time in the G/G/1 systems with arbitrary laws of input flow distribution and service time.
The practical significance of the work lies in the fact that the obtained results can be successfully applied in the modern theory of teletraffic, where the delays of incoming traffic packets play a primary role. For this, it is necessary to know the numerical characteristics of the incoming traffic intervals and the service time at the level of the first two moments, which does not cause difficulties when using modern traffic analyzers [10].
Prospects for further research are seen in the continuation of the study of systems of type G/G/1 with other common input distributions and in expanding and supplementing the formulas for average waiting time.