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In this article, we consider a stochastic model of wireless sensor networks (WSNs) in which each sensor node randomly and alternatively stays in an active mode or a sleep mode. The active mode consists of two phases, called the full-active phase and the semi-active phase. When a referenced sensor node is in the full-active phase of the active mode, it may sense data packets, transmit the sensed packets, receive packets, and relay the received packets. However, when the phase of the sensor node switches from the full active phase to the semi-active phase, it is only able to transmit/relay data. When the referenced sensor node is in a sleep mode, it does not interact with the external world. In this article, first, we develop a stochastic model for the sensor node of a WSN, and then we derive an explicit expression of the stationary distribution of the number of data packets in the sensor node. Furthermore, we figure out some important performance measures, including the sensor node's energy consumption for transmission, the energy consumption of the sensor operations, and the average energy consumption of the sensor node in a cycle of active and sleep modes. Also, a numerical analysis is provided to validate the proposed model and the results obtained. The novel aspects of our research are the development of a stochastic model for WSN with active and sleep features and the development of important analytical formulae for evaluating the energy consumption of a WSN. These results are expected to be useful as significant contributions to the fundamental theory of the design of various WSNs with active and sleep mode considerations.

A cognitive radio system with retrial possibility and an admission cost for second users (SUs) to join the retrial group is investigated in this paper. If the SU finds the primary user (PU) band unavailable, it must decide with a probability estimate to either enter a retrial group or give up its service and leave the system. SUs in the retrial group independently retry after an exponentially distributed random time until they successfully access the spectrum. When the PU arrives, the SU's service on the band is interrupted. This interrupted SU is then assumed to occupy the PU band immediately when the PU completes its service. First, the non-cooperative joining behavior of SUs that choose to maximize their benefit in a selfish distributed manner is investigated, and an inefficient Nash equilibrium is derived. Second, from the perspective of the social planner, the socially optimal joining strategy when SUs cooperate with each other is studied and the corresponding Nash equilibrium is derived exactly. Finally, the result that an individually optimal strategy in general does not yield the socially optimal is verified theoretically. Furthermore, to bridge the gap between the individually and socially optimal strategies, a novel strategy of imposing an admission fee on SUs to join the retrial group is proposed and investigated with the derivation of optimal value for the admission fee. The numerical analysis indicates that the proposed admission fee as an equilibrium strategy and the socially optimal strategy of SUs improve efficiency in utilization of the cognitive radio system.

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We consider an M/G/1 retrial queue with general retrial times, and introduce working vacations and vacation interruption policy into the retrial queue. During the working vacation period, customers can be served at a lower rate. If there are customers in the system at a service completion instant, the vacation will be interrupted and the server will come back to the normal working level. Using supplementary variable method, we obtain the stationary probability distribution and some performance measures. Furthermore, we carry out the waiting time distribution and prove the conditional stochastic decomposition for the queue length in orbit. Finally, some numerical examples are presented.

In a cognitive radio (CR) system, excessive access services for secondary users (SUs) lead to a substantial increase in congestion and the retrial phenomenon, both of which degrade the performance of CR networks, especially in overload conditions. This paper investigates the price-based spectrum access control policy that characterizes the network operator's provision to heterogeneous and delay-sensitive SUs through pricing strategies. Based on shared-use dynamic spectrum access (DSA), the SUs can occupy the dedicated spectrum without degrading the operations of primary users (PUs). The service to transmission of SUs can be interrupted by an arriving PU, while the interrupted SUs join a retrial pool called an orbit, later trying to use the spectrum to complete the service. In the retrial orbit, the interrupted SU competes fairly with other SUs in the orbit. Such a DSA mechanism is formulated as a retrial queue with service interruptions and general service times. Regarding the heterogeneity of delay-sensitive SUs, we consider two cases: the delay-sensitive parameter follows a discrete distribution and a continuous distribution, respectively. In equilibrium, we find that the revenue-optimal price is unique, while there may exist a continuum of equilibria for the socially optimal price. In addition, the socially optimal price is always not greater than the revenue-optimal price, and thus the socially optimal arrival rate is not less than the revenue-optimal one, which is contrary with the conclusion, i.e., the socially optimal and revenue-optimal arrival rates are consistent, drawn in the literature for homogeneous SUs. Finally, we present numerical examples to show the effect of various parameters on the operator's pricing strategies and SUs' behavior. INDEX TERMS Cognitive radio network, dynamic spectrum access, retrial queue, optimal pricing, strategic behavior.

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A novel model is developed and investigated for a duty-cycle wireless sensor network (WSN), where each sensor node can stay at one of four statuses (i.e., full-active phase, two semi-active phases each with different functions, and a sleep phase) to make the system more efficient. The explicit result of the joint probability for data packet numbers and the sensor phases is achieved by suitably constructing a multi-dimensional Markov process. Furthermore, various energy consumption measures, such as energy consumption per unit time in each of the three energy consumptions and energy consumption switching from one phase to another, are obtained. System performance measures, such as the average delay time of a data packet in the system, throughput of the system, and the probability of the sensor node staying at any of the phases, are also achieved. The numerical analyses are provided to validate the model and the analytic results. The proposed model and the analysis method are expected to be applied to the design and analysis of WSN models with various phases of the sensor. INDEX TERMS Energy consumption, performance analysis, system analysis and design, wireless sensor networks.