A Markovian approach for best-fit channel selection in cognitive radio networks

Ad Hoc Networks 12 (2014) 165–177

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A Markovian approach for best-fit channel selection in cognitive radio networks Suzan Bayhan ⇑, Fatih Alagöz Department of Computer Engineering, Bogazici University, Turkey

a r t i c l e

i n f o

Article history: Available online 5 September 2011 Keywords: Spectrum fragmentation Channel selection CTMC Markov model Cognitive radio Best-fit allocation

a b s t r a c t An efficient channel selection scheme in multi-user cognitive radio networks (CRN) is supposed to address two often conflicting objectives: enhancing the network-wide performance while satisfying the individual quality of service demands of cognitive radios (CRs). In this sense, best-fit channel selection (BFC) inspired from well-known classical bin-packing algorithms achieves better performance compared to the longest-idle time channel selection (LITC). BFC facilitates each CR, with the capability of primary channel idle time estimation, select the channel that is expected to be idle for sufficiently long duration for its traffic request. Unlike BFC, LITC favors the selection of the channel with the longest idle time although the channel is not needed and will not be used for such long duration by this CR. As a generalization of these two approaches, we introduce the p – selfish scheme in which a CR selects the longest channel with probability p. Hence, we also refer to p as degree of selfishness. In [1], we evaluate the performance of BFC and show that it improves performance of the CRN in terms of spectrum opportunity utilization and CR throughput, compared to the LITC. In this work, we present an analytic model for BFC using continuous-time Markov chains (CTMC). The performance improvement achieved by BFC is due the reduced spectrum fragmentation that is achieved by best-fit allocation. BFC can be considered as an implicit solution to spectrum fragmentation in time dimension. We study the CR performance in terms of spectrum opportunity utilization and probability of success under various degree of selfishness through the presented model and compare our results with the simulation results.  2011 Elsevier B.V. All rights reserved.

1. Introduction Dynamic spectrum access (DSA) has emerged as a solution to meeting the demand for wireless spectrum caused by the dramatic increase in wireless data volume and proliferation of innovative wireless technologies. Under the current scheme being used since early 1900s, known as static spectrum access, frequencies are allocated to licensed users for their exclusive use. This access scheme is not only far from meeting the needs of today’s communications, but also prevents the emergence of new and innovative ⇑ Corresponding author. E-mail addresses: [email protected] (S. Bayhan), alagoz@boun. edu.tr (F. Alagöz). 1570-8705/$ - see front matter  2011 Elsevier B.V. All rights reserved. doi:10.1016/j.adhoc.2011.08.007

technologies. However, DSA facilitates flexible access to the bands of primary users (PU) by cognitive radios (CRs) provided the PUs are not harmfully interfered. Therefore, it opens new opportunities for emerging technologies. Moreover, the best frequencies with favorable propagation characteristics can be utilized by CRs when they are not occupied by the actual owners, i.e. PUs. This new paradigm provides better spectrum efficiency and thereby can ease satisfying the demand for wireless spectrum. Channel selection in cognitive radio networks (CRNs) requires novel approaches other than those having been applied in traditional wireless networks. The reason is that wireless resources in CRNs are dynamically changing due to PU activity as opposed to the classical static spectrum access networks with quasi-static resources. Moreover,

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channel selection in cognitive radio ad hoc networks (CRAHNs) is more challenging compared to the centralized counterparts in which a base-station (BS) or similar centralized entity manages the allocation of resources. In CRAHNs, each CR decides on its own using various information it has already acquired with the goal of maintaining the highest performance. In networks composed of collaborative users, CRs exchange information regarding their local observations on the environment and also their own states. Besides, if CRs are cooperative, they act in a way that improves the network-wide performance. However, both collaboration and cooperation come at the expense of increased complexity, e.g. cost of message exchanges and control of the medium of this exchange [2]. Instead, CRs may implement non-collaborative and hence non-cooperative mechanisms in which each CR decides on its own utilizing only the local information. In this setting, a CR is selfish and acts with the goal of maximizing only its own performance. This selfish behavior may degrade the overall network performance; some CRs are satisfied with their performances while others suffer from low performance. This is a general issue in any type of resource allocation mechanism, which deals with the distribution of some restricted resources to a group of requesting users. In the CR domain, allocation or sharing of spectrum resources, e.g. frequency, bandwidth etc., requires careful attention for attaining high efficiency. The CRN throughput and spectrum opportunity utilization depend on the efficiency of the channel selection scheme. The orientation of spectrum opportunities, e.g. duration and frequency, is determined by the PU activities. Hence, it is imperative to utilize this information in order to have a better decision scheme. There are various schemes [3,4] that estimate and model the PU channel opportunities via observation and learning. A CR with knowledge on primary channel occupancies can decide on the channel to be accessed accordingly. This knowledge is mostly measured either in terms of the channel’s probability of being idle (pidle) or the estimated idle duration (Tidle) of the channel. In most of the works [5,6], the CR selects the channel with the largest pidle value or the longest Tidle value. For the latter, this is rational at first sight considering the CR as a single entity. However, in a multi-user CRN where the decision of a CR affects the others, this kind of access may not be optimal in network-wide sense. In this work, we argue that instead of such a selfish access scheme, referred to as longest idle time channel selection (LITC), each CR should select the channel with spectrum opportunity sufficiently long to meet its demand. Inspiring from the well-known best-fit bin-packing algorithm [7], we propose best-fit channel selection (BFC). In BFC, each CR with its self-awareness property is assumed to know how long its transmission will take. Moreover, we assume that each CR has the information on PU channel activities owing to the property of environment-awareness. The aim of BFC is to maximize the CR traffic that can be allocated into spectrum opportunities distributed over the PU channels. This problem can be defined as a variant of 0– 1 knapsack problem [8], in which the aim is to maximize the weights that are packed into finite-size bags. CR traffic durations are the weights and spectrum opportunities are

