Optimal Utilization of a Cognitive Shared Channel with a Rechargeable Primary Source Node
Abstract
This paper considers the scenario in which a set of nodes share a common channel. Some nodes have a rechargeable battery and the others are plugged to a reliable power supply and, thus, have no energy limitations. We consider two sourcedestination pairs and apply the concept of cognitive radio communication in sharing the common channel. Specifically, we give highpriority to the energyconstrained sourcedestination pair, i.e., primary pair, and lowpriority to the pair which is free from such constraint, i.e., secondary pair. In contrast to the traditional notion of cognitive radio, in which the secondary transmitter is required to relinquish the channel as soon as the primary is detected, the secondary transmitter not only utilizes the idle slots of primary pair but also transmits along with the primary transmitter with probability . This is possible because we consider the general multipacket reception model. Given the requirement on the primary pair’s throughput, the probability is chosen to maximize the secondary pair’s throughput. To this end, we obtain twodimensional maximum stable throughput region which describes the theoretical limit on rates that we can push into the network while maintaining the queues in the network to be stable. The result is obtained for both cases in which the capacity of the battery at the primary node is infinite and also finite.
cognitive network, stochastic energy harvesting, stability analysis, multipacket reception capability
I Introduction
\PARstartCognitive radio communication provides an efficient means of sharing radio spectrum between users having different priority [1]. The highpriority user, called primary, is allowed to access the channel whenever it needs, while the lowpriority user, called secondary, is required to make a decision on its transmission based on what the primary user does. The system considered in this paper is comprised of nodes that are either subject to energy availability constraint imposed by the battery status and stochastic recharging process or free from such constraint by assuming that they are connected to a constant power source.
In this paper, we consider the simple cognitive system of two sourcedestination pairs as shown in Fig. 1 and derive the maximum stable throughput region for a cognitive access protocol on the general multipacket reception channel model^{1}^{1}1When compared to collision channel model, it better captures the effects of fading, attenuation and interference at the physical layer. in which a transmission may succeed even in the presence of interference [2, 3, 4]. The secondary node can take advantage of such an additional reception capability by transmitting simultaneously with the primary. We adopt a similar cognitive access protocol proposed in [5], and also studied in [6], in which the secondary node not only utilizes the idle periods of the primary node, but also competes with the primary by randomly accessing the channel to increase its own throughput. However, the secondary user is still required to coordinate its transmission in order not to hamper the required throughput level of the primary link given the energy harvesting rate and this is done by appropriately choosing the random access probability.
To position our contribution with respect to the recent literature, we start a brief background review. In [7], the capacity of the additive white Gaussian noise channel with stochastic energy harvesting at the source was shown to be equal to the capacity with an average power constraint given by the energy harvesting rate. However, like most of informationtheoretic research, the result is obtained for pointtopoint communication with an always backlogged source. In [8], the slotted ALOHA protocol was considered for a network of nodes having energy harvesting capability and the maximum stable throughput region was obtained for bursty traffic. An exact characterization of the region was given in the paper for a twonode case over a collision channel. The analysis is not trivial even for such a simple network because the service process of a node not only depends on the status of its battery but also on the idleness or not of the other node. Note that the reason why the exact region is known only for the twonode and the threenode cases (even without energy availability constraints) is the interaction between the queues of the nodes [9, 10, 11, 12].
The initial study of a simple model involving only two sourcedestination pairs is not only instructive but also necessary. The reason is that the interaction between nodes causes considerable difficulties at the analytical level, and yet, reveals major insights at the conceptual level. In addition, we use the stochastic dominance technique and Loynes’ theorem [13] for the stability of stationary system to solve the problem. Also, as pointed out in [8], it is important to note that the ”service process” of the battery, i.e., the use of its energy, is independent of whether the transmission is successful or not.
The rest of the paper is organized as follows. In Section II, we define the stability region, describe the channel model, and explain the packet arrival and energy harvesting models. In Section III, we present the conditions for stability of the considered cognitive access protocol when the capacity of the battery at the primary node is assumed to be infinite. The proof of the result is given in Section IV which utilizes the stochastic dominance technique and arguments similar to those used in [8] and [10]. In Section V, we extend the result to the case when the capacity of battery is finite. As will be shown, the stability region for the case with finite capacity battery is a subset of that for the case with infinite capacity battery. For comparison’s sake, in Section VI, the result obtained in Section III is derived again for the case without multipacket reception capability, i.e., for a collision channel with additional probabilistic erasures. Finally, we draw some conclusions in Section VII.
