Cooperative Caching in Dynamic Shared Spectrum Networks

Transcription

1 Fina version appears in IEEE Trans. on Wireess Communications, 206. Cooperative Caching in Dynamic Shared Spectrum Networs Dibaar Das, Student Member, IEEE, and Ahussein A. Abouzeid, Senior Member, IEEE Abstract This paper considers cooperation between primary and secondary users in shared spectrum radio networs via caching. We first consider a networ with one channe shared between a singe macro (primary) base station and mutipe micro (secondary) base stations. Secondary base stations can cache some primary fies and thereby satisfy content requests generated from nearby primary users. For this cooperative scenario, we deveop two caching and scheduing poicies under which the set of primary and secondary request generation rates that can be supported increases from the case without cooperation. The first of these agorithms, Fixed Primary Caching Poicy (FPCP), provides more gain in the set of supportabe request generation rates. However under this agorithm primary pacet transmissions from secondary base stations have the same priority of access as secondary pacets and thus might suffer in terms of deay. In the second agorithm, Variabe Primary Caching Poicy (VPCP), primary pacet transmissions from the secondary base stations have higher priority of access than that of secondary pacets. We find that the set of request generation rate vectors for which a queues in the networ are stabe under each of these agorithms is greater than that under any non-cooperative agorithm. We conduct extensive simuations to compare the performance of both agorithms against that of an optima non-cooperative agorithm. Finay, we extend the anaysis to a networ with mutipe channes. Index Terms Cognitive radio, caching, sma base station, heterogeneous networ, cooperation, reay, queuing theory, Lyapunov drift, networ stabiity. I. I INTRODUCTION N recent years, dynamic shared spectrum or cognitive radio networs have been widey studied to efficienty use the often under-utiized spectrum []. In such networs secondary (i.e., unicensed) users opportunisticay transmit on a given channe if the primary (i.e., icensed) users of that channe are inactive. Recent wors have aso studied cooperation schemes where primary and secondary users assist in each other s transmissions. Such schemes can hep both types of users by reducing the duration of primary users transmission activity, thereby increasing secondary users transmission opportunities. Such cooperation has been studied from a physica-ayer information-theoretic perspective e.g., [2] [4]. Other recent wors, e.g., [5] [9], study networ ayer aspects of cooperation such as queuing and prioritized scheduing. Our wor in this paper beongs to the atter group of wors. Another soution that has been proposed to support increasing mobie networ traffic is the use of base stations with smaer coverage areas (commony caed sma-ces D. Das and A.A. Abouzeid are with the Department of Eectrica and Computer Engineering, Rensseaer Poytechnic Institute, Troy, NY, 280 USA e-mai: (dasd2@rpi.edu; abouzeid@ecse.rpi.edu). SB SB 2 Secondary user PB Primary user Fig.. A networ with primary base station (PB) co-existing with secondary base stations: SB and SB 2. The outermost circe and smaer circes indicates the transmission region of PB and SB, SB 2 respectivey. Number of secondary base stations (M) is 2, number of primary users (N ) and secondary users (N (s) ) are 7 and 5 respectivey, number of primary users within transmission range of SB (ϕ ) is 2, number of secondary users within transmission range of SB (ϕ (s) ) is 3. [0]). This eads to a higher spatia re-use of the spectrum. However, the imited capacity of bachau ins at the base stations may reduce the impact of this approach []. As a resut, caching popuar fies at nearby base stations has been proposed (e.g., in []) to reduce bac-hau usage as we as improve the deay performance for users. Caching is aso an attractive practica soution since storage-capacity is reativey inexpensive compared to other networ resources. In this wor we expore cooperation between primary and secondary networs via caching. We consider a primary networ consisting of a singe base station that serves primary users. Co-existing with the primary networ is a set of sma secondary base stations that serve secondary users. A schematic exampe of such a networ architecture is shown in Fig.. Content requests from users are queued at base stations. The base stations serve these requests by transmitting pacets corresponding to the requested fies from their cache if the fie is avaiabe in the cache. If the requested fie is not currenty present in its cache (i.e., if it is not cached ), then the base station fetches this fie, possiby after some deay, so as to satisfy the content request. We assume that content is fetched periodicay; we ca this period the cache-refresh period. Secondary base stations ocated coser to primary users may have better down-in channes than the primary base station. Therefore, if some of the content requests from primary users are served via these secondary base stations then it might free up spectrum resources for use by the secondary users. For the above networ scenario we study the ey probem of deveoping caching and scheduing poicies with performance guarantees. In particuar, we design agorithms under which

2 2 a centraized networ controer determines: (a) which fies to cache in every cache-refresh period and, (b) which fie requests (referred to as requests in the rest of the paper) to admit at a given base station and how to schedue transmissions in each time sot. The objective of the networ controer is to maintain the stabiity of the queues, i.e., to eep the ength of a queues in the networ bounded. For a networ with one channe we deveop two agorithms: Fixed Primary Caching Poicy (FPCP) and Variabe Primary Caching Poicy (VPCP), using Lyapunov-drift techniques [2]. The objective of FPCP is to increase the set of supportabe request generation rates whie ensuring that secondary base stations can serve at east one type of secondary fie request in every period. The FPCP agorithm soves this probem by having each secondary base-station cache a fixed set of primary fies and ony one secondary fie in every period. The objective of the VPCP agorithm is to increase the set of supportabe request