Flow Admission Control for Users Considering Dissatisfactions With Selection Errors and Loss Using Stochastic Evolutionary Game and Queueing Theory

Currently, streaming communications are widespread through YouTube and other media. Streaming communication focuses on real-time communication and requires flow admission control to ensure communication quality. Many existing studies have solved the system-side problems of streaming communications. However, few have considered user behavior, even though users also use the network. For example, if the selected bandwidth changes due to a selection error made by the user, the network traffic load should also change. However, previous studies have not taken into account the frustration caused by such errors in daily life. In this paper, we first model user bandwidth selection considering selection errors based on stochastic evolutionary game theory. Using the distribution of the derived user bandwidth selection, we propose a flow admission control method based on queueing theory. Furthermore, we define user dissatisfaction using the call loss probability derived by queueing theory, and model selection error behavior using this dissatisfaction. By numerical calculation, we drive the optimal threshold value that results in the highest overall user satisfaction when selection error is taken into account, thereby improving the overall user satisfaction.

SECTION I. Introduction

A. Background

In recent years, the use of smartphones and tablet devices has increased significantly due to the widespread use of the Internet in society. As a result, the amount of network traffic has increased and is expected to be about 396 exabytes (10 to the 18th power byte) per month in 2022 around the world [1]. One of the causes of the increase in traffic is users watching videos and music on the Internet such as on YouTube and other video distribution services using those devices. One of the common types of communication used for this is streaming communication. Streaming communication (flow) is a method in which packets are constantly flowing from a server, and the client downloads these packets in real time to watch a video or other media. Different from e-mail and file transfers, it is important to constantly ensure the quality of communication, because the communication is real-time. In addition, voice and video flows with different characteristics are transmitted simultaneously, which occupies a large amount of bandwidth resources. Therefore, QoS (Quality of Service) control is necessary to ensure communication quality [2]. In recent years, Internet Protocol, Version 6 (IPv6) [3] has also been introduced, and the flow label field [4] may facilitate the implementation of QoS-based flow routing mechanisms [5].

B. About Admission Control

In multiple traffic environments, where flows with different characteristics are accommodated in the same network, unlike conventional best effort communication, admission control that determines whether a flow is accepted or not is necessary to ensure communication quality depending on the characteristics of each flow. The previously mentioned flow labels, a feature of IPv6, can reduce the average processing load at the routers in the network. Reserving resources via flow labels is also expected to reduce frequent route changes [5]. For these reasons, admission control to reserve resources has increasingly received attention [6].

In a previous study, an admission control method was proposed that offloads two different types of flows in two links that accommodate the two different types of flows to ensure communication quality without compressing the communication bandwidth. While studies in other fields that consider user behavior have been increasing in recent years, this study focused on systems only and did not focus on users using the network [7], [8], [9]. Therefore, in [10], an admission control was proposed assuming that there are selfish users who will stop communicating depending on the amount of traffic. In the admission control method, broadband users are controlled by a threshold, and narrowband users receive warnings and can decide whether to stop communication by themselves. Moreover, a flow admission control method was proposed that assumes that there are users who operate the seek bar to watch a particular scene or skip it when watching a video or other content [11].

C. Purpose

All admission controls proposed in the past have considered only system problems, or even if they have considered user behavior, they have not taken into account irrational behavior that most people display in their daily lives, such as users making selection errors. For example, selection errors here could be due to misoperation when tapping on a smartphone or other device. In addition, misoperation when using smart glasses is also reported [12]. When a user makes selection errors, the type of traffic requested by the user changes, and the amount of traffic required for the system by the admission control changes. This is because traffic control needs to be based on the amount of demand. In addition, users may become dissatisfied as a result of making selection errors. Therefore, it is necessary to obtain a true traffic load that takes into account selection errors. A new traffic density that takes into account selection errors may increase the total user satisfaction by setting a new optimal threshold for the admission control. In addition, users may become dissatisfied due to selection errors or flow rejection. Our proposed method improves the level of satisfaction by using an optimal threshold that considers such dissatisfaction too.

Our method is divided into two types of methods. Method 1 is a method from [13], and Method 2 is a method that considers the flow blocking probability for the case where the user knows how much flow loss will occur. Method 1 involves a situation in which the user has just started watching a video and the degree of loss is unknown, while Method 2 involves a situation in which some time has passed since the user started watching a video and the degree of loss is known. These two types of situations are distinguished and modeled separately.

Therefore, we propose an admission control method that takes into account the bounded rationality of selection errors to solve the above problems and to take into account users who are close to those in daily life.

D. Contribution of This Study

The contributions of this study are summarized below.

• Derivation of the true traffic density when user selection errors are taken into account using stochastic evolutionary game theory.

  • Proposal and modeling of our method using stochastic evolutionary game theory.

  • Finding that the distribution of users due to selection errors is different.

• Derivation of flow blocking probability by admission control based on queueing theory.

  • Modeling using queueing theory and derive flow blocking probabilities.

• Combining of stochastic evolutionary game theory and queueing theory for model admission control that considers user selection errors.

  • Numerical analysis of total user satisfaction using flow blocking probabilities derived by modeling users with stochastic evolutionary game theory and then with queueing theory (Method 1).

  • Numerical analysis of total user satisfaction by modeling admission control with stochastic evolutionary game theory and queueing theory, taking into account the dissatisfaction level due to flow blocking probabilities in the gain matrix (Method 2).

  • Numerical analysis of total user satisfaction for other selection errors using the optimal threshold when the selection error is 0, showing the decrease in satisfaction with the threshold value.

E. Paper Structure

The organization of this paper is as follows. Related work is presented in Section II. Section III introduces the modeling of the proposed method. The numerical results and discussion are presented in Section IV, and Section V concludes the paper and discusses future directions.

