Modelling Analysis of Channel Assembling in CRNs Based on Priority Scheduling Strategy with Reserved Queue

Abstract

In cognitive radio networks, channel assembling allows secondary users (SUs) to expand network capacity and improve spectrum utilization. Scheduling strategies only based on heterogeneous service classification cannot guarantee the delivery priority of vital elastic services in special scenarios such as emergency rescue. Therefore, a priority scheduling strategy with reserved queue (Ps-rq) is proposed in this work. A static factor is defined to classify SUs into elastic services and real-time services based on message type, while a dynamic factor is defined to differentiate high-priority elastic services based on information validity, message correlation and message size. The high-priority users in the interrupted elastic services are placed in the reserved queue to ensure its services. Accordingly, the scheduling algorithm and the dynamic channel access process is presented. A continuous-time Markov chain analysis is conducted and all possible transition states, trigger events, transition rates and transition conditions of the system starting from a general state are derived. Furthermore, evaluation indexes of system performance are obtained. Study cases and simulation results prove that the proposed strategy can enhance network capacity, reduce blocking probability and forced termination probability for secondary users, and notably enhance the performance of high-priority elastic services. In addition, we analyze the characteristics of Ps-rq through a comprehensive comparison with four other schemes. The experiment proves that the Ps-rq strategy can effectively improve the service quality of the vital elastic services on the basis of providing fair scheduling.

1. Introduction

Traditional wireless communication networks are mainly based on static spectrum allocation. However, the licensed user always occupies a fixed frequency band, which will lead to an excessive waste of spectrum [1]. Cognitive radio networks (CRNs) allow secondary users (SUs) to explore and use the unused spectrum of primary users (PUs) through spectrum sensing to improve spectrum utilization [2,3].

Improving spectrum utilization requires cognitive radio systems to flexibly schedule and allocate available spectrum resources [4,5]. Dynamic spectrum access (DSA) can ensure SUs get access to spectrum gaps in the licensed spectrum bandwidth [6]. The flexibility of DSA is reflected in that the ongoing SUs can exit the channel once sensing the signals of the PUs, then spectrum adaptation can be performed through spectrum handoff or dynamic channel selection. Designing resource scheduling rules based on DSA that conform to the application environment can help improve the performance of the secondary network [7].

In CRNs, the adoption of channel assembling technology and spectrum adaptation technology in the process of DSA can significantly enhance the service efficiency of SUs. Channel assembling technology allows SUs to bind or aggregate one idle channel to more to increase the secondary network capacity. Combined with spectrum adaptation technology, the ongoing SUs can adaptively adjust the number of assembled channels according to the activities of other PUs and SUs to provide flexible scheduling. However, the modeling of the DSA process based on resource allocation is complex, so it is necessary to analyze and evaluate the performance of secondary networks to help design operation rules.

The service performance of SUs is inseparable from the resource scheduling and allocation method. The service flow in wireless communication can be divided into real-time service (such as audio and video) and elastic service (such as data) based on delay sensitivity. Real-time service usually has higher priority than elastic service, while elastic service has lower delay sensitivity, and its transmission rate can be adjusted adaptively according to the network environment.

Researchers proposed a variety of resource allocation methods for heterogeneous traffic to improve spectrum utilization [8,9,10,11,12,13,14]. The main technique is to aggregate the available spectrum resources into the spectrum pooling according to the dynamic network environment and then allocate them on demand. Among them, refs. [10,11,12,13] studied the resource scheduling problem of multi-channel cognitive radio networks with heterogeneous traffic based on channel assembling and used mathematical methods to establish the analysis model to evaluate the performance of secondary networks. Previous research [11] proved that arranging a variable number of assembled channels for elastic services and a fixed number of assembled channels for real-time services according to the operating environment can help improve spectrum efficiency.

In [15], the authors proposed that the adoption of the continuous-time Markov chain (CTMC) process can provide a modeling method for the performance analysis of CRNs with channel assembling, which has the advantage of avoiding the time synchronization problem of PUs and SUs on the same channel. In [11], the authors proposed a channel assembly strategy considering spectrum adaptation and heterogeneous services, and CTMC was used to evaluate the secondary network performance. However, the SU interrupted by the PU cannot continue the service after exiting the channel, which obviously increases the possibility of the SU being dropped. Therefore, ref. [12] proposed to establish queues for heterogeneous services and adopt a queue scheduling algorithm to improve network capacity on the basis of prioritization. Previous research [12,16] proved that establishing queues for SUs can improve network capacity and reduce the blocking probability of SUs. By considering a variety of factors that affect channel capacity, including a time-varying wireless link, a finite buffer, traffic classes and an adaptive modulation and coding (AMC) scheme, a channel assembling strategy with queues was proposed in [17] and the interaction between system parameters was investigated by establishing a CTMC analysis model. Based on channel assembling, an adaptive spectrum leasing algorithm was proposed in [18], and the DSA performance of SU traffic with access priority was discussed by developing the CTMC model. The above research shows that the adoption of channel assembling technology for heterogeneous services can improve the utilization of idle spectrum. In addition, the establishment of CTMC can help to model the scheduling dynamics of spectrum resources among users in the resource allocation problem. The aforementioned research presents a more rigorous and refined methodology for mathematically analyzing the performance of the secondary system.

In CRNs with multi-type services, resource allocation based on service classification can improve the transmission reliability of important information. However, the improvements usually result in the starvation problem of low-priority services due to a lack of service for a long time. Resource scheduling schemes that can alleviate the starvation problem of low-priority users were proposed in [19,20,21], and the secondary system performance was evaluated by establishing Markov analysis models with queues. In [20], channel binding technology with starvation mitigation was adopted, and a non-pre-emptive priority M/G/1 queuing model was used to depict the spectrum handoff process of SUs, so as to improve the resource utilization. A round robin priority (RRP) scheduling algorithm combining static and dynamic resource allocation was proposed in [21]. This algorithm obtained the transmission opportunities of low-priority users by temporarily suspending the high-priority queue. The purpose of this is to minimize the starvation of low-priority services in CRNs. In fact, the RRP algorithm in [21] adopts queue polling scheduling, but when the traffic of one queue is large, user scheduling only based on queue polling will greatly reduce the system performance [22]. The scheduling of heterogeneous services, based on service type division, can indeed offer a more convenient priority scheduling scheme for channel allocation and priority scheduling in CRNs during system initialization. The previous research, however, failed to address the transmission requirements of high-priority elastic users in practical applications, thus failing to provide a fundamental solution to the issue of service starvation caused by scheduling priorities.

Consequently, this motivates the investigation of our study. This paper investigates resource scheduling and allocation with service classification for CRNs that require ranking services in special scenarios. Some important elastic services need to be given higher priority in special CRN application scenarios, including emergency rescue, drones [23] and UAVs. Therefore, considering that scheduling schemes with hierarchical services often lead to the starvation problem of low-priority services due to long periods of unserved services, the goal of this study is to improve the delivery efficiency of important information based on providing as fair a distribution mechanism as possible.

