Method of Arrival Process Description for Packet Switch Konstantin E. Samouylova , Andrey K. Levakovb , Nikolai A. Sokolovc and Vitaliy S. Zaitcevd a Peoples’ Friendship University of Russia Mikluho-Maklaya St., 6, Moscow, 117198, Russia b Center Macro-regional Branch (Center MRF) of PJSC Rostelecom, “Comcity”, household 6, build. 1, Kievskoe shosse 22km, 108811, Moscow c Saint-Petersburg Branch of Central Science Research Telecommunication Institute, 11, Warshavskaya, St. Petersburg, 196128, Russia d The Bonch-Bruevich Saint-Petersburg State University of Telecommunications, 22/1 Prospect Bolshevikov, St. Petersburg, 193232, Russia Abstract The arrival process description is important problem for analysis of the packet switches that can be considered as queueing systems. Such queueing systems are the appropriate models for the study of the stochastic characteristics. The accuracy of the results of the model study is largely determined by the correctness of the description of the arrival process. This paper proposes a method to solve the problem of choosing a distribution type to describe arrival process at the teletraffic system input. The authors introduce a criterion for choosing the distribution type based on the error minimization in the estimates of the mean value and coefficient of variation in the delay times in the teletraffic system. Sometimes measurement results characterizing the arrival process are available. In this case, the correctness of the proposed distribution is also checked using the goodness-of-fit test. The paper provides case studies on how the proposed method can be applied to the process of choosing the distribution type in question. Keywords queueing system, packet switch, distribution function, average delay time, coefficient of variation, goodness-of-fit test, relative error 1. Problem Statement The stated problem can be considered a special case of estimating the distance between func- tions [1]. In the teletraffic theory, this problem has some peculiarities. In this article, an entity that should be serviced by the queueing system is called a request. Typical example of the request is IP packet that is processed by a packet switch. For systems with queues [2, 3], the reliable information about the requests arrival interval distribution at the input of the object Workshop on information technology and scientific computing in the framework of the X International Conference Information and Telecommunication Technologies and Mathematical Modeling of High-Tech Systems (ITTMM-2020), Moscow, Russian, April 13-17, 2020 Envelope-Open ksam@sci.pfu.edu.ru (K. E. Samouylov); levakov1966@list.ru (A. K. Levakov); nicksokolov@hotmail.com (N. A. Sokolov); zaitsev@protei.ru (V. S. Zaitcev) Orcid 0000-0002-6368-9680 (K. E. Samouylov) © 2020 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings http://ceur-ws.org ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org) 47 Konstantin E. Samouylov et al. CEUR Workshop Proceedings 47–56 under study is normally obtained based on measurements of packet traffic. If measuring is not possible, the proposed hypothesis should be thoroughly reasoned. Based on measurements, step function 𝐹 (𝑡) is formed, for which an interval of constant time 𝜏 is usually selected on the abscissa axis. In some cases, function 𝐴(𝑡) can be convenient to use if the initial distribution replacement simplifies the further model analysis. Function 𝐴(𝑡) is usually chosen from a set of known random value distributions [4]. For the analysis of most teletraffic models, it is sufficient to know the first and second moments of the request delay time in the system — 𝑆 (1) and 𝑆 (2) . Interestingly, most of the known relations [2, 3] rely on delay time coefficient of variation 𝑘𝑆 instead of the second moment. That’s why the proximity of values 𝑆 (1) and 𝑘𝑆 to the values obtained using measurements of function 𝐹 (𝑡) determines whether distribution 𝐹 (𝑡) has been chosen appropriately. Therefore, relative errors in evaluating values 𝑆 (1) and 𝑘𝑆 denoted below as 𝛿1 and 𝛿2 , respectively, should not exceed the predefined thresholds. We can choose value 𝜏 for function 𝐹 (𝑡) based on the considerations given in [5]. Values of function 𝐴(𝑡) at points that are multiples of 𝜏 are known. Figure 1 shows distribution functions 𝐹 (𝑡) and 𝐴(𝑡) up to value 8𝜏 on the abscissa axis. For example, parameter 𝑑3 defines the difference between two distributions at point 3𝜏. In this example, we observe the maximum difference between functions 𝐹 (𝑡) and 𝐴(𝑡) at point 8𝜏. F(t), A(t) 1 dmax F(t) A(t) d3 t/τ 0 1 2 3 4 5 6 7 8 Figure 1: Example of distribution functions 𝐹 (𝑡) and 𝐴(𝑡) The proximity of functions 𝐹 (𝑡) and 𝐴(𝑡) is normally measured by checking if they belong to the same distribution class using an appropriate goodness-of-fit test [6]. This approach does not allow us to make assertions about the values of errors in evaluating the characteristics at the teletraffic system output. Figure 2 shows the simplest model of a teletraffic system as a so-called “black box”. Function 𝐵(𝑡) represents the distribution of the request processing times in the teletraffic system. In other words, it defines the set of operations on functions 𝐹 (𝑡) or 𝐴(𝑡). Two distributions at the model output are of practical interest: request delay time 𝑆(𝑡) and time interval 𝐷(𝑡) at which the processed requests leave the teletraffic system. 48 Konstantin E. Samouylov et al. CEUR Workshop Proceedings 47–56 F(t) S(t) or B(t) B(t) and A(t) D(t) Figure 2: Teletraffic system modeled as a black box This paper only covers distribution 𝑆(𝑡). Moreover, the analysis of function 𝑆(𝑡) is limited to estimating values 𝑆 (1) and 𝑘𝑆 . 2. Choosing Distribution Based on Measurements Let us assume that the measurements were made correctly and we obtained step function 𝐹 (𝑡) with increment 𝑝𝑘 at point 𝑘𝜏. Some increments can evaluate to zero. Generally, index 𝑘 varies from zero to 𝑛, that is, for 𝑡 ≥ 𝑛𝜏, condition 𝐹 (𝑡) ≡ 1 is true. It is convenient to represent function 𝐹 (𝑡) as the Laplace-Stieltjes transform [7] denoted as 𝜑(𝑠). The first and second moments of the distribution, 𝐹 (1) and 𝐹 (2) , are determined according to the corresponding rules by differentiation of function 𝜑(𝑠). Standard deviation 𝜎𝐹 and coefficient of variation 𝑘𝐹 are calculated based on values 𝐹 (1) and 𝐹 (2) [4]. Approximating distribution 𝐴(𝑡) is normally chosen using the least squares method [8]. In some cases, it is reasonable to use the weighted least squares method [9]. In this case, normally, the following inequalities are true: 𝐴(1) ≠ 𝐹 (1) , 𝐴(2) ≠ 𝐹 (2) , 𝜎𝐴 ≠ 𝜎𝐹 and 𝑘𝐴 ≠ 𝑘𝐹 . These inequalities introduce additional errors in the evaluation of characteristics 𝑆 (1) and 𝑘𝑆 . A methodological approach based on the following operations can help us minimize these errors: • First, the most appropriate type of two-parameter distribution 𝐴(𝑡) is selected using a suitable goodness-of-fit test [6]. • Then, we determine such parameters of distribution 𝐴(𝑡), for which equations 𝐴(1) = 𝐹 (1) and 𝑘𝐴 = 𝑘𝐹 (or 𝜎𝐴 = 𝜎𝐹 if it simplifies the calculations) are true. So, the distribution parameters are calculated by solving a system of two equations. 3. Choosing Distribution Using Goodness-of-Fit Test Pearson’s chi-squared test [6], also known as 𝜒 2 , is often used to test the hypothesis that the sample belongs to the theoretical distribution 𝐴(𝑡). Some researchers prefer the Kolmogorov- Smirnov test for this purpose [10]. Some other tests may also be chosen. A goodness-of-fit test is an important step in solving the stated problem, which should be described in terms of “necessity and sufficiency” [11]. Replacing function 𝐹 (𝑡) with distribution 𝐴(𝑡) should be interpreted as a necessity but cannot be considered sufficient. Indeed, goodness- of-fit tests cannot provide numerical estimates of errors that occur when further operations are performed on distribution 𝐴(𝑡). Alternatively, if function 𝐹 (𝑡) and distribution 𝐴(𝑡) are not close to each other [1], an acceptable difference in characteristics 𝑆 (1) and 𝑘𝑆 can be unstable 49 Konstantin E. Samouylov et al. CEUR Workshop Proceedings 47–56 within the load range under study. Besides, this narrows the application scope of the proposed method for choosing distribution 𝐴(𝑡). However, we cannot insist that the condition, which was hereinafter treated as necessary, is indispensable. When solving some specific tasks, the mentioned tests may indicate that the chosen hypothesis is false while the evaluation accuracy of characteristics 𝑆 (1) and 𝑘𝑆 can be acceptable. A simple example, in which two moments are the same while distribution functions differ significantly, will be given below in this paper. Therefore, it appears that the obtained results will be still more valuable if we use proven laws of mathematical statistics. 