SECTION II. Related Works

Admission control of networks has been studied in the past [14], [15], [16], [17], [18], [19], [20], [21]. A survey paper has been proposed [22]. Recently, admission control has been applied to the 5G environment [23]. In a previous study, an admission control method was proposed based on POMDB (Partially Observable Markov Decision Process: a generalized Markov process), utilizing the characteristics of SDN (Software Defined Networking) networks [14]. It comprehensively optimizes the service performance of service nodes in the application layer and transmission links in the network layer and reduces the effects of link congestion on data transmission quality and QoS. In another previous study [15], the model of [24] was extended. An admission control in a two-link system was proposed that accommodates the flows of two different Poisson arrivals, where one link can offload a portion of its capacity that can be shared to support flows from the other link. In work [16], an admission control flowed MBTAC (measurement based traffic admission control: an admission control that efficiently manages resources on the network) was proposed that, continuously measures the bit rate of downlink and uplink traffic, accommodates new flows by calculating statistical indicators, and avoids overload conditions, to efficiently manage the resources of the IP network. This study compared and evaluated the proposed method with the [25] and [26] methods. In another previous study [27], a model was constructed in which real-time and non-real-time communications coexisted in an inter-sanitary link. They proposed and analyzed an admission control for real-time and elastic data traffic that moves, in a dynamic flow manner, the transmission capacity between requests in service in order to make more efficient use of limited transmission resources. In another previous study [28], the effectiveness of using the expected packet loss probability to determine the maximum number of VoIP flows to be accommodated was shown and compared with [29], [30] to reduce the flow blocking probabilities of VoIP flows, which are increasing in IP networks. Thus, admission control is needed in networks with various features and characteristics.

In addition, due to the recent spread of 5G and IoT, the number of 5G network admission controls [31], [32], [33] and IoT network admission controls [34], [35], [36] is increasing. In a previous study [32], the purpose was to solve the problem of current 5G networks, i.e., the lack of resources for interactive multimedia such as VR. For this purpose, a control method was proposed that uses mobile agents to collect information on applications and uses this information to offload non-interactive applications to alternative possible links if links that satisfy the traffic requirements of interactive applications are available. Another previous study [34] proposed a new mechanism that ensures sufficient bandwidth by adjusting the connection parameters of already accepted flows when a new flow cannot be accepted, and an adjustment method was considered that consumes the least amount of power to solve the problem of consuming extra energy. The previous study [36] proposed an admission control for IoT networks using 5G networks, which is a solution to the rapid increase in the number of connected objects and the significant increase in bit rate and energy consumption due to the advancement of IoT. Among IoT, VANETs used for vehicle-to-vehicle communication have recently been garnering attention [37], [38], and the technologies used for them [39] and admission control have been proposed [40]. Muhammad et al. adopted the multi-mediator method as an excellent solution to improve the routing scheme, to minimize network traffic, and to search for the shortest paths [40].

However, all of the admission controls described above solve problems in the system, but the problem here is that the user is not considered. A previous study [41] focusing on two admission controls [42], [43] compared the effectiveness of the controls and the QoE (Quality of Experience), which is the quality experienced by the end user when a flow is accepted. In addition, another previous study [11] proposed an admission control that considers users who use video navigation tools such as seek bars to skip to different time positions to watch a particular scene in a video. In [10], an admission control is proposed that assumes that some users are likely to stop connecting to streaming flows when the amount of network traffic increases. Broadband users are enforced by threshold values, while narrowband users receive warnings and can decide whether to stop by themselves.

However, these studies assume that users act rationally, so it is necessary to propose a new system that considers users’ irrational behavior. In this paper, we examine the total user satisfaction when admission control is performed in consideration of users’ selection errors as an irrational behavior. Then, we consider the total user satisfaction when admission control is performed, taking into account user dissatisfaction when a flow fails. In addition, we improve realism and show the necessity of considering selection errors.

SECTION III. Proposed Admission Control Method

A. Overview of our Admission Control Method

In this study, we propose the user of two methods that consider user selection errors, which have not been considered in existing studies. This admission control uses a technique called Server and Network Assisted DASH (SAND) [44], which was designed to enhance video delivery using Dynamic Adaptive Streaming over HTTP (DASH) technology. SAND enables communication between a client and a network node or between various network nodes. In addition, recently, an architecture called xStream [45] has been proposed that enables client applications to reflect network traffic conditions where cross-traffic of web traffic exists. In this study, we consider this technology to enable network and client collaboration.

The first method (Method 1) considers user dissatisfaction due to selection errors [13]. The second method (Method 2) considers the dissatisfaction due to selection errors and flow loss in the user’s selection. As shown in Figure 1, first, Method 1 models the stationary distribution of user selections using stochastic evolutionary game theory. Stochastic evolutionary game theory limits the number of users in the calculation. Therefore, we select active users for the theory and obtain a stationary distribution. Next, we model admission control using the stationary distribution with queueing theory, and we derive the call blocking probability. In modeling with queueing theory, we assume the existence of multiple stationary states that follow a stationary distribution. Finally, we derive the total user satisfaction using the call blocking probability. Method 2 first derives the call blocking probability using Method 1. By substituting the probability into the gain matrix in stochastic evolutionary game theory, the dissatisfaction caused by the probability is considered in the user’s selection. Then, we again obtain the stationary distribution of user selections using stochastic evolutionary game theory. Then, we model admission control using the stationary distribution with queueing theory to obtain a new call blocking probability. In the next subsection, we model our admission control method. Firstly, we show the notation for our model in TABLE 1.

TABLE 1 Symbols for This Article

FIGURE 1.

Flow of our proposed methods in this study.