Different from our previous studies [9], this paper considers a more general scheduling scheme based on heterogeneous services. The aim of this study is to propose a more comprehensive resource scheduling and allocation scheme for CRNs, which can guarantee the QoS of high-priority elastic services. More specifically, based on the polling scheduling, a new reserved queue is provided for the elastic services with high- priority that are interrupted by PUs, and the highest priority is arranged for the SUs in the reserved queue. Based on channel assembling and spectrum adaptation technology, the dynamic spectrum access process of heterogeneous services is analyzed and spectrum resource allocation among SUs is investigated through developing a CTMC. The major contributions of this study are as follows:

• This study proposes a priority scheduling strategy with reserved queue (Ps-rq strategy). The system model and service classification scheme of the proposed strategy are presented. By combining channel assembling and spectrum adaptation technology, a proportional priority allocation scheme based on Ps-rq strategy is given.
• The resource flow process of the secondary network is mapped on a CTMC. By employing stochastic analysis methods, the complex user activities in the CRN are associated with state transitions individually. The transition event, transition rate and transition condition that can trigger changes in the system states are provided, thereby offering a theoretical foundation for enhancing the performance of secondary systems through channel assembling.
• The procedure for dimension reduction of high-dimensional Markov chains is given. Based on this, the analytical results of the secondary network performance are investigated and numerically simulated. We present a comparison of the Ps-rq with four previous schemes, and it is proven through numerical verification that the proposed strategy ensures service quality for interrupted important elastic services while maintaining fair scheduling.

2. Priority Scheduling Strategy with Reserved Queue (Ps-rq Strategy)

This section provides a heterogeneous service classification scheme for CRNs with multi-type services. On this basis, a priority scheduling strategy with reserved queue (Ps-rq) is proposed. The purpose is to provide a fair scheduling for CRNs with multi-type services, which can guarantee the delivery reliability of important information while alleviating the starvation problem of low-priority queues.

2.1. System Model and Assumptions

Centralized cognitive radio networks (CRNs) consist of two types of radios, primary user (PUs) and secondary user (SUs). User scheduling is controlled by the corresponding base station. The central node is responsible for interacting with the surrounding environment, as well as managing spectrum allocation and scheduling decisions to other SUs through a control channel. The central node in centralized CRNs can enhance system performance by making efficient decisions and reducing interference among users. In contrast to distributed CRNs, centralized CRNs eliminate the need for further deliberation on aspects such as inter-user interference and node selfishness.

We assume that there are $M (M\in Z^{*})$ PUs sharing the spectrum bandwidth, and each PU can access only one channel. Compared with SUs, PUs have absolute channel access priority. SUs are equipped with the means for spectral energy detection, assuming that SUs have perfect sensing ability. And the sensing time required is so short that PUs will not reach the channel within this period. Once the PU is sensed, the ongoing SU will immediately release the assembled channels and exit and perform spectrum adaptation at the same time. Assuming that there is less time spent on spectrum adaptation, the channel switching will not affect the transmission of SUs.

2.2. Heterogeneous Service Classification

Supported by the system model, this section proposes a heterogeneous service classification scheme to satisfy the scheduling demands for CRNs with multi-type services including data, image, audio and video. For instance, delayed delivery or packet loss of vital information of patients during a medical emergency may be life threatening, and this part of the information is mainly an elastic service. However, a previous study [12] based on heterogeneous service classification did not consider important elastic services with high delivery priority under special scenarios. The transmission priority and reliability of important elastic services can hardly be ensured. On the basis of heterogeneous service classification, multi-type services in CRNs are also classified by dynamic factors, and the elastic services with high-priority are separated. The details are as follows:

• Static factor: message type. SUs in CRNs are classified into elastic services $(S{U}_e)$ and real-time services $(S{U}_r)$ according to delay sensitivity. Elastic services have the characteristics of short frame length, heavy traffic flow and high delay tolerance. Real-time services require the timeliness of data transmission. According to the tolerance of delay, the service priority is defined as follows:

  $$\mathrm{Priority}: S{U}_r>S{U}_e;$$
• Dynamic factor: elastic services in CRNs are reclassified based on three dynamic factors including information validity, message correlation and message size.

Information validity (Val) refers to the time remaining before the deadline. Messages with shorter times remaining have higher priority.

$$Val={\displaystyle \frac{\mathrm{The} \mathrm{time} \mathrm{remaining} \mathrm{before} \mathrm{the} \mathrm{deadline}}{T\mathrm{he} \mathrm{delivery} \mathrm{time} \mathrm{of} \mathrm{information} }};$$

(1)

Message correlation (Cor) refers to the type of message that needs to be delivered first.

$$\{\begin{array}{c}Cor=1,\\ Cor=0,\end{array}\begin{array}{c}\mathrm{Relevance} \mathrm{of} \mathrm{important} \mathrm{information}\\ Others\end{array};$$

(2)

Message size (Size) is inversely proportional to the message priority.

$$Size={\displaystyle \frac{\mathrm{The} \mathrm{message} \mathrm{size} \mathrm{of} \mathrm{the} \mathrm{SU}}{\mathrm{Average} \mathrm{message} \mathrm{length}}};$$

(3)

After combining the above three dynamic factors, elastic services are classified into high, medium and low priorities. The purpose is to identify the elastic services with high-priority $(S{U}_e^h)$, and then prioritize them over real-time services. The service priority is suggested as follows:

$$\mathit{Priority}: S{U}_e^h>S{U}_r>S{U}_e$$

Furthermore, a three-dimensional (3D) priority view of elastic services is presented in Figure 1. Takagi–Sugeno fuzzy control is used to prioritize elastic services [24]. The input value of the fuzzy inference system ranges from 0 to 1. The output of the system is priority and is divided into three fuzzy sets, whose domains are [−1, 1]. According to the fitness, the “If-then” rule is adopted to set 27 fuzzy rules, and the priority of elastic services are divided into high, medium and low priorities. For example, if Val = 0, Cor = 1 and Size = 1, then Priority = 1, which represents important information with a short remaining time and a small message size. From the input and output variables, Gaussian and trapezoidal membership functions are used to obtain the 3D priority view of elastic services.

