Multi-Strategy Dynamic Spectrum Access in Cognitive Radio Networks:Modeling,Analysis and Optimization

Yi Yang,Qinyu Zhang,＊,Ye Wang,Takahiro Emoto,Masatake Akutagawa,Shinsuke Konaka

1 School of Electronic and Information Engineering,Harbin Institute of Technology (Shenzhen),Shenzhen 518055,China

2 Department of Electrical and Electronic Engineering,Tokushima University,Tokushima,770-8506,Japan

Abstract:Dynamic spectrum access (DSA) based on cognitive radios (CR) technique is an effective approach to address the “spectrum scarcity” issue.However,traditional CR-enabled DSA system employs only single DSA strategy,which might not be suited to the dynamic network environment.In this paper,we propose a multi-strategy DSA (MS-DSA) system,where the primary and the secondary system share spectrum resources with multiple DSA strategies simultaneously.To analyze the performance of the proposed MS-DSA system,we model it as a continuous-time Markov chain (CTMC) and derive the expressions to compute the corresponding performance metrics.Based on this,we define a utility function involving the concerns of effective throughput,interference quantity on primary users,and spectrum leasing cost.Two optimization schemes,named as spectrum allocation and false alarm probability selection,are proposed to maximize the utility function.Finally,numerical simulations are provided to validate our analysis and demonstrate that the performance can be significantly improved caused by virtues of the proposed MS-DSA system.

Keywords:cognitive radio networks; dynamic spectrum access; multi-strategy; performance analysis; optimization

I.INTRODUCTION

Dynamic spectrum access (DSA) is an effective approach to mitigate the problem of spectrum scarcity caused by the current static spectrum allocation policy.With DSA,the spectrum utilization can be greatly increased [1].The concept of DSA can be implemented in many ways [1],[2].In this paper,we focus on the cognitive radio (CR)-enabled DSA system [3].In the CR-enabled DSA system,the spectrum resource is utilized in a hierarchical fashion as follows:the licensed users (also known as Primary Users,PUs) are authorized to access the spectrum,while the cognitive devices (also known as Secondary Users,SUs) can opportunistically access the licensed spectrum as long as they affect the communication activities of PUs insignificantly.The available spectrum resource for SUs is also known as “spectrum hole.” The spectrum usage of a CR-enabled DSA system is flexible and efficient [4],which makes a variety of popular wireless communications systems (e.g.,vehicular networks [5],cellular networks [6],White-Fi or super Wi-Fi [7]) be easily deployed.In the CR-enabled DSA system,the DSA strategy [8] (that determines how to dynamically share the spectrum resource between PUs and SUs) significantly affects the performance of wireless communications system.From the perspective of priority,the DSA strategies can be divided into three categories:Random Preemptive DSA (RPDSA) [9],[10],Collaborative Preemptive DSA (CP-DSA) [11],and Non Preemptive DSA (NP-DSA) [12].In the RP-DSA strategy,PUs have the highest priority to use the spectrum,and the communication activities of SUs are“transparent” to the PUs,that is,the spectrum unit not occupied by other PUs can be always allocated to a new arrival PU.The intuition behind the RP-DSA strategy is that it does not require any modification on the primary system,because the presence of SUs is completely oblivious to PUs.In the CP-DSA strategy,PUs collaboratively share spectrum with SUs.Specifically,a new arrival PU “prefers” to be served by spectrum not occupied by either other PUs or SUs,so as to avoid terminating the ongoing transmissions of SUs as possible.If no such spectrum available,spectrum unit serving an SU is allocated to the new arrival PU,the transmission of the SU is forced to terminate.In the NP-DSA strategy,PUs have a non-preemptive priority to access the spectrum,whereby the ongoing SUs yield to the coming PUs until they finish the current transmissions.To summarize,in the CP-DSA or the NP-DSA strategy,the transmissions of SUs may affect the transmissions of PUs.With these two strategies,it is suggested that SUs should pay more for spectrum to compensate the primary system than that in the RP-DSA strategy.In this paper,we only consider the RP-DSA and the CP-DSA schemes because they can avoid extra waiting delay and extra waiting buffer into the primary system.

In this paper,we have proposed a multi-DSA strategy system,which breaks the limit that cognitive radio networks can only work in single DSA strategy.

Most of existing CR related works focus on only one of the DSA strategies above,which is namely “single DSA strategy system,” and some fundamental issues remain open.First,the single DSA strategy system usually cannot satisfy the transmission requirements for both PUs and SUs.For example,in the RP-DSA strategy,due to the completely obliviousness to PUs,SUs suffer from the higher dropping probability,which makes real-time services/applications on SUs infeasible [11],and it is difficult to guarantee quality of service (QoS) for SUs.Compared to the RP-DSA strategy,in the CP-DSA and NP-DSA strategies,the secondary system is given higher priority to access spectrum for better QoS with higher payoff for spectrum leasing [12].However,other issues occur,such as the modification overhead for the primary system,and how to and when to charge the secondary system for the spectrum usage.Moreover,because SUs may face a dynamic network environment,including PUs' traffics,service types,and the number of shared channels,and so on [5],[13],only one DSA strategy limits flexibility for the secondary system to access spectrum.Take the CR-enabled vehicular network for example.