B. Process of Methods Considering User Selection Errors

1) Method 1

Our proposed admission control method maximizes the total user satisfaction while each user’s actions affect each other. For this purpose, the control method that intentionally causes flow loss depending on the situation by using a threshold. In this study, we use stochastic evolutionary game theory to model user behavior in the presence of users who have made selection errors and queueing theory to model admission control to derive a threshold that maximizes total user satisfaction. We show the details of the Method 1 in Algorithm 1.

Algorithm 1 Calculating $U(\boldsymbol{k}^{*})$ With Method 1

Require:

$A$ , $\varepsilon$ , $\beta _{i}$

Output:

$U(\boldsymbol {k}^{*})$ , $\boldsymbol {k}^{*}$

1:

for $n = 0$ to $N$ do

2:

for $i=1,2$ do

Derive the adaptive function $f_{i,n}$ for each bandwidth

3:

end for

4:

Derive the state transition probabilities $p_{n,n+1},p_{n,n-1}$ considering $\varepsilon$ which is the probability that users make selection errors

5:

end for

6:

Derive stationary distribution $\boldsymbol {\pi }_{1}$ for each number of users who choose broadband using $p_{n,n+1},p_{n,n-1}$

7:

for $i = 1,2$ do

8:

Assuming a flow loss network, derive $\boldsymbol {\rho }_{i}$ using $\boldsymbol {\lambda _{i}}$ following $\boldsymbol {\pi }_{1}$ and $\mu _{i}$

9:

end for

10:

for $k_{1}$ = 0 to $\lfloor {c_{0}}/{r_{1}}\rfloor$ do

11:

for $k_{2}$ = 0 to $\lfloor {c_{0}}/{r_{2}}\rfloor$ do

12:

for $i = 1,2$ do

13:

Derive the flow blocking probability $\boldsymbol {B}_{i}$ using $\boldsymbol {k}=(k_{1},k_{2})$ in accordance with the traffic density $\boldsymbol {\rho }_{i}$

14:

end for

15:

Derive the total user satisfaction $U(\boldsymbol {k})$

16:

end for

17:

end for

Modeling is conducted in the above flow, and the optimal threshold $\boldsymbol {k}^{*}$ is derived. In addition, to confirm how a user’s selection error affects the total satisfaction, we compare and discuss the maximum value $U(\boldsymbol {k}^{*})$ of total user satisfaction when the probability of selection errors is varied in this study.

2) Method 2

In our method 2, we derive the flow blocking probability by performing admission control once before the flow of the previous section. By putting this probability into the gain matrix of stochastic evolutionary game theory, we obtain a new user selection. We show the details of the Method 2 in Algorithm 2.

Algorithm 2 Calculating $U(\boldsymbol{k}^{*})$ With Method 2

Require:

$A$ , $\varepsilon$ , $\beta _{i}$

Output:

$U(\boldsymbol {k}^{*})$ , $\boldsymbol {k}^{*}$

1:

while do

2:

FLAG $\leftarrow ~0$

3:

for $n = 0$ to $N$ do

4:

for $i=1,2$ do

Derive the adaptivity function $f_{i,n}$ for each bandwidth

5:

end for

6:

Derive state transition probabilities $p_{n,n+1},p_{n,n-1}$ considering the $\varepsilon$ probability of user selection error

7:

end for

8:

Derive a stationary distribution $\boldsymbol {\pi }$ for each number of users who choose broadband using $p_{n,n+1},p_{n,n-1}$

9:

for $i = 1,2$ do

10:

Assuming a flow loss network, derive $\boldsymbol {\rho }_{i}$ using $\boldsymbol {\lambda _{i}}$ following $\boldsymbol {\pi }_{1}$ and $\mu _{i}$

11:

end for

12:

for $k_{1}$ = 0 to $\lfloor {c_{0}}/{r_{1}}\rfloor$ do

13:

for $k_{2}$ = 0 to $\lfloor {c_{0}}/{r_{2}}\rfloor$ do

14:

for $i = 1,2$ do

15:

Derive the flow blocking probability $\boldsymbol {B}_{i}$ for the admission control using the threshold $\boldsymbol {k}=(k_{1},k_{2})$ in accordance with the traffic density $\boldsymbol {\rho }_{i}$

16:

end for

17:

Derive the total user satisfaction $U(\boldsymbol {k})$

18:

Find the optimal threshold $\boldsymbol {k}^{*}=(k_{1}^{*},k_{2}^{*})$ when $U(\boldsymbol {k})$ is maximum

19:

if FLAG is 1 then

20:

break

21:

end if

22:

FLAG $\leftarrow ~1$

23:

end for

24:

end for

25:

Derive $E(B_{i})$ using $\boldsymbol {B}_{i}^{*}$ and $\boldsymbol {\pi }$

26:

Derive $A^{\mathrm {new}}$ using $E(B_{i})$ , $\alpha _{i}$ , $\alpha _{i}^{\text {error}}$ , $\beta _{i}$ , and $\beta _{i}^{\text {error}}$

27:

end while

We model the above process and derive the optimal threshold $\boldsymbol {k}^{*}$ . In addition, we compare the thresholds with those in the previous section, and we show that a new threshold is necessary when dissatisfaction due to flow loss is taken into account in the selection process.

Our method 1 does not take into account the dissatisfaction caused by flow loss during the user’s bandwidth selection phase. In reality, for example, when users select a video quality, their selection may be based on whether or not the video will stop due to network conditions. In this case, it is necessary to consider the dissatisfaction due to flow loss at the stage when users are making a selection. Therefore, we model user behavior using stochastic evolutionary game theory, which takes into account the dissatisfaction due to flow loss in equation (1) in Method 1. By doing so, we can consider modeling that takes into account dissatisfaction due to flow loss at the stage when the user selects a bandwidth. Then, admission control is modeled by queueing theory, and we derive the optimal threshold value that maximizes the total user satisfaction.

