Asymptotic Diffusion Method for Retrial Queues with State-Dependent Service Rate

Abstract

In this paper, we consider a retrial queue with a state-dependent service rate as a mathematical model of a node of FANET communications. We suppose that the arrival process is Poisson, the delay duration is exponentially distributed, the orbit is unlimited, and there is multiple random access from the orbit. There is one server, and the service time of every call is distributed exponentially with a variable parameter depending on the number of calls in the orbit. The service rate has an infinite number of values. We propose the asymptotic diffusion method for the model study. The asymptotic diffusion approximation of the probability distribution of the number of calls in the orbit is derived. Some numerical examples are demonstrated.

1. Introduction

Retrial queues [1] are appropriate models of communication systems, in which customers (clients, signals, calls, and data packages) do not leave the system in the case of busy servers (channel, control or computational device). They stay in a virtual place for some random time duration to attempt the service again. Such a feature is common for random access protocols in computer networks. Similar protocols are used for a FANET (flying ad hoc network) deployment. This type of network is young and is in search of a better protocol delivering quality of service requirements, taking into account moving nodes. Recently, various protocols for FANET networks have been suggested [2,3,4,5,6,7]. Most papers consider adaptive communication protocols based on classical ones. In [5], there is an example of FANET research using queuing theory methods. In their study, the authors considered the classical queuing model with FIFO queues. FANET consists of several UAVs (unmanned aerial vehicles), which collect information, and one control device, which processes information. UAVs can simultaneously transmit information, so it is appropriate to use random multiple access protocols [8]. Also, the transmission access time may be controlled in FANET.

In the current paper, we propose a mathematical model of a FANET node as a retrial queue with a service rate depending on the number of calls in the orbit. In terms of communication networks, retrials define the random access of UAVs, and the variable service rate has the mean of the increasing intensity of data processing or switching of a control device between UAVs. Note that this paper focuses on the quite complex mathematical model of retrial queues with dependent parameters and proposes a novel analytical method of its study. The application of the results of the mathematical modeling for FANETs will be considered in further research.

Queuing systems (QSs) with variable parameters are known in the literature. For instance, there are papers on state-dependent arrivals in queues [9,10,11,12]. Most of them study models with switching between two possible states of the arrival rate depending on some boundary value of the number of customers in the system. On the other hand, the service rate as a state-dependent parameter can be supposed. Papers devoted to the topic also take into account only varying threshold-based policies. In [13,14], the model with a switching service rate is considered. In [15,16], authors studied the same policy, taking into account the retrial behavior of customers. Also, a model with arrival and service rates defined by discrete functions depending on the number of customers in the orbit is used in [17,18]. The closest problem is studied in [19], where the authors considered a retrial queue with state-dependent service and arrival rates. However, the model in [19] differs from the current study: the retrial policy is FCFS, which means that only one customer in orbit has access to a server. In our research, a retrial queue with multiple access is considered the motivation of this feature is based on the practical problems described above. We use an asymptotic diffusion method from the class of asymptotic analysis methods of queuing theory [20,21]. We introduce a modification of the approach for the retrial queue under consideration.

Summarizing the above, in this paper, we propose a novel asymptotic diffusion method, which allows to study a quite complex retrial queuing model with a state-dependent service rate that has not been analytically investigated yet.

The rest of the paper is organized as follows. In Section 2, we describe the considered retrial queue, and the statement of the problem is formulated. Section 3 is devoted to the original asymptotic–diffusion analysis for retrial queues with the state-dependent service rate. In Section 4, we demonstrate some numerical experiments, which represent the behavior of the system for various values of the model parameters. Section 5 is dedicated to some concluding remarks.

2. Mathematical Model

Let us consider the following single-server retrial queue. Customers (in the other words, calls) arrive in the system according to a Poisson process with parameter $\lambda$. If a primary call finds a server free, it starts the service with an exponentially distributed service time with rate ${\mu }_i$. If the server is busy, the call goes to an orbit (some virtual place), where it stays during the random time distributed exponentially with rate $\sigma$. After the delay, the call makes an attempt to obtain the service again. The system structure is presented in Figure 1.