The available spectrum holes may vary both in temporal and spatial domains.Therefore,a DSA strategy with more flexibility is required to overcome the dynamic network environment issue for the secondary system.

Motivated by this,in this paper,we propose a CR-enabled system with multi DSA strategies,namely MS-DSA system,where the primary system and the secondary system share spectrum resource based on multiple DSA strategies.We partition spectrum into different spectrum pools.The spectrum allocation in each spectrum pool follows a DSA strategy.Meanwhile,in order to balance the interests of both PUs and SUs,we introduce a spectrum leasing and auction broker (SLAB) into the MS-DSA system.

Note that the SLAB is a central network entity that was first introduced in the DIMSUMnet framework [14] to improve spectrum utilization by considering the complexity and the agility requirement of the secondary system.When the primary system is underloaded,it entrusts parts of spectrum resources with the expected prices and the corresponding DSA strategies to the SLAB.The secondary system dynamically leases spectrums from the SLAB according to its demands.When the primary system and the secondary system reach an agreement (including the number of shared spectrum resources,the leasing prices,and the DSA strategies),the secondary system leases the spectrums from the SLAB.With the SLAB,the primary system can grant parts of its spectrum usage rights to the secondary system and get some extra profit from the secondary system.On the other hand,the secondary system pays the spectrum leasing fees only for opportunistic access.Both the primary and secondary systems have the incentive to participate in the MS-DSA system and multi DSA strategies with different usage priorities,which can be performed by leveraging the price mechanism.

In real situation,the spectrum sensing may be imperfect,including sensing error,false alarm (FA),and miss detection (MD) [15].An FA error causes SUs overlook spectrum opportunities and refrain from access,while an MD error leads to SUs accessing channels that are actually occupied by PUs,which results in collision with an active PU.In this study,we also put the spectrum sensing into consideration and use Newton method to obtain an optimal FA probability in the MS-DSA system.

We propose analytical models to investigate the performance of the MS-DSA system.In the analytical models,we model the interactive process between the primary system and the secondary system as a multi-dimensional continuous-time Markov chain (CTMC) that has been used as an effective tool in many DSA studies [11],[12],[13].The performance metrics in the analytical models include SU blocking rate,SU dropping rate,effective throughput,and interference quantity on PUs.Based on the performance metrics,we define an overall utility function to be elaborated later.We investigate spectrum allocation and an optimal spectrum sensing to maximize the overall utility for the MS-DSA system.We conduct simulation experiments to validate correctness of the analytical models.

The rest of this paper is organized as follows.In Section II,we describe the system model considered in this study.Then,the CTMC based model for the MS-DSA system is described in Section III.In Section IV,we derive the performance metrics of the MSDSA system.In Section V,we propose two approaches to optimize the MS-DSA system.Extensive simulations and discussions are provided to demonstrate the effectiveness of our approaches in Section VI,followed by the concluding remarks and the future works.

II.SYSTEM MODEL

In this section,we illustrate how to apply the multiple strategies into a DSA system,namely MS-DSA,as follows.A DSA system is composed of the three entities:the primary system,the secondary system,and the SLAB,as shown in Fig.1.The primary system owns licensed spectrum resources,such as 3G or 4G networks,and can temporarily lease parts of the licensed spectrum resources to the secondary system through the SLAB.The secondary system consists of a Secondary Access Point (SAP) and a group of SUs.The SLAB acts as a trading agent to balance the interests of both the primary system and the secondary system.The execution of the MS-DSA system consists of three stages:“Spectrum Leasing,” “Multi-Strategy Dynamic Spectrum Access,” and “Payoff.” The details of the three stages will be elaborated in the following subsections.

2.1 Spectrum leasing

In this study,we focus on the primary system that can simultaneously support the RPDSA and the CP-DSA strategies.Suppose that at timeT0,the primary system entrustsNunderutilized channel1In this paper,we use a spectrum unit to represent a basic physical transmission resource,e.g.,a resource block (RB) in a OFDMA-based system or a spread spectrum code-sequence in a CDMA-based system.for spectrum leasing by sending the Spectrum Leasing message (see Step 1.1 in figure 1) to the SLAB,which carries the parameters:N,[T0,Tp],ωRP,ωCP,andωINT.The period [T0,Tp] specifies the time period allowed for the secondary system to access theNspectrum units.TheωRPandωCPare the prices for the usage of a spectrum unit per time unit based on the RP-DSA strategy and the CP-DSA strategy,respectively.TheωINTis the punishment cost for interference (on each spectrum unit per time unit) resulted by the SUs.The determinations ofωRP,ωCP,andωINTwill be discussed later.