C. Modeling User Behavior Using Stochastic Evolutionary Game Theory

Stochastic evolutionary game theory is a theory that analyzes the long-term stable state of a strategy distribution, which implies the behavior of users in a social group where mutations occur stochastically and continuously [46]. In this study, we consider a flow to be selected as $i\in \mathcal {I}=\{1,2\}$ , and users select broadband flows ($i=1$ ) (strategy 1) or narrowband flows ($i=2$ ) (strategy 2) from a population of $N$ users. Then, we model the user’s selection with stochastic evolutionary game theory in order to consider the probability of a selection error being made.

Let the state $n$ be the number of users who select strategy 1, $n\in \mathcal {N}=\{0,1,\cdots,N\}$ . We consider optimal reaction dynamics in which each user in a population in state $n$ compares its gains with those of all other users and selects the strategy with the highest average gain. To perform such optimal reaction dynamics, we introduce a gain matrix $A$ that describes the gain for each strategy, $$\begin{align*} A=\begin{pmatrix}\alpha _{1} & \quad \alpha _{1}^{\text {error}} \\ \alpha _{2}^{\text {error}} & \quad \alpha _{2} \end{pmatrix}. \tag{1}\end{align*}$$ Here, the rows in equation (1) refer to the bandwidth selected by the user and the columns refer to the bandwidth actually requested by the user. Then, each user is assumed to choose either strategy 1 or strategy 2 necessarily. Assuming that the game is played with $N-1$ users except oneself, the average gain of strategy $i$ at state $n$ can be expressed by the following equations respectively [46], $$\begin{align*} f_{1,n}&=\frac {\alpha _{1}(n-1)+\alpha _{1}^{\text {error}}(N-n)}{N-1}, \tag{2}\\ f_{2,n}&=\frac {{\alpha _{2}^{\text {error}}}n+\alpha _{2}(N-n-1)}{N-1}. \tag{3}\end{align*}$$ This transition in user strategy selection corresponds to the Moran process of population genetics [47]. The probability of one user changing their choice each time is proportional to the adaptive function. Thus, the probability of choosing strategy 1 is given by $nf_{1,n}/\{nf_{1,n}+(N-n)f_{1,n}\}$ , and the probability of choosing strategy 2 is given by $1-nf_{1,n}/\{nf_{1,n}+(N-n)f_{1,n}\}=(N-n)f_{2,n}/\{nf_{1,n}+(N-n)f_{1,n}\}$ .

Also, let $\varepsilon$ be the probability that a user makes a selection error. The transition process of the number of users choosing strategy 1 is a Markov chain with state space $n=\{0,1,\cdots,N\}$ and state transition probabilities. Let $p_{n,n+1}$ be the state transition probability when the state is $n$ and one more user selects strategy 1, which means that the state transitions to $n+1$ . Also, let $p_{n,n-1}$ be the state transition probability when one less user selects strategy 1, which means that the state transitions to $n-1$ . $p_{n,n+1}$ and $p_{n,n-1}$ can be expressed as: $$\begin{equation*} p_{x,y}=0~~(|x-y|>1),~p_{0,1}=p_{N,N-1}=\varepsilon,\end{equation*}$$ for $i=1,\cdots,N-1$ , $$\begin{align*} p_{n,n+1}&=\frac {nf_{1,n}(1-\varepsilon)+(N-n)f_{2,n}\varepsilon }{nf_{1,n}+(N-n)f_{2,n}}\frac {N-n}{N},\\ p_{n,n-1}&=\frac {nf_{1,n}\varepsilon +(N-n)f_{2,n}(1-\varepsilon)}{nf_{1,n}+(N-n)f_{2,n}}\frac {n}{N}.\end{align*}$$

The Moran process is a birth-death process, so a stationary distribution can easily be shown explicitly. We show below the stationary distribution of the number of users who choose strategy 1, which is the stationary distribution $\boldsymbol {\pi }_{1}=(\pi _{1,0},\pi _{1,1},\cdots,\pi _{1,N})$ in all states for $n=1,\cdots,N$ : $$\begin{equation*} \pi _{1,n}=\pi _{1,n-1}\frac {p_{n-1,n}}{p_{n,n-1}}=\pi _{1,0}\prod _{m=0}^{n-1}\frac {p_{m,m+1}}{p_{m+1,m}}. \tag{4}\end{equation*}$$ Here, $\pi _{1,0}$ can be obtained by $\sum _{n\in \mathcal {N}}\pi _{1,n}=1$ . Also, there are two strategies in this study, so the stationary distribution $\boldsymbol {\pi }_{2}$ of the state of the user who chooses strategy 2 is equal to $\boldsymbol {\pi }_{1}$ . Here, $\boldsymbol {\pi }_{1}= \boldsymbol {\pi }_{2}= \boldsymbol {\pi }$ for symbol unification.

D. Total User Satisfaction

In the previous section, we used stochastic evolutionary game theory to derive a stationary distribution $\boldsymbol {\pi }$ of users who choose strategy 1 by selecting the type of demand flow as their strategy. On the other hand, in the admission control system, if $\boldsymbol {\lambda }_{i}$ is an average arrival rates vector for each bandwidth flow, $\boldsymbol {\pi }=(\pi _{0},\pi _{1},\cdots,\pi _{N})$ is the average arrival rate of users requesting broadband flows $\boldsymbol {\lambda }_{1}=(\lambda _{1,0},\lambda _{1,1},\cdots,\lambda _{1,N})=(0,1,\cdots,N)$ , the steady state probability $\boldsymbol {q}_{1}=\{q_{1,0},q_{1,1},\cdots,q_{1,N}\}$ for that average arrival rate can be considered equal to $\boldsymbol {\pi }$ . Similarly, if the average arrival rate of narrowband flows is $\boldsymbol {\lambda }_{2}=(\lambda _{2,0},\lambda _{2,1},\cdots,\lambda _{2,N})=(N,N-1,\cdots,1,0)$ , the steady state of $\boldsymbol {q}_{2}=\{q_{2,0},q_{2,1},\cdots,q_{2,N}\}$ can be considered equal to $\boldsymbol {\pi }$ .