4. Proposed Method for Choosing Distribution Measuring traffic multiple times at different packet switches shows that function 𝐹 (𝑡) belongs to a class of distributions that are defined on a limited interval. They are denoted with the lower index “𝑙” (the first letter in word “limited”). The lower index “𝑢” (the first letter in word “unlimited”) is used to denote distributions that take possible values along the entire positive semiaxis. The applied approximations 𝐴𝑢 (𝑡) introduce an error, which is usually very difficult to evaluate. Therefore, it is appropriate to look for an approximation to function 𝐹 (𝑡) in the 𝐴𝑙 (𝑡) class. A particular interest in the functions of the 𝐴𝑙 (𝑡) class is related to a beta distribution [12, 13]. It can be useful in studying functions 𝐴𝑙 (𝑡) with a high value of the coefficient of variation, which is typical of packet multiservice networks. Other distributions [4], such as parabolic, uniform, and others, are also relevant. When the derivative of function 𝐹 (𝑡) has several extrema, we can use a combination of two or more distributions that belong to the 𝐴𝑙 (𝑡) class. To get a conclusive estimate, it is sufficient to consider an example of using a beta distribution defined on the interval [0;1]. In this case, its density 𝑎(𝑥) is determined by the following relation [4]: Γ(𝑢 + 𝑣) 𝑢−1 𝑣−1 𝑎(𝑥) = 𝑥 (1 − 𝑥) . (1) Γ(𝑢)Γ(𝑣) Variable 𝑥 is a dimensionless value. It can be defined as time 𝑡 divided by value 𝑛𝜏. The following conditions are true for the distribution parameters in formula (1): 𝑢 > 0, 𝑣 > 0. The relations for calculating the mathematical expectation of the request arrival interval value 𝐴(1) and its coefficient of variation 𝑘𝐴 are given, for example, in [4]: 𝑢 𝑣 𝐴(1) = , 𝑘𝐴 = . (2) 𝑢+𝑣 √ 𝑢(𝑢 + 𝑣 + 1) It is evident that 𝐴(1) < 1. Having fixed value 𝐴(1) , we change the parameters of the chosen approximating distribution to obtain the necessary values for coefficient of variation 𝑘𝐴 . To calculate parameters 𝑢 and 𝑣, it is necessary to solve a system of two equations, which provides the following result: 2 )] [1 − 𝐴(1) ] [1 − 𝐴(1) (1 + 𝑘𝐴 2) 1 − 𝐴(1) (1 + 𝑘𝐴 𝑣= 2 , 𝑢= 2 . (3) 𝐴(1) 𝑘𝐴 𝑘𝐴 50 Konstantin E. Samouylov et al. CEUR Workshop Proceedings 47–56 Figure 3 shows an example of distribution 𝐹 (𝑥) obtained by measuring the packet traffic characteristics. The corresponding step function has five increments. The monotonically increasing curve corresponds to an approximation of the obtained dependence by function 𝐴(𝑥), which is a beta distribution of the first type with the following parameters: 𝑢 ≈ 0.052, 𝑣 ≈ 0.212. F(x), A(x) 1.0 A(x) 0.8 0.6 F(x) 0.4 0.2 x 0 0.2 0.4 0.6 0.8 1.0 Figure 3: Function 𝐹 (𝑥) and its approximation by distribution 𝐴(𝑥) Comparing two distributions with a 5% significance level according to Pearson’s chi-squared test showed that the beta distribution can be used in further research. Then, we should choose a teletraffic system model, which would allow us to evaluate errors in calculating values 𝑆 (1) and 𝑘𝑆 . In this paper, a packet switch is considered a teletraffic system. The request (IP packet) processing time can be safely considered a constant value [14, 15]. Therefore, in the Kendall’s notation [3], the model under study can be represented as 𝐵𝑒𝑡𝑎/𝐷/1. The designation “𝐵𝑒𝑡𝑎” in the first position specifies the nature of the arrivals process, as determined by the beta distribution. If the arrivals process is defined based on measurements, the letter “𝐺” [3] should be put in the first position. Later, we will describe the impact of request processing time distribution 𝐵(𝑡) on the evaluation accuracy of values 𝑆 (1) and 𝑘𝑆 . For this reason, for the sake of generality, the processing time for the 𝐵𝑒𝑡𝑎/𝐷/1 model is denoted below as moment 𝐵(1) . Values 