Before accessing the leased channels,the secondary system sends the Spectrum Leasing message to the SLAB (see Step 1.2 in figure 1).Suppose that the secondary system sends out the message atT1(whereT1∈[T0,Tp].The Spectrum Leasing message from the secondary system carries the parameters [T1,Ts],γRPandγCP.The [T1,Ts] specifies the time period when the secondary system will use the leased spectrum.TheγRPandγCPspecify the number of spectrum units the secondary requests based on the RP-DSA and the CP-DSA strategies,respectively.Note that to access the leased spectrum,the two conditions,[T1,Ts] ⊆ [T0,TP] and (γRP+γCP)≤Nmust hold.

Fig.1.System model.

Fig.2.The spectrum pools in MS-DSA system.

Upon receipt of the request from the secondary system,the SLAB checks whether the above two conditions hold at the same time.If so,the SLAB informs the secondary system with the leasing pricesωRPandωCPfor the RPDSA and the CP-DSA strategies,respectively.If the SLAB and the secondary system reaches the agreement (i.e.,the secondary system agrees the leasing price),the SLAB generates a leasing contract as shown in figure 1 (1.3) and sends the Temporary Spectrum License message to the secondary system (see Step 1.4 in figure 1).Note that how to bargain the leasing price is out of the scope of this paper,which can be found in the previous works [16].

2.2 Multi-strategy dynamic spectrum access

The N leased spectrum units are partitioned into three portions,includingγRPspectrum units (to be accessed based on the RP-DSA strategy) andγCPspectrum units (to be accessed based on the CP-DSA strategy),andγRES=N-(γRP+γCP) spectrum units (reserved for the PUs).Figure 2 shows an example.In this paper,the spectrum units allocated based on the RP-DSA strategy and the CP-DSA strategy are named as the RP spectrum units and the CP spectrum units,respectively.TheγRESspectrum units dedicated to the PUs are named as RES spectrum units.To simplify our discussion,we assume that each spectrum access from either PUs or SUs requests one spectrum unit.

As shown in figure 2,the primary system maintains a local database to record all spectrum units being occupied by the PUs in the three spectrum pools.To avoid collisions between PUs and SUs in the CP-DSA strategy [11],the local database also records the CP spectrum units occupied by SUs,which can be done by the secondary system's informing the primary system through the SLAB when an SU request is allocated with a CP spectrum unit.The secondary system also maintains a local database to record all spectrum units being occupied by SUs in the CP-DSA spectrum pool and RP-DSA spectrum pool,respectively.

The MS-DSA allocates spectrum units to PU spectrum access requests and SU spectrum access requests as follows (see figure 3 and 4):

As shown in Fig.3,for a PU spectrum access request,the primary system first checks with its local database whether there are idle spectrum units in the RES spectrum pool.If so,the request is allocated with one RES spectrum unit.Otherwise,the primary system checks with its local database with the status of the RP-DSA spectrum pool.

If in the RP-DSA spectrum pool,there are spectrum units that are not occupied by the PUs (i.e.,the number of spectrum units occupied by the PUs is smaller thanγRP),the primary system allocates an spectrum unit either be idle or be occupied by other US to the PU spectrum access request.In particular,if the PU allocated with a spectrum unit that has been occupied to a SU,the PU and the SU are served by the same spectrum unit for a while,because it is impractical for the secondary system to perform spectrum sensing all the time and vacate from the current transmission immediately.Otherwise (i.e.,all spectrum units are occupied by the PUs),the primary system checks with its local database with the status of the CP-DSA spectrum pool.

In the CP-DSA strategy [11],the primary system should avoid collisions between PUs and SUs.The primary system first checks whether there are idle spectrum units.If so,the PU is allocated with an idle spectrum unit.Otherwise,the primary checks whether there are spectrum units occupied only by SUs.If such spectrum units exist,the PU is allocated with one spectrum unit that has been allocated to the SU.Otherwise,the PU spectrum access request is blocked.

Fig.3.DSA process for a PU spectrum access request.

As shown in Fig.4,for the an SU spectrum access request,the secondary system is required to check the status of the spectrum pools with its local database,and then,performsspectrum sensing2In this study,we assume that the energy detection-based sensing [17] is adopted for the sensing operation due to its low complexity and the capability of fast detection.Moreover since the SU devices have usually small size and limited power,the sensing functionality is implemented in the SAP.to identify transmission opportunities and avoid interference on the PUs.Two types of spectrum sensing should be considered,namely “out-of-band sensing” and “in-band sensing”.Out-of-band sensing is referred to the process of sensing those RP and CP spectrum units not occupied by SUs,in order to discover the spectrum opportunities for the SU request,while the process of in-band sensing is referred to sensing those RP and CP spectrum units being in use by SUs,in order to avoid collisions caused by returns of PUs.