Here, if the average service time of each bandwidth flow is $1/\mu _{i}$ , the traffic density of each bandwidth can be considered to be $\boldsymbol {\rho }_{i}=(\rho _{i,0},\rho _{i,1},\cdots,\rho _{i,N})=(\lambda _{i,0}/\mu _{i},\lambda _{i,1}/\mu _{i},\cdots,\lambda _{i,N}/\mu _{i})$ , and its steady state probability can also be considered to be $\boldsymbol {\pi }$ .

Here, in the proposed admission control, the satisfaction $F_{i,n}$ of each bandwidth flow when it is selected on demand and the satisfaction $F_{i,n}^{\mathrm {error}}$ when it cannot be selected on demand due to a selection error are expressed as follows, respectively, obtained by the availability of the flow given some $\rho _{i,n}$ , $$\begin{align*} F_{i,n}&=(1-B_{i,n})\alpha _{i}+B_{i,n}\beta _{i}, \tag{5}\\ F_{i,n}^{\text {error}}&=(1-B_{i,n})\alpha _{i}^{\text {error}}+B_{i,n}\beta _{i}^{\text {error}}. \tag{6}\end{align*}$$ However, from equation (1), if the flow for bandwidth selected as requested is communicated, the user gets $\alpha _{i}$ satisfaction, and the user gets $\alpha _{i}^{\text {error}}$ satisfaction if the flow that was the result of a selection error and was thus not selected as requested is communicated. We also assume that the user gets $\beta _{i}$ satisfaction (dissatisfaction) if the flow selected for each band as requested is lost, and $\beta _{i}^{\text {error}}$ dissatisfaction if the flow that did not select as requested due to a selection error is lost. Thus, we can consider the difference in satisfaction when a user makes selection errors.

In this study, we define the total satisfaction $U$ as the sum of the satisfaction of each user [48]. In order to maximize $U$ , we search the appropriate threshold $\boldsymbol {k}=(k_{1},k_{2})$ that determines whether each bandwidth flow can be accommodated or not. The total user satisfaction $U$ can be expressed by the following equation: $$\begin{equation*} U(\boldsymbol {k})=\sum _{n\in \mathcal {N}}\pi _{n}\sum _{i\in \mathcal {I}}[\lambda _{i,n}\{(1-\varepsilon)F_{i,n}+\varepsilon {F_{i,n}^{\mathrm {error}}}\}]. \tag{7}\end{equation*}$$

Let $K$ be the set of parameters that affect $B_{i,n}$ . We consider finding the optimal parameter $\boldsymbol {k}^{*}=(k_{1}^{*},k_{2}^{*})\in {K}$ that maximizes the total user satisfaction $U$ when enough time has passed and the steady state of the system is $\boldsymbol {\pi }$ .

Thus, we can derive $\boldsymbol {k}^{*}$ as follows, $$\begin{align*} \boldsymbol {k}^{*}&= \mathop {\mathrm {arg max}}\limits _{ \boldsymbol {k}\in {K}}U, \\ &= \mathop {\mathrm {arg max}}\limits _{ \boldsymbol {k}\in {K}}\sum _{n\in \mathcal {N}}\pi _{n}\sum _{i\in \mathcal {I}}[\lambda _{i,n}\{(1-\varepsilon)F_{i,n}+\varepsilon {F_{i,n}^{\text {error}}}\}]. \tag{8}\end{align*}$$

To derive $\boldsymbol {k}^{*}$ , it is necessary to derive the flow blocking probability for each bandwidth traffic density $\rho _{i,n}$ . In this study, we derive the flow blocking probability by modeling with queueing theory.

E. Admission Control Method And Control Parameters

1) Method 1

In this study, we consider a hypothetical flow loss network [49] consisting of three links, 1,2, and 0 as shown in Figure 2, and we derive the flow blocking probability when admission control is performed by this network. As shown in Figure 2, let the capacity of links 1, 2, and 0 be $c_{1}, c_{2}$ , and $c_{0}$ and the requested bandwidth of the flows be $r_{i}$ ; classify them into two types: broadband with requested bandwidth $r_{1}$ and narrowband with requested bandwidth $r_{2}$ , on the basis of the selections determined in the stochastic evolution game theory in the previous section. Broadband flows pass through links 1 and 0, and narrowband flows pass through links 2 and 0. Therefore, $c_{1}$ is the capacity of the virtual link where only broadband flows can communicate, $c_{2}$ is the capacity of the virtual link where only narrowband flows can communicate, and $c_{0}$ is the capacity of the link where both broadband and narrowband flows can communicate. The average arrival rate of each bandwidth flow is assumed to be determined by stochastic evolutionary game theory, and the average arrival rate of each flow is assumed to follow a Poisson process with $\lambda _{i,n}$ [50], [51] when $n\in \mathcal {N}=\{0,1,\cdots,N\}$ states. Therefore, we assume $N+1$ types of steady states and derive the flow blocking probability for each of them. Also, we assume that the average service time has an invariant mean value for multiple steady states and follows an exponential distribution on average $1/\mu _{i}$ . The traffic density in each steady state is given by $\rho _{i,n}$ .

FIGURE 2.

Assumed flow loss network for this study.

A flow loss network is a network that is connected only when the required bandwidth exists in the link when a flow is accommodated and is flow loss otherwise. As described in the previous section, in the proposed admission control, the control parameter $\boldsymbol {k}=(k_{1},k_{2})$ limits the number of flows that can be accommodated by each requested bandwidth flow. Therefore, in the model shown in Fig. 2, by considering $\boldsymbol {k}=(k_{1},k_{2})$ , with $k_{1}$ corresponding to $c_{1}$ and $k_{2}$ corresponding to $c_{2}$ , we can consider a flow loss network considering the control parameter $\boldsymbol {k}$ .