2.3. Ps-rq Strategy

Based on the above heterogeneous service classification scheme, traffic with multi-type services entering the system is served by the priority scheduling strategy with reserved queue (Ps-rq strategy). As shown in Figure 2, the Ps-rq strategy consists of an information scheduling unit and a data transmission unit. Users entering the secondary system (SUs) are divided into the elastic service $(S{U}_e)$ and real-time service $(S{U}_r)$ according to a static factor through a packet classifier I. (Packet classifiers are employed for the purpose of parsing incoming packet types.) Different physical queues are arranged for $S{U}_e$ and $S{U}_r$, which are denoted $L_e$ and $L_r$. If the service of SUs is interrupted by a PU during the transmission, spectrum adaptation will be performed. Considering that $S{U}_r$ requires high timeliness and has jitter tolerance, if the spectrum adaptation fails, the interrupted $S{U}_r$ will be dropped and release the assembled channels, and it will not return back to the end of $L_r$. That is, $L_r$ has no feedback loop.

Different from $S{U}_r$, $S{U}_e$ has low requirements on timeliness and may contain vital signals. Thus, a feedback loop is arranged for $L_e$. The interrupted $S{U}_e$ will re-entered into packet classifier II and classified into high, medium and low priorities based on a dynamic factor. The purpose is to separate high-priority elastic services, mark them as $S{U}_e^h$, and put them into the reserved queue $L_e^h$. Meanwhile, the interrupted medium-priority elastic service $(S{U}_e^m)$ and low-priority elastic service $(S{U}_e^l)$ are removed from their marks and returned to the back of the queue $L_e$ (feedback loop). Assuming that each queue is isomorphic, the secondary users in the queue are waiting to be scheduled according to the order of first come first service (FCFS). The priority of the queue is set to $L_e^h>L_r>L_e$. SUs of different types can only line up in the corresponding queue. The service of SUs will be blocked if there is no space left in the queue. If it is full when the interrupted $S{U}_e$ and $S{U}_e^h$ enter the corresponding queue, the service will be forced to terminate.

2.4. Channel Allocation Based on Ps-rq Strategy

In the scheduling process, channel assembling technology is adopted to improve the service rate of the SUs by assembling one channel or more to SUs. Meanwhile, spectrum adaptation technology is used to allow adaptive allocation of the assembled channels. For CRNs with multi-type services, channel allocation based on the Ps-rq strategy is as follows:

The $S{U}_r$ and the marked $S{U}_e^h$ are assembled with a fixed number of channels to ensure the transmission efficiency. More specifically, the service of $S{U}_r$ can be started by aggregating $a$ channels, where $a\in Z*.$ Accordingly, $S{U}_e^h$ can start the service by aggregating $b$ channels, where $b\in Z*.$ The difference is $S{U}_e$ can flexibly adjust the number of assembled channels based on the spectrum adaptation technology. To be specific, the service of $S{U}_e$ can be started by aggregating $w$ channels, and the service rate can be improved by increasing the aggregated channels to the upper bound $v$, where $w, v\in Z^{*} and w\le v$. For the convenience of starting the service of $S{U}_r$ and the marked $S{U}_e^h$, the parameter setting of this study is based on $a=b<w.$ Suppose that the service rate of SUs increases linearly after the channel assembling technology is adopted, the service rate of $S{U}_r$ with $a$ assembling channels increases by $a$ times.

Channels are allocated to SUs based on proportional channel allocation with priority. According to the queue priority $L_e^h>L_r>L_e$, if there are $N_{free}$ idle channels, detect whether the three queues are empty. If none of them are empty, the idle channel is allocated to $S{U}_e^h$, $S{U}_r$ and $S{U}_e$ in the queue in proportion of $1:\alpha :\beta$, where $\alpha ,\beta \ge 1$. Here, the value of $\alpha , \beta$ can be determined according to the priority requirements and delay tolerance of $S{U}_e^h$, $S{U}_r$ and $S{U}_e$. Then, the remaining channels are allocated to $S{U}_e^h$ according to priority, followed by $S{U}_r$ and $S{U}_r$. If only two of the queues are not empty, the allocation process is the same as above. If only one is not empty, all $N_{free}$ channels are allocated to this queue. Furthermore, if there are still idle channels, the remaining channels are allocated to the ongoing $S{U}_e$. The above allocation process is realized in MATLAB, and the proportional channel allocation with priority is shown in Scheme 1. Here, the number of idle channels is still determined by SUs obtaining available idle channel information through spectrum energy detection technology to achieve opportunistic access.

3. Dynamic Channel Access Process

In CRNs, the arrival and departure behaviors of PUs and SUs result in a change in the system state. In order to analyze the resource flow process of the SUs in different system states, the dynamic channel access process based on the proposed Ps-rq strategy is investigated in this section through classifying the user activities in CRNs into four events: PU arrival, PU departure, SU arrival and SU departure.

• Event A: PU arrival

SUs in CRNs are classified into elastic services and real-time services according to delay sensitivity. Elastic services have the characteristics of short frame length, heavy traffic flow and high delay tolerance. Real-time services require the timeliness of data transmission. According to the tolerance of delay, the service priority is defined as follows:

When the PU arrives, if there is an idle channel in the system, it will be allocated to the newly arrived PU. In this case, the services of other SUs will not be affected. But if there are no idle channels in the system, the PU will interrupt any one ongoing channel of the SUs due to the higher data access priority. The interrupted SU will exit the channel and perform spectrum adaptation. That is, the ongoing $S{U}_e$ with the maximum number of channels will donate the channel to the interrupted SU until the assembled channels of $S{U}_e$ cannot support the service, then the ongoing $S{U}_e$ with the second maximum number of channels will donate its channel … until all of the $S{U}_e$ cannot provide the required number of channels of the interrupted SU. Then, the adaptation fails, which means that all the ongoing $S{U}_e$ cannot donate the channel for the interrupted SU. The $S{U}_e$ who fails the adaptation will enter packet classifier II and mark the priority through a fuzzy controller. If the $S{U}_e$ belongs to $S{U}_e^h$, it will enter the reserved queue $L_e^h$; otherwise, it will return back to the end of queue $L_e$. If the queue is full, the service of the SU will be forced to terminate.

If the PU pre-empts the channel of a $S{U}_r$ or a marked $S{U}_e^h$, spectrum adaptation will be performed in the same way. The service will continue when the adaptation is successful; otherwise, the interrupted $S{U}_r$ or $S{U}_e^h$ will be forced to terminate and release the assembled channel.

In particular, the channels released by the forcibly terminated $S{U}_e^h$ or $S{U}_r$ cannot be used to start the service of SUs in the queue due to the setting of $a=b<w$. However, the channels released by the forcibly terminated $S{U}_e$ can be used for the queuing $S{U}_e^h$ and $S{U}_r$ in $L_e^h$ and $L_r$. The channel allocation of SUs in the queue follows Scheme 1. If no SU in the queue uses the idle channel, these channels will be assembled to other SUs, obeying Rule 1.

Rule 1: The idle channel will be preferentially assembled to the ongoing $S{U}_e$ with the minimum aggregated channels until the $S{U}_e$ is assembled to the upper bound, $v$. If there are still idle channels, the remaining channels will be assembled to the ongoing $S{U}_e$ with the second minimum aggregated channels until all of the ongoing $S{U}_e$ reach the upper bound, $v$.