Note. In this section and further, we use the terms of queuing theory. As for practical problem, “calls” may mean data packages or some of the simplest tasks, a “server” may be the transmission channel or computing node, an orbit is a virtual place, and math abstraction is used for repeated requests.

The significant difference of the considered model is variable service rate ${\mu }_i$, which depends on the current value of stochastic process $i\left(t\right)$, which determines the number of calls in the orbit at moment t. For the mathematical research, it does not matter how ${\mu }_i$ changes. It may be a decreasing or increasing positive function of a discrete variable i.

Let process $i\left(t\right)$ define the number of calls in the orbit and $k\left(t\right)$ define the server state as follows:

$$k\left(t\right)=\left\{\begin{array}{c}0, \mathrm{if}\mathrm{a}\mathrm{server}\mathrm{is}\mathrm{free},\hfill \\ 1, \mathrm{if}\mathrm{a}\mathrm{server}\mathrm{is}\mathrm{busy}.\hfill \end{array}\right.$$

Two-dimensional stochastic process $\left\{k\right(t),i(t\left)\right\}$ is a continuous time Markov chain. We denote a probability that the server is in state k and there are i calls in the orbit by $P_k(i,t)=P\{k\left(t\right)=k,i\left(t\right)=i\}$. So, the system of Kolmogorov equations can be written for $i\ge 0$:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial P_0(i,t)}{\partial t}=-(\lambda +i\sigma )P_0(i,t)+{\mu }_i{P}_1(i,t),}\hfill \\ {\displaystyle \frac{\partial P_1(i,t)}{\partial t}=-(\lambda +{\mu }_i)P_1(i,t)+\lambda P_0(i,t)+\lambda P_1(i-1,t)+}\hfill \\ (i+1)\sigma P_0(i+1,t).\hfill \end{array}\right.$$

(1)

System (1) is a system of infinite number of differential equations, so its direct solving is quite hard. We propose the method of the asymptotic diffusion analysis [20] under the limit condition of a long delay $\sigma \to 0$ for System (1) solving.

Also, System (1) in the steady state can be solved numerically by a truncated method [1,22]. In this method, some boundary I is chosen so that probability $P\left(I\right)$ is close to zero, and the system of I equations is solved ($i=\overline{0,I}$). We solve System (1) in the general case for $i=\overline{0,\infty }$.

3. Asymptotic Diffusion Analysis

The approach of this paper continues our study in [20], where the methodology of the asymptotic diffusion analysis for the state-dependent infinite-server queuing system is proposed. The proposed asymptotic method lets us solve System (1) by the approximation of the process under study by a diffusion process under some limit condition. Here, we choose a limit condition of a long delay ($\sigma \to 0$).

3.1. First Order Asymptotics

First of all, we introduce $\sigma =\epsilon$ as an infinitesimal value. Also, the following denotations are used:

$$\sigma t=\epsilon t=\tau , i\sigma =i\epsilon =x, P_k(i,t)={\widehat{P}}_k(x,\tau ,\epsilon ).$$

Note that we use a linear interpolation of discrete variable i by continuous variable x; it is possible because $\epsilon$ is very small (in limit $\epsilon \to 0$). Also, the spline method of the necessary order is used for the existence of ${\widehat{P}}_k(x,\tau ,\epsilon )$ differentiability.

Let ${\mu }_i=\mu (i·\sigma )=\mu \left(x\right)$ be a positive differentiable function. Then, we have the following system of asymptotic equations:

$$\left\{\begin{array}{c}{\displaystyle \epsilon \frac{\partial {\widehat{P}}_0(x,\tau ,\epsilon )}{\partial \tau }=-(\lambda +x){\widehat{P}}_0(x,\tau ,\epsilon )+\mu \left(x\right){\widehat{P}}_1(x,\tau ,\epsilon ),}\hfill \\ {\displaystyle \epsilon \frac{\partial {\widehat{P}}_1(x,\tau ,\epsilon )}{\partial \tau }=-(\lambda +\mu \left(x\right)){\widehat{P}}_1(x,\tau ,\epsilon )+\lambda {\widehat{P}}_0(x,\tau ,\epsilon )+\lambda {\widehat{P}}_1(x-\epsilon ,\tau ,\epsilon )+}\hfill \\ (x+\epsilon ){\widehat{P}}_0(x+\epsilon ,\tau ,\epsilon ).\hfill \end{array}\right.$$

