Wireless Communications and Mobile Computing

Volume 2019, Article ID 7367028, 12 pages

https://doi.org/10.1155/2019/7367028

## Discrete-Time Analysis of Cognitive Radio Networks with Nonsaturated Source of Secondary Users

^{1}Universitat Politècnica de València, Spain^{2}University of Manitoba, Winnipeg, MB, Canada^{3}University of Pretoria, Pretoria, South Africa

Correspondence should be addressed to Vicent Pla; se.vpu@alpv

Received 22 June 2018; Accepted 22 November 2018; Published 2 January 2019

Guest Editor: Takayuki Ito

Copyright © 2019 Vicent Pla et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Sensing is a fundamental aspect in cognitive radio networks and one of the most complex issues. In the design of sensing strategies, a number of tradeoffs arise between throughput, interference to primary users, and energy consumption. This paper provides several Markovian models that enable the analysis and evaluation of sensing strategies under a broad range of conditions. The occupation of a channel by primary users is modeled as alternating idle and busy intervals, which are represented by a Markov phase renewal process. The behavior of secondary users is represented mainly through the duration of transmissions, sensing periods, and idle intervals between consecutive sensing periods. These durations are modeled by* phase-type* distributions, which endow the model with a high degree of generality. Unlike our previous work, here the source of secondary users is nonsaturated, which is a more practical assumption. The arrival of secondary users is modeled by the versatile* Markovian arrival process*, and models for both finite and infinite queues are built. Furthermore, the proposed models also incorporate a quite general representation of the resumption policy of an SU transmission after being interrupted by PUs activity. A comprehensive analysis of the proposed models is carried out to derive several key performance indicators in cognitive radio networks. Finally, some numerical results are presented to show that, despite the generality and versatility of the proposed models, their numerical evaluation is perfectly feasible.

#### 1. Introduction

The evolution and widespread deployment of wireless communications have generated an incessant and increasing demand for radio spectrum. This, combined with the static spectrum allocation policies that have been in place for quite some time, has led to a situation of spectrum scarcity (i.e., of unassigned frequency bands), at the same time, an important underutilization of a substantial part of the assigned bands.

Cognitive Radio (CR) is viewed as the enabling technology for dynamic spectrum access, which would allow solving the seeming paradox between spectrum scarcity and underutilization [1]. The basic idea of CR is to allow the unlicensed users, known as secondary users (SUs), to access licensed channels opportunistically when they are not in use by licensed users, known as primary users (PUs). This way, the interference that SUs produce to PUs should be kept to a minimum.

In this context,* white space* refers to spectrum that is not used by the PUs during a certain time interval at a specific location. The key to success for CR consists of effectively and efficiently sensing radio channels to detect* white space* when it occurs. An ideal, although unrealizable, sensing strategy would detect a portion of white space right after it starts and, likewise, would detect the end of it immediately after the transmission of a PU begins. Furthermore, such an ideal sensing strategy would only consume the minimum amount of energy needed for sensing. There has been a considerable long list of research papers over the years dealing with how to manage and implement CR. For details see [2] and other references therein.

As noted in [3], sensing is a major and challenging issue in CR networks. The choice of detection parameters poses a series of tradeoffs between achievable throughput, energy consumption, and interference caused PUs [4–13]. In general, if SUs spend more time on channel sensing, they obtain lower throughput, but the interference caused to PUs is also lowered. This is referred to as the* sensing-throughput tradeoff* (STT), which has been studied in a large number of papers (e.g., [4, 5, 9] and references therein). Furthermore, more channel sensing also raises the consumption of energy, which constitutes a critical aspect in certain scenarios (e.g., sensor networks). Consequently, a significant number of studies have considered energy efficiency a crucial part of spectrum sensing [7, 8, 10–13].

Several aspects come into play while considering sensing. Some examples are the duration of sensing periods, the width of the frequency band being scanned to search for unused channels, etc. For details see [2]. The vast majority of the models developed for studying CR networks, and specifically for spectrum sensing, assume that busy and idle times follow an exponential distribution (or geometric in discrete time); for example, see [14, 15] and all the above references to sensing studies. Using measurements, the authors of [16] showed that the channel idle time can be modeled by a lognormal distribution. This result was confirmed in [17], where it is shown that idle times follow a lognormal distribution for long durations, and a geometric distribution for short durations. Correlations between idle and busy times were also overlooked by most the previous papers. However, one could logically expect that some correlation exists between different intervals, as has in fact been noted in [17]. Our previous work in [2] was one of the first few ones to introduce correlation and also allow more general intervals.

The research on sensing strategies for CR networks is not new, and neither is the application of mathematical modeling for analysis and optimization of those strategies. However, the models presented in this paper embody a number of contributions which stem from the generality of the model assumptions. Our purpose in this paper is to propose a number of models that enable the analysis and evaluation of sensing strategies in cognitive radio networks under a broad range of conditions. More specifically, a Markov phase renewal process [18] is used to model the channel availability for SUs. This allows to consider a wide variety of distributions for the duration of idle and busy intervals and also to capture correlations between consecutive intervals. The authors of [19] proposed a similar model for the activity of PUs. However, correlations between different intervals were not considered in their model.

The behavior of SUs is represented mainly through the duration of transmissions, sensing periods, and idle intervals between consecutive sensing periods. These durations are modeled by* phase-type* distributions [18], which endow the model with a high degree of generality. In the literature of mathematical modeling, it is widely acknowledged that phase-type distributions offer an excellent compromise between applicability and tractability.