We present a flow chart, as shown in Figure 3, to show the dynamic channel access process based on the Ps-rq strategy on PU arrival. Similarly, other events including PU departure, SU arrival and SU departure can also use the flow chart to describe the channel access process.

• Event B: PU departure

The departure of a PU releases a channel. The released channel will be preferentially used for the SUs waiting in the queue. If the SUs in the queue do not use the idle channel, the idle channel can be used for the ongoing $S{U}_e$ whose assembled channels have not reach the upper bound with the channel assembling technology.

Specifically, the idle channel is preferentially allocated to $S{U}_e^h$ in $L_e^h$, obeying the rule of first come first service (FCFS). The service of $S{U}_e^h$ can be started only when $b-1$ free channels exist in the system. If the $S{U}_e^h$ in the queue do not use the idle channel, the channel can be assembled to $S{U}_r$ in $L_r$, and the service of $S{U}_r$ can be started only when $a-1$ free channels exist in the system. If both $S{U}_e^h$ and $S{U}_r$ in the queue do not use the idle channel, the channel can be assembled to $S{U}_e$ in $L_e$. Similarly, the service of $S{U}_e$ can be started only when $w-1$ idle channels already exist in the system. If no SU in the queue uses the idle channel, the channel will be assembled to the ongoing SUs in the system, obeying Rule 1.

• Event C: SU arrival

There are three kinds of SU arrivals in the system including $S{U}_e^h$, $S{U}_e$ and $S{U}_r$. If the SU arriving at the system is $S{U}_e^h$ and there are $b$ idle channels in the system, then the service of the newly arrived $S{U}_e^h$ can be started. Similarly, the service of the newly arrived $S{U}_e$ or $S{U}_r$ can be started up when the number of channels existing in the system reaches the minimum value $w$ or $a$. If the free channels in the system cannot support the service, the ongoing $S{U}_e$ will perform spectrum adaptation to donate the channel for the newly arrived SUs. If all of the ongoing $S{U}_e$ are unable to provide the required number of channels, the newly arrived SU will be ranked at the end of the corresponding queue. If the queue is full, the service of the SU will be blocked. However, since all the $S{U}_e^h$ entering the queue $L_e^h$ come from the interrupted $S{U}_e$, when $L_e^h$ is full, the service of $S{U}_e^h$ will be forced to terminate, that is, there is no blocked $S{U}_e^h$.

• Event D: SU departure

The departure of SU releases all aggregated channels. That is, the departure of $S{U}_e$ releases $k (w\le k\le v)$ channels; the departure of $S{U}_r$ releases $a$ channels; and the departure of $S{U}_e^h$ releases $b$ channels. Scheme 1 is used to allocate the channels released by SUs, where $N_{free}$ is the sum of the number of channels released by SUs and the number of the idle channels in the system. If there are still idle channels after all SUs waiting in the queue have been assembled, the idle channel will be allocated to the ongoing $S{U}_e$, obeying Rule 1.

4. CTMC Analysis and QoS Measures

In the channel access process, user activities trigger the change in system state. Continuous-time Markov chains (CTMCs) are used to set up a corresponding relationship between the possible state transition of the system and the triggering events that cause the change in the system state, which can depict the spectrum resource flow among multi-type SUs. Therefore, the performance analysis of the secondary system can be carried out by means of the CTMC. The activities of the SUs can be mapped in the multi-dimensional probability space. Then, the mathematical model of the proposed Ps-rq strategy can be established to obtain the QoS metrics through the steady-state solution and the performance of the secondary system can be evaluated.

4.1. Scheduling Dynamic Analysis and CTMC Modeling

Note that a general state is as follows:

$$x=\left\{l_w,\dots , l_v, g_a, g_b, l_{pu}, l_{eq}, l_{rq}, l_{hq}\right\}$$

where $l_m (w\le m\le v)$ is the number of $S{U}_e$ with $m$ aggregated channels, $g_a$ is the number of $S{U}_r$ with $a$ aggregated channels, $g_b$ is the number of $S{U}_e^h$ with $b$ aggregated channels, $l_{pu}$ is the number of PUs in the system and $l_{eq}$, $l_{rq}$ and $l_{hq}$ are the current queue lengths of $L_e$, $L_r$ and $L_e^h$, respectively.

Further, the set of feasible states of the system can be noted as

$$\begin{array}{c}S=\{l_w,\dots , l_v, g_a, g_b, l_{pu}, l_{eq}, l_{rq}, l_{hq}\ge 0; b(x)\le M; l_{eq}\le Q_{eq}; l_{rq}\le Q_{rq}; l_{hq}\le Q_{hq};\\ {\displaystyle \sum _{i=1}^{v-w}i{l}_{w+i}<w, if }{l}_{eq}>0\}\end{array}$$

(4)

Here, $Q_{eq}$, $Q_{rq}$ and $Q_{hq}$ are the total capacities of $L_e$, $L_r$ and $L_e^h$, respectively. ${\displaystyle \sum _{i=1}^{v-w}i{l}_{w+i}<w, if }{l}_{eq}>0$ means that if $L_e$ is not empty, the summation of the maximum number of channels that all ongoing $S{U}_e$ can donate will always be less than its minimum aggregation number. $b(x)$ is the total number of the utilized channels at state $x$ that can be described by:

$$b(x)=l_{pu}+a{g}_a+b{g}_b+{\displaystyle \sum _{m=w}^{v}m{l}_m}$$

(5)

Furthermore, $M-b(x)$ is the number of the idle channels in the system (that is, $N_{free}$ assumed in Scheme 1).

In order to analyze the transition rate from the general state $x$ to other states, we assume that the user arrival in the system obeys the Poisson distribution and the departure obeys the exponential distribution. Note that the arrival rate of PU is ${\lambda }_p$ and departure rate is ${\mu }_p$. The arrival rates of $S{U}_r$ and $S{U}_e$ are ${\lambda }_{sr}$ and ${\lambda }_{se}$, and the departure rates are ${\mu }_{sr}$ and ${\mu }_{se}$, respectively. Moreover, the scheduling with feedback loop of $S{U}_e$ may cause the external arrival of the services entering into $L_e$ and $L_e^h$ to no longer obey the Poisson distribution. To simplify our analysis, we assume that the setting of the feedback loop has little effect on the services in $L_e$ and $L_e^h$, and suppose that the exogenous arrival of SU services all obey the Poisson distribution.

Based on the above assumptions, by analyzing the channel access process, the CTMC model based on the Ps-rq strategy is established. All possible destination states, activities, transition rates and transition conditions of the system starting from the general state $x$ are developed.