We write decompositions for $(x+\epsilon ){\widehat{P}}_0(x+\epsilon ,\tau ,\epsilon )$ and $\lambda {\widehat{P}}_1(x-\epsilon ,\tau ,\epsilon )$ in $\epsilon$-neighborhood of the point x. Thus, we obtain the following equations:

$$\left\{\begin{array}{c}{\displaystyle \epsilon \frac{\partial {\widehat{P}}_0(x,\tau ,\epsilon )}{\partial \tau }=-(\lambda +x){\widehat{P}}_0(x,\tau ,\epsilon )+\mu \left(x\right){\widehat{P}}_1(x,\tau ,\epsilon ),}\hfill \\ {\displaystyle \epsilon \frac{\partial {\widehat{P}}_1(x,\tau ,\epsilon )}{\partial \tau }=-\mu \left(x\right){\widehat{P}}_1(x,\tau ,\epsilon )+\lambda {\widehat{P}}_0(x,\tau ,\epsilon )+x{\widehat{P}}_0(x,\tau ,\epsilon )+}\hfill \\ {\displaystyle \epsilon \frac{\partial (x·{\widehat{P}}_0(x,\tau ,\epsilon ))}{\partial x}-\epsilon \frac{\partial (\lambda ·{\widehat{P}}_0(x,\tau ,\epsilon ))}{\partial x}+O\left({\epsilon }^2\right),}\hfill \end{array}\right.$$

(2)

where $O\left({\epsilon }^2\right)$ is an infinitesimal value of order ${\epsilon }^2$.

Let us sum up equations of System (2):

$${\displaystyle \epsilon \frac{\partial {\widehat{P}}_0(x,\tau ,\epsilon )}{\partial \tau }+\epsilon \frac{\partial {\widehat{P}}_1(x,\tau ,\epsilon )}{\partial \tau }=\epsilon \frac{\partial (x·{\widehat{P}}_0(x,\tau ,\epsilon ))}{\partial x}-\epsilon \frac{\partial (\lambda ·{\widehat{P}}_0(x,\tau ,\epsilon ))}{\partial x}+O\left({\epsilon }^2\right).}$$

(3)

Taking $\epsilon \to 0$ in Equations (2) and (3), we obtain the following system:

$$\left\{\begin{array}{c}{\displaystyle -(\lambda +x){\widehat{P}}_0(x,\tau )+\mu \left(x\right){\widehat{P}}_1(x,\tau )=0,}\hfill \\ -\mu \left(x\right){\widehat{P}}_1(x,\tau )+(\lambda +x){\widehat{P}}_0(x,\tau )=0,\hfill \\ {\displaystyle \frac{\partial {\widehat{P}}_0(x,\tau )}{\partial \tau }+\frac{\partial {\widehat{P}}_1(x,\tau )}{\partial \tau }=\frac{\partial (x·{\widehat{P}}_0(x,\tau ))}{\partial x}-\frac{\partial (\lambda ·{\widehat{P}}_0(x,\tau ))}{\partial x}.}\hfill \end{array}\right.$$

(4)

Obviously, the solution has the form

$${\widehat{P}}_k(x,\tau )=\widehat{P}(x,\tau )·R_k\left(x\right),$$

where

$${\displaystyle R_0\left(x\right)=\frac{\mu \left(x\right)}{\lambda +x+\mu \left(x\right)}, R_1\left(x\right)=\frac{\lambda +x}{\lambda +x+\mu \left(x\right)}.}$$

(5)

Then we can write

$${\displaystyle \frac{\partial \widehat{P}(x,\tau )}{\partial \tau }=\frac{\partial {\widehat{P}}_0(x,\tau )}{\partial \tau }+\frac{\partial {\widehat{P}}_1(x,\tau )}{\partial \tau }=-\frac{\partial (a\left(x\right)·\widehat{P}(x,\tau ))}{\partial x},}$$

