Stable Throughput in a Cognitive Wireless Network

Stable Throughput in a Cognitive Wireless Network

Abstract

We study, from a network layer perspective, the effect of an Ad-Hoc secondary network with nodes accessing the spectrum licensed to a primary node. Specifically, a network consisting of one primary source-destination pair and secondary cognitive source-destination pairs that randomly access the channel during the idle slots of the primary user is considered. Both cases of perfect and imperfect sensing are considered and we adopt the SINR threshold model for detection to properly model the interference throughout the network. We first study the effect of the number of secondary nodes as well as secondary nodes’ transmission parameters such as the power and the channel access probabilities on the stable throughput of the primary node. If the sensing is perfect, then the secondary nodes do not interfere with the primary node and thus do not affect its stable throughput. However, if the sensing is imperfect, then the secondary nodes should control their interference on the primary node in order to keep its queue stable. It is shown that if the primary user’s arrival rate is less than some calculated finite value, cognitive nodes can employ any transmission power or probabilities without affecting the primary user’s stability; otherwise, the secondary nodes should control their transmission parameters to reduce the interference on the primary. It is also shown that in contrast with the primary maximum stable throughput which strictly decreases with increased sensing errors, the throughput of the secondary nodes might increase with sensing errors as more transmission opportunities become available to them. Finally, we explore the use of the secondary nodes as relays of the primary node’s traffic to compensate for the interference they might cause. Specifically, we introduce a relaying protocol based on distributed space-time coding that forces all the secondary nodes that are able to decode a primary’s unsuccessful packet to relay that packet whenever the primary is idle. In this case, for appropriate modulation scheme and under perfect sensing, it is shown that the more secondary nodes in the system, the better for the primary user in terms of his stable throughput. Meanwhile, the secondary nodes might benefit from relaying by having access to a larger number of idle slots becoming available to them due to the increase of the service rate of the primary. For the case of a single secondary node, the proposed relaying protocol guarantees that either both the primary and the secondary benefit from relaying or none of them does.

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1 Introduction

Restricting the spectrum access only to licensed users represents a highly inefficient resource utilization as actual measurements indicated that such resources remain idle for long proportions of time [1, 2, 3]. This observation as well as the development of sophisticated nodes capable of exploring licensed spectrum and adjusting their transmission parameters accordingly, motivated the idea of cognitive radio [4, 5, 6, 7] where the spectrum is made available to both licensed (also called primary) users as well as unlicensed (secondary/cognitive) users who opportunistically access the spectrum in such a way that the interference on the primary users is limited or even completely avoided.

Several approaches to cognitive radio operations have been suggested in the literature [8, 9, 6]. Two main paradigms exist for cognitive access, namely, spectrum sharing (SS) and opportunistic spectrum access (OSA). In spectrum sharing systems, secondary users are allowed to transmit concurrently with the primary users given some measures to keep the interference caused on primary users within an allowable range (usually within the primary node’s noise floor). Opportunistic spectrum access systems aim at avoiding concurrent transmissions between primary and secondary users by restricting the secondary users to access the channel only at unoccupied temporal, spectral or spatial holes. In order to achieve such goal, secondary users have to sense the channel at every slot to identify whether a primary transmission is ongoing or not[10, 11, 12]. Many cooperative sensing techniques have also been suggested [13] which efficiently mitigate the hidden terminal problem.

Cooperative communications was motivated by the effectiveness of space diversity in combatting fading, and hence single antenna users can benefit by the virtual MIMO effect induced by other nodes relaying their transmissions. Single relay channel has been studied from an information theoretic point of view in [14], but Shannon capacity was exactly characterized only for the case of physically degraded channels. The relay node in these works was dedicated to forward the source message, i.e., it does not have its own traffic. Later, cooperative protocols for two sources- two destinations case have been proposed and analyzed in [15] and distributed space-time codes for multiple relay scenarios have been developed in [16, 17]. However, the performance was based on information theoretic metrics such as capacity regions, achievable rates and outage probabilities. A network-level cooperative protocol for an uplink where a single pure cognitive relay is introduced to forward unsuccessful packets from source nodes during their idle slots has been proposed and analyzed in [18] with stable throughput and average delay as performance metrics under the assumption of perfect sensing. The assumption of pure relay has been relaxed in [19, 20] where the relay node is a source node having its own traffic but multi-relay case was not considered.

