Balancing Data Security and Blocking Performance with Spectrum Randomization in Optical Networks

Transcription

1 Balancing Data Security an Blocking Performance with Spectrum Ranomization in Optical Networks Saneep Kumar Singh, Wolfgang Bziuk, an Amela Jukan Technische Universität Carolo-Wilhelmina zu Braunschweig, Germany {saneep.singh, w.bziuk, arxiv: v [cs.ni] 8 Apr 206 Abstract Data ranomization or scrambling has been effectively use in various applications to improve the ata security. In this paper, we use the iea of ata ranomization to proactively ranomize the spectrum (re)allocation to improve connections security. As it is well-known that ranom (re)allocation fragments the spectrum an thus increases blocking in elastic optical networks, we analyze the traeoff between system performance an security. To this en, in aition to spectrum ranomization, we utilize an on-eman efragmentation scheme every time a request is blocke ue to the spectrum fragmentation. We moel the occupancy pattern of an elastic optical link (EOL) using a multi-class continuous-time Markov chain (CTMC) uner the ranom-fit spectrum allocation metho. Numerical results show that although both the blocking an security can be improve for a particular so-calle ranomization process (RP) arrival rate, while with the increase in RP arrival rate the connections security improves at the cost of the increase in overall blocking. I. INTRODUCTION Securing high ata rate applications in optical networks against physical layer attacks or unauthorize observation has long been subject of intense stuies [], [2]. In elastic optical networks (EONs), the security challenge can be mitigate in various ways, among others by ata scrambling along the coe, time an frequency omains [3]. The ata scrambling along the multiple imensions helps to resist the brute-force attacks, an makes it ifficult for the attacker to ecoe the ata. With the flexibility in assigning subcarriers, we believe that spectrum allocation in EONs also present a unique opportunity to provie optical layer security. If the spectrum ranomization process is regularly performe, then only a portion of a particular user s ata will be observe over a range of frequencies that is listene by an eavesropper. However, the ranom spectrum (re)reallocation fragments the spectrum, an as many previous stuies have shown, increases the blocking in EONs [4], [5]. [4] showe that the blocking probability ue to banwith fragmentation in EONs epens on the size of the available spectrum blocks on a link, an their alignment over ifferent links on the routing paths. In [5], we analytically showe that spectrum reallocation increases the overall blocking in an elastic optical link (EOL). To alleviate this problem, spectrum efragmentation scheme can be utilize to consoliate the free spectrum. During efragmentation, also some connections coul be isrupte ue to the retuning of transceivers an reconfiguration of optical switches. Although the isruption can be minimize using some techniques like make-before-break [6] or hitless efragmentation methos [7], [8], spectrum reallocation is not S 7 S 8 S 9 Fig S ( SR ), i 4 RaaS (SR4) {7,8,9,0} All possible state transitions from an into a RaaS state ue to RP. esirable. Therefore, fining a traeoff between ranom spectrum scrambling an the nee to efragment the spectrum to improve the performance is no easy task, which has not been aresse to ate. As we show, moeling a single EOL spectrum (re)allocation is analytically a challenge uner the spectrum contiguity constraint, an it gets more complicate in networks ue to an aitional spectrum continuity constraint (assuming no spectrum converter in EONs) [9] [3]. In this paper, we scramble/ranomize the spectrum usage pattern to secure elastic optical links, which we refer to as ranomization-as-a-service (RaaS) scheme. The RaaS can be performe proactively at ranom or perioic intervals, with mean RP interarrival time ( ). However, we also show that blocking gets worse ue to this ranomization process ( ). To this en, we utilize a efragmentation-as-a-service (DaaS) scheme, which is triggere every time a connection woul be blocke because the spectrum is scramble an fragmente. For the analytical moeling of the combine DaaS an RaaS, we moel an EOL using the multi-class