(6)

where

$$a\left(x\right)=-x{R}_0\left(x\right)+\lambda R_1\left(x\right).$$

(7)

In this way, we obtain a degenerate Fokker–Planck Equation (6) for the probability density of the limiting diffusion process $x\left(\tau \right)$ with drift coefficient $a\left(x\right)$. Thus, we can write

$$dx\left(\tau \right)=a\left(x\right)d\tau .$$

(8)

3.2. Second Order Asymptotics

In the second stage of the study, we suppose $\sigma ={\epsilon }^2$, where $\epsilon$ is an infinitesimal value, and make the following substitutions in System (1):

$$\tau ={\epsilon }^{2}t, i\sigma =i{\epsilon }^2=x\left(\tau \right)+\epsilon ·y, {\mu }_i=\mu (x+\epsilon y), P_k(i,t)={\widehat{P2}}_k(x,y,\tau ,\epsilon ).$$

We take into account Formulas (7) and (8). System (1) is rewritten as

$$\left\{\begin{array}{c}{\displaystyle {\epsilon }^2\frac{\partial {\widehat{P2}}_0(x,y,\tau ,\epsilon )}{\partial \tau }-\epsilon a\left(x\right)\frac{\partial {\widehat{P2}}_0(x,y,\tau ,\epsilon )}{\partial y}=}\hfill \\ -(\lambda +x+\epsilon y){\widehat{P2}}_0(x,y,\tau ,\epsilon )+\mu (x+\epsilon y){\widehat{P2}}_1(x,y,\tau ,\epsilon )+O\left({\epsilon }^3\right),\hfill \\ {\displaystyle {\epsilon }^2\frac{\partial {\widehat{P2}}_1(x,y,\tau ,\epsilon )}{\partial \tau }-\epsilon a\left(x\right)\frac{\partial {\widehat{P2}}_1(x,y,\tau ,\epsilon )}{\partial y}=}\hfill \\ -(\lambda +\mu (x+\epsilon y)){\widehat{P2}}_1(x,y,\tau ,\epsilon )+\lambda {\widehat{P2}}_0(x,y,\tau ,\epsilon )+\hfill \\ (x+\epsilon (y+\epsilon )){\widehat{P2}}_0(x,y+\epsilon ,\tau ,\epsilon )+\lambda {\widehat{P2}}_1(x,y-\epsilon ,\tau ,\epsilon )+O\left({\epsilon }^3\right),\hfill \end{array}\right.$$

(9)

where $O({\epsilon }^3)$ is an infinitesimal value of order ${\epsilon }^3$.

Summing up Equations (9), we obtain the following additional equation:

$$\begin{array}{c}{\displaystyle {\epsilon }^2\frac{\partial \widehat{P2}(x,y,\tau ,\epsilon )}{\partial \tau }-\epsilon a\left(x\right)\frac{\partial \widehat{P2}(x,y,\tau ,\epsilon )}{\partial y}=-(x+\epsilon y){\widehat{P2}}_0(x,y,\tau ,\epsilon )-}\hfill \\ \lambda {\widehat{P2}}_1(x,y,\tau ,\epsilon )+(x+\epsilon (y+\epsilon )){\widehat{P2}}_0(x,y+\epsilon ,\tau ,\epsilon )+\lambda {\widehat{P2}}_1(x,y-\epsilon ,\tau ,\epsilon )+O\left({\epsilon }^3\right).\hfill \end{array}$$

(10)

Writing decompositions of functions $\mu (x+\epsilon y)$, ${\widehat{P2}}_k(x,y+\epsilon ,\tau ,\epsilon )$ and ${\widehat{P2}}_k(x,y-\epsilon ,\tau ,\epsilon )$ in $\epsilon$-neighborhood of point x, we obtain the following system from Equations (9) and (10):