In the MS-DSA system,the secondary system first checks the status of the CP-DSA spectrum pools with the local database.If there exist spectrum units that are not occupied by the other SUs,the secondary system performs out-of-band sensing on the CP-DSA spectrum pool.If the sensing result shows that there are spectrum units not occupied by PUs and SUs,the SU is allocated with one spectrum unit in the CP-DSA spectrum pool.Otherwise (i.e.,all spectrum units are occupied by the PUs and other SUs),the secondary system checks the status of the RP-DSA spectrum pool with its local database,and then,performsspectrum sensing2to identify transmission opportunities and avoid interference on the PUs.Two types of spectrum sensing should be considered,namely “out-of-band sensing” and “in-band sensing”.Out-of-band sensing is referred to the process of sensing those RP and CP spectrum units not occupied by SUs,in order to discover the spectrum opportunities for the SU request,while the process of in-band sensing is referred to sensing those RP and CP spectrum is being in use by SUs,in order to avoid collisions caused by returns of PUs.

Fig.4.DSA process for a SU spectrum access request.

In the MS-DSA system,the secondary system first checks the status of the CP-DSA spectrum pools with local database.If there exist spectrum units that are not occupied by the other SUs,the secondary system performs out-of-band sensing on the CP-DSA spectrum pool.If the sensing result shows that there are spectrum units not occupied by PUs and other SUs,the SU is allocated with one spectrum unit in the CP-DSA spectrum pool.Otherwise (i.e.,all spectrum units are occupied by the PUs and other SUs),the secondary system checks the status of the RP-DSA spectrum pool with its local database.

If in the RP-DSA spectrum pool,there are spectrum units that are not occupied by other SUs,the secondary system performs out-ofband sensing on the RP-DSA spectrum pool.Based on the sensing results,the secondary system selects a clear spectrum unit from these spectrum units,and allocate it o the SU spectrum access request.Otherwise (i.e.,all RP-DSA spectrum units are occupied by the PUs and other SUs),the SU spectrum access request is blocked.

During this stage,the secondary system also performs in-band sensing on both RPDSA spectrum pool and CP-DSA spectrum pool,in order to detect reoccupations of the PUs on those spectrum units.If sensing results indicate that there exist SUs transmitting with PUs on the same spectrum units in RP-DSA spectrum pool and CP-DSA spectrum pool,SUs should vacate from current transmission immediately.

2.3 Payoff

When the leasing period [T1,Ts] is due,the secondary system will pay for the leasing cost and the interference punishment based on the spectrum leasing contract through the SLAB.At the end of the leasing period,withωCP(i.e.,the price for the usage of a CP spectrum unit per time unit) andωRP(the price for the usage of an RP spectrum unit per time unit),we have the total cost for the spectrum unit leasing for the secondary system can be expressed as

To evaluate the interference caused by SUs to PUs,we apply interference quantity concept defined in [18]:Suppose that there arenspectrum units occupied by PUs and SUs in a time unit.

We have the interference quantity in this time unit isn.Let ΓINTas the average interference quantity during the leasing period [T1,Ts].Then we define the interference punishment during the leasing period as

whereωINTis the punishment cost for interference (on each spectrum unit per time unit) caused by SUs.

The derivation of ΓINTis given in Section IV.

Applying Eqns.(1) and (2),the cost for the secondary system should pay for the leasing period is

III.ANALYTICAL MODEL FOR MS-DSA SYSTEM

In this section,we propose an analytical model to investigate the performance of the MSDSA system.We assume the PU (SU) access requests follow a Poisson process with rateλ p(λs),and a PU (an SU) will occupy a spectrum unit for a time period that has an exponential distribution with mean

The assumptions have been widely used in many DSA related studies [11],[12].We model the MS-DSA system as a continuous-time Markov chain (CTMC).The state space Ω and infinitesimal generator matrixQof the CTMC are elaborated in the following subsections.

3.1 State space Ω

In the MS-DSA system,each spectrum unit in the three spectrum pools can be in one of the four following states at a time unit:

PU_state (SU_state):the spectrum unit is occupied by a PU (an SU);

PU_SU_state:the spectrum unit is shared by a PU and an SU;

idle_state :the spectrum unit is neither occupied by PUs nor SUs.

The instantaneous states of the CP-DSA spectrum pool and the RP-DSA spectrum pool are represented by vectors [iCP,jCP,k CP,lCP],and,[i RP,j RP,k RP,lRP],respectively,whereiCP(iRP),jCP(jRP),k CP(kRP),lCP(lRP) are the numbers of spectrum units in PU_state,SU_state,PU_SU_state,and idle_state in the CPDSA spectrum pool (the RP-DSA spectrum pool),respectively.For the RES spectrum pool that is allowed to be accessed by only the PUs,each RES spectrum unit can be only in PU_state or idle_state,and the instantaneous state of the RES spectrum pool is [iRES,jRES].

The instantaneous state of the MS-DSA system can be expressed as [iCP,i RP,iRES,jCP,jRP,kCP,kRP,lCP,l RP,lRES].Given the values ofγCP,γRPandγRES,we havelCP=γCP-(iCP+jCP+kCP),lRP=γRP-(iRP+jRP+kRP)andlRES=γRES-iRES.Thus,a general state of the MS-DSA system can be simplified as a 7-tuple vector:

and the state space Ω is subject to

To simplify our discussion,we express the one-step state transition fromstos′ bys s x′=+ .The scalarwhere superscript “+”and “-” represent the operations “+1” and “-1”,respectively.For example,x=indicatess′=s+=

3.2 Inifinitestial generator matrix Q

In the MS-DSA system,an eventetriggering the state transition can bee PUA=(a PU spectrum access request event),e SUA=(an SU spectrum access request event),e PUD=(a PU departure event),ande SUD=(an SU departure event).The state transitions for the MSDSA system are described as follows.Suppose that the MS-DSA system is at states.

