A communication failure and repair mechanism with adjustable transmission rates for PU packets in CRNs

Abstract

As one key technology of future radio communication networks, cognitive radio networks (CRNs) can effectively handle the shortage of network resources. Two classes of users exist in traditional CRNs, namely, primary users (PUs) with higher priority and secondary users (SUs) with cognitive ability. In CRNs, during the communication process, packets need to travel through various servers, such as switches and routers, and these devices may fail at any time. We consider this type of problem to be a communication failure. The occurrence of communication failures trigger system repairs, which make the system return to regular work. In this paper, we present a communication failure and repair mechanism with adjustable transmission rates for PU packets in CRNs. We assume that PU packets can maintain low-speed transmission during the system failure state and resume high-speed transmission after the failure is repaired. We establish a three-dimensional Markov chain (3DMC) and build a queueing model based on discrete time. Through numerical experiments, we analyze some indicators’ impact on the system capability. In addition, compared with the traditional communication failure and repair mechanism, our proposed mechanism can reduce the blocking rate while considerably increasing the throughput of data packets.

1. Introduction

With rapid economic and technological progress, numerous applications need radio technology, and the demand for radio spectrums is increasing. To make effective use of wasted spectrums, cognitive radio networks (CRNs) have received more research and development [[1], [2], [3]]. Sorted by network access priority, CRNs mainly include two types of users: primary users (PUs) and secondary users (SUs) [4,5]. Cognitive radio (CR) can minimize the interference ratio between the two types of users in an effective way to make better use of the available spectrums [6]. PUs have higher channel usage rights so that they can preempt the spectrums of SUs. SUs dynamically access the spectrum without interrupting the transmission of PUs [[7], [8], [9], [10]]. This opportunistic occupancy mechanism of SUs in spectrum allocation strategies can greatly reduce spectrum holes and avoid wasting spectrum resources.

To enhance the performance of communication in CRNs, relevant researchers have conducted various deep studies on spectrum allocation strategies. In many studies, scholars assume that the communication process is failure free. In fact, it is impossible to have a completely failure-free situation. In CRNs, messages are passed between users through data packets. Devices, such as servers, switches and routers that the data packets pass through may fail at any time. In addition, some inherent unstable factors in the process of radio transmission can easily lead to a decline in system performance.

When considering communication failure, the more representative failure is a server failure, that is, the users cannot accept the corresponding service. In Ref. [11], the CRN system had two nonindependent subsystems, which were used to process requests from PUs and SUs, corresponding to the first service unit and the second service unit, respectively. The authors assumed that only the secondary server would fail randomly and discussed the change trend of system performance affected by the server failure. In Ref. [12], the authors considered a single-channel CRN with random service interruptions. Random service interruption means that failures happen randomly. The authors analyzed the performance of CR links subjected to periodic failures and outages through numerical experiments.

As mentioned above, it is very suitable to use queueing theory to solve network modelling problems and study spectrum allocation strategies for CRNs. Currently, the development of queueing theory is widely used in science, technology and engineering development, which is helpful for researching and implementing advanced technical design [13]. Queueing theory plays an important role in analyzing the throughput, blocking rate and other performance indicators for network users in CRNs [14].

In addition, some queueing models with variable service rates have been studied. In Ref. [15], the authors added a work vacation to an M/M/1 queue. The server maintained a lower service rate operation, rather than stopping work outright during the work vacation. In Ref. [16], exponential work vacations and vacation interruptions are applied to the M/G/1 queue. During vacations, the servers can work at a low speed, and if there are users in the system when the service is completed, the server will return to normal levels even if the vacation is not over yet. In Ref. [17], the authors analyzed the equilibrium strategy behavior of customers when there are Poisson failures and repairs in M/M/1 queues. When a partial failure appears, users cannot enter the system. The system continued to provide services to users already in the system at a low speed rather than completely stop working. The above studies show that the low-speed operation of the server in working vacations can effectively improve the service efficiency.

Conclusively, to enhance the performance of CRNs with failure, we propose a communication failure and repair mechanism with adjustable transmission rates for PU packets. References [[15], [16], [17]] illustrate that changing the service rate can be flexibly applied to the queueing model of the system. Therefore, according to the higher priority of PUs, we assume that PU packets can maintain a low-speed transmission rate during the system failure state so that PU packets can keep the transmission as much as possible, and it is restored to high-speed transmission immediately after the failure is repaired. The priority and real-time requirements of SU packets in CRNs are lower than those of PU packets, so SU packets' transmission will mainly be affected when a failure occurs. In Ref. [18], the authors summarized the source and classification of the failure of CRNs in detail and defined the failure rate as the number of failures in a specified period. In Ref. [19], the authors mentioned that when a CRN system failed, one solution was to terminate the service to the SUs forcibly. Therefore, in this paper, we assume that the SU packets entering the channel will randomly encounter failure, and these SU packets will be interrupted in transmission and enter the cache to wait at the same time. In addition, discrete time queueing systems are often applied in practice, especially in radio and communication networks [20,21]. According to the characteristics of today's digital development [22], we consider building a queueing model based on discrete time and solving the state transition probability matrices to dissect the proposed mechanism.