$$\left\{\begin{array}{c}{\displaystyle {\epsilon }^2\frac{\partial {\widehat{P2}}_0(x,y,\tau ,\epsilon )}{\partial \tau }-\epsilon a\left(x\right)\frac{\partial {\widehat{P2}}_0(x,y,\tau ,\epsilon )}{\partial y}=}\hfill \\ -(\lambda +x+\epsilon y){\widehat{P2}}_0(x,y,\tau ,\epsilon )+(\mu \left(x\right)+\epsilon y{\mu }^{\prime }\left(x\right)){\widehat{P2}}_1(x,y,\tau ,\epsilon )+O\left({\epsilon }^3\right),\hfill \\ {\displaystyle {\epsilon }^2\frac{\partial {\widehat{P2}}_1(x,y,\tau ,\epsilon )}{\partial \tau }-\epsilon a\left(x\right)\frac{\partial {\widehat{P2}}_1(x,y,\tau ,\epsilon )}{\partial y}=}\hfill \\ -(\lambda +\mu \left(x\right)+\epsilon y{\mu }^{\prime }\left(x\right))){\widehat{P2}}_1(x,y,\tau ,\epsilon )+\lambda {\widehat{P2}}_0(x,y,\tau ,\epsilon )+\lambda {\widehat{P2}}_1(x,y,\tau ,\epsilon )-\hfill \\ {\displaystyle \lambda \epsilon \frac{\partial {\widehat{P2}}_1(x,y,\tau ,\epsilon )}{\partial y}+\lambda \frac{{\epsilon }^2}{2}\frac{{\partial }^2{\widehat{P2}}_1(x,y,\tau ,\epsilon )}{\partial y^2}(x+\epsilon y){\widehat{P2}}_0(x,y,\tau ,\epsilon )+}\hfill \\ {\displaystyle \epsilon \frac{\partial ((x+\epsilon y){\widehat{P2}}_0(x,y,\tau ,\epsilon ))}{\partial y}+\frac{{\epsilon }^2}{2}\frac{{\partial }^2(x{\widehat{P2}}_1(x,y,\tau ,\epsilon ))}{\partial y^2}+O\left({\epsilon }^3\right),}\hfill \\ {\displaystyle {\epsilon }^2\frac{\partial \widehat{P2}(x,y,\tau ,\epsilon )}{\partial \tau }-\epsilon a\left(x\right)\frac{\partial \widehat{P2}(x,y,\tau ,\epsilon )}{\partial y}=}\hfill \\ {\displaystyle -\epsilon \frac{\partial }{\partial y}\left(\lambda {\widehat{P2}}_1(x,y,\tau ,\epsilon )-(x+\epsilon y){\widehat{P2}}_0(x,y,\tau ,\epsilon )\right)+}\hfill \\ {\displaystyle \frac{{\epsilon }^2}{2}\frac{{\partial }^2}{\partial y^2}\left(x{\widehat{P2}}_0(x,y,\tau ,\epsilon )+\lambda {\widehat{P2}}_1(x,y,\tau ,\epsilon )\right)+O\left({\epsilon }^3\right).}\hfill \end{array}\right.$$

(11)

Let the solution of System (11) have the following form:

$${\widehat{P2}}_k(x,y,\tau ,\epsilon )=\mathrm{\Pi }(y,\tau )(R_k\left(x\right)+\epsilon f_k(x,y,\tau ))+O\left({\epsilon }^2\right),$$

(12)

where $R_k\left(x\right)$ are defined in (5), and $f_k(x,y,\tau )$ and $\mathrm{\Pi }(y,\tau )$ are unknown functions.

Substituting Expression (12) into the first and the second equations of System (11) and making some transformations, we obtain

$$\left\{\begin{array}{c}{\displaystyle -a\left(x\right)R_0\left(x\right)\frac{\partial \mathrm{\Pi }(y,\tau )}{\partial y}\frac{1}{\mathrm{\Pi }(y,\tau )}=-y{R}_0\left(x\right)-(\lambda +x)f_0(x,y,\tau )+}\hfill \\ y{\mu }^{\prime }\left(x\right)R_1\left(x\right)+\mu \left(x\right)f_1(x,y,\tau )+O\left(\epsilon \right),\hfill \\ {\displaystyle -a\left(x\right)R_1\left(x\right)\frac{\partial \mathrm{\Pi }(y,\tau )}{\partial y}\frac{1}{\mathrm{\Pi }(y,\tau )}=}\hfill \\ {\displaystyle \left(-y{\mu }^{\prime }\left(x\right)R_1\left(x\right)-\mu \left(x\right)f_1(x,y,\tau )+(\lambda +x)f_0(x,y,\tau )+y{R}_0\left(x\right)\right)+}\hfill \\ {\displaystyle (x{R}_0\left(x\right)-\lambda R_1\left(x\right))\frac{\partial \mathrm{\Pi }(y,\tau )}{\partial y}\frac{1}{\mathrm{\Pi }(y,\tau )}+O\left(\epsilon \right).}\hfill \end{array}\right.$$