1) e PUA=:We consider the following five cases.

Case 1:iRES＜γRES.The PU request is allocated with an RES spectrum unit in idle_state.The state moves fromsto

Case 2:iRES=γRESandiRP+kRP＜γRP.The PU request is allocated with either an RP spectrum unit in idle_state with probabilityor an RP spectrum unit in SU_state with probabilityand the system state transits fromsto

Case 3:iRES=γRES,iRP+kRP=γRP,andiCP+jCP+kCP＜γCP.The PU request is allocated with a CP spectrum unit in idle_state,and the state transits fromsto

Case 4 :iRES=γRES,iRP+kRP=γRP,iCP+jCP+kCP=γCP,andjCP＞0.The PU request is allocated with a CP spectrum unit in SU_state,and the system state transits fromsto

Case 5:iRES=γRES,iRP+kRP=γRP,andiCP+kCP=γCP.The PU request is blocked.Le tRPUA(s,s′) be the transition rate from statestos′(s,s′∈Ω) triggered by event PUA.From Cases 1-5,we haveRPUA(s,s′) as (7) shown in the bottom at this page.

2) e SUA=:For an SU spectrum access request,the secondary system first performs spectrum sensing on the CP spectrum units that are not occupied by the other SUs,i.e.,out-of-band sensing on the CP-DSA spectrum pool.In this paper,two typical spectrum sensing errors are considered,namely FA error and MD error.An FA error indicateds that a spectrum unit is sensed as in PU_state or PU_SU_state,but in fact it is in idle_state or SU_state; an MD error means that a spectrum unit is sensed as in idle_state or SU_state,while it is in fact in PU_state or PU_SU_state.LetPfandPebe the FA probability and MD probability of the spectrum sensing on each spectrum units.

Without loss of generality,suppose that the secondary system totally identifies ()m n+ idle spectrum units in the CP-DSA spectrum pool,wheremis the number of rightly-sensed spectrum units that are in fact in idle_state (not FA error),andnis the number of wrongly-sensed spectrum units that are in fact in PU_state (MD error).Denoteandbe the conditional probability that the secondary system findsmrightly-sensed andnwrongly-sensed idle CP spectrum units withPfandPe,respectively,which can be written as

When (m+n) ＞ 0,we assume the SU access request is allocated with each sensed idle spectrum unit with probabilityLetandbe the probability that the SU request is allocated with a rightly-sensed and a wrongly-sensed CP idle spectrum unit,respectively.The value of them can be obtained by

When (m+n)=0,there is no CP spectrum unit available for SU request.Letbe the probability that all CP spectrum units are sensed to be busy,and we can get it by substitutingm=0,andn=0 into Eqns.(8) and (9),i.e.,

If sensing result indicates that all CP spectrum units are busy,the secondary system then senses the RP spectrum units that are not occupied by the other SUs,i.e.,out-of-band sensing on the RP-DSA spectrum pool.Denoteandbe the probability that the SU request is allocated with a rightly-sensed idle RP spectrum unit and a wrongly-sensed idle RP spectrum unit,respectively,andbe the probability that all RP spectrum units are sensed to be busy.Similar to the results in the CP-DSA spectrum pool,we can easily obtain thes probabilities.

In order to avoid interference on the PUs,the secondary system also performs sensing on those spectrum units being used by SUs in the RP-DSA spectrum pool and the CP-DSA spectrum pool,i.e.in-band sensing.Suppose that the secondary system totally identifies (uCP+uRP+vCP+vRP) spectrum units in collision,whereuCPanduRPare the numbers of rightly-sensed collision CP spectrum units and RP spectrum units that are in fact in PU_SU_state (no MD error),respectively;vCPandvRPare the numbers of wrongly-sensed collision CP spectrum units and RP spectrum units that are in fact in SU_state (FA error),respectively.

In this case,SUs on the sensed collision spectrum units are dropped from the current transmission immediately.Denotebe the probability thatuCPSUs are dropped from the rightly-sensed CP collision spectrum units,andbe the probability thatvCPSUs are dropped from the wrongly-sensed CP collision spectrum units.The value of them can be calculated by

In a similar way,we can also denote and get the value of the probabilitiesandrespectively.

According to the derivations above,we summarize the state transitions triggered bye SUA=.Consider the following six cases,where Cases 1-5 are based on the results of out-of-band sensing,and Case 6 is from the results of in-band sensing.