Ii System Model
We consider a timeslotted communication system consisting of two primary and secondary sourcedestination pairs of nodes, and , respectively, as shown in Fig. 1. Each source node has an infinite capacity buffer for storing arriving packets of fixed length. The secondary node is plugged to a reliable power supply, whereas the primary node is powered through a random timevarying renewable energy process and has a battery for storing energy which is assumed to be harvested in a certain unit from the environments. The capacity of the battery is denoted by . We first consider the case with and, after that, we relax to take any finite integer value. The slot duration is equal to the transmission time of a single packet and one unit of energy is consumed in each transmission. The packet arrival and energy harvesting processes are all modeled as independent Bernoulli processes of rate and per slot, respectively. The primary node is considered active if both and are nonempty at the same time. Similarly, the secondary is called active if is nonempty. Otherwise, they are called idle.
A shared channel is assumed and a transmission is said to be successful if the received signaltointerferenceplusnoiseratio (SINR) exceeds a certain threshold which depends on the modulation scheme, the target biterrorrate, and the number of bits in the packet (i.e., the transmission rate for a fixed packet duration). Denote by the probability that the transmission by source succeeds given that the sources in are transmitting simultaneously. Specifically, in our cognitive communication system in Fig. 1, the following success probabilities are of interest:
and it is assumed that and . Define and . In case that the simultaneous transmissions always fail, we have for all .
Denote by the length of at the beginning of time slot , the queue is said to be stable if
(1) 
Loynes’ theorem [13] states that if the arrival and service processes of a queue are strictly jointly stationary and the average arrival rate is less than the average service rate, then the queue is stable. If the average arrival rate is greater than the average service rate, then the queue is unstable and the value of approaches to infinity almost surely. The stability region of the system is defined as the set of arrival rate vectors for which the queues in the system are stable.
Iii Main Results
This section describes the cognitive access protocol and presents our main results concerning its stability. The proofs of the results are presented in the next section.
Iiia Description of the cognitive access protocol
The opportunistic cognitive access protocol proposed in [5] and also used in [6] is modified and studied again in the context of the energy harvesting environment. The energyconstrained primary node (see Fig. 1) transmits a packet whenever it is active. Note that the transmission by the primary node is independent of the secondary node . On the other hand, the transmission by the secondary node must be chosen in a careful manner in order not to impede the primary’s performance guarantees. Under our cognitive access protocol, node observes the status of and if is idle, i.e., either or is empty, it transmits with probability if its own packet queue is nonempty. Otherwise, if is active, transmits with probability to take advantage of the multipacket reception capability by transmitting along with the primary node although at the same time it risks impeding the primary node’s success. The design objective is to choose the transmission probability such that the secondary’s throughput is maximized while maintaining the stability of primary’s packet queue at given packet arrival and energy harvesting rates.
IiiB Stability Criteria
(2) 
(3) 
(4) 
Denote by the stability region of the system by considering all possible values of and define . Note that reflects the degree of multipacket reception capability. In the case of a collision channel in which and , . It is clear that increases as the multipacket reception capability improves.
Theorem III.1
The optimal achieving the boundary of the stability region is explicitly given in the following section. The subregion is depicted in Fig. 2 with solid line. Specifically, if , the line segments and correspond to the boundaries due to the inequalities (2) and (3), respectively. The subregion is also illustrated in the Fig. 3 with solid line. Note that when , is always contained in , i.e., , which is not necessarily true if .
Iv Analysis using Stochastic Dominance
Under the cognitive access protocol described in Section IIIA, the expressions for the average service rates seen by and are given by
(8) 
and
(9) 
Note that computing the average service rates and requires the specifications of a joint probability of doublets and , respectively. Since, however, , , and are all interacting, it is difficult to track them. We bypass this difficulty by utilizing the idea of stochastic dominance [10]. That is we first construct parallel dominant systems in which one of the nodes transmits dummy packets even when its packet queue is empty. The essence of the dominant system is to make the analysis tractable by decoupling the interaction between the queues. Since the queue sizes in the dominant system are, at all times, at least as large as those of the original system, the stability region of the dominant system inner bounds that of the original system. It turns out however that the stability region obtained using this stochastic dominance technique coincides with that of the original system which will be discussed in detail later in this section. Thus, the stability regions for both the original and the dominant systems are the same.