Our model of SUs also allows sensing errors, which can be misdetections and false alarms. For both of them, two different situations are distinguished, depending on whether the SU is only sensing or sensing and transmitting. This allows us to capture a broad range of SUs sensing capabilities.

However, most important in this paper is that we now have to introduce the arrival process of the SUs since the source is no longer saturated. We used the Markovian arrival process (MAP) to represent the SU arrival process. The MAP is a very versatile arrival process which can capture correlations and still allow modeling and computational tractability.

In our previous paper [2] on this class of problems, we assume that the source of SUs is saturated (i.e., there is always at least an SU waiting and ready to transmit). That assumption is not too realistic, even though it does give us an idea of the best we can achieve if there are always some SUs looking to transmit. In this current paper, we have relaxed that assumption. This makes the model more realistic while making it slightly more challenging as now we need to introduce an arrival process for the SUs. In addition, this new model creates a situation where a channel could be idle simply because there are no PUs or SUs needing to use it. These are the main contributions of this paper.

The rest of this paper is organized as follows. Section 2 introduces the model of the channel from the perspective of the SUs. The model of the secondary network, which includes the behavior of SUs, is described in Section 3. In Section 4 we describe the models for the complete system and their analysis when sensing is assumed to be ideal; this assumption is relaxed in Section 5. Some numerical results are presented in Section 6 to exemplify the capabilities of the proposed models and to show the feasibility of their numerical evaluation. Finally, the paper is concluded in Section 7.

#### 2. Channel Availability

We consider a single channel that can be idle or busy from the perspective of the SUs. The activity of PUs in this channel is described by a discrete-time Markov chain (DTMC) with state space . The channel is* busy* (b) if , and* idle* (i) if . We assume that during a time slot the condition of the channel changes at most once.

Let the matrix represent the transitions between busy states and represent the transitions from busy to idle states. Similarly, represents the transitions between idle states, and represents the transitions from idle to busy states. The matrices and are substochastic and of orders and , respectively.

Based on this, the transition matrix of can be written as

#### 3. Secondary Network

This section discusses SUs actions and how SUs interact with PUs through the channel status. Throughout this paper, sometimes we use “the SU” to refer to the set of all the SUs that can use the channel or to the SU that is at the head of the waiting line of SUs. We assume there is some coordination mechanism among SUs to perform channel sensing and to arrange channel access. Although this coordination mechanism is not trivial, its study is beyond the scope of this paper.

The SU can be in one of the following three modes:* sleeping*,* sensing*, or* transmitting*. The continuous period of time that the SU remains in one of these modes is called a* cycle* or* period*. Thus, we talk about, for example, a* sleeping period* or a* sensing cycle*. Next, we detail the characteristics of each type of cycle and how they alternate between them.

*(a) Sleeping*. The duration of a sleeping period is modeled by the phase-type distribution of order . A sleeping period is always followed by a sensing cycle.

*(b) Sensing*. During a sensing cycle, the SU performs a series of consecutive channel state measurements. If a measurement senses the channel as busy, the sensing period is interrupted and the SU enters the sleeping mode. The maximum number of measurements that would be taken is defined by the PH distribution with representation , of order . If the channel is sensed as idle in all the measurements of the cycle, the SU can initiate a transmitting period.

*(c) Transmitting*. During a transmitting cycle the SU attempts to transmit a message. The required transmission time (i.e., the number of time slots) to transmit the message is given by the PH distribution with representation of order . If the channel becomes busy and the SU is capable (can sense the channel while transmitting) of detecting it, the message transmission is interrupted and the SU switches to sleeping mode. If the transmission of the message is fully completed, the SU goes into sensing mode.

The SUs arrive according to a Markovian arrival process (MAP) represented by two substochastic matrices and of order . Then, the mean arrival rate, , is given as , where is the probability vector satisfying and and 1 is a column vector of ones of appropriate dimensions.

An SU that arrives and finds the channel busy waits in a buffer of size .

If an SU is in service when a PU arrives, the SU’s service is interrupted since the PU has preemptive priority. Now we specify the resumption policy followed by SUs when a message transmission was interrupted by PUs activity. This policy is described by matrix with elements . Suppose the SU’s service was interrupted in phase , let its service restart in phase with probability at resumption. It is clear that if it is a preemptive resume then , whereas if it is a preemptive repeat then . Hence, the matrix is a general representation.

#### 4. System Model I: Ideal Sensing

For the sake of clarity, we first assume that the SU receives perfect knowledge of the state of the channel. In Section 5 we relax that condition and assume that there could be errors in the sensing carried out by the SU.

Sensing takes place only when there is an SU in the system waiting to get access. Hence, when there is an SU in the system either it is receiving service because there is white space or it is waiting. The waiting could be because the channel is busy with PUs, there are other SUs ahead of it, or it is sensing the system for white space.

Here we assume that the time needed to perform a channel measurement cannot be neglected compared to the transmission time of a data unit. The length of a time slot is set so that a sensing measurement can be performed and know the result by the end of the slot. A conservative approach is applied to establish the outcome of the measurement: a sensing measurement taken during the time slot does not return* idle* as a result if the channel was busy at time instants or .

Although, as mentioned above, the duration of a single measurement cannot be neglected, in this section we assume that SUs can sense and transmit simultaneously. This could reflect instances in which SUs are equipped with two radios. The case in which SUs cannot transmit and sense simultaneously can be covered by the model with imperfect sensing described in Section 5.

##### 4.1. Head-of-the-Line (HoL) SU

Even though we are mainly interested in the nonsaturated case, first we consider the saturated case, from which we obtain the DTMC that constitutes the basis for the nonsaturated model.

Let the states of this system be classified as shown in Table 1.