Cognitive radio has been studied from an information theoretic point of view in [21, 22]. However, such formulation does not take into account the bursty nature of the traffic and mainly focuses on sophisticated coding techniques at the physical layer for relaying and interference nulling rather than network layer aspects such as stable throughput and delay analysis. The stable throughput of simple cognitive networks has been studied in In [23, 24, 25]. In [23], authors studied the stable throughput of a simple cognitive network consisting of one primary and one secondary source-destination pairs under the SINR threshold model for reception with and without relaying for perfect and erroneous sensing. However, such formulation does not capture the effect of the number of secondary nodes on the performance. Moreover, many simplifying assumptions were made such as neglecting the false alarm probability (which has huge impact for more than one secondary node as we will see later) and ignoring the case where secondary node can be successful when the primary transmits simultaneously. By having one secondary node, cooperation is restricted to single node relaying which is beneficial only if the average received SNR on the secondary source-primary destination link is larger than the average SNR on the primary source-primary destination link which is not guaranteed especially if the secondary source has to reduce its power to protect the primary from the interference due to sensing errors. This weakness is overcome in our work by using multinode relaying. In [24], a similar model was considered under general probabilistic reception model where the secondary node accesses the channel with probability one if sensed to be idle and with some optimized probability if sensed to be busy. In [25], a multiaccess (MAC) primary network with two primary transmitters with a single secondary node is considered. In [26], authors considered a more realistic model of a primary network consisting of a TDMA uplink with some dedicated cognitive relays deployed to help the primary, and secondary network consisting of an Ad-Hoc network with stable throughput as performance criterion. However, only the case of perfect sensing of the primary nodes was considered which is an idealistic assumption far from reality. The model for reception used is the collision channel model which is not a realistic assumption for currently used sophisticated receivers with multipacket reception (MPR) capability [27]; and in specific, it is not realistic for Ad-Hoc networks where different source-destination pairs can be largely separated in space and thus the interference of one on the other when transmitting simultaneously is likely to be small. Furthermore, only one relay helps the primary source node at a time, i.e., no multinode relaying was considered despite the presence of several dedicated relays in the system which largely limits the potential gain of relaying. In [28], the stable throughput of a network consisting of one primary link and a symmetric secondary cluster with physical layer enhancements and perfect sensing is considered. The secondary cluster is controlled via a central controller and communication within the cluster is assumed to be perfect with no overhead and only one secondary node is scheduled at each slot. Single node relaying has also been studied. However, the assumption of having a secondary cluster is not appropriate for Ad-Hoc networks where the presence of a central controller is not generally feasible and the secondary transmissions interfere.

This work is motivated by the fact that in a typical cognitive scenario, secondary users sense different licensed frequency bands to capture possible opportunities. We study from a network layer perspective via a cross-layer analysis the effect of a secondary Ad-Hoc network with transmitter-receiver pairs sharing the licensed band of a primary node on the stable throughput of the primary node. The throughput of the secondary network and the possibility of multinode relaying of the primary’s unsuccessful packets are also considered. We consider a primary network consisting of a single point to point link and a secondary network consisting of source-destination pairs that randomly access the channel. We adopt the physical interference model for reception (SINR threshold model) which captures the possibility of MPR capability in contrast with the oversimplified collision model. We study the effect of the interference of the secondary nodes on the primary node’s stable throughput rate, which may occur due to incorrect sensing or even with perfect sensing in the presence of malicious attacks[29]. Secondary queues are assumed to be saturated to avoid queueing interaction which is known to be a thorny problem and stability region is exactly known only for two and three nodes [30, 31, 32]. We then study the maximization of the secondary nodes’ sum throughput over the feasible set of transmission power and channel access probabilities that guarantees the primary protection. Having shown the detrimental effect of a large number of secondary nodes interfering with the primary because of imperfect sensing, we propose a relaying protocol that utilizes all the secondary nodes that can decode a primary unsuccessful packet to relay that packet by using orthogonal space-time block codes [33, 34, 35]. It is shown that under this protocol, the more secondary nodes present in the network, the more the primary node benefits in terms of its maximum stable throughput. Meanwhile, the secondary nodes might benefit from such relaying. The primary node benefits by having more nodes relaying its packets and the secondary might benefit by helping the primary emptying his queue and hence having access to a larger number of idle slots. It is also shown that for a network with a single secondary node, the proposed protocol guarantees that either both the primary and the secondary nodes benefit from relaying or none of them benefits.