Let $\boldsymbol {m}=(m_{1},m_{2})$ be the set of numbers of bandwidth flows that can be accommodated by each bandwidth flow, and define an equation $\Omega$ to determine $\boldsymbol {m}$ : $$\begin{align*} \Omega (\boldsymbol {x}) &=\{ \boldsymbol {m}; \boldsymbol {m}\geq \boldsymbol {0}, \boldsymbol {mR}\leq { \boldsymbol {x}}\}, \\ &=\{ \boldsymbol {m}; \boldsymbol {m}\geq \boldsymbol {0}, \\ & \quad (m_{1}r_{1},m_{2}r_{2},m_{1}r_{1}+m_{2}r_{2})\leq ((x_{1},x_{2},x_{0}))\}. \tag{9}\end{align*}$$ Here, we can obtain the number of $\boldsymbol {m}$ that can be accommodated in the network by assigning the link capacity available to $\boldsymbol {x}=(x_{1},x_{2},x_{0})$ , where $\boldsymbol {R}=(\boldsymbol {r}_{1}, \boldsymbol {r}_{2})^{T}$ is the vector of the requested bandwidth. $$\begin{align*} \boldsymbol {R}=(\boldsymbol {r}_{1}, \boldsymbol {r}_{2})^{T}=\begin{pmatrix} r_{1} &\quad 0 &\quad r_{1} \\ 0 &\quad r_{2} &\quad r_{2}\end{pmatrix}^{T}.\end{align*}$$ Then, we can obtain the number of $\boldsymbol {m}$ that can be accommodated in the network by assigning the link capacity available to $\boldsymbol {x}=(x_{1},x_{2},x_{0})$ . From these equations, we can derive $\Pi _{i,n}(\boldsymbol {m})$ , which is the distribution of the number of flows in the network for each bandwidth flow given the traffic density $\rho _{i,n}$ using the product form solution as follows, $$\begin{equation*} \Pi _{i,n}(\boldsymbol {m})= \frac {1}{G(\boldsymbol {c})}\prod _{j\in \mathcal {I}}\frac {\rho _{j,n}^{m_{j}}}{m_{j}!},\qquad \boldsymbol {m}\in \Omega (\boldsymbol {c}). \tag{10}\end{equation*}$$

Here, $G$ is shown below, $$\begin{equation*} G(\boldsymbol {x})=\sum _{ \boldsymbol {m}\in \Omega (\boldsymbol {x})}\prod _{i\in \mathcal {I}}\frac {\rho _{i}^{m_{i}}}{m_{i}!},\qquad ~~\boldsymbol {0}\leq \boldsymbol {x}\leq { \boldsymbol {c}}.\end{equation*}$$ Here, the flow blocking probability $B_{i,n}$ of a user $i$ can be expressed as follows, $$\begin{equation*} B_{i,n}=1-\frac {\sum _{ \boldsymbol {m}\in \Omega (\boldsymbol {c}- \boldsymbol {r_{i}})}\prod _{j\in \mathcal {I}}\frac {\rho _{j,n}^{m_{j}}}{m_{j}!}}{\sum _{ \boldsymbol {m}\in \Omega (\boldsymbol {c})}\prod _{j\in \mathcal {I}}\frac {\rho _{j,n}^{m_{j}}}{m_{j}!}}. \tag{11}\end{equation*}$$ However, for the sake of convenience, 0⁰ = 1.

Next, we discuss the flow blocking probability when the threshold $\boldsymbol {k}$ is considered. As mentioned in the previous section, $c_{0}$ refers to the actual capacity that can be accommodated, and $c_{1},c_{2}$ are those that can be freely changed for the purpose of admission control. We can think of $\boldsymbol {k}=(k_{1},k_{2})$ as the number of each bandwidth flow that can be communicated respectively, so we can write the link capacity $c_{1}$ and $c_{2}$ with the threshold value and the required bandwidth in the virtual network model. Therefore, $c_{1}$ and $c_{2}$ can be written using $k_{1}$ and $k_{2}$ , respectively: $$\begin{align*} c_{1}&=k_{1}r_{1},\quad ~0\leq {k_{1}}\leq {\lfloor \frac {c_{0}}{r_{1}}\rfloor }, \tag{12}\\ c_{2}&=k_{2}r_{2},\quad ~0\leq {k_{2}}\leq {\lfloor \frac {c_{0}}{r_{2}}\rfloor }. \tag{13}\end{align*}$$ Let $K$ be the set of all possible parameters.

In this study, we derive the flow blocking probability in Eq. (11) for all traffic densities $\rho _{i,n}$ and the total user satisfaction $U$ , and we derive the optimal threshold $\boldsymbol {k}^{*}$ by searching all thresholds.

2) Method 2

In Algorithm 2 of Method 2 described in the previous section, the flow blocking probability in Method 1 is included in the gain matrix of stochastic evolutionary game theory. Here we describe how to include the flow blocking probability in the gain matrix.