The remaining content of the paper consists of the following sections. Section 2 expounds the proposed communication failure and repair mechanism. According to the proposed mechanism, we build the system model and list the system's state transition probability matrices. Section 3 analyses and establishes some performance indicators. Section 4 depicts the change trend of each performance indicator with some variables through numerical experiments. We also compare the performance of our communication failure and repair mechanism with adjustable transmission rates for PU packets to the traditional failure and repair mechanism in CRNs with numerical results. Section 5 summarizes the full text.

2. System model

2.1. Communication failure and repair mechanism

There are PU packets and SU packets in the system to be transmitted in a single channel. PU packets that can preempt the channel have higher channel usage rights than SU packets. In each time slot, we assume that at most one packet enters the system, and the channel transmits one packet every time. In addition, the usage of the authorized channel should be detected first if an SU has packets to transmit. The state of the authorized channel and the information of the SU constitute a state information table. After receiving the state information table, the central controller will make a decision to determine whether the SU packet can access the authorized channel [23,24].

We propose a communication failure and repair mechanism. We stipulate that the system is in a normal state when no failure occurs at this time, and the system is in a failure state when there is a communication failure.

Considering the lower priority of SUs, we set a cache space for SU packets that are arranged according to the time order of arrivals in the cache. This means that the first packet arriving in the cache ranks first at the cache queue [25]. When the system state is normal, the SU packet ranking first in the cache enters the idle channel if there is no newly arrived PU packet. When the cache space is full, SU packets that need to enter the cache will be blocked. No cache is deployed for PU packets due to their transmission timeliness.

Fig. 1 shows the transmission actions of PU packets under the proposed communication failure and repair mechanism.

Fig. 1.

Transmission actions of PU packets under the communication failure and repair mechanism.

On the basis of the CRNs' working principles, the behavior of the system shown in Fig. 1 is expressed as follows. When a PU packet arrives in the system, the central controller in the system detects the channel to determine whether there are PU or SU packets in the channel. When the channel is empty, a newly arrived PU packet can enter the channel directly. The newly arrived PU packet leaves the system and is judged to be blocked if there is another PU packet in the channel. If an SU packet is in the channel, the newly arrived PU packet preempts the channel and conducts the transmission, and the SU packet whose transmission is interrupted enters the cache and waits. The reason for the above system behavior is that compared with SU packets, PU packets have higher channel usage right in CRNs, and PU packets can interrupt the transmission of SU packets to guarantee the quality of their transmission. In addition, we assume that a low transmission speed is maintained for PU packets when the system fails with the purpose of satisfying the relatively high real-time demands for the PU packets entering the channel. High-speed transmission is endowed to PU packets when the system is in a normal state. The central controller checks the failure and determines the PU packets’ transmission rate at the end of each time slot until the transmission is complete.

For SUs, during a normal state, the idle channel is allocated by the central controller to the incoming SU packet. Furthermore, when there is another packet in the channel, the central controller needs to check the status of the cache to determine whether the newly arrived SU packet can enter the cache. If a failure occurs, the system will turn into the failure state. The normal transmission of an SU packet will be suspended. During a failure state, the interrupted SU packets and newly arrived SU packets enter the cache. The system turns into the normal state after the failure is repaired, and the unfinished SU packets that rank first in the cache enter the channel.

2.2. Model building

In the system, the channel is abstracted as the service desk, PU packets are abstracted as the first type of customers, and SU packets are abstracted as the second type of customers [26]. Compared with SU packets, PU packets have higher channel usage rights. Meanwhile, the time axis is compartmentalized into equivalent slots. An early arrival discrete-time queueing model based on a single service desk with two types of customers is established. In this queueing model, the size of the SU packet cache is set to k.

We assume that the communication failure intervals and the system repair time both obey geometric distributions. The communication failure rate is p ($\bar{p}$ = 1 − p, 0 < p < 1) and the system repair rate is q ($\bar{q}$ = 1 − q, 0 < q < 1). The arrival interval and transmission time of data packets are independent of each other. The arrival intervals and the transmission time of packets all obey geometric distributions. The arrival rates of the two types of packets are λ₁ ($\bar{\lambda }$₁ = 1 − λ₁, 0 < λ₁ < 1) and λ₂ ($\bar{\lambda }$₂ = 1 − λ₂, 0 < λ₂ < 1). The transmission rates of the two types of packets are μ₁ ($\bar{\mu }$₁ = 1 − μ₁, 0 < μ₁ < 1) and μ₂ ($\bar{\mu }$₂ = 1 − μ₂, 0 < μ₂ < 1). Specially, the transmission time of PU packets follow a geometric distribution with a lower rate μ₃ ($\bar{\mu }$₃ = 1 − μ₃, 0 < μ₃ < 1) during the repair time.