Case 1:The SU request is allocated with a rightly-sensed idle CP spectrum with the probabilityand the system state transits fromstos′=(s+);

Case 2:The SU request is allocated with a wrongly-sensed idle CP spectrum unit with probabilityand the system state transits fromsto

Case 3:The SU request is allocated with a rightly-sensed idle RP spectrum unit with probabilityand the system state transits fromsto

Case 4 :The SU request is allocated with an wrongly-sensed RP spectrum unit with proba bilityand the system transits fromsto

Case 5:The SU request is blocked with probability

Case 6:uCPanduRPSUs are dropped from rightly-sensed collision CP spectrum units and RP spectrum units,andvCPandvRPSUs are dropped from wrongly-sensed collision CP spectrum units and RP spectrum units,with probability

and the system state transits froms′ tos′,whichs′ denotes:

LetRSUAbe the final transition rate triggered by event SUA.Applying Eqns (15) and (16),we haveRSUAas

3) e PUD=ande SUD=:With no PU and SU arrivals,any of PUs' and SUs' transmissions will be finished sucessfully.The transition rates triggered by evernts PUD and SUD can be written as

In summary,we can get the infinitesimal generator matrixQof the CTMC odel withPfandPeas (20) shown in the bottom at this page,whereQ(s,s′|Pf,Pe) represents the transition rate from statesto states′.

3.3 Steady-state probability

Assumption 1:(Rate Boundedness Assumption,RBA).The ratesv(s)=-Q(s,s),∀s∈Ω,are positive and bounded.

According to (20),it is observed that all ratev s(),∀∈Ωs,are linearly related to parameterλp,λs,up,andus.Hence,the RBA can be easily satisfied ifλp,λs,up,us＜∞.By using very deep mathematics it can be shown that under the RBA,a CTMC can be uniquely determined by the infinitesimal transition ratesQ.From a practical perspective,the case whenλp=+∞ andλs=+∞ is out of our scope.Detailed analysis for the limiting performance of the secondary system can be referred to [18].

Assumption 2:(Positive Recurrent Assumption,PRA).∀∈Ωscan be reached with probability 1 from any other state,and the mean recurrent time from statesto itself is finite.

According to (20),when (Pf,Pe) ∈ (0,1),any statescan transit to its neighbor states with positive finite transition rate,and therefore,the PRA holds.

Based on Markov ergodic theory [19],there exists a unique steady-state probabilityπ(s),∀s∈Ω for the MS-DSA CTMC model,ifAssumption 1andAssumption 2holds.The steady-state probability can be obtained by solving the balance equations as follows:

where Π={π(s),s∈Ω} is the row vector of the steady-state probability by lexicographically ordering the states,ande=[1,1,...,1]T.

Since Ω is finite,the steady-state probability can be computed by replacing the first linear equation of ΠQ=0 by Πe=1 and solving the resulting system of linear equations.Specifically,we form matrixQ1by replacing its first column bye=1 and form a row vectorb=[100...].So the balance equations become ΠQ1=b,and the steady-state probability can be obtained as Π=bQ1-1(the superscript “-1” denotes matrix inversion).

IV.PERFORMANCE ANALYSIS OF THE MS-DSA SYSTEM

Given the steady-state probability Π,the system performance metrics can be derived.

1) Blocking Rate:Blocking rate3is defined as the ratio of blocked SU arrivals to the total SU arrivals.Denote byPBlockthe blocking rate of SUs,which can be obtained as (22) shown in the bottom at this page.

2) Dropping Rate:Dropping rate is defined as the ratio of forcibly dropped SUs to the total SU arrivals.Denote byPDropthe dropping rate of SUs,which can be obtained as (23) shown in the bottom at next page.

3) Effective Throughput:The effective throughput of SUs' service is defined as the average number of successfully serviced SUs (that are neither be blocked nor be dropped) per unit time.Letηbe the effective throughput of SUs,which can be derived from (22) and (23) as follows:

4) Interference Quantity:Interference quantity is defined in [18] as the accumlated interference time to the ongoing PUs per unit time.Let ΓINTbe the interference quantity.According to the PASTA property [19],ΓINTcan be expressed as:

whereIs(t)=1 when the MS-DSA system works on statesat timet,andIs(t)=0 otherwise.

RemarkWhen spectrum sensing is accurate enough,i fandPSUI(s,s′|Pf,Pe) ≈ 0 otherwise.Thus,we can estimate the interference quantity by using ΓINT≈PDrop(1-PBlock) according to (23) and (25).This implies that the secondary system can estimate the interference quantity on the PUs by recording the number of blocked and dropped SUs when spectrum sensing is accurate enough.

5) Utility Function:Based on the above performance metrics,we present a utility function for the SU system as follows:

whereωSUis the service price for one completed SU service;ωINTis the price for punishment due to interference on the PUs;ωCPandωRPis the spectrum leasing price of each RP spectrum units and CP spectrum units per unit time,respectively.

V.OPTIMIZATION OF THE MS-DSA SYSTEM

5.1 Spectrum allocation for the MSDSA system

Define vectorγ={γDSA1,γDSA2,...} as spectrum allocation for each DSA strategy in the MSDSA system.Correspondingly,the optimal spectrum allocation refers to the solution of the vectorγto maximize utility functionU,i.e.,

It is difficult to findγ*by optimization approach,since whenever the number of spectrum unit for each DSA strategy changes,the state space will change according to (4),and all performance metrics should be re-derived.Therefore,the exhaustive method is adopted here with computation complexityO N()d,whereNis the number of total sharing spectrum anddis the number of DSA strategies.Note that the computation complexity is acceptable whendis finite and small enough.

Remark:WhenandγDSAj=∀≠0,j i,the MS-DSA system is reduced into a single DSA system.This implies that any single DSA system is a special case of the proposed MS-DSA system,and the performance of the MS-DSA system with the optimal spectrum allocation will not be inferior than that of any involved single DSA system.

5.2 Optimal FA probability selection for MS-DSA system

As aforementioned,the performance of spectrum sensing can be reflected by the ROC curve.Given sensing algorithm and SNR,Peis monotonically decreasing as a function ofPf.On one hand,this relationship implies that we cannot reduce both sensing errors simultaneously.On the other hand,it brings a new way to improve the performance of the MS-DSA system [15].Specifically,asPfincreases,spectrum sensing tends to be conservative such that the coming SU has lower risk of conflicting with the PUs,resulting in less transmission opportunities.On the contrary,a smallPfmakes spectrum sensing more aggressive to obtain more spectrum resources at some expense of interference on the PUs.

In this section,we attempt to find the optimal spectrum sensing strategy to maximize the utility function of MS-DSA system.Since there is a one-to-one mapping betweenPfandPe,the problem can be transformed into finding the optimal value ofPf,i.e.,

Rewrite the utility function in (26) as follows:

whereC=γCP⋅ωCP+γRP⋅ωRPindependent ofPf,r(s) is defined as the reward function with respect to statesand can be derived from (26):

and Π={π(s)},R={r(s)}T,∀∈Ωsis the row vector of the steady-state probability and the column vector of the reward function,respectively.

Due to the implicit relationship betweenPfandU,it is hard to obtain the closedform of the optimalPf.Therefore,we adopt a well-known Newton's method [20] that can approach to the stationary points of the differentiable functionUby iterating the value ofPfusing the following equation:

wheretis iteration times,andis the first and the second derivative ofUwith respect toPf.After finite iterations,the approximate solutions ofPfcan be obtained.

The first and second derivatives ofUwith respect toPfis necessary.However,it is impossible to have the derivatives of Π in terms ofPfbecause there is no explicit relation between them.Our idea to deal with this problem is to express the derivative of Π with the derivatives ofQandRwhich are explicit functions in term ofPf.

Theorem 1:In the MS-DSA system,the first-order and the second-order derivatives ofUwith respect toPfcan be obtained as follows:

whereQ#is the group invers ofQsatisfying

Proof :Refer to Appendix A.

VI.SIMULATION RESULTS AND DISCUSSIONS

6.1 Parameter setting

We assume the PU system totally hasN=10 spectrum units.The PUs arrival rateλp=5 min-1and PUs service rateup=1 min-1.Thus,the spectrum utilization can be calculated by

We assume that the ED-based spectrum sensing is adopted by the SAP.According to [17],the ROC relationship betweenPfandPecan be expressed:

where function

6.2 Analyses vs.simulations

In order to verify the analyses above,we first compare the theoretical results in Section V with the results obtained by event-based simulations.In this simulation,we select three spectrum allocations with (γCP=7,γRP=0,γRES=3),(γCP=0,γRP=7,γRES=3),and (γCP=2,γRP=5,γRES=3).Spectrum sensing parameters are assumed asPf=0.1,SNRdB=-22 .The event-based simulation time is 4×105min,which is shown to be sufficiently large to achieve the steady-state solutions.The SU arrival rateλs=2.5 min-1,and the service rateus=1 min-1.

Figure 5(a),5(b) and 5(c) show the comparisons of theoretical results and simulations results in terms of blocking rate,dropping rate and interference quantity on the PUs,respectively.In these figures,the dashed lines represent the analytical results,and the solid lines are obtained from simulations.Based on these curves,it is observed that along with simulation time elapsing,these performance metrics obtained from simulations eventually converge to the corresponding theoretical results after some fluctuation at the beginning,which indicates that the derivations in (22) (23) and (25) are correct and can be used to calculate the system performance for the proposed MSDSA system.

Fig.5.Theoretical results vs.simulations results.

6.3 Spectrum allocation for MS-DSA system

Figure 6(a),6(b),6(c) and 6(d) plot the performance metrics of the MS-DSA system with the variations of spectrum allocation.For givenγCP,PBlockdecreases as a function ofγRP.This is because the increase ofγRP.This is because the increase ofγRPmakes more spectrum resources available for SUs,thereby decreasing the probability that all spectrum units are occupied when SUs arrive.However,whenγRPincreases,the conflict risk between the PUs and SUs increases,which increasesPDrop.

With the increase ofγCP,bothPBlockandPDropreduces.This implies that when allocating more CP spectrum unit for US,the secondary system has a higher usage priority of the spectrum resources,which leads to an overall improvement for the SU system.

