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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.