• Event A: PU arrival

Upon the PU arrival, the state transitions of the secondary system starting from the general state $x$ to all possible triggering events are derived as shown in Figure 4, and the transition conditions for all possible states are listed in Figure 5. Except for state t₁ in the figure, there are no idle channels in the system that can be directly allocated to the arrived PU. Thus, there exists $M-b(x)=0$ in state t₂ to state t₃₅.

The arrival of the PU can pre-empt the service of any SUs. For state t₄ to state t₁₄ and state t₁₆ to state t₁₇ of service interruption in Figure 5, it indicates that the SUs in these states cannot perform spectrum adaptation, i.e., $\forall m (w<m\le v), l_m=0$. Since $L_r$ has no feedback loop, the service of the interrupted $S{U}_r$ will be forced to terminate, such as in state t₃₀ to state t₃₂.

In state t₃ and state t₁₅, there may be two cases when the SUs do not perform spectrum adaptation: there is no secondary user in the system who can perform adaptation, that is $\forall m (w<m\le v), l_m=0$, or the SUs in the system cannot perform spectrum adaptation and $\forall m (h=m=v), l_m>0$. Similar conditions must be met for states t₁₈, t₃₀ and t₃₃.

In addition, since $a=b<w$, $a=b\le w-1$. That is, the channels released by the interrupted $S{U}_e$ cannot start the service of $S{U}_e$ in $L_e$ but the service of $S{U}_r$ and $S{U}_e^h$ in $L_r$ and $L_e^h$ can be started, such as in state t₆ to state t₁₄. This is the same as the channels released by the forcibly terminated $S{U}_e$, such as in state t₂₁ to state t₂₉. However, the channels released by the interrupted or the terminated $S{U}_r$ and $S{U}_e^h$ cannot support the start of any service of SUs in the queue.

• Event B: PU departure

Upon the PU departure, the state transitions of the secondary system starting from the general state $x$ to all possible triggering events are derived in Figure 6, and the transition conditions for all possible states are listed in Figure 7.

In this case, if the queue $L_e^h$ is not empty and $b=1$, then the service of $S{U}_e^h$ in the queue can be started directly. If $b>1$, then $M-b(x)=b-1$ is required, that is, there exist $b-1$ channels in the system, such as in state t₁. This is the same as $S{U}_r$ such as that in state t₂. Since $a=b<w$, i.e., $w\ne 1$, the service of $S{U}_e$ in $L_e$ can only be started under the condition of $M-b(x)=w-1$, such as in state t₃.

• Event C: SU arrival

Upon the SU arrival, the state transitions of the secondary system starting from the general state $x$ to all possible triggering events are derived as shown in Figure 8, and the transition conditions for all possible states are listed in Figure 9.

The condition of the service of $S{U}_e$ can be started when its arrival is the sum of the number of the idle channels existing in the system and the channels that can be donated by the ongoing SUs are not less than the lower bound of $S{U}_e$, i.e., $M-b(x)+{\displaystyle {\sum }_{m=w+1}^v(m-w)l_m}\ge w$. Otherwise, it will be put into the queue $L_e$, provided that $l_{eq}<Q_{eq}$, such as in state t₃. $S{U}_r$ and $S{U}_e^h$ arrive in the same way, i.e., state t₆ and state t₉.

• Event D: SU departure

Upon the SU departure, the state transitions of the secondary system starting from the general state $x$ to all possible triggering events are derived as shown in Figure 10, and the transition conditions for all possible states are listed in Figure 11.

In this case, the total number of idle channels used by the ongoing SUs should not be greater than the number of channels released by the departing SUs, such as the condition of $k\ge {\displaystyle {\sum }_{m=w}^{v-1}(v-m)}{l}_m-(v-k)$ in state t₅. This is similar to that in state t₁₀ and state t₁₅.

4.2. QoS Metrics

Through CTMC modeling, any possible user activity may trigger the transition from any reachable state t_i to another reachable state t_j, as shown in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11. $q_{ij} (i, j\in S and i\ne j)$ represents the total transition rate, where $S$ is the system’s feasible state set. Let $\pi (x)$ be the steady-state probability of the system, which can be calculated from global balance equations and the normalization condition as follows.

$$\pi (x)Q=0, {\displaystyle \sum _x\pi (x)=1}$$

(6)

where $\pi (x)$ is the steady-state probability vector and $Q$ is the transition rate matrix generated by the transition rate $q_{ij}$. The diagonal elements of transition rate matrix $Q,$ $q_{ii}$ can be found as $q_{ii}=-{\displaystyle \sum _{j\ne i}{q}_{ij}}$. Thus, the QoS metrics of the secondary system based on the proposed Ps-rq strategy can be obtained.

• Network capacity refers to the average number of SU services completed in CRNs per time unit (unit: packet/unit time) [12]. Let $C_e , C_r$ be the capacities of $S{U}_e , S{U}_r$, respectively, we can get:

  $$C_r={\displaystyle \sum _{x\in S}a{g}_a{\mu }_{sr}\pi (x)}$$

  (7)

  $$C_e={\displaystyle \sum _{x\in S}{\displaystyle \sum _{k=w}^v(k{l}_k+b{g}_b){\mu }_{se}\pi (x)}}$$

  (8)

  where $x\in S$ is the set of all possible states in the systems contained in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11.
• Spectrum utilization refers to the ratio of the average number of utilized channels to the total number of available channels [25], which can be determined by:

  $$U={\displaystyle \sum _{x\in S}\frac{b(x)}{M}\pi (x)}$$

  (9)

  where $b(x)$ is the total number of available channels of state $x$, $M$ is the total number of channels in the system and $S$ is the set of all possible transition states contained in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11.
• Blocking probability refers to the probability that the newly arrived SU cannot be served due to all of the channels in the system being busy. Since the waiting queue is set for SUs in this study, the service of the newly arrived SU will be blocked when the corresponding queue is full. It should be noted that the $S{U}_e^h$ entering $L_e^h$ is the $S{U}_e$ interrupted by the PU, but not the SUs who are newly arriving in the system. Therefore, the $S{U}_e^h$ does not have the case of being blocked.