continuoustime Markov chain (CTMC) an inclue RaaS an DaaS states in aition to regular ata service states. Combining proactive RaaS an on-eman DaaS, we show that security is as goo as in the RaaS system, while blocking is better than if only a RaaS system use, with much lower reconfiguration time (than call holing times) an for a range of moerate loas. The rest of this paper is organize as follows. Section II presents the moel escription. In Section III, we present the overall blocking an the security analysis. Section IV evaluates the performance. Finally, we conclue the paper in Section V. II. MODEL DESCRIPTION The ranomization of spectrum allocation makes it more ifficult for an attacker to etect the spectrum assignment pattern an to successfully emoulate the spectrum, thus increasing the system security. At the same time, the ranomization of spectrum allocation might result in fragmente spectrum. This is illustrate in Fig., where the assignment

2 S 8 λ ( S ) p 7 S 2 µ Fig. 2. DaaS (S2) S 9 λ ( S ) p 0 2 µ S State transitions from an into a DaaS state for a eman of 3-slots. of spectrum slots to existing connections is ranomly reconfigure by entering the RaaS process (state SR 4 ), an as a result, one of the states S 7, S 8, S 9 an S 0 is chosen with equal probability. Notably, the ranomization feature can also be use to reallocate spectrum resources to consoliate free slots, which we call as a DaaS. This is illustrate in Fig. 2, where a request (say R i ) with eman 3 slots will be blocke in fragmente states S 8 an S 9 in a regular or RaaS system. In the combine RaaS-DaaS moel, however, the arrival of the request R i into such fragmente states will trigger DaaS process (state S 2 ), which reconfigures the existing connection(s) an finally, the system will move to efragmente states S 7 or S 0 with equal probability. A. Unerlining Assumptions We moel the occupancy pattern of an EOL using a multiclass CTMC [0] [3] with the following assumptions. i) The RaaS process is triggere for spectrum ranomization at exponential RP inter-arrival times T S with average rate. ii) The DaaS process is triggere when a request is blocke ue to the fragmentation of spectrum. iii) Call inter-arrival times T Ek an call holing times T Hk of class-k requests are inepenent an exponentially istribute with average rates λ k an µ k, respectively. iv) In RaaS an DaaS states, reconfiguration times T RT are exponentially istribute with rate µ. v) The services of all existing requests are interrupte uring the reconfiguration times of the DaaS an RaaS processes. vi) Further incoming requests are blocke uring the DaaS an RaaS perios, referre to as reconfiguration blocking. vii) We ifferentiate between resource blocking an blocking ue to fragmentation; in the case of resource blocking, the total number of free slots oes not satisfy the eman, while in the case of fragmentation blocking, there are enough free slots, but there is no sufficient amount of consecutive free blocks to satisfy the eman. Assumption (v) is isavantageous, because traffic interruption of some or all existing connections oes happen uring reconfiguration in the real system, though there are efficient techniques to aress this issue [6] [8]. Due to assumption (vi) we woul experience an increase in overall call blocking probability. However, the results show that although reconfiguration blocking is ae to the resource an fragmentation blocking (assumption vii), the overall blocking can be lower than a regular system without spectrum reallocation strategy, if the spectrum reallocation time is much lower than the normal service times of the connections. Notation C λ k (µ k ) TABLE I NOTATIONS AND THE PARAMETERS USED IN THE MODEL Description Total number of spectrum slots (or capacity units) Arrival (service) rate of class k calls, where k, 2,, K / Mean RP interarrival time (E[T S ] / ) /µ Mean reconfiguration time (E[T RT ] /µ ) n S i S ν SR ν p Sν i n k (S i) n(s i) a(s i, k) F S l (S i) (n, n 2,..., n K ), where n k is the number of class k calls Occupancy state for normal operation i,..., N SA Defragmentation state ν,..., N D Ranomization state ν,..., N R ( p SRν i ) transition probability from state S ν (SR ν) to target state S i N SA, N D, N R Number of class-k calls in state S i (n (S i),..., n K (S i)), realization of n in state S i Number of ifferent ways class-k call can be allocate in S i Size of l th fragment of free slots in state S i Number of regular, DaaS, an RaaS states respectively B. Mapping Transitions Between System States For the combine RaaS-DaaS moel, some of the notations an parameters are liste an escribe in Table I. Here, in aition to regular ata service states (S i ), we have RaaS an DaaS states, which are use for spectrum ranomization an efragmentation operations, respectively. Each RaaS (DaaS) state is associate with a non-overlapping set of states (S i ) with same connection pattern as SR ν (S ν ), where a connection pattern is efine by the number of connections per class, i.e. n(s j ) (n (S j ),..., n K (S j )). In general, we efine a set of states with same connection pattern as S j as follows. Γ(S j ) {S i n(s i ) n(s j ), i,..., N SA } () It shoul be note that a specific RaaS state SR ν is triggere (i.e., S i SR ν ) by states, with same connection pattern but ifferent spectrum occupancy pattern, belonging to the set Γ(SR ν ). Furthermore, after ranomization in the RaaS state SR ν, the transition SR ν S i can also be given by the set Γ(SR ν ). For example, in Fig., all states out of set Γ(SR 4 ) {S 7, S 8, S 9, S 0 } transit to a RaaS state SR 4 at rate. An, after ranomization process in SR ν, it transits back to a ranomly chosen system state S i Γ(SR 4 ) with rate p SR4 i µ µ /4, i {7, 8, 9, 0}. Due to the ranomization strategy, we assume equal transition probabilities p SRν i Γ(SR, where Γ(SR ν) ν is the number of elements in Γ(SR ν ). The mapping between a DaaS state to regular states, on the other han, is not straightforwar, since not all but only fragmente regular states S i with n(s i ) n(s ν ) trigger transitions into the DaaS state S ν. Similarly, from S ν transitions occur to only efragmente states S j satisfying n(s j ) n(s ν ), where in a efragmente state S j, all empty slots form a single block of free spectrum. Let us first efine a fragmente state, which can not allocate a large enough block F S l (S i ) of l consecutive free slots to a class-k request of eman k > l, even though it contains equal or more than

3 S RaaS (SR) ( SR ), i 2,..,6 S 2 S 6 µ λ / a( S ) 2, µ λ / a( S6, ) S 2 S 3 ( SR2 ), i,...,3 RaaS (SR2) S 3 µ λ S Fig. 3. S 5 S 4 λ 2 λ + λ 2 λ 2 ( S ), i DaaS (S) µ 2 λ2 µ λ S 5 S 4 ( SR3 ), i 4,5 µ λ / a( S, ) µ λ Partial state iagram with transitions leaing to spectrum ranomization an efragmentation in the RaaS-DaaS moel uner RF allocation policy. 2,6 RaaS (SR3) k number of free slots, as follows. if k > max l F S l (S i ) an fr(s i, k) k C j jn j (S i ) (2) 0 otherwise Using the above function, the set of fragmente states for class-k requests is efine as FB(k) {S i fr(s i, k), i,, N SA }, k. (3) Furthermore, the set of classes for which state S i is a fragmentation state is FI(S i ) {k S i FB(k)}, i,, N SA }. Resource blocking states, on the other han, o not have sufficient free slots to satisfy an incoming class-k request, i.e., RB(k) {S i k > C j jn j (S i ), i,..., N SA }. (4) Now using Eq. (), Γ(S ν ) comprises all states with connection pattern efine by DaaS state S ν. The subset FF(S ν, k) Γ(S ν ) FB(k) efines the set of states to be e-fragmente ue to an arriving class-k request, e.g. in Fig.2, for a class- arrival the set FF(S 2, ) inclues the fragmente states S 8 an S 9. Similarly, after reconfiguration, a DaaS state S ν will transit to a new target state S i FT(S ν ) with probability p Sν i, e.g. in Fig.2 from state S 2 to state S 0 with rate µ p S2 0. A target state is efine by the property, that all its free spectrum slots buil one consecutive block F S (S i ) irrespective of its spectral location. Thus, the set of target states is given as follows. FT(S ν ){S i F S (S i )C j jn j (S i ), S i Γ(S ν )} (5) Hence, the transition probability is given by p Sν i FT(S. ν) In Fig. 2, we have FT(S 2 ) {S 7, S 0 } with two possible target states, hence FT(S 2 ) 2 an p S2 7 p S2 0 /2. Now, we consier an example of an EOL with capacity C 7 slots uner ranom-fit (RF) policy, see Fig. 3. There are two ifferent classes of requests with emans 3 an 4 consecutive slots. Here, we show only a part of all states an transitions, where a new call of class k arrives in state S with eman 3. Uner RF policy, it allocates one of the possible assignments shown by the transitions from S to states {S 2, S 3, S 4, S 5, S 6 }. In the target states, we have n (S i ), i 2,..., 6 class- calls, which will leave F 4 free slots, thus there are a(s, k ) (n+f)! n!f! 5 ifferent ways to allocate the 7 free slots, an transitions occur with rate λ /5. Similarly, a new call of class k 2 arrives in state S 6 with eman 2 4 an can occupy only the possible state S 5. On the other han, transition rates out of states S i, i,..., 5, will occur ue to the eparture of a class- an/or class-2 requests with rate µ k, k, 2. As we can see, when a class-2 request (4 slots) arrives, blocking of the request ue to fragmentation shoul occur in the set of states FB(2) {S 3, S 4, S 5 }. In this simple example, we have FF(S, 2) Γ(S ) FB(2) FB(2), an the arrivals of such class-2 requests with rate λ 2 will trigger the transition from states S i FF(S, 2) to DaaS state S. Similarly, the set of states FB() {S 4, S 8, S 9 } are fragmente states for a class- call hanle by DaaS states S an S 2. Notice that we have FF(S, ) Γ(S ) FB() {S 4 }, an S 4 is also hanle by S for a class- arrival with rate λ. In total, this results into the transition from S 4 to S with joint rate λ + λ 2. The remaining states out of FB() are hanle by S 2 ue to the set FF(S 2, ) Γ(S 2 ) FB() {S 8, S 9 } as shown in Fig.2. In a similar form, all states out of the set Γ(SR ) {S 2, S 3, S 4, S 5, S 6 } will also transit to a RaaS state SR, an because Γ(SR ) 5 the process transits back to a ranomly chosen state S i Γ(SR ) with rate µ /5. It shoul be note that while the process is in DaaS or RaaS states, existing connections effecte by spectrum reconfiguration are interrupte for a time uration equal to the exponentially istribute reconfiguration time T RT with mean /µ. Due to the memoryless property of the exponential istribution, the remaining call holing times of interrupte calls are again exponentially istribute with mean /µ k. Thus, all transitions ue to call terminations after interruption are also moele with rates µ k. III. BLOCKING AND SECURITY ANALYSIS At first, we show how to erive the global balance equations (GBEs) of the Markov chain. Furthermore, we efine the following terms to calculate the transition rates of the system. if a class-k request is accepte, or A(S i, k) blocke ue to fragmentation in S i (6) 0 otherwise A(S i, k) etermines if a transition from state S i is possible ue to the arrival of a class-k call, an inclue the case that S i is a fragmente state for this class, i.e k FI(S i ). Aitionally,

4 T ± (S j, S i, k) etermines the cause of transition if it is ue to allocation (+) or eallocation (-), given as if S j S i ue to class-k T ± (S j, S i, k) arrival (eparture) in S j (7) 0 otherwise. To moel the ifferent many-to-one an one-to-many mapping between RaaS an regular states S i, we introuce an inicator function, which compares the connection pattern of states. { if n(s i ) n(sr ν ) δ(s i, SR ν ) (8) 0 otherwise. The above relation can also be use to efine the many-toone mapping from fragmente states S i to a DaaS state S ν (i.e., δ(s i, S ) with the restriction S i k FF(S, k). Similarly, one-to-many mapping from a DaaS state S ν to target efragmente states (in FT(S ν )) is efine as follows. { if S j FT(S ν ) σ(s ν, S j ) (9) 0 otherwise Finally, the GBE of each regular occupancy state S i uner the RF spectrum allocation policy can be obtaine by Eq. (0). ( K ) A(S i, k)λ k + n k (S i )µ k + π(s i ) k N SA j,j i k K ( λk T + ) (S j, S i, k) + T (S j, S i, k)µ k π(s j ) a(s j, k) ν N D + σ (S ν, S i ) µ π(s ν ) FT(S ν ) ν N R + δ (SR ν, S i ) µ π(sr ν ) Γ(SR ν ), i,, N SA (0) In Eq. (0), left han sie represents the output flow rate from the state S i incluing transitions to DaaS (taken into account by A(S i, k)) an RaaS states (rate ), while the right han sie represents input flow rate into the sate S i. More precisely, the secon line of Eq. (0) represents the input flows from other regular states S j, while the thir line of Eq.(0) efines the rate from exactly one DaaS state to state S i. The inicator function σ(s ν, S i ) selects a DaaS state only if S i FT(S ν ) i.e., if S i is a target efragmente state, an it can be reache after reconfiguration in S ν with rate µ FT(S. In the last line, the inicator δ (SR ν) ν), S i ) selects a correct RaaS state SRν with connection pattern equivalent to the state S i, where the factor Γ(SR ν ) takes into account, that the RaaS state only ranomly selects a target state S i with the probability / Γ(SR ν ). Similarly, the GBEs of a RaaS state SR ν an a DaaS state S ν can be given by Eq. () an Eq. (2), respectively. µ π(sr ν ) π(s j ), ν, 2,, N R () S j Γ(SR ν) K µ π(s ν ) k λ k δ (S j, S ν ) π(s j ), ν,..., N D (2) S j FB(k) In Eq. (2), the function δ (S j, S ν ) selects the corresponing fragmente states S j FB(k) with relate rate λ k. As an example, the GBE for state S 4 in Fig.3 (which is the only one of the fragmentation states for both classes) is given by (λ +λ 2 +µ + )π(s 4 ) λ /5π(S )+µ /5π(SR ). The input rate λ /5 (µ /5) is ue to transition S S 4 (SR S 4 ), as explaine in Sec. II-B. For the output rate, a eparture of an existing class- connection takes place with rate µ, an RP requests arrive with rate. Furthermore, we have two ifferent classes of requests that will be blocke ue to fragmentation, i.e. S 4 FF(S, ) an S 4 FF(S, 2), an thus will cause transition to a DaaS state S with total rate λ +λ 2. Similarly, we can write the GBE for a RaaS state SR 3, which is reache from a set of states Γ(SR 3 ) {S 4, S 5 } when a RP request arrives with rate in one of these states. After ranomization in SR 3, system will return back to one of the possible states in Γ(SR 3 ). However, the total output flow rate from SR 3 is still µ. Hence, the GBE of state SR 3 is given by µ π(sr 3 ) (π(s 4 ) + π(s 5 )). The GBE of a DaaS state S can also be written as µ π(s ) λ 2 π(s 3 )+(λ +λ 2 )π(s 4 )+λ 2 π(s 5 ). In this example, we have δ(s j, S ) for regular state S j FB(), which selects a state S 4 for class-, as well as S j FB(2) which selects states S 3, S 4, S 5 for class 2 arrivals. Uner the stationary conition, the state probabilities π [π(s ), π(s 2 ),, π(s NSA ), π(sr ),, π(sr NR ), π(s ),, π(s ND )] can be calculate by solving π Q 0 subject to i π(s i) + ν π(sr ν) + ν π(s ν), where N N SA + N R + N D is the total number of states, an Q is the transition rate matrix. The blocking per class-k ue to the unavailability of resources, i.e., resource blocking (RB), an ue to fragmentation, i.e. fragmentation blocking (FB) can be calculate using Eq. (3) an Eq. (4), respectively. RB(k) π(s i ) (3) F B(k) π(s i ) (4) S i RB(k) S i FB(k) In a RaaS system (I), i.e., without DaaS states an associate transitions, reconfiguration blocking can be given as follows. N R RCB I π(sr ν ) (5) ν On the other han, the reconfiguration blocking in the combine RaaS-DaaS system (II) is given as follows. N R N D RCB II π(sr ν ) + π(s ν ) (6) ν ν It shoul note that RaaS an DaaS states are blocking states for all classes of requests. Finally, the overall blocking in the RaaS system (I) an the combine RaaS-DaaS system (II) can be given by the Eq. (7) an Eq. (8), respectively. BP I RCB I k + λ k(rb(k) + F B(k)) k λ (7) k

5 S i Observation Winow [6, 9] Fig. 4. An example of eavesropping in spectrum winow [6, 9]. BP II RCB II + k λ k(rb(k) + F B(k)) k λ k (8) In a regular system i.e., without DaaS an RaaS states, the GBEs can be simplifie. In Eq. (0), all transitions from an into DaaS (S ν ) an RaaS (SR ν ) states are not present, which can simply be moele by setting µ 0 an 0. Furthermore, Eq. (2) an () have to be omitte. In the regular system, in aitional to the blocking ue non-availability of resources, blocking also occurs ue to the fragmentation. Hence, the overall blocking for regular system can be calculate as, i.e., BP regular k λ k(rb(k) + F B(k)) k λ (9) k A. Security Analysis In this subsection, we show how spectrum ranomization can improve the connections security. Consier a scenario, where an eavesropper gets access to a transponer with fixe capacity f 0 f n to spoof the ata of a particular user. It shoul be note that if RaaS is not performe within the lifetime of the user s connection allocate over f 0 f n, then all ata volume of the connection is observe. Therefore, in this case, security of those ata volumes epens if the eavesropper can ecoe the ata or not. However, in our moel, only a fraction of the user s ata will be observe by the attacker if RaaS is performe frequently, which enhances the ata security on top of the encryption. RaaS states are inclue for this purpose, which ranomly reassign subcarrier slots to connections. We assume that RaaS is triggere at exponential RP inter-arrival times T S with average rate. An, after ranomization, it returns to one of all possible states out of set Γ(SR ν ) having same connection pattern (but ifferent occupancy patterns) as shown in Fig. an Fig. 3. Let us calculate the total number of possible rearrangements of connections in a state S i. If the state S i contains n k (S i ) connections of class k with eman k, an there are E(S i ) C K k n k(s i ) k empty slots. Then, for the RF policy an without istinguishing the same class of connections, the number of all possible rearrangements of connections of S i with spectrum contiguity constraint can be given as follows []. ( E(S i ) + ) K k n k(s i )! R n (S i ) E(S i )! K k n (20) k(s i )! Now, let us assume that an eavesropper can observe any part of spectrum with slot-with W, where W C, an the observe winow is uniformly istribute over the entire spectrum. For example in Fig. 4, the probability (p W ) that at any point in time the observe winow lies in the range [6, 9] is /, since there are (4-4 +) ifferent ways of observation across spectrum with C 4 an W 4. In general, for a uniform istribution, p W C W +. Furthermore, an attack (eavesrop) is efine as successful if the observe connections within the spectrum winow (n in ) before an after a ranomization process remains same, an the probability of a successful attack for an observation winow of size W (i.e., P W SA ) is given as follows. [ P W SA N SA i2 P W SA(S i ) where P W SA(S i ) ] π(s i ) NSA j2 π(s, j) C W + p W Rj n W (S i ) R n (S i ) j (2) In Eq. (2), PSA W (S i) is the successful attack probability given that the system is in state S i, an it is given by the expecte value of the ratio of Rn j W (S i ) an R n (S i ). This ratio gives the successful attack probability for a given winow [j, j + W ] an state S i. Here, Rn j W (S i ) efines the possible number of rearrangements of connections of n(s i ) which results in same n in in the winow [j, j + W ] before an after the ranomization. Consier example in Fig. 4, where an attacker listens to a connection (with eman 3- slots) within its observation winow [6, 9] when the system is in state S i. Here, there is a single connection for every class with eman k {2, 3, 4}, k, 2 an 3. Therefore, for a given connection pattern n(s i ) (,, ) an 5 empty slots, the ranomization process will result in one of all possible rearrangements i.e., 280 ( (5+3)! 5! ). It shoul be note that there is only one connection with k 3-slots insie the observation winow. Therefore, for an attack to be successful, the insie connection pattern n in (S i ) (0,, 0) shoul be same as before an after the ranomization. In general, the number of such rearrangements (Rn j W (S i )) is not straightforwar. However, for a small-scale EOL, we can calculate the number of such rearrangements by counting all states that comprise same connection pattern n, an same n in. In Fig. 4, it is easy to see that the number of rearrangements insie the winow [6, 9] i.e., Rn 6 in (S i ) 2 (Eq. (20)); an there are 6 possible rearrangements of connections an free slots outsie the winow (i.e., Rn 6 out (S i ) 6, explaine latter). Therefore, there are Rn 6 W (S i ) 32 possible spectrum occupancy patterns out of 280, where a 3-slots connection within the winow [6, 9] can be successfully observe even after the ranomization. Now, we show in three steps how to fin the number of rearrangements outsie the winow, which is similar to a partition problem a NP har problem [4]. Firstly, form a multiset E of elements as (for each free slot) an eman k (for connections in n out (S i )). Seconly, fin the number of possible ways of partitioning of E into two subsets E an E 2 such that e i E e i C L an e i E 2 e i C R, where C L an C R are the number of slots to the left an to the right of the winow, respectively. Finally, for each partition set fin the possible permutations of the elements. In Fig. 4, we have four free slots an two connections with emans 2 an 3 4 (i.e., n out (S i ) (, 0, ) for K 3) outsie the winow. Here, E {,,,, 2, 4}, an C L C R 5.

6 Different types of Blocking Overall Blocking Probability Fraction of Observe ata ($) Winow Size 4 Winow Size 8 Winow Size 2 Winow Size 6 Simulation Resouce(6 S 7 0) Regular,6 7 0 S 0.5 Fragment(6 7 0) S 6,7 0 S 0.4 Resouce(6 S,7 0) 6,7 00 S Fragment(6 S,7 0) 6 S 2, Reconfig(6,7 0) S 6 2,7 00 S 0.2 Simulation Simulation (a) Offere Loa (Erlangs) (b) Offere Loa (Erlangs) (c) RP arrival rate (6 S ) Fig. 5. Evaluation of RaaS moel with C 20: (a) blocking parts in a Regular an RaaS system; (b) overall blocking in a Regular an RaaS moels for various reconfiguration rate (µ ) an RP arrival rates ; an (c) fraction of observe ata in RaaS moel for various at a fixe µ 00. There are two partitions {e (,,, 2), e 2 (, 4)} an {e (, 4), e 2 (,,, 2)} which sum to 5 each. Therefore, the total number of permutations of partition set elements is 2 4 3! 2! 6, thus R6 n out (S i ) 6. The average number of RP performe uring a lifetime of a connection (N r ) can be calculate as the ratio of the average holing time of a connection an the average RP interarrival time i.e., N r /µ k / µ k, where we restrict N r N for nµ k, n {, 2, }. Until RP is triggere for the first time, an attacker observes ata successfully with probability. Therefore, the amount of ata successfully observable before RP is b, where b is the average ata rate observe in a winow of size W. Similarly, until the 2 n RP, successfully observable ata is b [+P W SA successfully observable ata until N th r ]. In general, the amount of RP is given as follows. D W SA b [ + P W SA + + [P W SA] Nr ] b [P SA W ]Nr PSA W (22) Now, the total amount of ata transmitte uring the mean b holing time /µ k is µ k. Therefore, the fraction of successfully observe ata is given as the ratio of successful observe ata an total transmitte ata, an given as follows. Λ [,W ] µ k [P SA W ] µk PSA W IV. NUMERICAL RESULTS TABLE II PARAMETERS USED FOR THE NUMERICAL RESULTS EOL capacity C 20 C 00 Demans ( k ) {4, 6, 8} {5, 0, 5} Reconfiguration rate (µ ) {0, 00} {00, 000} RP arrival rate ( ) {, 2,, 0} {, 2,, 0} (23) In this section, we present the analytical an verifying Monte Carlo (MC) simulation results for a small-scale EOL with capacity C 20 spectrum slots. Since the number of states increases exponentially with the number of slots, we present MC simulation results for a large-scale EOL with capacity C 00 slots. The parameters use for the two ifferent scenarios are liste in Table II. Arrival rates are uniformly istribute i.e., λ k λ K, where λ is the total arrival rate. We assume mean holing time as one unit for all classes of requests [0] [2]. Loa on the link is calculate as K λ k k k µ k. We consier ranom-fit spectrum allocation for the performance evaluation ue to the nature of our moel which ranomizes the spectrum irrespective of the spectrum allocation policy. In Fig. 5, we evaluate our RaaS moel (i.e., with RaaS states, but without DaaS states) against a regular system without spectrum reallocation. Fig. 5a shows the blocking parts ue to the lack of resources (RB, Eq. 3), fragmentation (FB, Eq. 4) an reconfiguration (RCB, Eq. 5). As expecte, in a regular an RaaS systems an for traffic loa range shown here, the blocking is ominate by fragmentation an not by resource unavailability. In Fig. 5b, blocking in our RaaS moel is always higher than the regular system irrespective of the an µ. However, the overall blocking in RaaS system comes closer to the blocking in regular system when µ 00. The reason is that when mean reconfiguration time (E[T RT ] /µ ) reuces (i.e., rate is increase from 0 to 00), then the number of requests arriving uring reconfiguration also reuces, an hence the RCB reuces. It shoul be note that when the average RP interarrival time ecreases (i.e., increases), then blocking also increases significantly. The reason is that RaaS states are calle frequently an ranomization oes result in fragmente spectrum. However, with the increase in the average RP arrival rate ( ), the fraction of observe ata ecreases exponentially (in Fig. 5c), which means security also increases exponentially. Here, the fraction of observe ata is plotte (Eq. (23)) at fixe link loa of 20 Erlangs. Aitionally, we see that when the size of the observation winow W increases, the fraction of observe ata also increases ue to the increase in the probability of success (Eq. (2)). Note that when winow size (W ) equals to capacity (C), full spectrum can be observe with probability (Eq. (2)), an the attacker will observe 00% of ata. It shoul be note that RaaS moel provies security at the cost of a slight increase in blocking. Therefore, now we present

7 Overall Blocking Probability Blocking Gain (%) Fraction of Observe Data ($) Winow Size W 0 Winow Size W 30 Winow Size W 50 Winow Size W Regular,6 S , S 6 S, S, , S 6 S 2, S 2, , S 6 2, S (a) Offere Loa (Erlangs) (b) Offere Loa (Erlangs) (c) RP arrival rate (6 S ) Fig. 6. Evaluation of the combine RaaS-DaaS moel with C 00: (a) overall blocking in a Regular an RaaS-DaaS moels; (b) Blocking gain (in %) in RaaS-DaaS moel as compare to a Regular system; an (c) fraction of observe ata in RaaS-DaaS moel for various at a fixe µ 00. in Fig. 6 performance of the combine RaaS-DaaS moel to show how the DaaS process impacts blocking an security for a large-scale EOL with capacity 00 slots. As compare to regular system an RaaS system, the overall blocking in our combine RaaS-DaaS moel highly epens on the loa (in Fig. 6a). To unerstan this, we plot blocking gain against the loa for various an µ in Fig. 6b. Here, blocking gain is efine as the change in blocking (in %) in our combine moel as compare to the regular system. First, we see that at lower loas blocking is higher irrespective of the an µ. This is ue to the fact that λ k, an therefore the system spens more time in RaaS states which blocks all incoming requests. At moerate loas, blocking gain is positive an increasing with loa. However, note that with the increase in, the loa (connection arrival rate) shoul also increase to get the positive gain. Finally, at higher loas gain again ecreases ue to increase in fragmentation. At last, the security in the combine moel (Fig. 6c) is as goo as in the RaaS moel (Fig. 5c) ue to the fact that efragmentation is not just shifting the spectrum usage towars one en, but also ranomizes the connections while keeping free slots intact. The most important finings of this paper are: i) in RaaS moel, increase in blocking can be minimize as compare to the regular system without reconfiguration epening on the mean reconfiguration time (E[T RT ] /µ ); ii) however, if the mean reconfiguration time is much lower (e.g. µ 000) than the mean holing time E[T Hk ], the overall blocking in our combine RaaS-DaaS moel is lower than the regular system without reconfiguration, for a range of moerate loas; an iii) the security can be improve by many fols with only a smaller RP arrival rate. Base on our finings, network operator can tune the reconfiguration times an RP arrival rate for a particular loa value of the network in orer to experience the higher gain from the our moel. V. CONCLUSION In this paper, we examine the effect of spectrum reallocation on the connections blocking an the associate security. 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