Figure 6(d) shows the ΓINFas a function of spectrum allocation.It observed that ΓINFhas a similar tendency toPDrop.Both ΓINFandPDropcan be seen as the effect caused by the conflicts between the PUs and SUs that are manifested on the PU system and SU system,respectively.Moreover,the value of ΓINFapproximatesPDrop×(1-PBlock),which confirms our Remark in Section IV.

Fig.6.Performance Metrics vs.Spectrum Allocation.

Besides,compared with the RP-DSA and the CP-DSA,the MS-DSA system can reach more points in terms ofPBlock,PDrop,ΓINFandηby adjusting the spectrum allocation.This means the proposed MS-DSA system is more flexible to meet diverse QoS requirements.

Table I shows the variations of utility functionUas a function of users' traffic loads.It is seen that the MS-DSA system with optimal spectrum allocation outperforms both RP-DSA and CP-DSA in terms ofUin any case.When users' traffic load is light (λpandλsis small),the system prefers to use RP spectrum units due to the cheap spectrum price.Whenλpincreases from 0.5 min-1to 3 min-1,the MSDSA system reduceγRPin order to degrade the interference on the PUs slightly,and replace RP spectrum units by CP spectrum units asλpfurther increases from 3.5 min-1to 5 min-1.With the increase ofλs,the SU system have to purchase more spectrum resources in order to the increasingly SU arrivals.Similar to the increase ofλp,the SU system gradually replaces RP spectrum units by CP spectrum units in order to avoid conflicts with the PUs.

Fig.7.Utility Function vs.Pf.

6.4 FA probability selection for MSDSA system

Figure 7 shows the utility functionUas a function ofPfwith different SNRs,wheremin-1,u p=us=1 min-1,γ=(7,0,3) as an example according to Table I.As shown figure 7,for given parameters.Uis concave with respect toPf.This proves that the proposed Newton method is reasonable and reliable.In addition,we notice that the optimalPfincreases when SNR degrades.This is because that lower SNR leads to the decrease of the accuracy of spectrum sensing.In this case conservative sensing strategy with higherPfis proper.Meanwhile,it is also noticed that with the decrease of interference punishmentωINT,the optimalPfbecomes small.The rationality is that,when higher interference is tolerable for the PUs,a more aggressive sensing strategy is preferred by the SU system to find more transmission opportunities.

6.5 Joint spectrum allocation and FA probability selection

At least,we verify the MS-DSA system with optimized spectrum allocation and FA probability selection in a dynamic network environment.The simulation scenario is divided into five periods with different users' traffic loads and SNRs,and each period lasts about 180 minutes,as illustrated in figure 8.From the results,we can see that the proposed MS-DSA system outperforms the existing single DSA system in term ofU,even though the optimal spectrum allocation and FA probability are drived under stationary condition.

Table I.Utility function with optimal spectrum allocation.

VII.CONCLUSIONS

In this paper,we have proposed a multi-DSA strategy system,which breaks the limit that cognitive radio networks can only work in single DSA strategy.Based on the Markov Theory,we have presented a CTMC model and the corresponding analytical solutions to the performance metrics for the MS-DSA system.We have further discussed how to optimize the MS-DSA system by spectrum allocation and false alarm probability selection.Numerical results have demonstrated that the proposed MS-DSA system can extend the adjustable range of performance metrics to a large extent,which consequently results in more flexibility to meet QoS requirements for the SU system,as well as more potentials to improve the system performance.In addition,compared with the existing single DSA strategy systems,the proposed MS-DSA system can achieve better utility in a dynamic network environment.In the future work,we will study how to jointly optimize spectrum leasing and accessing for the MS-DSA system.

ACKNOWLEDGMENT

This work was supported in part by the National Natural Sciences Foundation of China (NSFC) under Grant 61525103,the National Natural Sciences Foundation of China under Grant 61501140 and the Shenzhen Fundamental Research Project under Grant JCYJ20150930150304185.

Appendix A

Proof of Theorem 1

By taking the derivative of (29),we can get the first derivative ofUwith respect toPfas follows:

where ∇Ris an explicit function in terms ofPfand can be calculated directly.

Taking the derivative of the balance equations Π=Q0,we can get

By multiplying both sides of (37) on the right byQ#,and using ∇Π=e0,we can get

Fig.8.Comparisons of MS-DSA system and single DSA system in a dynamic network envrionment.

whereQ#is the group inverse ofQsatisfyingQ#=(Q-eΠ)-1+eΠ,and has the propertyQ#Q=QQ#=1 -eΠ .

Substituting (38) into (36),and then (32) can be obtained as,

Then,taking the derivative of (39) with respect toδ,we can get

Note thatQQ#=Q#Q=1-eΠ,so we can get

and

SinceQ#and (I-eΠ) are exchangeable,i.e.Q#(I-eΠ)=(I-eΠ)Q#.By multiplying both side of (42) on the left byQ#,we can get

Substituting (43) into (40),and then we can get