Then, we consider the equations in the limit by $\epsilon \to 0$:

$$\left\{\begin{array}{c}{\displaystyle -(\lambda +x)f_0(x,y,\tau )+\mu \left(x\right)f_1(x,y,\tau )=}\hfill \\ {\displaystyle y(R_0\left(x\right)-{\mu }^{\prime }\left(x\right)R_1\left(x\right))-a\left(x\right)R_0\left(x\right)\frac{\partial \mathrm{\Pi }(y,\tau )}{\partial y}\frac{1}{\mathrm{\Pi }(y,\tau )},}\hfill \\ {\displaystyle (\lambda +x)f_0(x,y,\tau )-\mu \left(x\right)f_1(x,y,\tau )=-y(R_0\left(x\right)-{\mu }^{\prime }\left(x\right)R_1\left(x\right))+}\hfill \\ {\displaystyle (-a\left(x\right)R_1\left(x\right)-x{R}_0\left(x\right)+\lambda R_1\left(x\right))\frac{\partial \mathrm{\Pi }(y,\tau )}{\partial y}\frac{1}{\mathrm{\Pi }(y,\tau )}.}\hfill \end{array}\right.$$

(13)

We obtain two equal equations, so we solve one of them. Let us present functions $f_k(x,y,\tau )$ in the following form:

$$f_k(x,y,\tau )=C{R}_k\left(x\right)+y{g}_k\left(x\right)-\frac{\partial \mathrm{\Pi }(y,\tau )}{\partial y}\frac{1}{\mathrm{\Pi }(y,\tau )}·{\varphi }_k\left(x\right),$$

(14)

where $C=const$, ${\varphi }_k\left(x\right)$ and $g_k\left(x\right)$ are defined by the following expressions:

$$\begin{array}{c}{\displaystyle g_0\left(x\right)=R_0^{\prime }\left(x\right)=\frac{{\mu }^{\prime }\left(x\right)(\lambda +x)-\mu \left(x\right)}{{\lambda +x+\mu \left(x\right))}^2},}\\ {\displaystyle g_1\left(x\right)=R_1^{\prime }\left(x\right)=\frac{\mu \left(x\right)-{\mu }^{\prime }\left(x\right)(\lambda +x)}{{\lambda +x+\mu \left(x\right))}^2},}\\ {\displaystyle {\varphi }_0\left(x\right)=-\frac{a\left(x\right)R_0\left(x\right)}{\lambda +x+\mu \left(x\right))}, {\varphi }_1\left(x\right)=\frac{a\left(x\right)R_0\left(x\right)}{\lambda +x+\mu \left(x\right))},}\end{array}$$

(15)

which are obtained by substituting (14) into System (13).

For deriving the formula for function $\mathrm{\Pi }(y,\tau )$, we substitute Expression (12) into the third equation of System (11). After some transformations, we obtain the following equation:

$$\begin{array}{c}{\displaystyle \frac{\partial \mathrm{\Pi }(y,\tau )}{\partial \tau }-a\left(x\right)\frac{\partial \mathrm{\Pi }(y,\tau )}{\partial y}(f_0(x,y,\tau )+f_1(x,y,\tau ))=}\hfill \\ {\displaystyle -\frac{\partial }{\partial y}\left(\mathrm{\Pi }(y,\tau )(\lambda f_1(x,y,\tau )-y{R}_0\left(x\right)-x{f}_0(x,y,\tau ))\right)+\frac{1}{2}\frac{{\partial }^2}{\partial y^2}\left(\mathrm{\Pi }(y,\tau )(x{R}_0\left(x\right)+\lambda R_1\left(x\right)\right).}\hfill \end{array}$$