Iva The first dominant system: secondary node transmits dummy packets
Construct a hypothetical system in which the secondary node transmits dummy packets when its packet queue is empty. Hence transmits with probability 1 whenever is idle and with probability if is active. As a result, the average service rate of in (8) reduces to
(10) 
Since transmits with probability 1 whenever it is active, if is saturated^{2}^{2}2Note that in describing the service rates in (8) and (9), it is assumed that the corresponding packet queue is nonempty. This is simply because if the queue is empty, the ”server” becomes idle., is modeled as a decoupled discretetime system with arrival and service rates and , respectively. It follows from Little’s theorem that is nonempty for a fraction of time [14]. Consequently, we have
(11) 
For satisfying , i.e., when in this dominant system is stable, we now obtain the average service rate of . We note from (9) that the probability of being active, i.e., , needs to be specified beforehand. For this, we take an approach similar to the one used in [8]. The approach utilizes a simple property of a stable system, that is the rate of what comes is equal to the rate of what goes out. Given the fact that is active, the average number of packets out of is given by . Because the average number of packets into is and, because it satisfies , the fraction of active slots must be
(12) 
After some manipulation, the average service rate of can be obtained from (9) as
(13) 
By applying Loynes’ theorem, we find that the stability condition for the dominant system is given by
(14) 
when
(15) 
An important observation made in [10] is that the stability conditions obtained by using stochastic dominance technique are not merely sufficient conditions for the stability of the original system but are sufficient and necessary conditions. The indistinguishability argument applies to our problem as well. Based on the construction of the dominant system, it is easy to see that the queues of the dominant system are always larger in size than those of the original system, provided they are both initialized to the same value. Therefore, given , if for some , the queue at is stable in the dominant system then the corresponding queue in the original system must be stable; conversely, if for some in the dominant system, the node saturates, then it will not transmit dummy packets, and as long as has a packet to transmit, the behavior of the dominant system is identical to that of the original system because the action of dummy packet transmissions is employed increasingly rarely as we approach the stability boundary. Therefore, we can conclude that the original system and the dominant system are indistinguishable at the boundary points.
The portion of the stable throughput region by the first dominant system is given by the closure of the rate pairs described by (14) and (15) as varies over . To obtain the closure of the rate pair, we first fix and maximize as varies over . By replacing by and by , the boundary of the stability region for fixed can now be written as
(16) 
for . Differentiating with respect to yields,
(17) 
where is defined in Section IIIB. It can be observed that the denominator is strictly positive and the numerator can be positive or negative depending on the value of .

If , the derivative is nonpositive for all feasible and, thus, is a decreasing function of in the range of all possible values of . Therefore, and the stability region is given in (6).
Note that for the first dominant system the value of is upper bounded by the term .
IvB The second dominant system: primary node transmits dummy packets
In the previous section, we obtained the stability region of the first dominant system which yields one part of the stability region of the original system. To finalize the analysis, consider the complementary dominant system in which the primary node transmits dummy packets whenever its packet queue is empty, and the secondary node behaves exactly as in the original system. Even in the dominant system, however, cannot transmit if its battery is empty. Therefore, the average service rate of in (9) reduces to
(18) 
Since transmits with probability 1 whenever its battery is nonempty, is modeled as a decoupled discretetime system with arrival rate and service rate 1. Consequently, (18) becomes
(19) 
From Little’s theorem, the probability that is nonempty for some is given by
(20) 
Because in this dominant system is decoupled, i.e., independent, from the rest of the system, we can rewrite the average service rate of in (8) as
(21) 
Plugging (20) into (21) and, after some manipulations, we find the stability condition for this dominant system is given by
(22) 
for
(23) 
The indistinguishability argument at saturations holds here as well.
To specify the boundary of the stability region which is the closure of the rate pairs over feasible , we follow the same methodology as in the previous section. By replacing and by and , respectively, the boundary for fixed is written as
(24) 
for . It is not difficult to see that its first derivative with respect to is given as
(25) 
where . Since is always nonpositive under our assumption, is a nonincreasing function of . Therefore, the optimal value of maximizing is 0 but this is valid only if the condition is met. At , it becomes . Substituting into (24) yields (7). If , we obtain . By substituting into (24), we obtain (4). Note that in obtaining the stability region for this dominant system, it is assumed that . At , the optimal transmission probability of the secondary node is which gives the upper bound on in (4).
V The Case with Finite Capacity Battery
We now consider a realistic scenario in which the primary node is equipped with a battery whose capacity is finite. The harvested energy units can be stored only if the battery is not fully charged.
(26) 
(27) 
(28) 
Theorem V.1
The stability region of the cognitive multiaccess system with finite battery is described by
(29) 
where the subregion is described as follows:

If ,
(30) The optimal achieving the boundary is zero.
The subregion is described as with
(31) 
and as given by (28). The optimal achieving the boundaries of the subregions and are obtained as and , respectively. {proof} In the dominant system in which the primary node transmits dummy packets when its queue is empty, is decoupled from the remaining of the system and modeled as a discretetime system with arrival and service rates and , respectively. We know in that case that is always ergodic and nonempty with
(32) 
with strictly less than . If , is nonempty with probability 1 which is not of our interest since we can rule out the role of the battery in the steadystate. The rest of the proof is similar to that of Theorem III.1.