generation rates such that (a) secondary base stations serve primary user requests with high priority and (b) minimize the number of primary fies cached at secondary base stations. The motivation behind the first constraint is to ensure that primary user requests have higher priority of service even at secondary base stations. The motivation behind the second constraint is to ensure that secondary base stations ony cache as many primary fies as is required to maintain stabiity of the networ. The VPCP agorithm soves this probem by having each secondary base station cache variabe sets of primary fies in each period based on a penaty parameter. We then prove that the set of primary and secondary request generation rate vectors for which a queues in the networ are stabe under each of these agorithms is greater than that under any non-cooperative agorithm. Unie the FPCP agorithm under VPCP agorithm the secondary base-stations can serve more than one type of secondary fie requests in each cache refresh period. Simuation resuts show that this agorithm, with proper seection of a penaty parameter, tends to have better deay performance than the FPCP agorithm when request generation rates are ow. In such scenarios it is not necessary, for the purpose of system stabiity, to cache as much primary fies as possibe in every secondary base-station. In Section VII we aso show that the deay performance of VPCP is not guaranteed to be better than FPCP for any choice of the penaty parameter. However, the guaranteed stabiity region of the networ under VPCP is ess than that under FPCP. Finay, we consider a networ with mutipe orthogona channes where a secondary base stations are non-interfering to each other (i.e., they can simutaneousy transmit when primary base station is not transmitting) and deveop an agorithm for this scenario, referred to as MutiChanne Caching Poicy (MCCP). There is a significant body of wors on caching in noncognitive wireess networs (e.g. [], [3], [4] and the references therein). Caching in cognitive networs has been studied recenty in [5] and [6]. However these wors did not consider primary-secondary cooperation. Our periodic cacherefresh poicy and the use of Lyapunov drift to deveop a scheduing poicy is motivated by [7] and [8]. However, their resuts do not directy appy in the cognitive networ setting since here primary users have higher priority of channe access than secondary users. The novety of our wor ies in showing how Lyapunov drift techniques can be used to design efficient joint cooperative caching and scheduing agorithms in dynamic shared spectrum networs. The difficuty of designing such agorithms ies in higher priority of channe access for primary users. Whie a simpe Lyapunov drift-based agorithm tries to serve the queues with higher bacogs first, in our mode the queues in the primary base station woud have to be served before those at the secondary base stations even when the former have reativey ower bacog. The difficuty of deveoping caching and scheduing agorithms is even higher when primary user requests are served with higher priority even at secondary base stations, a scenario considered in the VPCP agorithm. Such compications resuting from higher priority of service for primary users are addressed in this wor, and up to our nowedge, have not been addressed before. The paper is organized as foows. In Section II we describe the system mode for a singe channe case. In Section III, we define an achievabe capacity region consisting of supportabe request generation rate vectors (i.e., request generation rate vectors for which a queues in the networ are stabe). Using this region we measure the stabiity performance of the FPCP, VPCP and MCCP agorithms. In Section IV we present the FPCP agorithm and show that under FPCP a queues are stabe for every request generation rate vector within the achievabe capacity region. In Section V we present the VPCP agorithm and find a guaranteed stabiity region for it. Section VI presents the mutichanne networ mode, an achievabe capacity region and the MCCP agorithm that stabiizes the networ for a request generation vectors within this region. In Section VII we present simuation resuts. Section VIII concudes the paper. Due to ac of space, we provide proofs of Lemmas, 2 and Theorems, 2 and 3 in our technica report [9]. II. SYSTEM MODEL The networ consists of a macro-ce wherein a singe primary base station PB serves N primary users: PU,...,PU N. It aso contains M secondary sma-ces SC,..., SC M associated with secondary base stations (SB): SB,...,SB M respectivey. There are N (s) secondary users: SU,...,SU N (s). Exacty one sma base station serve each secondary user. We consider a discrete time mode. Every request is served by successfuy transmitting C pacets of equa size. A base station attempts a new pacet transmission ony at the beginning of a time sot; it can attempt at most one such transmission at any time sot. If a pacet transmission fais, the pacet is re-transmitted at another time sot. Next, we present detais about the transmission mode, interference mode, caching and scheduing mode. We summarize important notations in Tabe I. A. Transmission Mode for Primary and Secondary Base stations We mode a hybrid access cognitive femtoce system wherein SBs transmit ony in sots when PB is not transmitting and can serve nearby primary users. A base stations transmit

3 3 TABLE I LIST OF NOTATIONS N (N (s) ) number of primary (secondary) users PU i (SU i ) i-th primary (secondary) user M number of sma ces t r,j j th sot in r th period F ( F (s) ) set of primary (secondary) fies D set of primary avaiabiity matrices λ i (λ (s) i ) request generation rate from PU i (SU i ) D primary avaiabiity matrix A f,i (t)(a(s) f,i (t)) requests generated at sot t by PU i(su i ) PB primary base station F i (F (s) i ) i-th primary (secondary) fie SB i i-th secondary base station µ X (t) transmission rate offered to PB for pacets of fie f under poicy X at sot t µ X f, (t) transmission rate offered to SB for pacets of fie f under poicy X at sot t U (s) (t)(u f,i f,i (t)) U P i queue ength at sot t for primary (secondary) fie f in SB i (t) queue ength at sot t for primary fie f in PB (P (s) i ) probabiity of i-th primary (secondary) fie being requested, given primary (secondary) fie request at fixed power. A SBs have identica transmission range. We consider the case where some primary users have poor downin channe from PB than from adjacent SBs. This situation can occur (e.g., due to shadowing or near-far effect) in severa rea ife scenarios such as in the case of macroce edge users that are far away from PB but happen to be coser to an SB. We are interested in studying the best case gains obtained by offoading primary user traffic to nearby SBs. Hence, for simpicity, we assume non idea transmissions from PB to primary users, but idea transmission from SBs to users within their transmission range: A) At every sot, a transmission from PB to a primary user ocated within and out of transmission range of an SB succeeds with probabiity p and p respectivey (where 0 < p, p ). At any sot an SB transmits to a user within its transmission range ony if PB is not transmitting simutaneousy; the transmission succeeds with probabiity. In the rest of the paper, for simpicity, we have assumed that p equas p. However, the anaysis can be easiy extended for any other vaue of p. Each primary user is served by at most one SB, i.e., no two sma-ces contain the same primary user. We denote the set of primary and secondary users in SC i, i {,..., M}, as ϕ i and ϕ (s) i respectivey. A secondary users and base stations are within the transmission range of PB and share a singe channe. B. Caching Mode Every time an SB fetches a set of fies for caching it incurs some overhead cost. We mode this cost by requiring that SBs ony cache fies periodicay. Higher frequency of caching refects higher cost. A cache-refresh period is defined as the number of time sots between two successive caching events. A cache refresh period may aso represent the frequency with which contents in fies become outdated thereby requiring newer versions to be fetched. It consists of T time sots with the first caching event being at time sot t =. Henceforth, we refer to the cache refresh period simpy as period whenever there is no confusion. C. Content Requests We consider the case where primary and secondary networs are different, and hence have statisticay different content requirements. This can occur, for exampe, if the The case where the users have homogeneous content requirements can be studied in a simiar fashion. sma-ces serve an industria or academic environment whie users of the macro-ce are the genera pubic who are typicay interested in video content. We denote the ibrary of fies requested by primary users as F and individua fies in the set as F,...,F. Simiary we denote the ibrary F of fies requested by secondary users as F (s) and individua fies in the set as F (s),...,f(s) F (s). The sets F and F (s) are mutuay excusive. Henceforth we wi ca fies in F and F (s) as primary and secondary fies respectivey. Simiary, we ca pacets corresponding to primary and secondary fies as primary and secondary pacets respectivey. A fies are of equa size. We denote the set of primary and secondary fies cached by base station SB, {,..., M} at time sot τ as (τ) and H(s) (τ) respectivey where H (τ) F and (τ) F (s). Users request fies according to a fixed popuarity distribution. Given a primary user requests a fie, the requested fie is F i with probabiity P i where i {, 2,..., F }. Simiary, given a secondary user requests a fie, the fie H H (s) F (s) i is requested with probabiity P (s) i. Without oss of generaity, we assume fies are indexed according to their popuarity, i.e., P i P (s) i+ and P j P (s) j+ i {, 2,..., F } and j {, 2,..., F (s) }. for every At every time sot, primary user PU m, m {,..., N }, requests a fie with probabiity λ m. Note that the probabiity that PU m requests fie F i therefore equas λ m P i. Simiary, at every time sot, secondary user SU, {,..., N (s) }, requests fies with probabiity λ (s). A request generation processes are identica and independenty distributed (iid) from time sot to time sot. We denote primary and secondary request generation rate vectors:(λ,..., λ ) T and (λ (s) N,..., λ(s) ) T as λ and N (s) λ (s) respectivey. D. Service Discipine for Primary Users A request from a primary user is served by either a secondary base station (if the primary user is ocated in a sma-ce and the associated secondary base station contains the fie) or by the primary base station. Every base station maintains queues for every fie. Queues store pacet requests that need to be satisfied in response to a request. Whenever a base station admits request for a primary fie f F, C pacet requests 2 are queued at the queue for fie f. Within each 2 Reca, each fie request is served by successfuy transmitting C pacets.

4 4 queue pacet requests are satisfied in a first-come first-serve (FCFS) manner. In order to focus ony on the effect of cooperative caching by the secondary networ, we assume the primary base station can cache a the F primary fies. On the other hand, the size of cache at each secondary base station is finite. Each such cache can store at most B fies where 0 B min( F, F (s) ). A secondary base station ony admits a primary request if the requested fie is currenty present in its cache 3. We denote a request generated from PU i, i {,..., N }, at time sot t for fie f (where f F ) by variabe A f,i (t). This variabe equas C if such a request is indeed generated at t; otherwise it equas 0. We denote the queue ength of primary fie f (where f F ) in SB, {,..., M}, at time sot t as U f, (t). If PU i is within transmission range of a secondary base station SB containing the fie f requested by PU i at t, then SB may admit this request 4 ; C pacet requests are queued at SB and U f, (t) is incremented by C. Otherwise, this request wi be served by PB and the ength of the associated queue, denoted as U (t), wi be incremented by C. The system begins at time sot t= with a queues being initiay empty. On successfu transmission of a pacet corresponding to primary fie f (where f F ) from PB at t, U (t) is decremented by. We denote the transmission rate offered to base station PB and SB for pacets of fie f, where f F F (s), under caching and scheduing poicy X at time sot t by the binary variabes µ X f,0 (t) and µx f,(t) {0, } respectivey. Therefore the queue ength at PB under poicy X evove as: U (t + ) = U (t) µx (t)i PB (t) + A f,i (t) + + i:pu i M j= ϕ j M = M = i:pu i ϕ i:pu i ϕ ) A f,i ( (t) Îf,(t) A f,i (t) ( χ f, (t) ) Îf, (t) f F, () where I PB (t) is an indicator variabe representing whether the transmission from PB to the primary user at time sot t was successfu or not, χ (t) is an admission-contro variabe f, representing whether SB (where {,..., M}) admits request at time sot t for cached primary fie f H (t) and Îf,(t) is an indicator variabe representing whether SB (where {,..., M}) contains a primary fie f at time sot t or not. The variabe I PB (t) equas if the transmission is successfu at t and is zero otherwise; χ f, (t) equas if 3 Secondary base stations do not admit requests for uncached primary fies because doing so wi iey deteriorate deay performance of primary users. Note that secondary base stations cannot serve such requests at east unti the next cache refresh period. But PB can aways serve such requests as its cache aways contains every primary fie. 4 The admission contro decision is taen by the networ controer. The controer may decide not to admit a primary request at a secondary base station even when the requested fie is present in the atter s cache. SB admits request for f at t and is zero otherwise. The variabe Îf,(t) equas if SB contains fie f at t and is zero otherwise. No transmission-rate is offered to PB when a its queues are empty. The queue ength of primary fies at secondary base stations evove under poicy X as: { U f, (t + ) = max + } f, (t) µx f,(t), 0 U i:pu i ϕ A f,i (t)χ f, (t) {,..., M}, f H (t). (2) E. Service Discipine for Secondary Users Requests from a secondary user is submitted and aways admitted by the unique secondary base station associated with that user. Simiar to the case of primary fies, each secondary base station maintains a queue for every secondary fie. We denote a request generated from SU j, j {,..., N (s) }, at time sot t for a secondary fie f (where f F (s) ) by a variabe A (s) f,j (t). It equas C if such a request is indeed generated at t; otherwise it equas 0. For every {,..., M} we denote the queue ength in SB at t of fie f as U (s) f, (t). The ength of this queue is incremented by C every time SB receives a request for fie f. It is decremented by every time SB transmits a pacet of f. The queue evove under poicy X as, { } U (s) f, (t + ) = max U (s) f, (t) µx f,(t), 0 + A (s) f,j (t) j:su j ϕ (s) {,..., M}, f F (s). (3) We refer to the queues corresponding to primary and secondary fies as primary and secondary queues respectivey. We denote the vector of a primary and secondary queues in the networ at sot t as U (t) and U (s) (t) respectivey. In this paper we use the notion of strong stabiity of queues, defined as foows. Definition : A discrete time queue U(t) is strongy stabe if im sup t t t E[ U(τ) ] <. The networ is strongy stabe τ=0 if every queue in the networ is strongy stabe. F. Interference Mode We represent the set of a secondary base stations that can transmit simutaneousy by an activation vector of ength M. An activation vector E is binary and its m th, m {,..., M}, component corresponds to SB m. It is set to if SB m is transmitting in that time sot; otherwise it is set to zero. The set of a feasibe activation vectors is denoted as Ẽ. No secondary base station can transmit when PB is transmitting. III. ACHIEVABLE CAPACITY REGION In this section we describe an achievabe capacity region Λ corresponding to the system mode in Section II. The region contains every request generation rate vector for which a queues in the networ are stabe considering ony a restricted set of caching and scheduing agorithms. First, we specify this restricted set of agorithms that satisfy a certain caching constraint; this constraint is usefu for tractabe anaysis. Sec-

5 5 ond, we introduce a new term caed the primary avaiabiity matrix which represents the set of primary fies currenty cached at secondary base stations. Third, we obtain the set of feasibe transmission rates for secondary base stations given the probabiity with which each primary avaiabiity matrix is used in each period. Fourth, we use the set of feasibe transmission rates for secondary base stations to describe the region Λ simiar to the description of capacity region in [2]. Fifth, we find a simper description of the region Λ in Lemma. Sixth, for the networ with non-interfering secondary base stations we compare the region Λ with the actua capacity region in Lemma 2. The actua capacity region consists of a request generation rate vectors for which every queue is stabe, considering a stationary caching and scheduing agorithms, not just the restricted ones. Seventh, we discuss extending our anaysis for unequa fie sizes. Finay, we discuss maximum supportabe request generation rates for both types of users for a symmetric networ. In the description of the region Λ we ony consider the set of caching and scheduing agorithms that satisfy the foowing constraint: C: Each secondary base station has at east one secondary fie in its cache in every sot. We consider ony this restricted set of agorithms instead of a caching and scheduing agorithms because it aows for tractabe anaysis and is not a huge imitation. In particuar, it aows us to determine the stabiity constraints at secondary base stations without accounting for probabiity of the event: the networ controer offers transmission rate to a secondary base station which has no cached secondary fie. Obtaining probabiity of such events is cumbersome but is required to anayze stabiity performance of agorithms which do not satisfy constraint C. Using constraint C does not distort the system mode significanty from that in rea environments for a coupe of reasons. First, this constraint guarantees each secondary base station at east has one secondary fie to transmit in any sot. Since the main goa of a secondary base station is to serve secondary users, in a rea ife system, a constraint simiar to C (e.g., one that requires each secondary base station to cache at east 2 or some other non-zero number of secondary fies per period) might be required to baance the abiity of a secondary base station to transmit mutipe type of secondary fies in a period versus increasing transmission opportunities to secondary users. Second, we show that the performance gap (in terms of supportabe rates) between poicies satisfying C and other stationary poicies corresponds to the rate of requests for the B th most popuar primary fie. For arge B and F, as in rea systems (e.g., B = 00 and F = 000 []), this rate is sma. We quantify this gap in Lemma 2 for the networ in which a secondary base stations are non-interfering to each other. Now we introduce the primary avaiabiity matrix. A primary avaiabiity matrix is a binary matrix that indicates the avaiabiity of primary fies in SBs at any given time sot. Each such matrix contains the same number of rows and coumns as the tota number of primary fies and number of SBs respectivey. The (, j) th component of a primary avaiabiity matrix D (where {,..., F }, j {,..., M}), denoted as D,j, equas if the fie F is present at the cache of SB j ; otherwise it equas zero. Given a primary avaiabiity matrix D, we indicate whether or not a primary fie f (where f F ) requested by PU i, i {,..., N }, is present in a nearby secondary base station by the binary variabe Q(i, f D ). In particuar, for every primary fie F, the variabe Q(i, F D ) equas if there exists j such that (s.t.) D = and PU i ϕ j ; it is set to 0 otherwise. Since,j for tractabe anaysis we consider ony agorithms that satisfy constraint C, we consider ony the set of primary avaiabiity matrices for which the sum of every coumn is ess than B. This is because each coumn of the primary avaiabiity matrix denotes the set of primary fies cached at an SB and under C each SB can have upto (B ) primary fies in its cache at any sot. We denote this set as D. We find the set of feasibe transmission rates for SBs, given a probabiity distribution over the set of primary avaiabiity matrices in D, in the foowing manner. Consider the vector q = (q D ) where the term q D denotes the iid probabiity with which the primary avaiabiity matrix D D is seected in each period. For every D the term q D shoud be non-negative and ess than (i.e., 0 q D ) and the sum of a q D terms shoud equa (i.e., = ). Ceary, due to D D q D assumption A, given a set of primary fies cached at SBs, transmission opportunities for SBs are maximized when each SB admits a requests of a cached primary fie. Then, the rate of primary pacet transmissions by a SBs, equas C N = λ D D F = P Q(, F D )q D. For a stabe system the rate of primary pacet transmissions by PB is the rate of a primary pacet requests minus the rate of a primary pacet requests served by SBs, i.e., N C N = λ D D F = = Cλ ( ) P Q, F D q D. The probabiity that PB is not transmitting in a given time sot is therefore, N N F Cλ C λ P Q (,F = = D D = p D ) q D We denote as Γ(q) the set of feasibe secondary transmissionrate vectors for a given vector q. The set Γ(q) is defined in average sense as the convex hu of a feasibe activation vectors when PB is not transmitting, mutipied by the probabiity that PB is not transmitting, as Γ(q) = C N = { λ N = Cλ p D D F = ( P Q, F p ) D q D }. conv(ẽ) (4)

6 6 where conv of a set of vectors is the set of a possibe convex combinations of its eements. We use the set of feasibe transmission rate vectors for SBs to define the region Λ. The region Λ is the set of a possibe request generation rate vectors: (λ, λ (s) ) for which there exists vector q, (R,..., R M ) T Γ(q) and variabe π 0 s.t. π 0 p (5) C + i:su i ϕ (s) j D D where N π 0 = C = λ (s) i :PU ϕ j = λ N λ F = D D P λ D,j q D R j j M, (6) F = P Q(, F D )q D. (7) The term π 0 in (7) represents the rate of successfu pacet transmissions by PB corresponding to the vector q. The right hand side (RHS) of (7) is the product of the rate of primary user requests served by PB and C, the number of successfu pacet transmissions per served request. The constraint (5) is the stabiity constraint at PB: the rate of pacet transmissions by PB cannot exceed. Constraint (6) represents the stabiity requirement at SBs. The eft hand side (LHS) of (6) represents tota arriva rate into a the queues in a given SB; the RHS of (6) represents a feasibe transmission rate offered to that base station. The region Λ contains the set of λ and λ (s) supportabe without cooperation as the a-zero primary avaiabiity matrix (i.e., one whose every component is 0) is in the set D. We now simpify the description of the region Λ. Intuitivey, we observe that SBs create maximum transmission opportunities for themseves by transmitting as much primary traffic as possibe. This means that the region Λ can be described by ony considering poicies in which each SB, in every period, caches exacty B most popuar primary fies and admits a user requests for those fies. We state this formay in Lemma. Lemma : The region Λ can be defined by considering, in (5) and (6), ony the vector q for which q D = and q D = 0 for every primary { avaiabiity matrix D D where Dn,j, if n B, j M = (8) 0, otherwise. Now we compare the achievabe capacity region Λ with the genera capacity region Λ gen for a networ in which a SBs are non-interfering. The region Λ gen consists of a request generation rate vectors for which the networ is stabe under any stationary scheduing and caching agorithm, and not just the restricted set of agorithms that satisfy constraint C. Ceary, Λ gen Λ. In particuar, we compare the tota secondary request generation rate that can be maximay supported by each SB under the restricted set of agorithms and that under a stationary agorithms. We quantify the performance gap, measured in terms of supportabe secondary request generation rates, between these two sets of poicies as foows. For a given λ, et λ (s) j,max,ach (λ ) and λ (s) j,max,gen (λ ) denote the maximum tota secondary request generation rate that can be supported (i.e., a queues in the networ are stabe) at base station SB j, j {,..., M}, under the restricted set of poicies and under a poicies respectivey. Then λ (s) j,max,ach (λ ) is ower bounded as foows. Lemma 2: Consider a networ in which a SBs are noninterfering. For any primary request generation rate vector λ ( which is present in Λ (i.e., there exists λ (s) such that λ, λ (s)) Λ) we have for every j {,..., M}, λ (s) j,max,ach (λ ) λ (s) j,max,gen (λ ) ( p ) F (s) n= P n (s) C (s) n λ (s) M n= n j :PU ϕ j :PU ϕ n λ P B λ P B. (9) p Extension to case of different fie sizes The anaysis can be extended to the case of unequa fie sizes as foows. First, in the system mode, the queue evoution wi change. For every new request, the queue size wi be incremented by a parameter different from C corresponding to the size of the fie. Second, we wi need to redefine the achievabe capacity region by modifying constraint C. For exampe, we can redefine C so that any secondary fie can aways be stored at an SB independent of state of queues for primary fies. Third, we wi need to modify (4)-(7) that define the region Λ. In particuar, we repace each occurrence of the terms Cλ i and Cλ (s) (where i N, N (s) ) in (4)-(7) with terms F m= P m C m λ i and where C m and C n (s) denote the number of pacets required to serve fie F m and F (s) n respectivey ( m F, n F (s) ). Fourth, we wi modify Lemma based on the assumption used in modified C. For exampe, suppose SBs aways reserve a fixed amount of cache space for secondary fies. Then we can restate Lemma as one in which each SB aways stores the most popuar primary fies in its remaining cache space. Using Lemma we can derive the counterparts of FPCP, VPCP and MCCP with performance guarantees simiary as in this paper. Gain due to Cooperation In order to quantify the gain due to cooperation we consider a symmetric networ where every primary (respectivey secondary) user generate requests at same rate, a sma ces contain equa number of primary (respectivey secondary) users, there is no primary user which is outside a sma ces and a secondary base stations are non-interfering. Given SU n (where n N (s) ) requests content at rate λ (s) n, the maximum request generation rate for PU m (where m N ) with cooperation, denoted as λ m,coop(λ (s) n ) can be obtained, using

7 7 (4)-(7) and Lemma, as, λ m,coop(λ (s) λ(s) n ρ n ) = max{( CN M ) p ( B = P ) + B M = P, 0} () and that without cooperation, denoted as λ m,no-coop(λ (s) n ) can be obtained as, λ m,no-coop(λ (s) n ) = max{p( CN λ(s) n ρ ), 0} M where ρ = N (s) is the ratio of number of secondary to primary N users. For fixed N both λ m,coop(λ (s) n ) and λ m,no-coop(λ (s) n ) decreases with increasing ρ. This is expected as higher ρ impies arger number of secondary users transmitting at rate λ (s) n eading to ower supportabe λ m,coop. Cooperation benefits a primary user for given λ (s) n if λ m,coop(λ (s) n ) λ m,no-coop(λ (s) n ). Simiary, given PU m requests content at rate λ m, the maximum request generation rate for SU n with cooperation, denoted as λ (s) n,coop(λ m ) is, λ (s) n,coop(λ m ) = max{ M CN (s) Mλ m ρp B ( = P ) λ m ρ B = n,no-coop(λ P, 0} and that without cooperation, denoted as λ (s) m ) is, λ (s) n,no-coop(λ m ) = max{ M Mλ m CN (s) ρp, 0}. Again, λ (s) n,coop(λ m ) and λ (s) n,no-coop(λ m ) decreases with increasing number of secondary users. This is expected as higher number of secondary users means ower transmission opportunities per user. Ceary, cooperation benefits a secondary user for given λ m if λ (s) m ) λ (s) m ). n,coop(λ n,no-coop(λ IV. FIXED PRIMARY CACHING POLICY In this section we present FPCP, an agorithm that stabiizes a queues in the networ for every request generation rate vector in the interior of the achievabe capacity region Λ. The agorithm is constructed using Lyapunov drift techniques that are widey used to deveop efficient scheduing agorithms in communication networs. In order to construct the agorithm we first find the set of fies cached at the SBs in each period. Given the set of fies currenty cached, in FPCP we schedue pacet transmissions according to a modified bacpressure poicy. We formay state the agorithm in Tabe 2 and anayze its stabiity performance in Theorem. Finay, we briefy discuss a drawbac of the FPCP agorithm. We obtain the set of fies cached at the SBs in each period as foows. First, reca, according to Lemma, the achievabe capacity region Λ can be described by considering ony those poicies under which, in every period, each SB caches exacty B most popuar primary fies and aways admits requests corresponding to those fies. Therefore, in the construction of FPCP we consider ony those poicies. Next we refer to the aggregate of a primary queues in SB as the aggregate primary queue in SB for every {,..., M}. Ceary, the ength of the aggregate primary queue in SB, denoted as U (t), is the sum of the ength of a primary queues i.e., U (t) = U f, f F (t). We then appy the Lyapunov drift technique to find the set of cached secondary fies in every period. We denote as t r,j the j th sot in r th period (where r {, 2,..., } and j {,..., T }) and form the Lyapunov function L (r) = M = { f F (s) (U (s) f, (t r,)) 2 + (U (t r,)) 2 }. We find the set of secondary fies to be cached at each SB in the r th period by minimizing the upper bound of the conditiona drift (r) = E[L (r + ) L (r) U (t r, ), U (s) (t r, )]. We perform this minimization over a poices X under which each secondary base station caches exacty B most popuar primary fie in every period. Therefore under FPCP, for every stricty positive integer r, in the r th period each base station SB (where {,..., M}) can cache ony one secondary fie, denoted as f (r). The expression of f (r) is obtained as, f (r) argmax U (s) f, (t r,) =, 2,..., M. (2) f F (s) Given the set of fies cached at each base station in any period, in FPCP we schedue pacet transmissions from base stations according to a modified bacpressure poicy. At any time sot when PB is not transmitting, we schedue pacet transmissions by soving a max-weight optimization probem of instantaneous queue engths of the cached fies in a base stations. Tabe 2: FPCP Agorithm For every stricty positive integer r foowing steps are performed in the r th period: ) Caching Scheme: At the beginning of the period, every secondary base station caches the B most popuar primary fies. In addition, every secondary base station SB (where {,..., M}) caches the secondary fie f (r). 2) Scheduing for PB: At any time sot t PB transmits a pacet corresponding to the Head-of-Line (HOL) pacet request at the queue of highest ength (in case of ties, pic one arbitrariy). Mathematicay, µ FPCP (t) = if U F f and is 0 otherwise. χ f, (t) > 0 and U (t) U (t) for every f 3) Request Admission and Scheduing Poicy at SBs: Each secondary base station SB, {,..., M}, admits every request for a cached primary fie from a primary user within its transmission-range, i.e., for a t, (t) equas if and ony if (iff) f {F,..., F B }. If PB is not transmitting at t r,j, then obtain the set of base stations that can transmit at t r,j, denoted as EFPCP (t r,j), by soving the foowing max-weight ( probem: ( { max EFPCP(t r,j ) argmax E Ẽ U (t r,j) ) T E. (3) } ), U (s) M f (r),(t r,j) = Suppose, according to EFPCP (t r,j) some SB is aowed to transmit at this time sot. Then we determine which pacet to transmit from SB in the foowing manner.

8 8 We transmit the pacet corresponding to the HOL pacet request at the queue of secondary fie f (r) if U (s) f (r),(t r,j) is greater than or equa to U (t r,j). Otherwise transmit the pacet corresponding to the HOL pacet request at the aggregate primary queue in SB. Mathematicay, given th component of EFPCP (t r,j) is, µ FPCP, (t r,j ) =, µ FPCP f (r),(t r,j) = 0 if U (t r,j) U (s) f (r),(t r,j); µ FPCP, (t r,j ) = 0, µ FPCP f (r),(t r,j) = otherwise. Here, µ X, (τ) denotes the transmission rate offered to SB, under poicy X, to transmit a pacet from its aggregate primary queue at time sot τ. Note that FPCP does not require nowedge of request generation rates. We show that FPCP stabiizes a queues in the networ for every request generation rate vector in the interior of the achievabe capacity region: Theorem : FPCP stabiizes a queues in the networ for each (λ, λ (s) ) Interior(Λ). Whie the FPCP agorithm stabiizes every queue for a request generation rate vectors in the interior of Λ, it aows each SB to cache ony one secondary fie per period. Our simuation resuts in Section VII show that this can adversey affect the deay performance of secondary users, especiay when primary request generation rates are ow and it is not necessary to cache B primary fies per SB per period in order to stabiize the queues. Hence, in Section V we present the VPCP agorithm which dynamicay varies the number of cached primary fies at SBs in every period according to instantaneous queue engths. However, the guaranteed stabiity region under VPCP is ower than that under FPCP. V. VARIABLE PRIMARY CACHING POLICY In this section we present VPCP, an agorithm under which the networ controer dynamicay determines the number of primary fies to be cached in each period. Furthermore, primary pacet requests are satisfied ahead of secondary ones at every SB. We first provide an overview of the motivation behind construction of this agorithm; in the foowing subsections we discuss the construction methodoogy and present a guaranteed stabiity region under VPCP. The VPCP agorithm addresses the difficut probem of serving primary pacets at each base station ahead of secondary ones whie aso attempting to minimize the number of cached primary fies and maintain stabiity of a queues. The difficuty of the probem ies in the couped nature of the evoution of primary and secondary queues. In particuar, scheduing decisions for secondary pacets from any base station depend on whether or not the primary queues in that base station are empty. Evoution of primary queues in turn depend on cooperative caching decisions since any base station can ony transmit pacets of a cached primary fie. If we view the primary queue engths as the system state, then the probem is a constrained Marov Decision Probem where the system state itsef evoves according to the contro decisions. One way to sove such probems efficienty is by using renewa frame based optimization techniques. Such techniques can sove certain constrained Marov Decision Probems without suffering from the pitfas associated with conventiona soutions to those probems, e.g., requiring extensive nowedge of system dynamics, arge convergence times [9]. Renewa frame based techniques are based on partitioning the time-ine into distinct coection of contiguous time sots caed frames and then maing contro decisions in the beginning of each new frame. Length of each frame is variabe and depends on the contro decisions taen at the beginning of the frame. In Section V-A we discuss how we partition the timeine into renewa frames in our mode. However, we cannot use the renewa frame based optimization techniques directy due to some restrictions imposed by the system mode. In particuar, in renewa frame based methods, contro decisions (e.g. caching) are made ony at the beginning of every variabe ength frame. On the other hand, according to our system mode, caching decisions tae pace at the beginning of every fixed ength cache-refresh period which may not occur at the beginning of a new frame. We address the chaenge of appying renewa frame method to this scenario by constructing the VPCP agorithm in two steps. First, as discussed in Section V-B, in each period we estimate, the number of primary fies that woud be cached at SBs by a renewa frame based caching and scheduing poicy had the beginning of the period coincided with beginning of a new frame. This renewa frame based method maes caching decisions ony at the beginning of a frame by minimizing the ratio of a drift pus penaty expression over a frame, and the expected ength of the frame. Then, as discussed in Section V-C, we use those estimates to construct VPCP by first determining the actua number of primary fies cached at each SB in every period and then the scheduing poicy. Because of this two-step construction process, VPCP is not an optima renewa frame based optimization poicy, i.e., it does not provide any optimaity guarantee on the average number of primary fies cached at SBs. In order to render tractabe anaysis of its stabiity performance, the VPCP agorithm is constructed under certain restrictions. First, under this agorithm ony a fixed set of non-interfering SBs, denoted without oss of generaity as SB,..., SB G respectivey (where G M), cooperate with the primary networ by caching primary fies, admitting and serving primary requests. Secondy, the poicy VPCP is constructed such that it satisfies constraints C (described in Section III) as we as two additiona scheduing constraints described in Section V-B. In Section V-D we present a guaranteed stabiity region for this agorithm. A. Renewa Frame Techniques for Cognitive Networs In this subsection we discuss how we appy the renewa frame technique. We define the system state to be the sum of the ength of a primary queues in the networ, simiar to the way primary user s channe occupancy process was used as system state in [9] 5. Defining the system state in this way is usefu because scheduing decision for secondary pacets depends on whether this system state is zero or not. Each 5 Note that in [9] the authors did not study cooperative caching in cognitive radio networs.

9 9 expression under poicy X: T 2T 3T 4T Time-sot E[ M t U (s) f, G ( t r +, r,) µ X f, (τ) U(s) ( t r,)] Frame Frame 2 Frame 3 V ˆn (r = f F ) (s) τ= t r,. E[( t = r+, t r, ) U (s) ( t r,)] Non-empty Empty primary (4) primary request-queue request-queue Minimizing the expression in (4) refects the tradeoff Fig. 2. Partition of time-ine into frames. Shaded region shows time sots between stabiity of queues versus caching ess primary fies at for which at east one primary queue in the networ is non-empty. An arrow SBs. Caching more primary fies at SBs minimizes the fraction indicates beginning of a new period. term in (4), whie it increases the penaty term: V G ˆn (r ). frame begins when the system state changes, i.e., the sum of a primary queue engths transitions from zero to a non-zero vaue. Each frame consists of two distinct phases, one in which at east one primary queue in the networ is non-empty, and one in which a primary queues in the networ are empty. Fig. 2 shows an exampe of such partition. B. Caching Decisions by a Renewa Frame Based Optimization Poicy In this subsection we estimate the number of primary fies, ˆn (r), that woud be cached at each cooperative secondary base station SB (where {,..., G}) at the beginning of the r th period (where r {, 2,..., }) under a renewa frame based optimization poicy. The optimization poicy tries to minimize the average number of primary fies cached whie maintaining stabiity of a queues. The poicy aso transmits primary pacets with higher priority than secondary ones from each cooperative SB. The poicy assumes a new frame began at time sot t r,, irrespective of the actua ength of primary queues, and subsequent caching decisions tae pace ony at the beginning of every frame. In the rest of this sub-section, we assume those assumptions are indeed true and that the r th frame (where r {, 2,..., }) began at time sot t r,. Next we discuss how we obtain the ˆn (r) variabes. We obtain the ˆn (r) variabes by minimizing the ratio of a conditiona drift pus penaty expression to expected frame ength as foows. First, we consider the Lyapunov function of secondary queue engths at beginning of each frame, L 2 (r ) = M f, ( t r,)) 2 and its conditiona (U (s) = f F (s) drift, 2 (r ) = E[L 2 (r + ) L 2 (r ) U (s) ( t r,)]. Here, t n,j denotes the j th, j {,..., t n+, }, time sot in the n th, n {, 2,..., }, frame. Next we form the sum of 2 (r ) and a penaty term: V E[ G t r +, ˆn (r ) U (s) ( t r,)], where V is = τ= t r, a positive constant and ˆn (r ) denotes the number of primary fies cached at SB in the r th frame. The penaty parameter V refects the trade-off between queue stabiity and number of primary fies cached. High V impies ess primary fies shoud be cached and vice-versa. We then obtain a simpified upper bound to this sum by considering ony those λ for which the primary networ can stabiized even without cooperation. Finay, we obtain the ˆn (r) variabes by finding the set of primary fies cached at base stations in the r th frame, and the scheduing scheme used throughout the r th frame, that minimizes the ratio of this upper bound to that of the expected frame-ength. By simpifying this ratio we obtain the foowing = In order to precisey define a guaranteed stabiity region for the VPCP agorithm (described in Section V-C), we minimize the expression in (4) by considering ony caching and scheduing agorithms that satisfy C and two scheduing constraints C2 and C3 that we state next. C2: At every sot PB transmits a pacet iff it has at east one non-empty primary queue and a cooperative SBs have ony empty primary queues. A cooperative SB transmits a secondary pacet ony if it has a empty primary queues. Constraint C2 ensures that primary pacets are transmitted ahead of secondary pacets from each cooperative SB. Furthermore, this constraint aso simpifies construction of a guaranteed stabiity region for VPCP as it ensures that the events of primary pacet transmission from cooperative SBs are independent from base station to base station. C3: No non-cooperative SB simutaneousy transmits secondary pacets when some cooperative SB is transmitting a primary pacet. Constraint C3 simpifies construction of a guaranteed stabiity region for VPCP by ensuring the foowing event cannot occur: concurrent transmission of a secondary and primary pacet from a non-cooperative SB and a cooperative SB respectivey. In Appendix C of [9] we find an approximate soution to the probem of minimizing the expression in (4) by considering ony poicies that satisfy constraints C, C2, C3 and under some additiona assumptions. In particuar, we obtain an estimate of the number of most popuar primary fies, ˆn (r), that ought to be cached at each cooperative secondary base station SB in the r th period if the expression in (4) is minimized. C. Description of the VPCP Agorithm In this subsection we use the ˆn (r) variabes to construct the VPCP agorithm. First, we discuss the probem with simpy caching ˆn (r) most popuar primary fies at each cooperative base station SB, {,..., G}, in the r th period. Then we discuss how the caching scheme in VPCP resoves this probem. Finay, we state the VPCP agorithm in Tabe 3. We cannot simpy cache ˆn (r) most popuar primary fies at each cooperative base station SB, {,..., G}, in the r th period for every r {, 2,...}. This is because we cacuated ˆn (r) estimates were assuming that a new frame began at time sot t r, regardess of the actua ength of primary queues at t r,. Hence, it is possibe that at t r,, the queue ength in SB for some primary fie f, not one of the ˆn (r) most popuar primary fies, is non-zero. Not caching fie f in the r th period can ead to oss of transmission opportunities for SB since a