𝐵(1) should be chosen in such a way as to investigate the dependence of errors 𝛿1 and 𝛿2 on model load 𝜌. According to the above-mentioned designations, load 𝜌 is defined by the relation of 𝐵(1) to 𝐴(1) . In this paper, the load range of 0.1 ⩽ 𝜌 ⩽ 0.9 is selected based on two considerations. The load of less than 0.1 is of no practical interest in terms of compliance with the Quality of Service targets. The load greater than 0.9 is not typical of the teletraffic system’s operating processes and requires additional research. The proposed range of load change 𝜌 is sufficient to analyze the operation modes of a model being a teletraffic system, when the object under study functions in standard conditions, which allow us to assume that the number of waiting places in a queue is unlimited. This hypothesis is true if the real capacity of the buffer memory is rated for the loss probability at the approximate level of 0.001, as stated by the International Telecommunication Union Standardization Sector in Recommendation Y. 1541 [16]. The validity of this assumption was established in [17]. 51 Konstantin E. Samouylov et al. CEUR Workshop Proceedings 47–56 For the model under study, the values of errors 𝛿1 and 𝛿2 in the given range did not exceed 1.5%. This is quite acceptable for the tasks in the telecommunication network design. Several similar models with other step function types have shown acceptable estimates for errors 𝛿1 and 𝛿2 in the given range of the load change. The change in distribution 𝐵(𝑡), which allows us to analyze the change in values 𝛿1 and 𝛿2 when the coefficient of variation of the request processing time is increased to 2.0, showed that the corresponding errors are within the same range. It means that the obtained estimates of errors 𝛿1 and 𝛿2 are almost invariant with the request processing time distribution. The type of function 𝐹 (𝑥), for which Pearson’s chi-squared test discards the hypothesis that this function is similar to beta distribution 𝐴(𝑥), was chosen artificially using variations in values of increments 𝑝𝑘 . The analysis of such functions 𝐹 (𝑥) and 𝐴(𝑥) showed that errors 𝛿1 and 𝛿2 begin to grow significantly and often exceed 20%. Generally, this value is not considered acceptable for the analysis of the teletraffic system characteristics. The results confirm an intuitive conclusion that the positive result of the goodness-of-fit test should be considered a “necessary” condition for applying the proposed method for choosing function 𝐴(𝑡). This statement is based on the understanding that the goodness-of-fit test is indicative of a relatively small distance between the functions [1]. However, as the analysis was limited to using only one distribution, it does not allow us to apply this statement to all types of functions 𝐴(𝑡). If we limit the types of function 𝐴(𝑡) to the distributions that passed the goodness-of-fit test, it will meet the “beauty in science” criterion [18]. 5. Errors Related to Different Types of Distributions The condition that the two moments of functions 𝑆(𝑡) and 𝐴(𝑡) should be equal can be met for several types of the approximating distribution. This brings up the question about the preferred type of distribution 𝐴(𝑡). It may happen that some distributions will show very close values of errors 𝛿1 and 𝛿2 . This assumption is based on the results given in [19]. This monograph presents a graph of Hurst exponent 𝐻 dependence [20] on coefficient of variation 𝑘𝐴 for two types of distributions, a gamma distribution and Weibull distribution [4]. The mentioned graph is reproduced in Figure 4. According to the graphs, the maximum deviation of the corresponding dependencies does not exceed 5%. H 1.0 Gamma distribution 0.9 0.8 Weibull distribution 0.7 0.6 0.5 kA 1.0 1.5 2.0 3.0 5.0 10.0 Figure 4: Relationship between coefficient of variation 𝑘𝐴 and Hurst exponent 𝐻 52 Konstantin E. Samouylov et al. CEUR Workshop Proceedings 47–56 Errors of type 𝛿1 and 𝛿2 for different distributions 𝐴(𝑡) but with identical first and second moments, respectively, are of practical interest. The variance or the coefficient of variation can be used instead of the second moment if this simplifies the required