We assume that the packets arrive at the beginning of timeslot (t, t⁺) and leave at the end of timeslot (t⁻, t). Suppose that at t⁺, the total number of packets in the system is C_t, the amount of PU packets in the system is S_t, and the system state is W_t. (W_t is 0 when the system state is normal and W_t is 1 when the system state is failure); then, {(C_t, S_t, W_t): t ≥ 0} composes a three-dimensional Markov chain (3DMC). The state space R is represented as follows:

$$\mathit{R}=\left\{\left(\mathrm{0,0},c\right):c=\mathrm{0,1}\right\}\cup \left\{\left(a,b,c\right):a=\mathrm{1,2},\cdots ,k+1,b=\mathrm{0,1},c=\mathrm{0,1}\right\}$$ (1)

In Eq. (1), a represents that there are a user packets in the current system, where b are PU packets, and c represents the system state.

2.3. Model analysis

It is specified that the system is in w state when there are w packets in it. According to the cache capacity k of SUs, the minimum system state is 0, and the maximum system state is k+1. In the system, Q_w,v is the probability matrix transferring from the w state to the v state, the total state transition probability matrix Q is represented in the form of matrix blocks in Eq. (2), and each matrix block is a one-step transition probability submatrix.

$$\mathit{Q}=\left(\begin{array}{cccccc}{\mathit{Q}}_{\mathrm{0,0}}& {\mathit{Q}}_{\mathrm{0,1}}& {\mathit{Q}}_{\mathrm{0,2}}& & & \\ {\mathit{Q}}_{\mathrm{1,0}}& {\mathit{Q}}_{\mathrm{1,1}}& {\mathit{Q}}_{\mathrm{1,2}}& {\mathit{Q}}_{\mathrm{1,3}}& & \\ & {\mathit{Q}}_{\mathrm{2,1}}& {\mathit{Q}}_{\mathrm{2,2}}& {\mathit{Q}}_{\mathrm{2,3}}& {\mathit{Q}}_{\mathrm{2,4}}& \\ & & & \ddots & \ddots & \ddots \\ & & & {\mathit{Q}}_{k,k-1}& {\mathit{Q}}_{k,k}& {\mathit{Q}}_{k,k+1}\\ & & & & {\mathit{Q}}_{k+1,k}& {\mathit{Q}}_{k+1,k+1}\end{array}\right)$$ (2)

The block matrices of Q are expressed in Eqs. (3), (4), (5), (6), (7), (8), (9), (10), (11), (12), (13), (14), respectively.

• (1)If the system state remains at 0, the submatrix is denoted as Q_0,0.

$${\mathbf{Q}}_{\mathrm{0,0}}=\left(\begin{array}{cc}\bar{p}{\bar{\lambda }}_1{\bar{\lambda }}_2& p{\bar{\lambda }}_1{\bar{\lambda }}_2\\ q{\bar{\lambda }}_1{\bar{\lambda }}_2& \bar{q}{\bar{\lambda }}_1{\bar{\lambda }}_2\end{array}\right)$$ (3)

• (2)If the system state changes from 0 to 1, the submatrix is denoted as Q_0,1.

$${\mathbf{Q}}_{\mathrm{0,1}}=\left(\begin{array}{cccc}\bar{p}{\bar{\lambda }}_1{\lambda }_2& p{\bar{\lambda }}_1{\lambda }_2& \bar{p}{\lambda }_1{\bar{\lambda }}_2& p{\lambda }_1{\bar{\lambda }}_2\\ q{\bar{\lambda }}_1{\lambda }_2& \bar{q}{\bar{\lambda }}_1{\lambda }_2& q{\lambda }_1{\bar{\lambda }}_2& \bar{q}{\lambda }_1{\bar{\lambda }}_2\end{array}\right)$$ (4)

• (3)If the system state changes from 0 to 2, the submatrix is denoted as Q_0,2.

$${\mathbf{Q}}_{\mathrm{0,2}}=\left(\begin{array}{cccc}0& 0& \bar{p}{\lambda }_1{\lambda }_2& p{\lambda }_1{\lambda }_2\\ 0& 0& q{\lambda }_1{\lambda }_2& \bar{q}{\lambda }_1{\lambda }_2\end{array}\right)$$ (5)

• (4)If the system state changes from 1 to 0, the submatrix is denoted as Q_1,0.

$${\mathbf{Q}}_{\mathrm{1,0}}=\left(\begin{array}{cc}\bar{p}{\bar{\lambda }}_1{\bar{\lambda }}_2{\mu }_2& p{\bar{\lambda }}_1{\bar{\lambda }}_2{\mu }_2\\ 0& 0\\ \bar{p}{\bar{\lambda }}_1{\bar{\lambda }}_2{\mu }_1& p{\bar{\lambda }}_1{\bar{\lambda }}_2{\mu }_1\\ q{\bar{\lambda }}_1{\bar{\lambda }}_2{\mu }_3& \bar{q}{\bar{\lambda }}_1{\bar{\lambda }}_2{\mu }_3\end{array}\right)$$ (6)

• (5)If the system state changes from w to w−1, the submatrix is denoted as Q_w,w−1, where 2 ≤ w ≤ k.

$${\mathbf{Q}}_{w,w-1}=\left(\begin{array}{cccc}\bar{p}{\bar{\lambda }}_1{\bar{\lambda }}_2{\mu }_2& p{\bar{\lambda }}_1{\bar{\lambda }}_2{\mu }_2& 0& 0\\ 0& 0& 0& 0\\ \bar{p}{\bar{\lambda }}_1{\bar{\lambda }}_2{\mu }_1& p{\bar{\lambda }}_1{\bar{\lambda }}_2{\mu }_1& 0& 0\\ q{\bar{\lambda }}_1{\bar{\lambda }}_2{\mu }_3& \bar{q}{\bar{\lambda }}_1{\bar{\lambda }}_2{\mu }_3& 0& 0\end{array}\right)$$ (7)

• (6)If the system state remains at w, the submatrix is denoted as Q_w,w, where 1 ≤ w ≤ k − 1.

$${\mathbf{Q}}_{w,w}=\left(\begin{array}{cccc}\bar{p}{\bar{\lambda }}_1\left({\lambda }_2{\mu }_2+{\bar{\lambda }}_2{\bar{\mu }}_2\right)& p{\bar{\lambda }}_1\left({\lambda }_2{\mu }_2+{\bar{\lambda }}_2{\bar{\mu }}_2\right)& \bar{p}{\lambda }_1{\bar{\lambda }}_2{\mu }_2& p{\lambda }_1{\bar{\lambda }}_2{\mu }_2\\ q{\bar{\lambda }}_1{\bar{\lambda }}_2& \bar{q}{\bar{\lambda }}_1{\bar{\lambda }}_2& 0& 0\\ \bar{p}{\bar{\lambda }}_1{\lambda }_2{\mu }_1& p{\bar{\lambda }}_1{\lambda }_2{\mu }_1& \bar{p}{\bar{\lambda }}_2\left({\lambda }_1{\mu }_1+{\bar{\mu }}_1\right)& p{\bar{\lambda }}_2\left({\lambda }_1{\mu }_1+{\bar{\mu }}_1\right)\\ q{\bar{\lambda }}_1{\lambda }_2{\mu }_3& \bar{q}{\bar{\lambda }}_1{\lambda }_2{\mu }_3& q{\bar{\lambda }}_2\left({\lambda }_1{\mu }_3+{\bar{\mu }}_3\right)& \bar{q}{\bar{\lambda }}_2\left({\lambda }_1{\mu }_3+{\bar{\mu }}_3\right)\end{array}\right)$$ (8)

• (7)If the system state changes from w to w+1, the submatrix is denoted as Q_w,w+1, where 1 ≤ w ≤ k − 1.

$${\mathbf{Q}}_{w,w+1}=\left(\begin{array}{cccc}\bar{p}{\bar{\lambda }}_1{\lambda }_2{\bar{\mu }}_2& p{\bar{\lambda }}_1{\lambda }_2{\bar{\mu }}_2& \bar{p}{\lambda }_1\left({\lambda }_2{\mu }_2+{\bar{\lambda }}_2{\bar{\mu }}_2\right)& p{\lambda }_1\left({\lambda }_2{\mu }_2+{\bar{\lambda }}_2{\bar{\mu }}_2\right)\\ q{\bar{\lambda }}_1{\lambda }_2& \bar{q}{\bar{\lambda }}_1{\lambda }_2& q{\lambda }_1{\bar{\lambda }}_2& \bar{q}{\lambda }_1{\bar{\lambda }}_2\\ 0& 0& \bar{p}{\lambda }_2\left({\lambda }_1{\mu }_1+{\bar{\mu }}_1\right)& p{\lambda }_2\left({\lambda }_1{\mu }_1+{\bar{\mu }}_1\right)\\ 0& 0& q{\lambda }_2\left({\lambda }_1{\mu }_3+{\bar{\mu }}_3\right)& \bar{q}{\lambda }_2\left({\lambda }_1{\mu }_3+{\bar{\mu }}_3\right)\end{array}\right)$$ (9)

• (8)If the system state changes from w to w+2, the submatrix is denoted as Q_w,w+2, where 1 ≤ w ≤ k − 1.

$${\mathbf{Q}}_{w,w+2}=\left(\begin{array}{cccc}0& 0& \bar{p}{\lambda }_1{\lambda }_2{\bar{\mu }}_2& p{\lambda }_1{\lambda }_2{\bar{\mu }}_2\\ 0& 0& q{\lambda }_1{\lambda }_2& \bar{q}{\lambda }_1{\lambda }_2\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$$ (10)

• (9)If the system state remains at k, the submatrix is denoted as Q_k,k.

$${\mathbf{Q}}_{k,k}=\left(\begin{array}{cccc}\bar{p}{\bar{\lambda }}_1\left({\lambda }_2{\mu }_2+{\bar{\lambda }}_2{\bar{\mu }}_2\right)& p{\bar{\lambda }}_1\left({\lambda }_2{\mu }_2+{\bar{\mu }}_2\right)& \bar{p}{\lambda }_1{\bar{\lambda }}_2{\mu }_2& p{\lambda }_1{\bar{\lambda }}_2{\mu }_2\\ q{\bar{\lambda }}_1{\bar{\lambda }}_2& \bar{q}{\bar{\lambda }}_1& 0& 0\\ \bar{p}{\bar{\lambda }}_1{\lambda }_2{\mu }_1& p{\bar{\lambda }}_1{\lambda }_2{\mu }_1& \bar{p}{\bar{\lambda }}_2\left({\lambda }_1{\mu }_1+{\bar{\mu }}_1\right)& p{\bar{\lambda }}_2\left({\lambda }_1{\mu }_1+{\bar{\mu }}_1\right)\\ q{\bar{\lambda }}_1{\lambda }_2{\mu }_3& \bar{q}{\bar{\lambda }}_1{\lambda }_2{\mu }_3& q{\bar{\lambda }}_2\left({\lambda }_1{\mu }_3+{\bar{\mu }}_3\right)& \bar{q}{\bar{\lambda }}_2\left({\lambda }_1{\mu }_3+{\bar{\mu }}_3\right)\end{array}\right)$$ (11)

• (10)If the system state changes from k to k+1, the submatrix is denoted as Q_k,k+1.

$${\mathbf{Q}}_{k,k+1}=\left(\begin{array}{cccc}\bar{p}{\bar{\lambda }}_1{\lambda }_2{\bar{\mu }}_2& 0& \bar{p}{\lambda }_1\left({\lambda }_2{\mu }_2+{\bar{\mu }}_2\right)& p{\lambda }_1\left({\lambda }_2{\mu }_2+{\bar{\mu }}_2\right)\\ q{\bar{\lambda }}_1{\lambda }_2& 0& q{\lambda }_1& \bar{q}{\lambda }_1\\ 0& 0& \bar{p}{\lambda }_2\left({\lambda }_1{\mu }_1+{\bar{\mu }}_1\right)& p{\lambda }_2\left({\lambda }_1{\mu }_1+{\bar{\mu }}_1\right)\\ 0& 0& q{\lambda }_2\left({\lambda }_1{\mu }_3+{\bar{\mu }}_3\right)& \bar{q}{\lambda }_2\left({\lambda }_1{\mu }_3+{\bar{\mu }}_3\right)\end{array}\right)$$ (12)

• (11)If the system state changes from k+1 to k, the submatrix is denoted as Q_k+1,k.

$${\mathbf{Q}}_{k+1,k}=\left(\begin{array}{cccc}\bar{p}{\bar{\lambda }}_1{\bar{\lambda }}_2{\mu }_2& p{\bar{\lambda }}_1& 0& 0\\ 0& 0& 0& 0\\ \bar{p}{\bar{\lambda }}_1{\bar{\lambda }}_2{\mu }_1& p{\bar{\lambda }}_1{\mu }_1& 0& 0\\ q{\bar{\lambda }}_1{\bar{\lambda }}_2{\mu }_3& \bar{q}{\bar{\lambda }}_1{\mu }_3& 0& 0\end{array}\right)$$ (13)

• (12)If the system state remains at k+1, the submatrix is denoted as Q_k+1,k+1.

$${\mathbf{Q}}_{k+1,k+1}=\left(\begin{array}{cccc}\bar{p}{\bar{\lambda }}_1\left({\lambda }_2{\mu }_2+{\bar{\mu }}_2\right)& 0& \bar{p}{\lambda }_1& p{\lambda }_1\\ 0& 0& 0& 0\\ \bar{p}{\bar{\lambda }}_1{\lambda }_2{\mu }_1& 0& \bar{p}\left({\lambda }_1{\mu }_1+{\bar{\mu }}_1\right)& p\left({\lambda }_1{\mu }_1+{\bar{\mu }}_1\right)\\ q{\bar{\lambda }}_1{\lambda }_2{\mu }_3& 0& q\left({\lambda }_1{\mu }_3+{\bar{\mu }}_3\right)& \bar{q}\left({\lambda }_1{\mu }_3+{\bar{\mu }}_3\right)\end{array}\right)$$ (14)