The newly arrived $S{U}_e$ will be blocked if both of the following conditions are met: (1) the sum of the total number of idle channels in the system and the total number of channels that can be donated by the ongoing $S{U}_e$ are not enough to start the service of the $S{U}_e$ (that is the lower bound of channel assembling is not reached) and (2) the queue, $L_e$, is full. Thus, the blocking probability of $S{U}_e$, $P_b^e$, can be obtained by:

$$P_b^e={\displaystyle \sum _{A_e}\pi (x)}$$

(10)

where the set $A_e$ is the trigger condition for which $S{U}_e$ is blocked:

$$A_e=\{x\in S, l_{eq}=Q_{eq}, M-b(x)+{\displaystyle \sum _{k=w+1}^v(k-w)l_k}<w\}$$

Similarly, the newly arrived $S{U}_r$ will be blocked if both of the following conditions are met: (1) the sum of the total number of idle channels in the system and the total number of channels that can be donated by the ongoing $S{U}_e$ are not enough to start the service of the $S{U}_r$ and (2) the queue, $L_r$, is full. Then, the blocking probability of $S{U}_r$,$P_b^r$, can be determined by:

$$P_b^r={\displaystyle \sum _{A_r}\pi (x)}$$

(11)

where the set $A_e$ is the trigger condition for which $S{U}_e$ is blocked:

$$A_r=\{x\in S, l_{rq}=Q_{rq}, M-b(x)+{\displaystyle \sum _{k=w+1}^v(k-w)l_k}<a\}$$

• Forced termination probability refers to the probability that the SUs being served are forced to terminate due to the arrival of a PU. As the set of the feedback loop of $S{U}_e$, the $S{U}_e$ interrupted by the PU can return to the queue $L_e$ and $L_e^h$ through packet classifier II. If the corresponding queue is full, the service will be forced to terminate.

Accordingly, the service of $S{U}_e^h$ is forced to terminate when the following three conditions are met: (1) there are no idle channels in the system; (2) the ongoing $S{U}_e$ cannot perform spectrum adaptation to share channels with the interrupted SUs and (3) the queue, $L_e^h$, is full. To be specific, states t₃₃ to t₃₅ in Figure 4 indicate that the PU pre-empts any $S{U}_e^h$ with $b$ aggregated channels, and the $S{U}_e^h$ is forced to terminate. The forced termination probability of $S{U}_e^h$, $P_f^{eh}$, can be calculated by the ratio of the average forced termination probability to the arrival rate of $S{U}_e^h$, ${\lambda }_{se}$, as follows:

$$P_f^{eh}={\displaystyle \sum _{B_e^h}\frac{{\lambda }_{p}b{g}_b}{(M-l_{pu}){\lambda }_{se}}\pi (x)}$$

(12)

where $B_e^h$ is the set of conditions for which $S{U}_e^h$ is forced to terminate:

$$B_e^h=\{x\in S, M=b(x), l_{pu}<M, g_b>0, l_{hq}=Q_{hq}, \{or\begin{array}{c}\forall m(w<m\le v), l_m=0\\ \forall m(w=m=v), l_m>0\end{array}\}$$

Similarly, the service of $S{U}_e$ is forced to terminate when the following three conditions are met: (1) after being pre-empted, the number of remaining channels of $S{U}_e$ is insufficient to start the service, that is the $S{U}_e$ with $w$ aggregated channels is pre-empted; (2) the ongoing $S{U}_e$ cannot perform spectrum adaptation to share channels with the interrupted SUs and (3) the queue, $L_e$, is full. On this basis, states t₁₈ to t₂₉ in Figure 4 indicate that the PU pre-empts any $S{U}_e$ with $w$ assembled channels, and the $S{U}_e$ is forced to terminate. The forced termination probability of $S{U}_e$, $P_f^e$, can be obtained by:

$$P_f^e={\displaystyle \sum _{B_e}\frac{{\lambda }_{p}w{l}_w}{(M-l_{pu}){\Lambda }_e}\pi (x)}$$

(13)

where ${\Lambda }_e$ represents the arrival rate of $S{U}_e$ that is not blocked:${\Lambda }_e={\lambda }_{se}(1-P_b^e)$ and B_e is the set of conditions for which SU_e is forced to terminate:

$$B_e=\{x\in S, M=b(x), l_{pu}<M, l_w>0, l_{eq}=Q_{eq}, \{or\begin{array}{c}\forall m(w<m\le v), l_m=0\\ \forall m(w=m=v), l_m>0\end{array}\}$$

Since the feedback loop is not set for the queue $L_r$, if the following two conditions are met, the service of $S{U}_r$ is forced to terminate: (1) there are no idle channels in the system to continue its service and (2) the ongoing $S{U}_e$ cannot donate the channels for it. Based on this, states t₃₀ to t₃₂ in Figure 4 indicate that the PU pre-empts any $S{U}_r$ with $a$ aggregated channels, and the $S{U}_r$ is forced to terminate. The forced termination probability of $S{U}_r$, $P_f^r$, can be obtained by:

$$P_f^r={\displaystyle \sum _{B_r}\frac{{\lambda }_{p}a{g}_a}{(M-l_{pu}){\Lambda }_r}\pi (x)}$$

(14)

where ${\Lambda }_r$ represents the arrival rate of $S{U}_r$ that is not blocked:${\Lambda }_r={\lambda }_{sr}(1-P_b^r)$ and $B_r$ is the set of conditions for which $S{U}_r$ is forced to terminate:

$$B_r=\{x\in S, M=b(x), l_{pu}<M, g_a>0, \{or\begin{array}{c}\forall m(w<m\le v), l_m=0\\ \forall m(w=m=v), l_m>0\end{array}\}$$

5. Performance Analysis of the Proposed Ps-rq Strategy

5.1. Model Characteristics Comparison and Analysis

In this section, we compare the proposed Ps-rq strategy with two classical channel allocation schemes (reference [12], 2014) as well as two novel ones (reference [18], 2020 and reference [9], 2022), aiming to illustrate the characteristics of the proposed strategy. The comparison of model characteristics is shown in Figure 12.

The mentioned research models, as depicted in Figure 12, aim to enhance the capacity of the secondary networks. By employing the random analysis method and establishing the corresponding continuous-time Markov chain for systematic analysis, performance metrics are derived to evaluate the effectiveness of the research schemes. However, the mentioned studies have faced challenges in distinguishing service types, as demonstrated in [9,18], or encountered limitations in implementing differentiated service applications due to network environment constraints, such as that in [12].

These limitations serve as the inspiration for the strategy outlined in this paper. The Ps-rq strategy proposed in this paper aims to provide a more equitable differentiation service by addressing the issue of starvation. To achieve this, we employ packet classifiers I and II to perform two levels of service differentiation. The purpose of packet classifier I is to divide the arriving SUs into $S{U}_e$ and $S{U}_r$. Considering that if the $S{U}_e$ interrupted by the PU is all queued for re-service, it will inevitably lead to the starvation problem of another queue. The model proposed in this paper, therefore, considers the service requirements of the partially interrupted high-priority $S{U}_e$ and incorporates packet classifier 2 to ensure fair service provision while offering differentiated services, which represents a notable aspect of our research framework. However, the complexity of the proposed scheduling strategy remains challenging to circumvent in light of service classification refinement. Next, we will conduct a detailed performance analysis of the research Ps-rq strategy presented in this paper.