the bags in classic knapsack problem. As in 0–1 knapsack problems, fractional assignment is not possible in our case. That is, once a CR discovers a sufficiently long opportunity, it accesses the channel and completes its transmission. In case there is no sufficiently long opportunity, the packets are dropped and the transmission attempt fails. In our previous works [9], we analyzed BFC via extensive simulations. We compared its performance to that of LITC under various types of traffic activities, increasing number of CRs and degree of selfishness. Simulation studies showed that BFC improves performance of the CRN in terms of spectrum opportunity utilization and CR throughput. In this work, we aim to present an analytic model for BFC using continuous-time Markov chains (CTMC). Markov based modeling has been extensively used in the literature [10–14]. For instance, [10] models a generalized PU and SU spectrum sharing system considering the effect of SU sensing errors. State space constructed via sensing at discrete time instants results in a DTMC model. State space (number of PUs and number of CRs in the system) observed by SUs, so called detected state space, differs from the true state space in number of PUs due to sensing errors. Effect of various parameters such as sensing periodicity, tradeoff between number of channels and sensing cost are analyzed and best operating regions are determined. Similarly, works in [11,13,14] examine the effect of imperfect sensing utilizing CTMC models. [13] compares the access scheme with proportional fairness criterion and with the throughput maximizing criterion via a Markov model. Moreover, performance improvement (i.e. decrease in waiting time) by adding buffering capability to the CRs are determined. Similarly, [12] derives optimal access policies for CRs via CTMC modeling. The remainder of the paper is organized as follows. The next section presents the system model and basics of BFC. Additionally, it provides brief information on spectrum fragmentation which is the underlying phenomenon related to the performance improvement enabled by BFC. Section 3 introduces the corresponding analytic model which utilizes the basic theory of CTMCs. Subsequently, Section 4 examines how accurately this model captures the essential features of BFC or a channel selection scheme in general by a comparison of analytic model with the simulation results. Finally, Section 5 concludes the paper. 2. System model and assumptions 2.1. Network model We consider a system composed of a primary network (PN) and a decentralized cognitive radio network (CRN). The PN is abstracted as a system consisting of primary channels. Hence, rather than modeling the PUs explicitly, we model their effect on the primary channels, i.e. the primary channel traffic patterns. The set of channels is represented by A ¼ fC 1 ; C 2 ; . . . ; C M g; C i standing for the primary channel i. The CRN consists of a number of CRs that is in the interference range of each other. This assumption is for eliminating the challenge of modeling the dynamics of spectrum opportunities due to the spatial variations.

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Traffic of each channel and CRs are modeled as twostate Markov processes: one state representing the activity times and the other for inactivity times. For the PU channels, these states are referred to as busy and idle states whereas for the CRs they are on and off states, respectively. The state sojourn times are exponentially distributed with mean values a⁄ and b⁄, ⁄ 2 {PU, CR}. Both PUs and CRs apply a random access scheme in which PUs access the band whenever they have packets to transmit, whereas the CRs follow the rules of channel selection policy. Fig. 1 illustrates the system under consideration for seven CRs (N = 7) and four PUs (M = 4). As the figure depicts, channel selection can be considered as a function of spectrum opportunities and CR traffic attempt duration that maps to a PU channel (Ci), or 1 (failure). 2.2. Best-fit and p-selfish channel selection Basic operation of the proposed channel selection policy [1] is summarized as follows: the CR checks the occupancy state of each channel in A based on its knowledge on the primary channel traffic pattern. The CR with its self-awareness property, knows how long its next transmission will take. In order not to prevent any interference to the PN, primary channels with spectrum holes longer than this value, Ton, are added to the candidate channels list. Next, the CR prepares its channel search sequence B out of this candidate channels that can meet its transmission time demand. As the occupied primary channels have spectrum holes with zero length, they are not considered in the candidate list. If the candidate channel list is empty ðB ¼ ;Þ, the CR’s traffic transmission request cannot be satisfied and it is dropped. We call this operation approach as a conservative approach and call it transmit-or-drop policy. We assume that the CRs know the traffic pattern of primary channels. How this information is acquired is beyond the scope of this work. However, there are various modeling and estimation approaches existing in the literature. Please refer to [3,6,15,16] for related modeling approaches. Best-fit channel selection (BFC) aims to provide best-fit between the spectrum holes and the CR traffic requests. In this approach, channels in B are sorted in increasing CRs 1

Channel selection

Ton 2 3

f(TA,Ton)

order according to their spectrum hole duration, that is the first element in B is channel with the shortest spectrum opportunity. We call BFC as non-selfish since each CR selects the opportunity that is sufficiently long instead of the longest one. As a generalization of selfishness issue, we introduce degree-of-selfishness (p) to represent how much a CR is willing to cooperate. Simply, it is the probability that a CR will select the longest spectrum opportunity. In this regard, in p-selfish channel selection each CR in each channel selection decision probabilistically selects the longest opportunity channel. Hence, 1-selfish scheme represents the total selfishness to which we also refer as longest idle time channel selection (LITC). Similarly, BFC is the 0-selfish scheme. Channel search sequence determines the order of channels to be sensed by a CR till locating an opportunity. However, since we assume perfect knowledge, the channel expected to be idle (busy) is idle (busy), i.e. probability of miss detection (Pm) and false alarms (Pf) are 0. Hence, the first element is selected as the transmission channel. However, this scheme can be extended to the non-zero Pm cases and CR can continue sensing following the order specified by B. We have evaluated the performance of the proposed policy with increasing degree of selfishness. Simulation results in [1] show that increase in selfishness degrades CRN performance in terms of CR throughput, probability of successful transmission, and queueing delay. In other words, BFC (p = 0) outperforms LITC (p = 1). This performance improvement of BFC over LITC is due to the fact that BFC facilitates the spectrum allocation in such a way that spectrum fragmentation is reduced. In the next section, we provide brief information on spectrum fragmentation issue which have not been examined in details in the literature. 2.3. Spectrum fragmentation in channel selection As CRs occupy the spectrum opportunities and release them upon completion of their transmission, available spectrum becomes increasingly divided into discrete fragments. Although being free, the fragmented spectrum may be effectively unusable due to the cost of using such

Spectrum opportunities at time t: TA= BUSY IDLE

C3

5

7

C1 C2

{Ci, -1}

4

6

PU channels A={C1,C2,C3,C4}

C4 Time of CR traffic arrival: t

Time PU activity: BUSY PU in-activity: IDLE Fig. 1. An example with N = 7, M = 4.