It can be seen in Q that the 3DMC {(C_t, S_t, W_t): t ≥ 0} is aperiodic, irreducible and normally recurrent. Remember that π_a,b,c is the steady-state distribution of the 3DMC. π_a,b,c is expressed in Eq. (15).

$${\pi }_{a,b,c}=\underset{t\to \infty }{\lim }P\left\{C_t=a,S_t=b,W_t=c\right\},\left(a,b,c\right)\in \mathit{R}$$ (15)

The steady-state probability vector is expressed as Π_a when the system state is a, then:

$$\{\begin{array}{c}\left({\mathbf{\Pi }}_0,{\mathbf{\Pi }}_1,\cdots ,{\mathbf{\Pi }}_k,{\mathbf{\Pi }}_{k+1}\right)\mathit{Q}=\left({\mathbf{\Pi }}_0,{\mathbf{\Pi }}_1,\cdots ,{\mathbf{\Pi }}_k,{\mathbf{\Pi }}_{k+1}\right)\\ \left({\mathbf{\Pi }}_0,{\mathbf{\Pi }}_1,\cdots ,{\mathbf{\Pi }}_k,{\mathbf{\Pi }}_{k+1}\right)\mathit{e}=1\end{array}$$ (16)

In Eq. (16), e is a column vector, and all numbers in this column vector are 1. We can obtain the results of Π by numerical calculation.

3. Performance indicators

3.1. Performance indicators for PU packets

PU packets' blocking means the situation where arriving PU packets cannot get into the channel. The amount of PU packets blocked per timeslot is called PU packets’ blocking rate B_p, which is expressed in Eq. (17).

$$B_p=\sum _{i=1}^{k+1}\left({\pi }_{i,\mathrm{1,0}}{\lambda }_1{\bar{\mu }}_1+{\pi }_{i,\mathrm{1,1}}{\lambda }_1{\bar{\mu }}_3\right)$$ (17)

PU packets' throughput T_p refers to the number of successfully transmitted PU packets per timeslot, which is expressed in Eq. (18).

$$T_p={\lambda }_1-B_p$$ (18)

3.2. Performance indicators for SU packets

SU packets' blocking means the situation where incoming SU packets cannot enter the channel and the cache is also full. The corresponding SU packets' blocking rate B_s represents the quantity of SU packets blocked per timeslot, which is expressed in Eq. (19).

$$B_s={\pi }_{k,\mathrm{0,0}}{\lambda }_2\left(\bar{p}{\lambda }_1{\bar{\mu }}_2+p{\bar{\mu }}_2\right)+{\pi }_{k,\mathrm{0,1}}{\lambda }_2\left(\bar{q}+q{\lambda }_1\right)+{\pi }_{k+\mathrm{1,0,0}}{\lambda }_2\left[p+\bar{p}\left({\bar{\mu }}_2+{\lambda }_1{\mu }_2\right)\right]+{\pi }_{k+\mathrm{1,1,0}}{\lambda }_2\left[p+\bar{p}\left({\bar{\mu }}_1+{\lambda }_1{\mu }_1\right)\right]+{\pi }_{k+\mathrm{1,1,1}}{\lambda }_2\left[\bar{q}+q\left({\bar{\mu }}_3+{\lambda }_1{\mu }_3\right)\right]$$ (19)

SU packets' interruption includes two cases, both of which can cause transmission interruptions of ongoing SU packets. One case is the preemption of the channel by PU packets, and the other is the occurrence of system failures. If the cache reserved by the system is full, the SU packets whose transmissions are interrupted can only leave the system. We assume that SU packets' preemption loss rate L_s1 represents the amount of SU packets lost because of preemptive interruptions per timeslot, which is expressed in Eq. (20), and SU packets' failure loss rate L_s2 represents the number of SU packets lost because of failure interruptions per timeslot, which is expressed in Eq. (21).

$$L_{s1}={\pi }_{k+\mathrm{1,0,0}}\bar{p}{\lambda }_1{\bar{\mu }}_2$$ (20)

$$L_{s2}={\pi }_{k+\mathrm{1,0,0}}p{\bar{\mu }}_2$$ (21)

SU packets' throughput T_s refers to the amount of SU packets successfully transmitted per timeslot, which is expressed in Eq. (22).

$$T_s={\lambda }_2-B_s-L_{s1}-L_{s2}$$ (22)

4. Experimental results

Through experiments, we analyze the performance of the proposed communication failure and repair mechanism more clearly and intuitively. The numerical experiment corresponding to each performance indicator is carried out in MATLAB. Some fixed parameters are set as follows: PU packets' normal transmission rate is μ₁ = 0.7, SU packets' transmission rate is μ₂ = 0.6, and PU packets' failure transmission rate is μ₃ = 0.3. We consider that a CRN system with too high of a communication failure rate p or too low of a system repair rate q is meaningless for practical application, so we set p in lower levels and q in higher levels in the following experiments.

4.1. Performance analysis for PU packets

Influenced by the communication failure rate p, the change curve of PU packets' blocking rate B_p is displayed in Fig. 2, and the change in PU packets’ throughput T_p is displayed in Fig. 3.

Fig. 2.

The change trend in PU packets' blocking rate B_p.

Fig. 3.

The change trend in PU packets' throughput T_p.

Through Figs. 2 and 3, we can observe that with the increase of communication failure rate p, PU packets' blocking rate B_p increases and PU packets’ throughput T_p declines. As p increases, the time for PU packets to maintain a low transmission rate increases, and the time for PU packets to stay in the channel correspondingly shows an upwards trend. As the number of newly arrived blocked PU packets grows, B_p will increase, and T_p will naturally decrease.

It can be seen in Figs. 2 and 3 that when the system repair rate q increases, the PU packets’ blocking rate B_p will decrease and throughput T_p will increase. This is because as q increases, the time that the system state is normal increases, the transmission speed of PU packets is accelerated, and the residence time of PU packets in the channel reduces, so PU packets will encounter less blocking and the corresponding throughput will increase.

In addition, in Figs. 2 and 3, the PU packets' blocking rate B_p and throughput T_p will both increase when the PU packets' arrival rate λ₁ grows. This is because blocked PU packets increase with incoming PU packets. The number of PU packets arriving increases, and the behavior of preempting the channel of SU packets increases correspondingly, so more PU packets are transmitted successfully, and T_p will increase. In addition, the SU packets’ arrival rate λ₂ has no effect on B_p and T_p. This is because for a PU packet, whether there is an SU packet in the channel has no impact on its transmission.

4.2. Performance analysis for SU packets

The change trend of SU packets' blocking rate B_s with communication failure rate p is shown in Fig. 4. It can be seen in Fig. 4 that when the communication failure rate increases, the SU packets' blocking rate B_s increases. As p grows, the probability of SU packets coming into the cache increases. Due to limited cache space, more newly arrived SU packets will be blocked.

Fig. 4.

The change trend in SU packets' blocking rate B_s.

Fig. 4 shows that when the system repair rate q increases, the SU packets' blocking rate B_s decreases. As q grows, the system repair time decreases, and the possibility for SU packets entering the channel to be transmitted increases, so B_s decreases.

Fig. 4 shows that the SU packets' blocking rate B_s increases with the PU packets' arrival rate λ₁. A larger λ₁ means that the possibility that the channel is occupied by PU packets will grow; therefore, more SU packets depart from the system because they are blocked. In addition, as SU packets' arrival rate λ₂ grows, B_s will also increase. This is because the capacity of the cache is stable, and a larger λ₂ means that more SU packets cannot enter the channel. These SU packets can only be blocked and leave the system.

Figs. 5 and 6 show the change trends of SU packets' preemptive loss rate L_s1 and SU packets' failure loss rate L_s2 with communication failure rate p.

Fig. 5.

The change trend in SU packets' preemptive loss rate L_s1.

Fig. 6.

The change trend in SU packets' failure loss rate L_s2.

As Figs. 5 and 6 show, when the communication failure rate p increases, we find that SU packets' preemptive loss rate L_s1 and SU packets' failure loss rate L_s2 both show upwards trends. As p increases, the number of SU packets whose transmissions are suspended by failures increases; thus, as the cache space decreases, the SU packets that cannot enter the cache will increase.

In Figs. 5 and 6, it was found that when the system repair rate q increases, SU packets' preemptive loss rate L_s1 and SU packets' failure loss rate L_s2 of SU packets both show downwards trends. As q increases, the free cache space will increase as SU packets in it can return to the channel. Therefore, more SU packets whose transmissions are interrupted can enter the cache, and the loss rates will decrease.

It can be seen in Figs. 5 and 6 that SU packets' preemptive loss rate L_s1 and SU packets' failure loss rate L_s2 both show upwards trends when PU packets' arrival rate λ₁ increases. As λ₁ increases, the possibility of PU packets preempting the channel grows, and more SU packets will be lost. The chance of SU packets coming into the cache because of PU packets' preemption increases, and then the cache space is reduced, so more SU packets whose transmissions are interrupted due to failures cannot enter the cache and can only leave the system. In addition, when the SU packets' arrival rate λ₂ increases, the SU packets' preemptive loss rate L_s1 and the SU packets' failure loss rate L_s2 both increase. A higher λ₂ means a greater possibility of SU packets entering the channel, so SU packets will encounter more channel preemptions or failures, and the resulting losses will also increase accordingly.