calculations. Let us consider three types of distribution 𝐴(𝑡). The first and second types are the same as the distributions shown in Figure 4. The third type is a hyperexponential distribution [4]. All the described functions belong to the 𝐴𝑢 (𝑡) family. For all three distributions, 𝐴(1) = 1 and 𝑘𝐴 = 2. Table 1 shows the values of the skewness and kurtosis [4], which significantly differ for the distributions under study. Table 1 Characteristics of three types of distributions 𝐴(𝑡) Parameter Gamma distribution Weibull distribution Hyperexponential distribution skewness 4.00 5.58 5.87 kurtosis 24.00 57.75 49.17 However, the pattern of change in the three curves that represent the density graphs of the distributions under study, has a common nature, which is illustrated in Figure 5 and confirmed by the 𝜒 2 test. In other words, functions 𝑓𝑢1 (𝑥), 𝑓𝑢2 (𝑥) and 𝑓𝑢3 (𝑥) are close to each other [1]. For this reason, any distribution can be chosen as an approximating dependence if values 𝐴(1) and 𝑘𝐴 are the same. fu1(x), fu2(x), fu3(x) 0.5 Probability density function of 0.4 hyperexponential distribution 0.3 Probability density function of Weibull distribution 0.2 Probability density function 0.1 of gamma distribution x 0 2 4 6 8 10 Figure 5: Probability densities for three distributions 𝐴𝑢 (𝑡) The last statement was verified by modeling a teletraffic system of type 𝐺/𝐷/1 [3]. As in the previous experiment, the load change was selected in the range of 0.1 ⩽ 𝜌 ⩽ 0.9. The values of errors 𝛿1 and 𝛿2 do not exceed 11%, which is quite acceptable for solving most of the practical problems. It seems appropriate that, among the alternative functions of type 𝑓𝑢𝑗 (𝑥), we select a function with skewness closest to a similar value obtained by measuring the parameters of the approximated distribution. This statement relies on the use of the skewness in the relation in order to evaluate the quantile of the IP packet delay time, as recommended in [16]. 53 Konstantin E. Samouylov et al. CEUR Workshop Proceedings 47–56 If the measurements do not match the approximating distribution but they have identical values 𝐴(1) and 𝑘𝐴 , there may be significant errors in further analysis of the teletraffic models. Figure 6 shows an example of two such functions for distributions 𝐴𝑙 (𝑡). It should be emphasized that, while two densities are clearly different, they have the same values 𝐴(1) and 𝑘𝐴 . Moreover, both distributions have the same values of the skewness coefficient, which is equal to zero due to the symmetry of functions 𝑓𝑙1 (𝑥) and 𝑓𝑙2 (𝑥). fl1(x), fl2(x) 2.5 Probability density function 2.0 of parabolic distribution 1.5 Probability density 1.0 function of arcsine distribution 0.5 x 0 1.0 2.0 3.0 4.0 | Figure 6: Probability densities for two distributions 𝐴𝑙 (𝑡) This example with functions 𝑓𝑙1 (𝑥) and 𝑓𝑙2 (𝑥) demonstrates a radical divergence of distri- butions where values 𝐴(1) and 𝑘𝐴 are the same. For these two functions, error 𝛿1 is small. It amounts to a few percent for the given range of the model load. The situation with error 𝛿2 is different: the error almost reaches 100% when the model is under high load. This indicates that comparing only the mean values of random variables can yield false results. 6. Discussion of Results and Further Research Directions The proposed method of choosing distribution 𝐴(𝑡) is characterized by very high accuracy in evaluating the indicators of the quality of service for the multiservice traffic service presented as a set of IP packets. This method is similar to the procedure proposed in [21] for the analysis of the stochastic characteristics of models with queues. This fact also indicates that the proposed method for calculating characteristics 𝐴(1) and 𝑘𝐴 is acceptable. However, it should be noted that, in theory, there could be some specific models with lower accuracy in evaluating the indicators of the quality of service for the multiservice traffic. Therefore, from this perspective, additional research is necessary to solve the following three tasks. The first task is to establish the relations between the values of errors 𝛿1 and 𝛿2 and values 𝑑𝐼 , or, possibly, only 𝑑𝑚𝑎𝑥 . 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