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fragmentation challenge, whereas the latter can combine up to k noncontiguous channels thus tackles the fragmentation issue. However, there is a trade-off between reduced fragmentation and increased overhead that is linear with k. Hence, it is crucial to identify the impact of the parameter k on network performance. Work in [18] introduces the time-spectrum block concept to represent the time during which a CR uses a specific frequency band, and formalizes the spectrum allocation problem. Authors show that finding a feasible schedule, i.e. allocation with non-overlapping time-spectrum blocks, is NP-hard. Hence, heuristics are proposed. However, the introduced approach can better fit to delay-tolerant traffic since lower bandwidth blocks spanning a longer time duration are preferred [17]. For an elaborate analysis of spectrum fragmentation, we refer the readers to recent work on theoretical analysis of channel fragmentation process utilizing the fundamentals of analysis of disk-allocation algorithms [20]. In our work, we consider 1-agile radios and provide a Markovian framework for the channel selection scheme in [1]. The proposed scheme suppresses fragmentation in time dimension at MAC layer by enabling the CRs share spectrum effectively.

small chunks of the spectrum [17]. Fig. 2 depicts this phenomenon in a network consisting of five frequencies F = {f1, f2, f3, f4, f5}. The left figure depicts the change in occupancy of each frequency with time whereas the right figure is a snapshot of the former at a specific time t. As the figures show, spectrum opportunities are distributed over various frequencies with various sizes. Moreover, the left figure also depicts the fragmentation in time dimension whereas the right figure shows how the spectrum is fragmented in frequency dimension. At t, non-adjacent f2 and f4 are idle. In such a case, if a CR does not possess the hardware to utilize these two frequencies simultaneously, it will transmit through only either f2 or f4. However, if they were adjacent, CR would be able to tune its hardware to transmit over these bands. If not tackled seriously, spectrum fragmentation leads to inefficient use of the spectrum opportunities and thereby results in significant detrimental effects on CRN performance. Hence, resource allocation schemes should define precautions in order to overcome fragmentation. Various mechanisms in physical layer (PHY) and medium access control (MAC) layer exist in the literature [17,18]. Providing a physical layer solution to fragmentation, current OFDMA based CRs can utilize these noncontiguous fragments by defragmenting via spectrum aggregation at the expense of increased complexity and spectrum overhead. Channel aggregation requires implementation of guard bands at the boundaries in order to prevent interference between these adjacent channels. [19] proposes two solutions, one at the PHY and one at the MAC layer. At the PHY, CRs transmit combining noncontiguous multiple frequencies to a single higher bandwidth block by using OFDMA. At the MAC, the receiving and transmitting CR pairs synchronously adjust their frequency. In other words, they periodically perform online defragmentation by moving their communication to other bands. Various methods in selection of these new bands are examined. Similar to our work, authors also conclude that best-fit spectrum allocation outperforms all other heuristic approaches, namely worst-fit and first-fit. Defragmentation process proposed in this work may result in disrupting CR communications while moving communications to alternate bands. Work in [17] presents another PHY layer solution and analyzes two kinds of radios, namely 1-agile and k-agile radios. The former can only use single frequency channel with adaptable bandwidth hence faces the spectrum

Frequency

3. Analytic modeling of BFC by Markov chains In this part, we present Markovian model of the proposed channel selection scheme. BFC is proposed for distributed networks in which time synchronization is very challenging. Hence, BFC operates in a random access manner lacking the discrete time-slot operation. Therefore, it has a continuous time nature. Moreover, spectrum opportunities are not discrete. Considering these two aspects, continuous time continuous state space models seem to be more appropriate to the nature of our proposal. However, this approach may lead to state space explosion. Therefore, in this work we prefer a simplified continuous time discrete state space model as described in the following sections. 3.1. State space definition In a system with M primary channels and N CR users, let NPU(t) and NCR(t) stand for the stochastic processes representing the number of PUs and CRs actively transmitting

Spectrum fragments

f1

f2 f3

f4 f5

f1

PU traffic

f2

Spectrum fragment

f3

Frequency

f4 f5 t

Time

At time t, spectrum opportunities may be dispersed over a number of noncontiguous frequency channels.

Fig. 2. Spectrum may be fragmented in CRNs in time (left) and frequency (right) dimensions. Discontinuous OFDMA is proposed as a direct solution in PHY layer for the frequency dimension at the cost of increased hardware complexity. As alternate remedies, upper layer solutions without requiring additional hardware processing manage resource allocation accordingly.

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in the system at time t, respectively. Let X(t) = Si,j = {NPU(t) = i, NCR(t) = j} denote the state of the system at time t. Number of channels that are occupied by PUs is i and total number of CRs transmitting in any of the channels is j. The set of primary channels that are occupied by PUs and CRs are denoted by CPU and CCR , respectively. Cidle ðCbusy Þ is the set of idle (busy) channels. The set of all primary channels is C where C ¼ Cbusy [ Cidle and Cbusy ¼ CPU [ CCR . Note that CPU \ CCR ¼ ;. The set of CRs that are in transmission is denoted by N CR whereas the set of CRs not transmitting is N idle . The set of CRs is N where N ¼ N CR [ N idle and N CR \ N idle ¼ ;. The state space S ¼ fS0;0 ; S0;1 ; . . . ; Si;j ; . . . ; SM;0 g consists of the states Si,j with 0 6 i, j and (i + j) 6 M and j 6 min (N, M). Since simultaneous transmission in a channel is not permitted (i.e. CPU \ CCR ¼ ;), maximum number of transmitting users (CRs and PUs) in the system is restricted to the number of primary channels, M. The idle system is represented by S0,0 in which all channels are idle whereas Si,Mi represents the cases all channels are occupied. All channels can be occupied either by only CRs (S0,M if M 6 N), or only PUs (SM,0) or both types of users exist in the system. Let the set of these states in which all channels are occupied be S full ¼ fSi;Mi g, and the set in which all channels are idle be S empty ¼ fS0;0 g. Since we assume that the CRs are aware of the PU traffic, they do not collide with the PUs. Besides, number of CRs in the network is N, therefore maximum number of CRs in transmission is bounded by the following: min (N, M). Hence, total number of states jSj is:

jSj ¼

M X ðminðM  i; NÞ þ 1Þ:

ð1Þ

i¼0

In practical systems, number of CRs is greater than the number of primary channels, i.e. N P M, leading to min (M  i, N) = M  i. Hence, jSj becomes: M X ðM þ 1ÞðM þ 2Þ : jSj ¼ ðM  i þ 1Þ ¼ 2 i¼0

ð2Þ

In this model (as in [10,14,21]), instead of the state of each channel (whether occupied by a CR, occupied by a PU, or unoccupied), the total number of CRs and PUs transmitting in the system is represented. Hence, channel-based analysis is not possible. If the state of each channel were to be modeled individually, the state space R would consist of the following M-tuples: R(t) = (R1, R2, . . . , Ri, . . . , RM) where the state of Ci can be Ri={occupied by PU (0), occupied by CR (1), idle (2)}. S can be considered as a compact set that represents a group of states in R such that Si,j cor   M Mi responds to different states in R; i and j i j being subject to the same restrictions as before. In this alternate model, the number of states is calculated as follows:

8 M M for M 6 N > < ðjRi jÞ ¼ 3    M minðMi;NÞ jRj ¼ P : P M Mi > for M > N : i j i¼0 j¼0

ð3Þ

However, the comparison of the state space sizes of R and S shows that jRj grows significantly faster with increasing N and M compared to jSj. For instance, jRj is 59,049 whereas jSj is 66 with N = 20 and M = 10. Rather than this model which preserves state information on the individual channels, in order to keep the system analytically tractable, we prefer the former simplified compact model. In this work, we assume that all PU channels are identical, e.g. have the same traffic occupancy distributions. Thus, this simplification would not make a significant difference in our analysis as it would in case of nonidentical PU channels. 3.2. PU channel and CR model In CRN, PU channel traffic is mostly modeled as an on– off process with exponentially distributed state durations [21,22]. We also use this modeling approach as our PU traffic model. Consider a system with M independent and identical channels. Let U define the duration that a PU channel is occupied until the PU completed its transmission. The channel remains idle for a period of D. The first period is referred to as on state while the second period is the off state. Fig. 4 depicts the state of a single channel changing with time. Successive on state durations U are iid. exponentially distributed with l. Similarly, successive off state durations D are iid. exponentially distributed with k. Moreover, U and D are independent. Let X(t) be the random variable denoting the number of primary channels that are occupied by PUs (in on state) at time t. {X(t), t P 0} is a CTMC with the following state space S = {0, 1, 2, . . . , M} [23]. Fig. 5 depicts the states of S and the transition among these states. Given U  exp (l) and D  exp (k), nonzero elements of transition rate matrix Q are defined in (4). 2 6 6 6 6 6 Q ¼6 6 6 6 6 4

Mk

3

Mk

7 7 7 7 7 7: 7 7 7 7 5

l ðl þ ðM  1ÞkÞ ðM  1Þk ..

.

..

.

ðilÞ ðil þ ðM  iÞkÞ ðM  iÞk .. .. . . Ml

ðMlÞ

ð4Þ

CRs operating in on–off manner are modeled similarly. In the following section, we validate our model is a CTMC using these process models. 3.3. CTMC model validation Recall that a CTMC must possess two properties: state sojourn time in any state is exponentially distributed (memoryless property) and time to transition to other states are mutually independent [24]. We check if our model exhibits these properties. Let the system be in state Si,j at time t and e denote the event occurring at time t + Dt. With the effect of the event e the system changes state to Sk,l. e can be one of the following: (1) PU arrival, (2) PU departure, (3) CR arrival, and (4) CR departure. Arrows in Fig. 3 mark transitions caused by these events for the state Si,j. The following defines the state transitions from Si,j.

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NPU

M,0

i+j=M System utilizes the whole capacity. All channels are occupied.

N
i+1,j

6 1 i,j-1

i,0

4

3

i,j 2

i,j+1

i,M-i

5 0,N+1

i-1,j

0,0

0,j-1

0,j

0,K

0,j+1

0,M

0,K+1

NCR

Idle system

Fig. 3. State space S for M primary channels (M 6 N). The state space reduces to the one that only includes the states on the left of the line marked with N < M. The shaded states are missing in CRNs with N < M.

On U1

U2 Time

Off D1

D2

Fig. 4. A PU channel with two states: on (occupied by PU) and off (idle). U and D are iid. exponential random variables with rate l and k, respectively.

Mλ

0

1 µ

i iµ

λ

(M-i)λ

(M-i+1)λ

M-1 (i+1)µ

M Mµ

Fig. 5. In a CRN with M channels, number of channels that are occupied by PUs can be modeled by a CTMC.

 Si,j to Si+1,j: State transition to Si+1,j happens in case of a PU arrival. Since there are (M-i) primary channels not being used by the PUs, there can be an arrival in any of these channels. Hence the first arrival results in Si+1,j. Say the earliest arrival is in channel k at time t + Tk. Hence Ti,jji+1,j can be defined as follows:

T i;jjiþ1;j ¼ minðT k : k 2 C n CPU Þ:

ð5Þ

Since all Tk are exponentially distributed and mutually P  Mi k¼1 kk . We assume all

independent, T i;jjiþ1;j  exp

channels are identical, therefore Ti,jji+1,j  exp((M  i)kPU).

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This derivation can also be directly accessed from Fig. 5 and through formula in (4).  Si,j to Si1,j: Similarly, state transition to Si1,j happens at t + Ti,jji1,j if one of the PUs in service completes its service. The first PU completing its service in channel k releases the channel. Service time Tk is exp(lk), and it is the minimum of i PU channels ðjCPU j ¼ iÞ. The state transition time to Si1,j is defined as follows:

T i;jji1;j ¼ minðT k : k 2 CPU Þ:

ð6Þ

Hence, Ti,jji1,j  exp (ilPU).  Si,j to Si,j+1: In case of a CR arrival, system may enter the state Si,j+1. Since there are N CRs in the network and j are already in transmission, an arrival can happen due to the remaining (N  j) CRs. However, in order to move to Si,j+1, the arriving CR must find an idle channel that is longer than its transmission time. For now, assume that CR can find such a channel with probability Pcs. Hence the first arrival is the event that triggers this state change.

T i;jji;jþ1 ¼ minðT k : k 2 N idle Þ:

ð7Þ

Ti,jji,j+1 is exponential with rate parameter (N  j)kCRPcs.  Si,j to Si,j1: The first CR completing its transmission at t + Tk releases the channel k and results in a new state Si,j1. Duration of stay in Si,j until this transition is Ti,jji,j1 and it is defined as follows:

T i;jji;j1 ¼ minðT k : k 2 CCR Þ:

ð8Þ

From above, it is derived that Ti,jji,j1  exp (jlCR).  Si,j to Si,j: The system does not change state in two cases: – A CR arrival occurs but the CR cannot find an opportunity that is sufficiently long for its transmission although there are some idle channels (arrow 5 in Fig. 3) or – A CR arrival occurs but all channels are occupied (arrow 6 in Fig. 3). For the former, the transition rate is the total CR arrival rate ((N  j)kCR) multiplied by probability of failure in channel selection (1  Pcs). Probability of success (and failure) in channel selection (Pcs) depends on the channel selection scheme and the state of the system X(t) = Si,j. For the latter, rate of transition is total CR arrival rate. Since self transitions are not allowed in CTMCs by definition (i.e. qii = 0 "i), such transitions must be represented in a different way. Hence, we revised our model (arrow 5 and arrow 6 in Fig. 3) as in Fig. 6. In this new model, we represent the failure of an arriving CR’s channel selection attempt by the transition arrow 5. For the cases corresponding to transitions shown by arrow 5, derivation of T i;jji;jfail is similar to Ti,jji,j+1. T i;jji;jfail is exponential with rate parameter (N  j)kCR(1  Pcs). For the remaining cases (arrow 6), we also extend the state space with Si;ðMiÞfull states. Similar to the previous extension, this state represents the cases where the previous CR transmission attempt has failed. However, as opposed to the previous case where there are idle but unsatisfactory channels, it has failed since all channels in the system are already occupied (full). In these states, NPU = i and NCR = M  i. Such transitions are experienced when a CR arrival event occurs

but finds the system full. We represent time to this transition as T i;Miji;ðMiÞfull , and it is also exponential with rate parameter ((N  j)kCR). Let S fail and S f stand for the set of all these added fail states and full states, respectively. With the addition of these two types of states, the new state space can be represented as a two-layered system. One layer stands for the ordinary states whereas the other consists of states in S fail [ S f as depicted in Fig. 7. State transitions are illustrated for a fail state, however it applies to the full states by replacing the transition marked as 5 with transitions marked as 6. Rate of transition k0 depicted in the figure equals to that of k where k 2 1, . . . , 6. Expanding our model with Si;jfail for each state in S n S full and Si;ðMiÞfull for each state in S full , now the state space S has twofold states resulting in jSj ¼ ðM þ 1ÞðM þ 2Þ. As discussed above, each T SjS0 where S; S0 2 S is an exponential random variable and independent of all others. Let kT S;S0 denote the rate parameter for T S;S0 . State P sojourn time T S  expð 8S0 2S kT S;S0 Þ since it is the minimum of all T S;S0 . This completes the verification that our model is a CTMC. 3.4. Transition rate matrix Q The transition probability matrix P defines the probability of change from Si,j to Sk,l. The steady state probability vector p ¼ ½ps ð8s 2 SÞ is obtained by solving the following system:

pP ¼ p; X ps ¼ 1:

ð9Þ ð10Þ

s2S

In the CTMC model validation section, we define all transitions from state Si,j and corresponding transition rates. Before obtaining P = [P(i,jjk,l)] from these rates, we need to define the probability of success in channel selection (Pcs). Pcs is the probability that a CR with transmission request at a time can find an appropriate opportunity. As explained before, if CR can find an appropriate opportunity, the channel selection is completed with a success, or completed with a failure otherwise. Pcs depends on the channel selection scheme, i.e. BFC, LITC or p-selfish. Theorem 1. Theconditional channel   selection  success probabilities for BFC P Bcs ði; jÞ and LITC P Lcs ði; jÞ at state Si,j are given by

PBcs ði; jÞ

¼1

PLcs ði; jÞ ¼ 1 

kBG kBG þ 2kCR on kLG kLG þ 2kCR on

!MðiþjÞ ;

ð11Þ

;

ð12Þ

!MðiþjÞ

B L where kCR on ; kG and kG stand for the parameter of CR on-time distribution, parameter of gap size if BFC is applied as the channel selection scheme and gap size parameter for LITC, respectively.

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i+j=M i+1,j

6

i,(M-i)full

1 4

3 i,j

i,j-1

i,j+1

i,M-i

6 2 5 5 i-1,j

i,jfail

The last channel selection attempt failed. Similar to state (i,j), in this state NPU=i NCR=j

The last channel selection attempt failed since all channels are already occupied.

Fig. 6. State transitions with self transition (arrow 5 and arrow 6) in Fig. 3 are removed.

i,jfail 1'

Fail or full states 4'

3'

5'

5

i+1,j

i,j1

1

4 i,j

2' 2

3

i,j+ 1

i-1 ,j

Fig. 7. Two-layered representation of the state space, one layer has elements from Sfail [ Sf whereas the other is composed of the states in S n ðSfail [ Sf Þ.

Proof. Let T CR on and G denote the random variables representing the CR transmission time duration and PU channel CR idle durations, respectively where T CR on  expðkon Þ. Effective size of PU channel idle duration is the duration that a PU channel is going to be idle observed by a CR at the CR arrival instant. Fig. 8 depicts the single channel case. The CR arrives at TCR and remaining idle time of the channel is Gend  TCR. The channel opportunity has started at Gstart and ends at Gend. For this CR to be satisfied with the channel, the effective size be longer than the requested  must  transmission time T CR on . Assuming that the arrival probability of the CR during a spectrum opportunity is uniform,

the CR arrival occurs at the middle of the spectrum gap on average. If spectrum opportunity size is G = Gend  Gstart, TCR equals to Gstart + (Gend  Gstart)/2. Therefore, expected effective size is G  G/2 = G2. Hence, probability of success in channel selection is simply interpreted as follows:

  G : Pcs ¼ Pr T CR 6 on 2

ð13Þ

We can generalize this finding to the M channel case as in Eq. (14). Pcs(i, j) denoting the success probability at state Si,j is the probability that at least one of the channels in Cidle with jCidle j ¼ M  ði þ jÞ has sufficiently long opportunity.

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If spectrum gap duration at each channel is iid. exponentially distributed with mean k1 G ; P cs equals to the following:

Pcs ði; jÞ ¼ 1 

kG

parameters for M = 8 and N = 10. In the figure, only the states in the first layer (in Fig. 7) are depicted since states in the second layer have the same Pcs values as their counterparts in the first layer. States are enumerated based on their (i, j) values from left to right and down to up according to their location in Fig. 3, starting with S0,0, S0,1, . . . , S0,8, S1,0, . . . , S8,0. Note that change in Pcs follows a pattern and repeats it. Each single pattern corresponds to a row of S in Fig. 3. Remember that in each row, the left to right move represents an increase in NCR, number of active CRs, resulting in decrease in Pcs. Similarly, move to next pattern in Fig. 11 corresponds to a move in up direction in S in Fig. 3. In words, NPU, number of active PUs increases by 1. Regarding the spectrum opportunity duration lengths, success probabilities in the first case where k1 PU ¼ 2:5, are lower than that of the second case with k1 PU ¼ 10. Regarding the access schemes, BFC and LITC seem to have very similar Pcs. However, BFC has always higher values which lead it to outperform the latter in general. The data points on the x-axis stand for the states in S full , hence Pcs are zero. In our model, we use Pcs(i, j) values collected from the simulations. Using the state transitions defined in Section 3.3 and Pcs, we can define the rate of transition from Si,j to Sk,l (denoted by Q(i,jjk,l)) as follows:

!MðiþjÞ ð14Þ

;

kG þ 2kCR on

PBcs ði; jÞ and PLcs ði; jÞ are derived by replacing the gap size parameter (kG) with kBG for BFC and with kLG for LITC as follows:

PBcs ði; jÞ ¼ 1 

PLcs ði; jÞ

¼1

kBG

!MðiþjÞ

kLG kLG þ 2kCR on

ð15Þ

;

kBG þ 2kCR on !MðiþjÞ

:

ð16Þ



Initially, spectrum opportunity (gap) at each channel is exponentially distributed with parameter kPU. However, as the CRs access the bands, gap size distribution changes. Our analysis on the spectrum gap size collected from simulations with a distribution fitting tool [25] shows that effective size of gaps observed by CRs at the CR arrival instant is exponential with parameter kG. Moreover, BFC results in longer gaps (thereby larger effective size) compared to LITC resulting in kBG < kLG . Fig. 9 depicts the mean fragment size for both BFC and LITC with the increasing mean PU spectrum opportunity duration. As seen in Fig. 9, BFC has always longer mean fragment size compared to the selfish scheme LITC. Since we do not make any differentiation among channels, we make this analysis considering gaps from all channels as a single source. On the other hand, as the Fig. 10 depicts, the mean fragment size caused by the fragmentation process for each channel may yield different fragment sizes. However, this variance is marginal and it is ignored for the sake of simplicity. Please note that Pcs depends on the kG parameters. However, these values can be derived by analysis of the simulation results and are computed offline after the completion of simulations. Fig. 11 illustrates the state dependent Pcs values for BFC and LITC under two k1 PU

8 ðM  iÞkPU > > > > > ilPU > > > > > < ðN  jÞkCR Pcs jlCR > > > ðN  jÞkCR ð1  Pcs Þ > > > > > ðN  jÞkCR > > : 0

for k ¼ i þ 1; i < M; l ¼ j for k ¼ i  1; i P 1; l ¼ j for k ¼ i; l ¼ j þ 1; l < M for k ¼ i; l ¼ j  1; j P 1 : for k ¼ i; l ¼ jfail ; i þ j < M for k ¼ i; l ¼ jfull ; i þ j ¼ M ow:

ð17Þ

Pcs values are critical in determining the steady states. Hence, we compute these values by averaging Pcs derived in various ways such as probability in terms of number of attempts and analytic values calculated through Eq. (14) by setting kG > kPU. Inserting Pcs values in Eq. (17) to the transition rate matrix and setting main diagonal eleP ments Q ði;jji;jÞ ¼  ðk;lÞ;ðk;lÞ–ði;jÞ Q ði;jjk;lÞ , we derive Q = [Q(i,jjk,l)].

G: Spectrum opportunity duration

Gstart : Gap start time

Gend : Gap end time

Effective size

TCR : CR traffic arrival CR

Ton : CR requested traffic transmission time Fig. 8. Probability of finding an appropriate channel depends on the CR traffic request size, PU spectrum opportunity sizes, and the state of the system, Si,j.

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S. Bayhan, F. Alagöz / Ad Hoc Networks 12 (2014) 165–177

Next, we derive P = [P(i,jjk,l)] by normalizing each row and setting the main diagonal entries to 0 [26]. Now, since the constructed Markov chain is irreducible, we can solve the linear system of equations in Eqs. (9) and (10), and find the steady state probability distributions. A Markov chain’s irreducibility can be tested using basic graph algorithms. Simply, Markov chain is represented as a directed graph, so testing the irreducibility is possible via finding the strongly connected components of this graph. If there is a single component, then the MC is irreducible [27,28].

14 BFC LITC

G

Mean fragment size (λ−1)

12 10 8 6 4 2 0

3.5. Performance parameters 1

2

3

4

5

6

7

Mean PU off time

8

9

10

In the following equations pi,j denotes the stationary probability of state Si,j.

(λ−1 ) PU

 Average number of CRs transmitting in the network

Fig. 9. Change in mean gap size for BFC and LITC with increasing PU opportunity duration, M = 8, N = 10.

X

E½NCR  ¼

ðjpi;j Þ:

ð18Þ

8Si;j 2S

14

 Average number of PUs transmitting in the network

12 G

Mean gap size (λ−1)

X

E½NPU  ¼

BFC

ðipi;j Þ:

ð19Þ

8Si;j 2S

10

 Average number of channels occupied 8

LITC

E½C busy  ¼

X

ði þ jÞpi;j :

ð20Þ

8Si;j 2S

6

 Average CR throughput is the total CR throughput divided by the number of CRs measured in seconds (e.g. airtime or successful transmission duration). Simply, it is throughput for each CR over M channels for the simulated time duration (Tsim).

4 2 0

Ch 1 Ch 2 Ch 3 Ch 4 Ch 5 Ch 6 Ch 7 Ch 8

RCR ¼

Channel index Fig. 10. Channel based fragmentation analysis for BFC and LITC, k1 PU ¼ 10, M = 8, N = 10.

E½NCR   T sim : N

ð21Þ

 Average probability of successful transmission (ps) is the probability that a CR traffic request can find an

State: (7,0)

State probability of success, Pcs (i)

1 0.9 State:(0,0)

0.8 0.7 State: (1,5)

0.6

State: (2,4)

0.5 0.4

BFC λ

0.3

LITC

−1 =2.5 PU

−1

0.2

BFC λPU=10 LITC

0.1

State: (0,8)

State: (1,7)

State: (2,6)

System is at capacity, i.e. all channels are occupied.

0

State index (i) 1 1 Fig. 11. State dependent Pcs values. M = 8, N = 10 and k1 PU ¼ 2:5 and kPU ¼ 10. For very long spectrum opportunity durations, i.e. kPU ¼ 10, Pcs values are almost 1 meaning that the CR will certainly acquire a channel. For shorter spectrum opportunity case k1 PU ¼ 2:5, success probabilities are lower.

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S. Bayhan, F. Alagöz / Ad Hoc Networks 12 (2014) 165–177

ps ¼

E½NCR  : N  DCR

ð22Þ

 Spectrum opportunity utilization (H): It is the ratio of spectrum opportunities that CRs used for transmission to the total spectrum opportunities through all M channels. DPU is the PU duty cycle.

E½NCR  H¼ : Mð1  DPU Þ

ð23Þ

0.02

Error in steady state probability (Δ)

appropriate spectrum opportunity and can achieve transmission in the selected opportunity. DCR is the CR duty cycle.

Δ =π’−π i

0.015

i

i

0.01 0.005 0 −0.005 −0.01 −0.015

State (i) 4. Evaluation of the analytic model Fig. 12. Error in steady state probability distribution for N = 5. Di ¼ p0i  pi .

5.5

Average number of transmitting CRs

In order to measure how good the introduced model matches to the real system, we compare our analytic model with the model defined by a system-level simulator. A discrete-event simulator is developed to mimic the considered model and operation of channel selection schemes. Simulation runs are collected from a number of independent runs (10 independent runs) for ensuring the statistical goodness. Since we simplified our continuous time continuous space model into a compact model that only considers the number of PUs and CRs in the network, we expect some deviation between the performance results of these two models. However, if the presented analytic model has the power more-or-less to present the real world (simulated) case, the performance evaluation of analytic model will match that of the simulated model. To this aim, we analyze three cases: low, moderate and heavy load CR traffic cases. In all cases, we set number of channels M = 10, kCR = 1, lCR = 1, kPU = 0.4, and lPU = 2. Under these parameters, setting N = 5 leads to low load case with 0.3 CR traffic load. We represent the moderate and heavy load cases by setting N = 10 and N = 15 corresponding to the CR traffic load 0.6 and 0.9, respectively. As the derivations in Section 3.5 show, performance metrics are directly computed from steady state probability vector. Hence, as a first step, we compare the steady state probabilities derived from analysis pi,j with the ones derived from the simulations p0 i,j [29]. In order to ensure the system is in steady state, simulation time is set to sufficiently long duration. Fig. 12 plots the error in steady state distribution where Di;j ¼ p0i;j  pi;j . The zero-line (values exactly on x-axis) shows the perfect match between p and p0 whereas points much above/below this line show high deviation between the analytic and simulated model. These deviations result into errors in computation of performance values. The errors lie in interval [0.010, 0.015]. In this case jSj = 102 while for N = 10 and N = 15, jSj = 132. Fig. 13 depicts the change in average number of transmitting CRs and PUs in the system. At a first glance, it can be seen that selfishness degrades the CRN performance leading less CRs to be able to capture a channel for transmission. Fig. 13 shows that there is almost a perfect match for low load case in average number of transmitting CRs. The error is around 0.7–4% depending on the degree of selfishness. For moderate and high load cases, analytic results

5 4.5 4 Analytic, CRs, N=5 Simulation, CRs, N=5 Analytic, CRs, N=10 Simulation, CRs, N=10 Analytic, CRs, N=15 Simulation, CRs, N=15

3.5 3 2.5 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Degree of selfishness (p) Fig. 13. Comparison of average number of transmitting CRs in analytic model and simulations.

deviate from the simulations with an error rate between 0.5–6% and 3–8%, respectively. Regarding PUs, as expected, number of PUs does not change with p. PUs are independent of the CRN and their performance is not affected by the access mechanism. Figs. 14 and 15 depict the corresponding ps and H values for each load case. As the derivations in Section 3.5 show, ps and H are linear functions of NCR. Hence, the error in computation of NCR is carried to these metrics. Therefore, these two metrics follow the same trend as NCR depicted in Fig. 13. Examining the results, we see that there is a better match between analytic model and simulations under low load while the error increases in moderate and heavy load scenarios. In low load case, the error is around 0.1–3% whereas it is 0.7–6% in moderate and 3–8% in heavy load case. These results also apply to the analysis of H. The increase in deviation between the two models is due to the increased state space size. With larger state space, the error depicted in Fig. 12 increases on the average. This may be mitigated with much longer simulations.

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S. Bayhan, F. Alagöz / Ad Hoc Networks 12 (2014) 165–177

Fig. 16 depicts the performance of BFC and LITC for  1 increasing CR mean on duration 1=kon CR or lCR . As the CR packets get longer (increasing CR on duration), the success probability decreases. This is expected since fractional transmission is not enabled in the system under consideration. Therefore, transmission requests exceeding the spectrum opportunities at the time of transmission attempt fail. Hence, CR packet durations should be kept significantly lower than the mean PU spectrum opportunity duration. However, note that smaller packet size results in higher overhead. In our scheme, such an analysis is not included. However, it should be considered in the design of practical communication systems.

1 0.95

Probability of success (ps)

Simulation, N=5

Analytic, N=5

0.9 0.85

Analytic, N=10 Simulation, N=10

0.8 Analytic, N=15

0.75 0.7 0.65

Simulation, N=15

0

0.2

0.4

0.6

0.8

1

Degree of selfishness (p) Fig. 14. Comparison of probability of success (ps).

Spectrum opportunity utilization (Θ)

0.65 0.6 0.55 0.5 0.45 Analytic, N=5 Simulation, N=5 Analytic, N=10 Simulation, N=10 Analytic, N=15 Simulation, N=15

0.4 0.35 0.3 0.25

0

0.2

0.4

0.6

0.8

1

Degree of selfishness (p) Fig. 15. Comparison of spectrum opportunity utilization in analytic model and simulations (H).

BFC LITC

0.9

s

Probability of success (p )

1

0.8

0.7

5. Conclusions In our earlier work [1], we strictly relied on the simulation results and showed that BFC enhances the CRN performance by effectively sharing the spectrum among CRs. In this work, we introduced a Markovian approach for theoretical analysis of BFC. We first provided an overview on spectrum fragmentation concept that is crucial to disclose the performance superiority of the BFC scheme over longest idle time selection (LITC) scheme. Fragmentation is a well-examined topic in the scope of disk-allocation algorithms, however it still remains to be explored for CRNs. In CRNs, spectrum becomes fragmented due to the nature of DSA, e.g. channels with adaptive bandwidths as opposed to fixed-bandwidth channels in classical wireless networks. Implications of fragmentation on the CRN performance is crucial to be clarified in order to have better resource allocation schemes. Hence, our work can be considered as an effort towards this goal. Next, we introduce our Markov based model, define the derivations and performance metrics. Finally, we evaluate our model by comparing it with the outcomes of simulations presented in [1]. Our model is a simplified one that models a continuous time continuous state space model with a continuous time discrete state space model. Despite this simplification, our results show that it captures the basics of BFC. As future work, we aim to define a more elaborate model with continuous state space model that can better capture the operation principles of proposed channel selection without requiring Pcs values to be utilized that are computed in simulations. However, it is quite challenging to discover such a scheme with low-complexity that overcomes the exponential state space explosion. Moreover, for a more realistic CRN, effect of errors in spectrum opportunity time estimation and PU detection impairments, i.e. non zero Pm and Pf values, must be incorporated in the model.

0.6

Acknowledgments 0.5

0.4

1

1.5

2

2.5

3

3.5

4

4.5

5

−1

CR mean on duration (μCR) Fig. 16. Probability of success with increasing CR on duration for kPU = 5.

This work is supported by the State Planning Organization of Turkey (DPT) under the TAM Project with Grant No. DPT-2007K 120610, and the Scientific and Technological Research Council of Turkey (TUBITAK) with Grant No. 109E256. The authors thank H. Birkan Yilmaz for his valuable discussions and feedback on the Markovian analysis.

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Suzan Bayhan received her BS and MS degrees in Computer Engineering both from the Bogazici University, Istanbul, Turkey in 2003 and 2006, respectively. Currently she is a PhD candidate in the same department, and works as a researcher at Telematics Research Center (TAM) in the Bogazici University. She received the EMEA Google Anita Borg Scholarship in 2009, which is awarded to the outstanding women studying computer science. Her research interests are cognitive radio networks and satellite communications.

Fatih Alagöz is an associate professor in the Computer Engineering Department of Bogazici University. During 2001–2003, he was with the Department of Electrical and Computer Engineering, United Arab Emirates University. He obtained his B.Sc. degree in electrical engineering in 1992 from Middle East Technical University, Turkey, and his D.Sc. degree in electrical engineering in 2000 from George Washington University, Washington, DC. His current research areas include wireless/ mobile/satellite networks and UWB communications. He has edited various books and published more than 100 scholarly papers in selected journals and conferences.