Fig. 7 shows the change trend of SU packets' throughput T_s with communication failure rate p. It can be seen in Fig. 7 that when the communication failure rate p increases, the throughput of SU packets declines. As p increases, the time that SU packets occupy the channel drops, and fewer SU packets complete the transmission.

Fig. 7.

The change trend in SU packets' throughput T_s.

As shown in Fig. 7, as the system repair rate q shows an upwards trend, SU packets' throughput T_s increases. SU packets can enter the channel only in the normal system state. Therefore, as q increases, the time for maintaining the normal system state increases, and the probability for SU packets to receive normal transmission also increases; then, T_s increases.

In Fig. 7, SU packets' throughput T_s will decrease as PU packets' arrival rate λ₁ increases. As λ₁ increases, more SU packets are lost because of being preempted the channel, so fewer SU packets are transmitted successfully. In addition, Fig. 7 shows that T_s shows an upwards trend when the SU packets' arrival rate λ₂ increases. More incoming SU packets will increase the time that the channel takes to transmit SU packets; therefore, more SU packets complete transmission.

4.3. System performance comparison

Through numerical results, we contrast the representation of the proposed communication failure and repair mechanism with adjustable transmission rates for PU packets to the traditional failure and repair mechanism in CRNs. In the traditional mechanism, when a communication failure occurs, the transmission of user packets in the channel will stop, and packets are not allowed to enter the system. In our mechanism, for the purpose of ensuring timeliness and transmission quality, PU packets can still maintain a low-speed transmission rate when in a failure system state. In addition, SU packets can enter the cache to wait. Some fixed parameters are set as follows: λ₁ = 0.3, λ₂ = 0.3, μ₃ = 0.3, and q = 0.9.

Figs. Fig. 8, Fig. 9, Fig. 10, Fig. 11 demonstrate the performance comparison in PU packets' throughput T_p, PU packets' blocking rate B_p, SU packets' throughput T_s and SU packets' blocking rate B_s.

Fig. 8.

Performance comparison in PU packets' throughput T_p.

Fig. 9.

Performance comparison in PU packets' blocking rate B_p.

Fig. 10.

Performance comparison in SU packets' throughput T_s.

Fig. 11.

Performance comparison in SU packets' blocking rate B_s.

In Figs. Fig. 8, Fig. 9, Fig. 10, Fig. 11, we can see that compared with the traditional failure and repair mechanism, our mechanism effectively reduces the blocking rate of user packets as well as improving the throughput of the system.

In addition, Figs. Fig. 8, Fig. 9, Fig. 10, Fig. 11 compare the performance measures with different transmission rates of user packets. When PU packets' normal transmission rate μ₁ increases, PU packets' blocking rate B_p and SU packets' blocking rate B_s will decline, and PU packets' throughput T_p and SU packets' throughput T_s will increase at the same time. When μ₁ becomes larger, the time for PU packets to complete transmission decreases, the channel is prone to be idle, fewer PU and SU packets are blocked, and more PU and SU packets transmit successfully. As shown in Figs. 8 and 9, since PU packets have a higher channel usage right than SU packets, the changes in μ₂ do not influence B_p and T_p. Moreover, it can also be observed in Figs. 10 and 11 that when SU packets' transmission rate μ₂ increases, B_s decreases and T_s increases. As μ₂ increases, the time taken by SU packets to complete transmission decreases, the chance of the channel remaining idle increases, fewer SU packets are blocked, and more SU packets transmit successfully.

5. Conclusions

In this paper, we present a communication failure and repair mechanism with adjustable transmission rates for PU packets in CRNs. When a failure occurs in the system, PU packets will maintain low-speed transmission, while SU packets will be interrupted and enter the cache. After the failure is repaired, PU packets will resume a high-speed transmission, and the SU packets ranking first in the cache queue can access the channel. First, the modelling analysis was carried out. Then, the formula representations of its performance indicators were established. In addition, the changes in performance indicators with the changes in different parameters were obtained through numerical experiments. By comparing our mechanism with the traditional failure and repair mechanism, it was found that our mechanism can considerably improve the packets' throughput and reduce the packets' blocking rate.

Declarations

Author contribution statement

Yuan Zhao, Qi Lu: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Wrote the paper.

Zhisheng Ye, Kang Chen: Performed the experiments; Analyzed and interpreted the data; Wrote the paper.

Funding statement

This work was supported by the National Natural Science Foundation of China [grant number 61701097] and the Natural Science Foundation of Hebei Province [grant number F2016501073], China.

Data availability statement

Data will be made available on request.

Declaration of interest’s statement

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.