The subregion is depicted in Fig. 2 with dotted line. Specifically, if , the line segments and correspond to the boundaries due to the inequalities (26) and (27), respectively. The subregion is also plotted in Fig. 3 with dotted line. One can easily observe from the figures that the stability region for the case with finite capacity battery is always a subset of that for the case with infinite capacity battery. Also, note that as , the stability region for the finite battery case approaches to that for the infinite battery case.
Vi Collision Channel with Probabilistic Erasures
For the completeness of our discussion, we present the stability conditions for the collision channel case with probabilistic erasures. The stability region is given by:
(33) 
where
(34) 
and
(35) 
The proof is omitted for brevity. It is trivial to observe that and, thus, . The optimal achieving the boundaries is always . It is intuitive that the welldesigned cognitive access protocol will not allow the secondary node to access the channel when the primary node is transmitting. This is because such simultaneous transmissions would definitely result in a collision. The stability region is depicted in the Fig. 4. Since the stability region is identical with the subregion for the case of with multipacket reception capability in Fig. 2(b), the stability region for the collision case is a subset of that for the case with the multipacket reception capability.
Vii Conclusion
We employed an opportunistic multiple access protocol that observes the priorities among the users to better utilize the limited energy resources. Owing to the multipacket reception capability, the secondary node not only utilizes the idle slots but also can take advantage of such an additional reception by transmitting along with the primary node by randomly accessing the channel in a way that does not adversely affect the quality of the communication over the primary link. Consequently, at a given input rate of the primary source, we could choose the optimal access probability by the secondary transmitter to maximize its own throughput and this maximum was also identified. The result is obtained for both cases when the capacity of the battery at the primary node is infinite and also finite. This initial research provides some insights on how to run such a network of nodes having different energy constraints. Extending the approach proposed here to more realistic environments with multiple set of sourcedestination pairs, although highly desirable, presents serious difficulties due to the interaction between the nodes.
References
 [1] Q. Zhao and B. M. Sadler, “A survey of dynamic spectrum access,” IEEE Signal Processing Magazine, vol. 24, no. 3, pp. 79–89, May 2007.
 [2] S. Ghez and S. Verdú, “Stability property of slotted aloha with multipacket reception capability,” IEEE Transactions on Automatic Control, vol. 33, no. 7, pp. 640 – 649, Jul. 1988.
 [3] Q. Z. L. Tong and G. Mergen, “Multipacket reception in random access wireless networks: from signal processing to optimal medium access control,” IEEE Communications Magazine, vol. 39, no. 11, pp. 108–112, Nov. 2001.
 [4] V. Naware, G. Mergen, and L. Tong, “Stability and delay of finiteuser slotted aloha with multipacket reception,” IEEE Transactions on Information Theory, vol. 51, no. 7, pp. 2636–2656, Jul. 2005.
 [5] B. Rong and A. Ephremides, “On opportunistic cooperation for improving the stability region with multipacket reception,” Proceedings of NETCOOP, LNCS, vol. 5894, pp. 45–59, 2009.
 [6] S. Kompella, G. D. Nguyen, J. E. Wieselthier, and A. Ephremides, “Stable throughput tradeoffs in cognitive shared channels with cooperative relaying,” Proceedings of IEEE INFOCOM 2011.
 [7] O. Ozel and S. Ulukus, “Informationtheoretic analysis of an energy harvesting communication system,” in Proceedings of IEEE PIMRC, Sep. 2010.
 [8] J. Jeon and A. Ephremides, “The stability region of random multiple access under stochastic energy harvesting,” To appear in the proceedings of IEEE ISIT 2011.
 [9] B. S. Tsybakov and V. A. Mikhailov, “Ergodicity of a slotted aloha system,” Problems of Information Transmission, vol. 15, no. 4, pp. 301–312, 1979.
 [10] R. Rao and A. Ephremides, “On the stability of interacting queues in a multiaccess system,” IEEE Transactions on Information Theory, vol. 34, no. 5, pp. 918–930, Sep. 1988.
 [11] W. Szpankowski, “Stability conditions for some distributed systems: Buffered random access systems,” Advances in Applied Probability, vol. 26, no. 2, pp. 498–515, Jun. 1994.
 [12] W. Luo and A. Ephremides, “Stability of N interacting queues in randomaccess systems,” IEEE Transactions on Information Theory, vol. 45, no. 5, pp. 1579–1587, Jul. 1999.
 [13] R. M. Loynes, “The stability of a queue with nonindependent interarrival and service times,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 58, no. 3, pp. 497–520, 1962.
 [14] L. Kleinrock, Queueing Theory, Volume I: Theory. New York: Wiley, 1975.