Using Expression (14), the following equation is derived:

$$\begin{array}{c}{\displaystyle \frac{\partial \mathrm{\Pi }(y,\tau )}{\partial \tau }=\frac{\partial \left(\mathrm{\Pi }(y,\tau )y{a}^{\prime }\left(x\right)\right)}{\partial y}+\frac{1}{2}\frac{{\partial }^2}{\partial y^2}\left(\mathrm{\Pi }(y,\tau )b\left(x\right)\right),}\hfill \end{array}$$

(16)

where $b\left(x\right)=\lambda R_1\left(x\right)+x{R}_0\left(x\right)-2{\varphi }_0(\lambda +x)$.

Thus, we obtain Fokker–Planck Equation (16) for probability density function $\mathrm{\Pi }(y,\tau )$ of diffusion process $y\left(\tau \right)$ with drift coefficient $y{a}^{\prime }\left(x\right)$ and diffusion coefficient $b\left(x\right)$.

In this way, the solution has the following form:

$$dy\left(\tau \right)=y{a}^{\prime }\left(x\right)d\tau +\sqrt{b\left(x\right)}dw\left(\tau \right),$$

where $w\left(\tau \right)$ is a Wiener process.

3.3. Result of Asymptotic Diffusion Method

For composing the results of the first and the second asymptotics, we introduce the stochastic process

$$z\left(\tau \right)=x\left(\tau \right)+\epsilon y\left(\tau \right).$$

Obviously, the process satisfies the following equation:

$$dz\left(\tau \right)=a\left(z\right)d\tau +\sqrt{\sigma b\left(x\right)}dw\left(\tau \right).$$

Denoting the probability density function of diffusion process $z\left(\tau \right)$ by $\pi (z,\tau )$, we write the Fokker–Planck equation for $z\left(\tau \right)$:

$${\displaystyle \frac{\partial \pi (z,\tau )}{\partial \tau }=\frac{\partial }{\partial z}\left(a\left(z\right)\pi (z,\tau )\right)+\frac{\sigma }{2}\frac{{\partial }^2}{\partial z^2}\left(b\left(z\right)\pi (z,\tau )\right).}$$

(17)

In the steady state, the solution of Equation (17) is given by

$${\displaystyle \pi \left(z\right)=\frac{c}{b\left(z\right)}exp\left\{\frac{2}{\sigma }{\int }_0^z\frac{a\left(x\right)}{b\left(x\right)}dx\right\},}$$

(18)

where $c=const$ can be found from the normalization condition.

Expression (18) defines the asymptotic probability density function of the number of calls in the orbit in the considered retrial queue.

Note that the method of the asymptotic analysis lets us obtain an analytical formula for the Fokker–Planck equation solution (density of the probability distribution under the study). This is an advantage of the proposed method because a simulation or a computer calculation of differential equations is not necessary.

4. Numerical Examples

For demonstrating the proposed asymptotic diffusion method application area, we present the comparison of the asymptotic and the empirical (calculated by simulation) distributions for different values of the model parameters.

Asymptotic probability distribution $P\left(i\right)$ of the number of calls in the orbit is calculated as follows:

$${\displaystyle P\left(i\right)=\frac{\pi \left(\sigma i\right)}{{\sum }_i\pi \left(\sigma i\right)},}$$

where $\pi \left(\sigma i\right)$ are defined by Formula (18) in Section 3.

In the first numerical example (Figure 2), we compare the asymptotic and the simulated distributions for different $\sigma$. The arrival and service rates are the following:

$$\lambda =1, \mu \left(x\right)=x+1,$$

where $x=\sigma i$.

In Figure 2, the comparison of the distributions is demonstrated.

We also compare the approximate and simulated distributions in the case of non-linear function $\mu \left(x\right)$. For instance, we obtain the following results (Figure 3) for $\mu \left(x\right)=exp\left(x\right)$.