The paper is organized as follows: In section II, we describe the network and channel models. In section III, we introduce basic definitions and theorems concerning the queue stability that will be used throughout the paper. Section IV studies the throughputs of the primary and secondary nodes in the perfect sensing case which serves as an upper bound on the performance, while, in section V, we analyze the effect of erroneous sensing on the primary’s and secondary’s throughputs. In section VI, we propose and analyze the relaying protocol to benefit of the large population of secondary nodes. Section VII presents the numerical results and in section VIII, we conclude the paper.

2 System Model

Figure 1: System Model

The system consists of one primary link and a secondary network consisting of N secondary cognitive source-destination pairs forming an interference network as shown in Figure 1. All the nodes use the same frequency band for transmission. This situation arises when that band is licensed to the primary node, while to improve the spectral efficiency, some secondary nodes are allowed to access that band in an opportunistic way. All nodes have buffers of infinite capacity to store their packets to be transmitted. Time is slotted with one packet duration equal to the slot duration. Arrival process to the primary source node is assumed to be stationary with an average rate packets/slot, while secondary source nodes are assumed to be saturated. Throughout the paper, we designate the primary node by the subscript P and the ith secondary node by the subscript i with . The th source node is denoted by and the th destination node is denoted by , . The ith source node transmits at power , .

2.1 Channel Model:

The distance between node and node is denoted by , where . For instance, denotes the distance between the primary source and the th secondary destination node. Path loss exponent is assumed to be equal to () throughout the network. The link between the pair of nodes is subject to stationary block fading with fading coefficients which are independent over slots and mutually independent among links. All nodes are subject to independent additive white complex Gaussian noise with zero mean and variance . Under the adopted threshold model for reception, node is able to successfully decode a packet if the received signal-to-interference plus noise ratio (SINR) remains larger than some threshold throughout the packet duration [36]. The threshold depends on the modulation scheme, the coding and the target BER set by the receiving node as well as other features of the detector structure. Upon the success or failure of a packet reception at a node, an Acknowledgment/Non-Acknowledgment (ACK/NACK) packet is fedback to the corresponding transmitter. The ACK/NACK packets are assumed to be instantaneous and error free which is a reasonable assumption for short length ACK/NACK packets that have negligible delay, and small error rate achieved by using low rate codes on the feedback channel.
Under this model, the received signal at the th node of the transmitted signal by the th node in the presence of an interfering set of nodes at time slot is given by:

(1)

where is the transmitted packet by the th node at time slot and is of unit power while is the additive white complex Gaussian noise at node .
In this case, the success probability of the th node transmission at the receiving node is given by:

(2)

2.2 Multiple Access Protocol:

Both the primary and the secondary users transmit over the same frequency band, and hence, secondary users are restricted to use the idle slots of the primary. The primary node has the priority for transmission. At the beginning of each slot, the secondary nodes sense the channel and only if a slot is detected to be idle, do they access the channel in a random access way. The th secondary node will transmit in a slot with a probability whenever that slot is detected to be idle. We assume that there is sufficient guard time at the beginning of each slot to allow the secondary nodes to sense the channel.

According to the cognitive radio principle, secondary nodes should be “transparent” to the primary in the sense that their transmissions should not affect some performance criterion (here, the queueing stability) of the primary node. If the sensing is perfect, the secondary nodes never interfere with the primary and can employ any transmission parameters (power/channel access probability) that maximize their sum throughput without affecting the stability of primary. However, if the sensing is not perfect, the secondary nodes must limit their interference on the primary by controlling their transmission parameters to achieve that goal while maximizing their opportunistic throughput. We discuss the constraints on the secondary transmission parameters in case of imperfect sensing in section IV.

