# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2018195

## Performance analysis and optimization for cognitive radio networks with a finite primary user buffer and a probability returning scheme

 1 School of Computer and Communication Engineering, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China 2 Department of Intelligence and Informatics, Konan University, Kobe 658-8501, Japan

* Corresponding author: Yuan Zhao

Received  October 2017 Revised  January 2018 Published  December 2018

In this paper, in order to reduce possible packet loss of the primary users (PUs) in cognitive radio networks, we assume there is a buffer with a finite capacity for the PU packets. At the same time, focusing on the packet interruptions of the secondary users (SUs), we introduce a probability returning scheme for the interrupted SU packets. In order to evaluate the influence of the finite buffer setting and the probability returning scheme to the system performance, we construct and analyze a discrete-time Markov chain model. Accordingly, we determine the expressions of some important performance measures of the PU packets and the SU packets. Then, we show numerical results to evaluate how the buffer setting of the PU packets and the returning probability influence the system performance. Moreover, we optimize the system access actions of the SU packets. We determine their individually and the socially optimal strategies by considering different buffer settings for PU packets and different returning probabilities for SU packets. Finally, a pricing policy by introducing an admission fee is also provided to coincide the two optimal strategies.

Citation: Yuan Zhao, Wuyi Yue. Performance analysis and optimization for cognitive radio networks with a finite primary user buffer and a probability returning scheme. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018195
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##### References:
System actions of PU packets and SU packets
Average queue length $E_{PU}$ of PU packets
Throughput $\theta_{PU}$ of PU packets
Blocking rate $\beta_{SU}$ of SU packets
Average delay $\delta_{SU}$ of SU packets
Function $F_I(\lambda_2)$ of the individual net benefit
Function $F_S(\lambda_2)$ of the social net benefit
Notations for the system model
 Symbol Explanation $\lambda_1$ Arrival rate of the PU packets $\lambda_2$ Arrival rate of the SU packets $\mu_1$ Transmission rate of the PU packets $\mu_2$ Transmission rate of the SU packets $K_1$ Capacity of the PU buffer $K_2$ Capacity of the SU buffer $q$ Returning probability for the interrupted SU packets $P_n$ Number of PU packets in the system at the instant $t=n^+$ $S_n$ Number of SU packets in the system at the instant $t=n^+$
 Symbol Explanation $\lambda_1$ Arrival rate of the PU packets $\lambda_2$ Arrival rate of the SU packets $\mu_1$ Transmission rate of the PU packets $\mu_2$ Transmission rate of the SU packets $K_1$ Capacity of the PU buffer $K_2$ Capacity of the SU buffer $q$ Returning probability for the interrupted SU packets $P_n$ Number of PU packets in the system at the instant $t=n^+$ $S_n$ Number of SU packets in the system at the instant $t=n^+$
Numerical results for the individually and socially optimal strategies
 $K_1$ $K_2$ $q$ $\lambda_i$ $r_i$ $\lambda_s$ $r_s$ min max min max 0 5 0.4 0.26 0.27 0.52 0.54 0.18 0.36 2 5 0.4 0.15 0.16 0.30 0.32 0.10 0.20 0 5 0.8 0.30 0.31 0.60 0.62 0.19 0.38 2 5 0.8 0.20 0.21 0.40 0.42 0.12 0.24 0 8 0.8 0.32 0.33 0.64 0.66 0.22 0.44 2 8 0.8 0.23 0.24 0.46 0.48 0.15 0.30
 $K_1$ $K_2$ $q$ $\lambda_i$ $r_i$ $\lambda_s$ $r_s$ min max min max 0 5 0.4 0.26 0.27 0.52 0.54 0.18 0.36 2 5 0.4 0.15 0.16 0.30 0.32 0.10 0.20 0 5 0.8 0.30 0.31 0.60 0.62 0.19 0.38 2 5 0.8 0.20 0.21 0.40 0.42 0.12 0.24 0 8 0.8 0.32 0.33 0.64 0.66 0.22 0.44 2 8 0.8 0.23 0.24 0.46 0.48 0.15 0.30
Numerical results for the admission fee
 $K_1$ $K_2$ $q$ $\lambda_s$ $f$ 0 5 0.4 0.18 1.9650 2 5 0.4 0.10 1.7917 0 5 0.8 0.19 5.8778 2 5 0.8 0.12 5.4328 0 8 0.8 0.22 6.3116 2 8 0.8 0.15 5.9168
 $K_1$ $K_2$ $q$ $\lambda_s$ $f$ 0 5 0.4 0.18 1.9650 2 5 0.4 0.10 1.7917 0 5 0.8 0.19 5.8778 2 5 0.8 0.12 5.4328 0 8 0.8 0.22 6.3116 2 8 0.8 0.15 5.9168