For the quantification of the study results, we use the Kolmogorov distance:

$$\delta =\underset{i\ge 0}{max}\left|\sum _{l=0}^i\left[\tilde{P}\left(l\right)-P\left(l\right)\right]\right|,$$

where $P\left(l\right)$ is the asymptotic probability distribution, and $\tilde{P}\left(l\right)$ is the corresponding empirical one (based on the simulation). Values of the Kolmogorov distances for the presented example are in

Table 1. The Kolmogorov distances $\delta$ for different $\sigma$ and $\mu \left(x\right)$.

	$\mathit{\sigma }=1$	$\mathit{\sigma }=0.1$	$\mathit{\sigma }=0.01$
$\mu \left(x\right)=x+1$	0.036	0.017	0.005
$\mu \left(x\right)=exp\left(x\right)$	0.085	0.013	0.004

.

Figure 2 and Figure 3 demonstrate that the asymptotic distribution is quite close to the simulated one for not very small $\sigma$ values. The means of the distributions differ from each other by less than 2%.

Note that in the examples above, the retrial queuing system is in a steady state because the service rate is defined by an increasing function. Let us present the results for different values of $\lambda$ (

Table 2. The Kolmogorov distances for different $\lambda$.

$\mathit{\lambda }$	0.5	1	1.5	5	10
$\delta$	0.017	0.011	0.008	0.004	0.003

, Figure 4). Here, we suppose $\mu \left(x\right)=exp\left(x\right)$, $\sigma =0.1$.

From Table 1 and Table 2, we can conclude that the proposed method of the asymptotic diffusion analysis has high enough accuracy for $\sigma \le 0.1$ and different $\lambda$.

One more example is for $\mu \left(x\right)=x/(x+1)$. We demonstrate the distributions for $\lambda =0.97$ and $\lambda =1$ (Figure 5).

As you can see, in the left example, the system is in a steady state, unlike the right one. The reason is a value of drift coefficient $a\left(x\right)$ (Formula (7) in Section 3):

• If $a\left(x\right)>0$, the system is in a non-stationary mode;
• If equation $a\left(x\right)=0$ has one root and $a(\infty )<0$, the system is in a steady state. It is easy to show that $\mu (\infty )>\lambda$ is a stability condition;
• If equation $a\left(x\right)=0$ has two roots, the system is a non-stationary mode, but there is a pseudo steady-state period [20].

Thus, the proposed method lets us evaluate the length of the period of the stable operations in a real node or evaluate the parameters for the system stability operation (if we can set them).

5. Conclusions and Discussion

In this paper, the retrial queue with the state-dependent service rate is considered. The values of the service rate are defined by a continuous positive function depending on the number of calls in the orbit. The obtained formula for drift coefficient $a\left(x\right)$ allows making the conclusion about the steady state of the studied retrial queue. We also show that the approximation of the stationary probability distribution of the number of calls in the orbit is accurate enough.

Numerical analysis of the results shows the following properties of the system behavior:

• If function $\mu \left(x\right)$ is monotonically increasing and $\mu (\infty )>\lambda$, the steady state always exists;
• If function $\mu \left(x\right)$ is monotonically decreasing, there is no stationary regime in the system (however, there can be a pseudo steady-state period [20]);
• The accuracy of approximation grows with the decreasing of the retrial rate ($\sigma \to 0$) and with the increasing of the arrival rate (the last is an unexpected result for us).

In this way, the novelty of the paper lies in the developing of a new asymptotic method for the quite complex model of retrial queues with state-dependent parameters, which let us obtain some formulas, which can be used for the analytical calculation of the system performance characteristics for any values of the parameters. Moreover, we solve the problem in the general case, without any conditions on the service rate function (i.e., it may be increasing or decreasing, limited or unlimited). Such a model has more application examples since the service increase or degradation often occurs in real systems.

The proposed asymptotic diffusion method belongs to the class of asymptotic analysis methods, which can be also applied for more complex retrial queues, i.e. with non-Poisson arrival processes [21]. In the future, we plan to apply the study approach for a retrial queuing system with state-dependent rates with non-Poisson arrivals (e.g., Markov arrival process) or heterogeneous customers. Also, we plan to apply the obtained results for the real data of FANET communications.