Using equations (4) and (11), the expected values of flow blocking probability $E(B_{i})$ for each band can be determined as: $$\begin{equation*} E(B_{i})=\sum _{n\in \mathcal {N}}\{\pi _{i,n}B_{i,n}+(1-\pi _{i,n})B_{i,n}\}. \tag{14}\end{equation*}$$

Using the flow blocking probability calculated above, and the satisfaction at acceptance $\alpha _{i},\alpha _{i}^{\text {error}}$ and dissatisfaction at blocking $\beta _{i},\beta _{i}^{\text {error}}$ , we define the elements of a new gain matrix $A^{\text {new}}$ : $$\begin{align*} A^{{\text {new}}}&=\begin{pmatrix}\alpha _{1}^{\text {new}} & \quad \alpha _{1}^{\text {error,new}} \\ notag \\ \alpha _{2}^{\text {error,new}} &\quad \alpha _{2}^{\text {new}}\end{pmatrix}, \\ \alpha _{i}^{\text {new}}&=(1-E(B_{i}))\alpha _{i}+E(B_{i})\beta _{i}, \\ \alpha _{i}^{\text {error,new}}&=(1-E(B_{i}))\alpha _{i}^{\text {error}}+E(B_{i})\beta _{i}^{\text {error}}. \tag{15}\end{align*}$$

By adapting stochastic evolutionary game theory by putting equation (15) into equation (1), we can consider dissatisfaction due to flow loss from a user’s selection.

SECTION IV. Numerical Analysis

A. Setting Numerical Parameters

Let $r_{1}=2$ Mbps be the requested bandwidth for broadband flows, $r_{2}=1$ Mbps for narrowband flows, and $c_{0}=50$ Mbps for the total bandwidth. Also, the number of users of this game is $N=40$ . We assume 41 different average arrival rates for broadband and narrowband flows $\lambda _{1}=(0,1,\cdots,40)$ and $\lambda _{2}=(40,39,\cdots,0)$ in the queueing model used to derive the flow blocking probabilities. As mentioned in the previous section, the steady state probability per average arrival rate follows $\boldsymbol {\pi }$ , and if the processing rates of broadband and narrowband flows are $\mu _{1}=\mu _{2}=1$ flows/s, the traffic density for each band is $\rho _{1}=(0,0.02,0.04 \cdots,1.56,1.6)$ and $\rho _{2}=(0.8,0.78,0.76,\cdots,0.02,0)$ . In this section, we numerically analyze how the total user satisfaction derived from these 41 types of steady state traffic changes with the chosen threshold value. However, in the previous section, the threshold for broadband flows $k_{1}$ took the range $k_{1}>{c_{0}}/r_{1}$ , and the threshold for narrowband flows $k_{2}$ took the ranges $k_{2}>{c_{0}}/r_{2}$ , and $r_{1}k_{1}+r_{2}k_{2} < {c_{0}}$ . In the analysis in this section, we considered combinations of all possible thresholds. The gain matrix of Method 1 and the initial values of the gain matrix of Method 2 are unified gain matrices, which are $\begin{aligned} A=\begin{pmatrix} 4 & 1 \\ 1 & 2 \end{pmatrix} \end{aligned}$ . We set $\alpha _{i}$ under the assumption that satisfaction is proportional to the bandwidth obtained [52]. A satisfaction of a user who changed to a narrowband bandwidth caused by a tap error is low because of the lower quality of the image. Moreover, under the pay-as-you-go pricing system [53], users who change to a broadband bandwidth are less satisfied than users who request to narrowband bandwidth because they have to pay more because of the higher quality of the image. For these reasons, $\alpha _{i}^{\text {error}}$ is set lower than $\alpha _{i}$ . This gain matrix implies that when both broadband and narrowband flows are obtained as requested, the user who obtains the broadband one is more satisfied. If the flow cannot be obtained as requested, it means that the gain of both is low. In addition, the dissatisfaction due to flow loss is $\beta _{1}=\beta _{2}=-1$ for each band.

B. Relationship Between Selection Error and Total Use Satisfaction With Optimal Threshold

Figure 3 shows the relationship between selection error and total user satisfaction with the optimal threshold [10] is used in Method 1 when the selection error is 0. From the figure, total user satisfaction increases from $\varepsilon =0$ to 0.2, and decreases after $\varepsilon =0.2$ . On the other hand, when the optimal threshold is used when selection error is not considered [10], total satisfaction decreases when selection error is raised. As shown in Figure 4, the average arrival rate of each flow changes every selection error. Thus, the total user satisfaction changes with the selection error. This means that the call loss probability also changes with the change in average arrival rate. Figure 4 shows the steady-state distribution for each epsilon, and we can see that the higher the epsilon, the more users select narrow-band flows. In other words, when $\varepsilon =0$ in Method 1, since almost all users select the broadband flow based on the gain matrix, the result shows high call loss probability and low total user satisfaction. However, as the number of users who make a wrong choice increases, the number of users who select the narrow bandwidth increases, and the allocated bandwidth decreases, resulting in a low call loss probability and a high total user satisfaction. As the number of users who make mistakes increases, the number of users who connect in the narrow bandwidth increases, and their dissatisfaction increases, resulting in a decrease in total satisfaction.

FIGURE 3.

Relationship between selection error and total user satisfaction when the optimal threshold [2] [10] is used in Method 1 when the selection error is 0. From the graph, it can be seen that total user satisfaction increases from 0 to 0.2 for the selection error and decreases after 0.2. On the other hand, when the optimal threshold is used without considering the selection error, it can be seen that the higher the selection error, the lower the total satisfaction. The reason for this difference in characteristics is that the arrival rate of each flow changes with the selection error, as shown in Figure 4.

FIGURE 4.

Distribution of the number of users who select broadband flow when $\varepsilon =0.2$ and 1 with Method 1.

