III. RESEARCH METHODOLOGY




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III. RESEARCH METHODOLOGY. 
It should be noted that there are cases when the application of the solution of
the Lindley integral equation (IE) by the spectral method is fairly easy to 
implement for exponential and hyperexponential distributions [8]. Based on the
available studies of traffic and its distributions with the intensity of packet arrival λ=
0.6 and 0.8, two special cases of MSM were considered: systems P1/W1/1 and 
P2/W1/1, where the symbols P and W correspond to Pareto distributions f1(x) and 
Veybul f2 (x), which have the form 
( )
(1) 
( )
(
)
α-1 
( (
)
α
), (2) 
where α is the shape parameter; β is a scale parameter. It is known that for
MSM G/G/1 the spectral method for solving Lindley is possible, If for arbitrary 
distributions of time intervals between incoming packets and packet processing time 
intervals f1(x) and f2(x), an approximation in the form of a sum of decaying 
exponentials is used 
exp1 
( ) ∑
k
e (3) 
exp2 
( )
= ∑
k
e (4) 
where α
k 
, α
k 
, b
k
 are coefficients by approximate procedures [8]. The proposed method 
made it possible to find the resulting function of the network packet waiting time in the 
queue for o6 service for each of the considered MSMs (see figure). In addition, with the 
help of this method, an estimate was made of the average characteristics of the average 
waiting time and the length of the queue for service. For the P1/W1/1 system, the following 

parameters were set as the time intervals between P1 packets: α=0,39 and β=7,07*107; and 
for packet lengths W – parameters: α=0.78 and β=1500. After carrying out calculations for 
the variant with the intensity of packet arrival α= 0.6, it was obtained: 
FIG. I. the graphs of the functions of the distribution of the waiting time in the queue w
1;2(t) 
with the intensity of the arrival of packets λ = 0,6 and 0,8 
- average waiting time for an application in the queue 
𝑡=7,437ms; 
queue length 
𝑄=4,46 packets. For the P2/W1/1 system, the time intervals between packets: 
P2 were defined by parameters 123123123 and the packet lengths W – parameters: α=0, 78 
and β=1500. After carrying out calculations with the packet arrival rate λ = 0.8, the 
following was obtained: 
- average waiting time for an application in the queue 
𝑡=1.612 ms;
- queue length 
𝑄=1,29 packets. 

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