5.2. Numerical Simulation and Analysis

To illustrate the performance of the secondary system based on the proposed channel assembling strategy, in this section, numerical simulation results of elastic services and real-time services are presented based on MATLAB. The simulation results are based on the procedure for the dimension reduction of high-dimensional Markov chains. In order to verify the developed model, we present the simulation results of the secondary system performance under varying primary user arrival rates. The basic parameters of the numerical simulation are set as follows.

The number of the PUs in the system is set to 6 $(M=6)$, and the numerical experiment is carried out in a CRN with six channels sharing the spectrum bandwidth. The arrival rate of the PU is set as ${\lambda }_p$ = 1, and the departure rate is set as ${\mu }_p$ = 0.5. The arrival rate and departure rate of the elastic service are set as ${\lambda }_{se}$ = 2 and ${\mu }_{se}$ = 1, respectively, and that of the real-time service are set as ${\lambda }_{sr}$ = 1 and ${\mu }_{sr}$ = 1. Note that the unit of these parameters is “sessions per time unit”. By assigning specific values to these parameters, such as 0.5 sessions/sec and flow length in bits, the capacity can be quantified in terms of kbps or Mbps.

The dynamic parameters of the system $(a,b,w,v,Q_{eq},Q_{rq},Q_{hq})$ in the spectrum access process are set as $a$ = $b$ = 1, $w$ = 2, $v$ = 4, $Q_{eq}$ = 2, $Q_{rq}$ = 1 and $Q_{hq}$ = 1. The proportional partition coefficient is $\alpha :\beta =1:1$. Based on the above system settings, the general state $x$ of the system in Section 4.1 can be specified:

$$(\stackrel{3}{l_2} \stackrel{2}{l_3} \stackrel{1}{l_4} \stackrel{6}{g_1} \stackrel{6}{g_1} \stackrel{6}{l_{pu}}\stackrel{2}{Q_{eq}} \stackrel{1}{Q_{rq}} \stackrel{1}{Q_{hq}})$$

(15)

where $l_i (i=2, 3, 4)$ represents $S{U}_e$ who aggregates $i$ channels and $g_a$, $g_b$ represent $S{U}_r$ with $a$ aggregated channels or $S{U}_e^h$ with $b$ aggregated channels $(a=b=1)$; the superscript of each position indicates the upper limit of the number of users at the corresponding position, such as $\stackrel{2}{l_3}$, which indicates that there are at most two $S{U}_e$ assembling three channels in the system at the same time, $\stackrel{6}{l_{pu}}$, which means that at most six PUs exist in the system at the same time and $\stackrel{2}{Q_{eq}}$, which indicates that the maximum capacity of the queue $L_e$ is 2.

Numerical simulation is based on the parameterized general state. The procedure for the dimension reduction of high-dimensional Markov chains is presented as follows. Since the general state assumed in this paper is high-dimensional, it is difficult to directly generate the transition rate matrix $Q$. Therefore, we consider downgrading the markers of high-dimensionality. Theoretically, the size of the matrix $Q$ is $2592\times 2592$, which is generated by the product of the superscripts of Equation (15). Take any general state $I:(1 0 0 1 1 1 0 0 0)$ existing in the system to show the downgrade process as follows. The state $I:(1 0 0 1 1 1 0 0 0)$ indicates that the system has an ongoing $S{U}_e$ with two aggregated channels, an ongoing $S{U}_r$ with one aggregated channel, an ongoing $S{U}_e^h$ with one aggregated channel and an ongoing PU. Therefore, there are five flows existing in the system in this state. Accordingly, state $I:(1 0 0 1 1 1 0 0 0)$ can be marked as state $I_{5, -}$, where the first position of the subscript represents the total number of co-existing flows in the system, and the second position is the counting in order, and the upper limit is the sum of the possible combination states with five flows in the system. If a user arrival occurs in the system in this state, such as a PU arrival, then the corresponding state transition is $(1 0 0 1 1 1 0 0 0)\to (1 0 0 1 1 2 0 0 0)$. In this case, there are six flows co-existing in the system and the transition process is marked as $I_{5, -}\to I_{6, -}$. If a user departure occurs in the system in this state, such as an $S{U}_e^h$ departure, then the corresponding state transition is $(1 0 0 1 1 1 0 0 0)\to (0 1 0 1 0 1 0 0 0)$. In this case, there are four flows co-existing in the system and the transition process is marked as $I_{5, -}\to I_{4, -}$. According to Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11, the possible transition events and transition rates of the system in any general state can be obtained, and then the transition rate matrix $Q$ can be developed. Furthermore, the steady-state probability $\pi (x)$ can be solved and the performance metrics of the secondary system including its network capacity, spectrum utilization, blocking probability and forced termination probability can be further simulated. We use MATLAB to solve the Markov model and to run simulations. On this basis, four study cases are presented and the channel assembling performance based on the proposed Ps-rq strategy is analyzed.

• Case 1: capacity of secondary network vs. arrival rate of PUs

Figure 13 shows the relationship between the network capacity of $S{U}_e$ and $S{U}_r$ and the arrival rate of PUs. As can be seen from the figure, the capacity of the secondary network decreases with the increase in the arrival rate of PUs. When the number of PUs entering the system increases, the absolute data access priority will reduce the total number of available remaining channels in the system, so the number of SUs who can accept the services in the system will also decrease accordingly. As a result, the network capacity of $S{U}_e$ and $S{U}_r$ decreases with the increase in the arrival rate of PUs.

However, it can be seen from Figure 13 that increasing the total capacity of the queue of SUs increases the capacity of the secondary network. At ${\lambda }_p=0.2$, the capacity of the queue of $S{U}_e$, $Q_{eq}$, increases from 0 to 4, and the secondary network capacity increases by 4.8%; the capacity of the queue of $S{U}_r$, $Q_{rq}$, increases from 0 to 4, which is a 9.43% increase in secondary network capacity. After the queue capacity is increased, the number of SU flows that can coexist in the system will increase, thus increasing the probability that SUs will be able to accept the service. By comparing Figure 13, it is not difficult to find that the gradient descent of the network capacity of $S{U}_r$ is less than that of $S{U}_e$. This is because $S{U}_r$ has a higher service priority than $S{U}_e$ (except for the interrupted $S{U}_e^h$), so it is easier to start the service for the newly arrived $S{U}_r$ than $S{U}_e$.

• Case 2: spectrum utilization vs. arrival rate of SU_e

The relationship between spectrum utilization and the arrival rate of SUs is shown in Figure 14 with the arrival rate of $S{U}_e$ as the x-axis. It can be seen from the figure that the spectrum utilization increases with the increase in the arrival rate. When $Q_{rq}=2, Q_{eq}=4$, the spectrum utilization rate is up to 92.3%. With the increase in the arrival rate of SUs, the idle spectrum is fully utilized. When the spectrum utilization becomes saturated gradually, the gradient will decrease. In addition, it is not difficult to find that increasing the queue capacity can increase the secondary network capacity, so spectrum utilization is correspondingly improved.