Throughout the work, we consider both perfect and imperfect sensing. Perfect sensing is an optimistic case and only serves as an upper bound on performance. We also consider both cases of asymmetric network with arbitrary fading distribution and the symmetric network with Rayleigh fading where . We will find that most results apply for both cases but in the asymmetric case, some results are not in closed form while the symmetric case is easier to analyze and get intuition from.
The symmetric case with Rayleigh fading that we will consider is characterized as follows:

  • , for .

  • , for .

  • , for .

  • , for .

  • , for .

  • , for .

  • , for .

  • , for .

  • , for .

This geometry, for instance, arises whenever the secondary sources lie on a circle and secondary destinations, along with primary source-destination pair lie on a line passing by the center of that circle and perpendicular to its plane.

3 Queue Stability

We adopt the definition of stability used by Szpankowski in [32].
Definition 1:
A multidimensional stochastic process is stable if for every x the following holds:

(3)

where is the limiting distribution function and x is componentwise limit.
If a weaker condition holds, namely,

(4)

then the process is called substable.
Under the saturation assumption of the secondary source queues, there is no issue of stability except for the primary queue.
The primary source queue evolves as:

(5)

where:
is the length of primary source queue at the beginning of time slot . and are the arrival and the service processes at the primary source queue in time slot respectively and =max.
Throughout the paper, we use the following lemma [37, 32] sometimes referred to as Loynes’ Theorem:
Lemma 1:
For a queue evolving as in (5). Let the pair be strictly stationary process (i.e. and are jointly stationary), then the following holds:
(i) If E E, then the queue is stable in the sense of the definition in (3).
(ii) If E E, then the queue is unstable and almost surely.
The arrival process at the primary source queue is stationary by assumption and is independent of its service process, hence a necessary and sufficient condition for stability of the primary source queue is that E.

4 Perfect Sensing Case

In this case, the secondary nodes are able to perfectly identify the primary idle slots where they can access the channel from the busy slots where they must remain silent to avoid interfering with the primary. In this case, the primary gets its maximum possible service rate. Clearly, this is an ideal situation serving as an upper bound on the performance.

4.1 Primary Queue

Theorem 1:
The stability condition for the primary queue in the perfect sensing case is given by:

(6)

For Rayleigh fading, we have that .

Proof.

The service process of the primary node is given by , where denotes the event of no outage at the primary destination node in slot and 1 is the indicator function which takes the value of one if its argument is true and zero otherwise. This event depends on the fading process on the link which is stationary and hence, is stationary.
The average service rate of the primary queue in this case, that we denote by , is given by:

(7)

For the Rayleigh fading case, is exponentially distributed with mean , and the results follows.
Finally, by using Loynes’ theorem (Lemma ), we can get the stability condition of the primary node as given in (6). ∎

4.2 Secondary Queues

Source node of the secondary cluster transmits with probability - independently of the other secondary nodes- whenever a slot is detected to be idle.
Theorem 2:
In the perfect sensing case, the throughput region of the secondary network is given by:

(8)

where:

which is -for the Rayleigh fading case- equal to:

Proof.

Let be the event that only nodes in set of secondary nodes transmit in slot and let be the event of no outage on the link in slot when all nodes in the set transmit.
The departure process of the th secondary node can be written as:

(9)

By using the fact that if the primary queue is stable, then the process is stationary [32, 31]; it can be easily shown that then, the process is stationary. Hence, we drop the time indices. By Little’s law [38], it follows that: .
Given a set of secondary nodes transmitting in a slot, the probability that the secondary destination node is able to successfully decode the th secondary source node transmission is given by:

(10)

For the case of Rayleigh fading (Refer to Appendix A for the proof), is given by:

(11)

where:
.
By independence of the events in (9), the throughput rate of the th secondary source node is given by:

(12)

Finally, the throughput region of the secondary network is obtained by taking the union over all possible transmission probability vectors as in (8). ∎

Next, we consider the symmetric case introduced in section(II). In that case, the probability of success of th secondary node in the presence of other interfering transmissions is given by (Refer to Appendix A for the proof):

(13)

In this case, the throughput rate of the th secondary node is given by:

(14)

We note that due to perfect sensing, secondary nodes do not interfere with the primary. Hence, they can transmit at their maximum power in order to maximize their throughput rate without affecting the stability of the primary queue and hence satisfying the cognitive principle of being transparent to the primary. This is not necessary true in the case of imperfect sensing as we will see in the next section.
Next, we calculate the optimum transmission probability at which secondary nodes should transmit in order to maximize their throughput. A very small will limit the interference between secondary nodes but will at the same time reduce the throughput while a large value of causes much interference between secondary nodes and hence also leads to a degradation in their throughput. By setting , we get , where . Thus, for small number of secondary nodes , it is beneficial to transmit with probability one, while for a large value of , secondary nodes should backoff to limit the interference on each other. Moreover, it is beneficial for both primary and secondary nodes that the primary node transmits at its maximum power and hence getting its maximum service rate and meanwhile maximizing the fraction of idle slots available to the secondary nodes. This may not be true if the sensing is not perfect because of the interference between primary and secondary nodes, so secondary nodes may suffer from degradation of their throughput if the primary node increases its transmission power.

5 Imperfect Sensing Case

Due to fading and other channel impairments, secondary nodes can encounter errors while sensing the channel and hence there is some possibility that they interfere with the primary node leading to a possible drastic reduction of its stable throughput. In this section, we quantify the effect of non ideal sensing on the throughput of primary and secondary nodes.

5.1 Channel Sensing

Two errors may occur at the secondary nodes while sensing the channel, namely, false alarm and misdetection errors. All subsequent results are applicable for any sensing method as they are given in terms of general false alarm and misdetection probabilities at the th secondary node. It should be noted that for a particular detector, and are related by its receiver operating characteristics (ROC)[39].

False Alarm Event

False alarm occurs whenever the primary node is idle but is sensed to be busy. Clearly, false alarm errors do not affect the stable throughput of the primary but degrade the throughput of the secondary nodes.

Misdetection Event

It occurs when the primary node is busy but is sensed by some secondary nodes to be idle. Those secondary nodes will simultaneously transmit with the primary causing some interference at the primary destination. If the interference is strong enough, it may lead to instability of the primary queue.
Note that by the independence of the fading processes between nodes, the misdetection and false alarm events are independent between secondary nodes.

5.2 Primary Queue Analysis

Theorem 3:
In the imperfect sensing case, the stability condition of the primary queue is given by:

(15)

where and are given by (17), (18).

Proof.

Let be the set (possibly empty) of secondary nodes that had misdetection probabilities at time slot . A subset of these nodes will choose to transmit at that time slot.
The service process of the primary node can then be expressed as:

(16)

where denotes the event that only nodes in the set commit an error in detection of the primary node in time slot ; is the event that only nodes in the set transmit at time slot and is the event of no outage on the link in the presence of an interfering set of secondary nodes. The process is clearly stationary, thus we drop the time indices subsequently.
By independence, the probability that only nodes in the set have misdetection is given by:

(17)

The service rate at the primary node given a set of transmitting nodes -as defined above- is given by:

(18)

Noting that , we can get the average service rate at the primary queue as given in (18), and hence by Loynes’ theorem, the proof is complete. ∎

Note that from equation (18), unless is the empty set.
Using that , we get that equation (15) is a convex combination of terms less than or equal to , hence given in equation (15) is strictly less than , which is an expected result due to secondary interference.
Next, we specialize to the symmetric case with Rayleigh fading. In this case, by symmetry, the probability of misdetection is the same for all the secondary nodes,i.e., .

Let be the success probability of primary node given secondary concurrent transmissions, then by similar analysis as in Appendix A, we get as:

(19)

where:

(20)

By symmetry, the average service rate of the primary queue is given by:

(21)

The effect of imperfect sensing is shown in the multiplication of by a term less than one.
For stability, and applying Loynes’ theorem, we should have:

(22)

The primary user will choose its arrival rate independently of the secondary network. In the imperfect sensing case, , and hence, the secondary nodes should limit their transmission power and/or transmission probabilities to limit the interference on the primary node and hence ensuring that its arrival rate be less than to avoid the instability of its queue.