Next, Figure 5 shows the total user satisfaction when comparing Method 2 and the conventional method. The first major difference from Figure 2 is the degree of decrease in total user satisfaction. In Method 2, total user satisfaction drops slightly when $\varepsilon =0.2$ , while in the conventional method, it drops significantly there. In Method 2, the gain matrix is $\begin{aligned} A=\begin{pmatrix}1.9622 & 0.1849 \\ 1 & 2 \end{pmatrix} \end{aligned}$ when $\varepsilon =0$ because the call loss probability affects the gain matrix. Therefore, all users who do not make mistakes will try to select a narrow bandwidth. On the other hand, when $\varepsilon =0.1$ , which causes a few mistakes, more users are accommodated with broadband flows, thus the total satisfaction is higher. However, when $\varepsilon =0.2$ , the number of users requesting broadband increases significantly, thus the call loss probability increases and the total user satisfaction decreases. This call loss causes the number of users with narrow bandwidth flows to increase when $\varepsilon$ = 0.3. Since the bandwidth used is reduced, the call loss ratio is lowered, and the satisfaction level increases again. When the selection error rate is greater than 0.3, the total user satisfaction decreases because the number of narrowband flows accommodated increases. In other words, when user error behavior increases, the call loss probability changes. Therefore, the selected flows also change, and the total user satisfaction changes. From the above numerical calculations, we show that total user satisfaction can be improved by setting the optimal threshold value that takes into account the selection error and call loss probability.

FIGURE 5.

Relationship between total user satisfaction at optimal threshold and selection error and between total user satisfaction at optimal threshold when selection error is 0 and selection error for Method 2. The first major difference from Figure 3 is the degree of decrease in total user satisfaction. In Method 2, total user satisfaction drops slightly when $\varepsilon =0.2$ , while in the conventional method, it drops significantly there. In Method 2, the gain matrix is changed when $\varepsilon =0$ because the call loss probability affects the gain matrix. Therefore, all users who do not make mistakes will try to select a narrow bandwidth. However, when $\varepsilon =0.2$ , the number of users requesting broadband increases significantly, thus the call loss probability increases and the total user satisfaction decreases. In other words, when user error behavior increases, the call loss probability changes. Therefore, the selected flows also change, and the total user satisfaction changes.

FIGURE 6.

Distribution of the number of users who select broadband flow when $\varepsilon =0.1$ , 0.2, 0.3, 0.4, and 1 with Method 2.

SECTION V. Conclusion

In this paper, we propose a method for admission control in which users choose between two types of bandwidths considering selection errors, and we evaluate total user satisfaction in an environment where there were users who made selection errors by using stochastic evolutionary game and queueing theory. Our method 1 that considers the difference in satisfaction due to user selection errors shows that the total user satisfaction was higher when there was a small number of users who made selection errors than when there were no selection errors as in the conventional studies, and that there was a new optimal threshold at that time. Similarly, our method 2 that considers the dissatisfaction due to flow loss in user selection shows that the total user satisfaction is higher when there is a small number of users who made selection errors than when there are no users who made selection errors as in conventional studies. It is also found that there existed a new optimal threshold at that time. In addition, we can improve the total user satisfaction by using optimal threshold.

In the future, we will propose an admission control method that considers selection errors in three or more multiple bandwidths suitable for the 5G environment. In addition, since the values of the gain matrix are hypothetical ones that were assumed to be perceived by the user, it is necessary to conduct an actual investigation. Stochastic evolutionary game theory can analyze a finite number of users. When the number of flows in the analysis increases, the scale of the derived distribution also large, and the types of stationary states to be treated in queueing theory also increase. In such cases, statistical multiplicity effects may simplify the model. Also, in the case of link failures, the statistical multiple effect does not work, so there are many advantages to the mathematical method. However, we have not yet considered link failures and will study this point in the future.

Appendix

If equation (10) holds, then equation (11) follows [54]. The flow blocking probability is the probability that a flow is not accepted into the system. However, since the generation process of flows in band $i$ is a Poisson process, the steady state distribution of the state seen by the flows in band $i$ generated in the steady state is equal to the steady state. On the other hand, if the system state $\boldsymbol {n}$ at the time the flow occurs in band $i$ is contained in $\Omega (\boldsymbol {c}- \boldsymbol {r}_{i})$ , this flow is accepted by the system. Therefore, the flow blocking probability $B_{i}$ of band $i$ is given by the following. $$\begin{equation*} B_{i}=1-\sum _{ \boldsymbol {n}\in \Omega (c-r_{i})}\pi (\boldsymbol {n}) \tag{16}\end{equation*}$$ This is consistent with equation (11). Therefore, in the following, we consider equation (10).

Assume that service times follow an exponential distribution. If the link capacity $\boldsymbol {c}$ is infinite, there is no interference between bands, and the number of flows in the system for each band can be modeled as M/M/$\infty$ . Therefore, the stationary distribution $\pi ^{(\infty)}(\boldsymbol {n})$ can be expressed as follows when the link capacity $\boldsymbol {c}$ is infinite. $$\begin{equation*} \pi ^{(\infty)}(\boldsymbol {n})=\prod _{i=1,2}e^{-\rho _{i}}\frac {\rho _{i}^{n_{i}}}{n_{i}!}\quad ~\boldsymbol {n}\geq \boldsymbol {0} \tag{17}\end{equation*}$$

Here, given a stochastic process $X_{i}(t)$ representing the number of flows in band $i$ , $X_{i}(t)$ is a reversible continuous-time Markov chain. Furthermore, since they are independent of each other, the process $\boldsymbol {X}(t)=(X_{1}(t),X_{2}(t))$ of the number of flows is also an invertible continuous-time Markov chain.

When the link capacity is $\boldsymbol {c}$ , the state space $\Omega (\infty)=\{ \boldsymbol {n}; \boldsymbol {n}\geq \boldsymbol {0}\}$ corresponding to the infinite bandwidth case can be considered as a reversible continuous-time Markov chain broken towards the state space $\Omega (\boldsymbol {c})$ . Therefore, when the link capacity is $\boldsymbol {c}$ , the flow number process is a reversible continuous-time Markov chain, and its stationary distribution is expressed by the following equation. $$\begin{equation*} \pi (\boldsymbol {n})=\frac {\pi ^{(\infty)}(\boldsymbol {n})}{\sum _{m\in \Omega (c)}\pi ^{(\infty)}(\boldsymbol {m})} \tag{18}\end{equation*}$$ This equation is consistent with equation (10).