• Case 3: blocking probability vs. arrival rate of PUs

Figure 15 shows the relationship between the blocking probability of SUs and the arrival rate of PUs. In general, the blocking probability of SUs increases with the arrival rate of PUs. The idle channels are fully utilized when the number of the PUs in the system increases due to the absolute data access priority, resulting in the increase in the blocking probability of SUs. Since $S{U}_r$ has a higher priority than $S{U}_e$, the blocking probability of data access is lower than that of $S{U}_e$.

It appears that both curves are relatively stable, and the difference in blocking probability is no more than 0.1. It is illustrative of the fact that the proposed Ps-rq strategy can make $S{U}_e$ and $S{U}_r$ have a stable performance on the basis of reducing the blocking probability of high-priority users $S{U}_e^h$. The adoption of differentiated services provides a more equitable compromise scheduling without causing over-starvation of low-priority users $\left(S{U}_e\right)$. However, the available idle spectrum existing in the secondary system decreases with the increase in the arrival rate of PUs, resulting in the blocking probability of $S{U}_r$ and $S{U}_e$ gradually approaching 1.

By comparing the blocking probability of $S{U}_e$ and $S{U}_r$ under different queue capacities, we can find that a larger capacity improves the reduction of blocking probability. For example, for $S{U}_r$, when the arrival rate of $PU$ is ${\lambda }_p=0.6$, the blocking probability is reduced by 10.82% when the queue capacity is $Q_{rq}=2, Q_{eq}=4$ compared with $Q_{rq}=0, Q_{eq}=0$. And for $S{U}_e$, when ${\lambda }_p=0.6$, the blocking probability is reduced by 11.04% under the same value. This explains that the system can provide services for more flows with the increase in the queue capacity, which provides more possibilities for the speculative access of SUs. By comparison, it can be observed that the increase in queue capacity can significantly improve the performance of $S{U}_e$. The reason is that a feedback loop is set for the service of $S{U}_e$ in the proposed Ps-rq strategy and a larger queue capacity is installed at the system setup.

• Case 4: forced termination probability vs. arrival rate of PUs

Figure 16 shows the relationship between the forced termination probability of SUs and the arrival rate of PUs. As is depicted in the figure, the forced termination probability of SUs increases with the arrival rate of PUs, which is also determined by the data access priority of PUs. Since a reserved queue is set for the interrupted $S{U}_e^h$, the forced termination probability of $S{U}_e^h$ is the lowest in Figure 16, such as at the maximum difference value, ${\lambda }_p=0.8$, when $Q_{rq}=1, Q_{eq}=2, Q_{hq}=1$, the forced termination probability of $S{U}_e^h$ is 29.23% lower than that of $S{U}_e$ and 42.5% lower than that of $S{U}_r$. However, due to the setting of the feedback loop for $S{U}_e$, $S{U}_r$ has a higher forced termination probability than $S{U}_e$. Therefore, the interrupted $S{U}_e$ can go back to the queue and will not be forced to terminated until the queue is full.

Real-time services are more sensitive to delay, and packet loss caused by slight jitter does not cause serious interference to real-time voice/image quality. Therefore, no feedback loop is adopted for $S{U}_r$. The total channel occupancy rate can be reduced in this way, but at the expense of increasing the forced termination probability of $S{U}_r$. Taken overall, the proposed Ps-rq strategy can differentiate the service priorities of heterogeneous services and provide fair scheduling for them. On this basis, the service quality of the high-priority elastic service, $S{U}_e^h$, that is interrupted by PUs can be improved.

To further demonstrate the superiority of the proposed strategy in enhancing the performance of SUs, a comparison is provided between the Ps-rq strategy and the non-Ps-rq strategy regarding forced termination probability, as depicted in Figure 17. The comparison results all occurred under the initial conditions $Q_{eq}$ = 2 and $Q_{rq}$ = 1. As can be seen from the figure, whether for $S{U}_e$ or $S{U}_r$, when the strategy is not adopted, the probability of forced termination for SUs increases significantly. The performance analysis of $S{U}_e^h$ is specific to the strategy proposed in this paper; thus, there is no comparison of the forced termination probability of $S{U}_e^h$ when not adopting Ps-rq. The figure illustrates that the increase in the forced termination probability for $S{U}_e$ is greater than that for $S{U}_r$ in the absence of Ps-rq. Especially when the arrival rate of PU is small, $S{U}_e$ has more opportunities to access the channel. It is evident from the proposed strategy that the excellent result can be attributed to the improved performance of $S{U}_e^h$.

6. Conclusions

This study designs and proposes a priority scheduling strategy with reserved queue (Ps-rq strategy) for CRNs. This strategy adopts the service classification scheme combining static factors (message type) and dynamic factors (information validity, message correlation and message size), which can differentiate the elastic services with high-priority based on the heterogeneous service classification in secondary services. By setting a new reserved queue for vital elastic services, the proposed strategy can improve the service priority of high-priority elastic services interrupted by PUs. In addition, Takagi–Sugeno fuzzy control is used to prioritize elastic services, and a 3D priority view of elastic services is presented. On this basis, a proportional channel allocation scheme with priority that is combined with channel assembling technology and spectrum adaptation technology is provided based on the Ps-rq strategy. By dividing user activities in CRNs into four events including PU arrival, PU departure, SU arrival and SU departure, the flow of resources in the secondary network is investigated, and the dynamic channel access process based on the Ps-rq strategy is given. By establishing the CTMC, the resource flow process in the secondary network is mapped to the high-dimensional state space, and all possible transition states, transition rates and transition conditions triggered by the four kinds of events that start from a general state are derived. After downgrading the high-dimensional markers, the steady-state solution of the system can be obtained by acquiring the transition rate matrix, and a series of performance evaluation indexes such as secondary network capacity, spectrum utilization, blocking probability and forced termination probability are further obtained.

Furthermore, we compare the proposed Ps-rq strategy with four channel assembling schemes to demonstrate its suitability for specific scenarios involving the transmission of elastic traffic. By combining the procedure for dimension reduction of high-dimensional Markov chains, we numerically simulate the analytical results of performance indexes using MATLAB and provide four study cases. The simulation results indicate that, in the same dimension, as the buffer capacity increases, the $S{U}_e$’s network capacity increases by 4.8% and the blocking probability decreases by 11.04%, while the $S{U}_r$’s network capacity increases by 9.43% and the blocking probability decreases by 10.82%. And the forced termination probability of high-priority elastic services concerned in this paper is 29.23% lower than that of $S{U}_e$ and 42.5% lower than that of $S{U}_r$. Numerical results show that the proposed strategy can help improve the service priority of high-priority elastic services that are interrupted by PUs in CRNs and reduce the forced termination probability on the basis of providing fair scheduling.