It is straightforward to establish the following properties about given by (21).
Proposition 1:
The primary node service rate in the non-ideal sensing case, as given by (21) satisfies:

From proposition , we can draw some important conclusions: property (i) states that the effect of non ideal sensing at the secondary nodes is the degradation of the service rate of the primary licensed node due to the interference from secondary nodes on the primary. Such negative effect does not occur in the perfect sensing case where the service rate of the primary node is independent of secondary transmissions. Properties (ii),(iii) and (vi) reveal that unless i.e. either or , primary node cannot achieve its maximum service rate achieved in the case of perfect sensing. Also, for fixed , which is the case of interest, secondary nodes have a maximum power, possibly infinite if is small enough, at which they can transmit without affecting the stability of the primary node. Moreover, even if the interference of the secondary nodes is very high (case of ), due to the random access of the secondary nodes to the channel, primary node can still achieve a portion of its maximum service rate because there is a positive probability that none of the secondary nodes will transmit in a given slot. Finally, properties (iv), (v) and (vii) suggest that for fixed and , secondary nodes can control their interference level on the primary by adjusting their transmission probabilities which is sometimes easier to implement than power control due to hardware complexity and non linearity of the power amplifiers needed for power control over wide range.
For to be satisfied, we can solve for minimum value of and for the maximum value of to get the maximum possible transmission power and maximum possible transmission probability of the secondary nodes while remaining “transparent” to the primary, i.e., without affecting its stability.
By using equation (21) and proposition , we obtain:

(23)

Hence:

(24)

For fixed primary transmission power , we can calculate the maximum transmission power allowed at secondary nodes as:

(25)

From equations (24) and (25), we conclude that for fixed , if , secondary nodes can transmit at any desired chosen probability without affecting the stability of the primary while they have to backoff to reduce their interference on the primary node if . On the other hand, for fixed transmission probability , if , secondary nodes can transmit at any power without affecting the stability of the primary node while there exists a finite maximum allowed power if . This can be understood by noting that is the probability that none of the secondary source nodes transmit in a slot while the primary is busy, attracting the attention to the benefit of using random access as a multiple access protocol in the secondary network. Note that, in practical situations, the transmit power of a node is also limited by the power amplifier used, but we ignore this aspect here.

5.3 Secondary Queues

A secondary node gets a packet served in the imperfect sensing case, either if the primary node is idle with no false alarm occurring at that secondary node and that node transmits and is successful, or if the primary node is busy with an incorrect detection of the primary node occurring at that secondary node and the secondary node transmits and is successful.
Theorem 4:
The throughput of the th secondary source node in the imperfect sensing case is given by:

(26)

where:
and are the average throughput rates of the secondary node given that the primary node is idle, and busy respectively. and are given by (32), (33).

Proof.

By the saturation assumption of the secondary queues, the average service rate of the primary node is independent of the secondary nodes’ queue states (i.e. there is no queueing interactions) and hence, the probability that the primary node is idle = 1- Probability that primary node is busy = .
The departure process at the th secondary source node is given by:

(27)

where , is the event that only the nodes in set have a false alarm in slot whenever the primary source node is idle, is the event that only the nodes in set have a misdetection of the primary node in slot whenever the primary is busy, is the event that only nodes in set transmit at time slot . The event is the event of no outage on the th secondary source-destination link when the set of nodes has false alarm and nodes in the set of secondary nodes transmit simultaneously at time slot , while the event is the event of no outage on the th secondary source-destination link when the set of nodes has misdetection of the activity of the primary node, and nodes in the set of secondary nodes transmit simultaneously.
Since the process is stationary, we can drop the time indices.
The secondary node departure rate can be written as:

(28)

If the primary node is idle, then the probability that a set of secondary nodes has false alarms while all other secondary nodes do not is:

(29)

then we can write:

(30)

where:

(31)

where . Hence,

(32)

Similarly:

(33)

where:

(34)

Finally, the throughput region of the secondary nodes is given by:

(35)

where is given by (28) for . ∎

Next, we specialize to the case of symmetric secondary cluster introduced in section (II). In this case, and for all . The average throughput at the th secondary node can be written as: