Multiuser Cognitive Access of Continuous Time Markov Channels: Maximum Throughput and Effective Bandwidth Regions

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1 Multuser Cogntve Access of Contnuous Tme Markov Channels: Maxmum Throughput and Effectve Bandwdth Regons Shyao Chen, and Lang Tong School of Electrcal and Computer Engneerng Cornell Unversty, Ithaca, NY Emal: Abstract The problem of sharng multple channels owned by prmary users wth multple cogntve users s consdered. Each prmary user transmts on ts dedcated channel, and ts occupancy s modeled by a contnuous tme Markov process. Each cogntve user s capable of sensng one channel at a tme and t transmts accordng to a slotted structure. The transmssons of cogntve users on each channel are subject to a prescrbed collson constrant. Under tght collson constrants, the maxmum throughput regon s obtaned by a polcy referred to as Orthogonalzed Perodc Sensng wth Memoryless Access (O). Characterzatons of the maxmum throughput regon are also provded when the collson constrants are loose. It s shown that the O polcy acheves the maxmum sum-rate under all collson constrants when the number of cogntve users equals to that of the prmary users. Inner and outer bounds for the effectve bandwdth regon are formulated as a par of convex optmzatons. When there are only two channels, corner ponts (the sngle user scenaro) of the optmal effectve bandwdth regon are also obtaned. Index terms Cogntve rado networks, dynamc spectrum access, opportunstc multaccess, effectve bandwdth, Constraned POMDP, queueng networks. I. INTRODUCTION We consder the problem of sharng N channels owned by prmary users wth K N cogntve users. In the context of herarchcal overlay cogntve networks [1], the prmary users transmt at wll, oblvous to the presence of cogntve users. The cogntve users are allowed to access the channels only f they constran ther nterference below prescrbed levels. The cogntve users n ths settng are capable of channel sensng. We assume that cogntve users can only sense and transmt on one channel at a tme, and ther access to channels are dstrbuted wthout a central control. The cogntve users must dscover transmsson opportuntes n channels owned by prmary users, coordnate among themselves to share these opportuntes, and transmt wthn the gven collson constrants. For multple cogntve users, the general performance measure s a vector, wth each component characterzng the performance acheved by the correspondng cogntve user. In Ths work s supported n part by the Natonal Scence Foundaton under Contract CCF-6357 and the Army Research Offce MURI Program under award W911NF ths paper, we focus on throughput and effectve bandwdth as measures of performance. In partcular, we are nterested n characterzng the maxmum throughput and effectve bandwdth regons for the K-cogntve user multaccess network. Such characterzatons provde an analytcal bass to allocate resources among cogntve users; they may also be used by the prmary users to determne far prces for the cogntve access. A. Summary of results We present several new results n ths paper on the characterzaton of throughput and effectve bandwdth regons. Frst, we show that, when the collson constrants are tght, the optmal multuser cogntve access s acheved by a smple polcy referred to as Orthogonalzed Perodc Sensng wth Memoryless Access (O), frst proposed n [2]. By tght collson constrants we mean that the maxmum nterference (to be defned n Secton II) from the cogntve users on each channel must be kept below a small threshold, for whch we provde a closed form expresson. See Theorem 2 n Secton IV. As we relax the collson constrants, O no longer gves the largest throughput regon, but t always acheves the maxmum sum-throughput when K = N. To characterze the maxmum throughput when the collson constrants are loose, we may need to consder mxed polces nvolvng tme sharng. To ths end, we consder two sngle user polces. The frst s the Perodc Sensng wth Memoryless Access () polcy from whch OPS- MA s based. was frst proposed n [3], [4] and was also ndependently consdered n [5]. has recently been shown to be optmal under tght collson constrants [2], [6]. The second sngle user polcy s generalzed from a myopc polcy orgnally proposed for slotted prmary user cogntve network by Zhao, Krshnamachar, and Lu [7]. Ths polcy, heren referred to as the ZKL polcy, has a smple round robn structure and s shown to be optmal for dentcal and postvely correlated Markov channels [7], [8]. In ths paper, we adapt the ZKL structure for the contnuoustme Markov channels wth collson constrants. The resultng polcy, referred to as, employs ZKL for channel sensng and memoryless probablstc transmsson for access. We show that when all channels are dentcal wth equal collson constrants, s optmal under all collson

2 2 constrants. See Theorem 1. When the channels have dfferent collson constrants or channels are not dentcal, there s no defntve orderng between and. Ths means that, when usng tme sharng to mx polces, specfc channel and constrant parameters must be taken nto account. To establsh the effectve bandwdth regon, we derve nner and outer bounds as a par of convex optmzatons. We also show that, when there s one cogntve user and two dentcal prmary channels wth equal constrants, s optmal and s strctly suboptmal. B. Related work The results presented n ths paper appear to be the frst that characterze the maxmum throughput and effectve bandwdth regons for a multuser cogntve network. Our results are generalzatons from the correspondng sngle cogntve user problem consdered recently n [6], [2], [9]. In [6], [2], t s shown that when the collson constrants are tght, the Perodc Sensng wth Memoryless Access () polcy s optmal, and the maxmum nterference levels for whch s optmal are also derved. The problem of multuser cogntve access s also consdered n [2], and the O access polcy s proposed as a heurstc way to generalze PS- MA to a multuser settng. Here the optmalty of O s establshed formally. In [1], a dstrbuted multuser cogntve access scheme based on the ALOHA s consdered. Whle the scheme n [1] does not acheve the maxmum throughput regon, t does not requre pre-arranged orthogonalzed sensng. A consderable amount of work exsts when there s only one cogntve user [1]. In [11], [12], Zhao et al. consder the case when the prmary users follow a slotted Markovan transmsson structure. It s partcularly sgnfcant that the myopc polcy (ZKL) proposed n [7] s optmal for dentcal and postvely correlated Markovan channels [7], [8]. One of the earlest cogntve access polces for contnuous tme Markov channels s presented n [3], [4]. By fxng a perodc sensng polcy, the authors of [3], [4] propose the optmal access polcy based on the framework of constraned Markov decson processes. As part of benchmark comparsons, the polcy was proposed as a lower bound 1. Around the same tme as [3], Arkbar and Tranter also propose a perodc sensng polcy [5] but wth determnstc transmssons. As such, the approach n [5] does not provde a guarantee to meet the requred collson constrants. The fundamental lmts and the structure of cogntve access of a sngle contnuous tme Markovan channel s nvestgated n [13], [14], [15] where the authors derve the form of optmal transmsson polcy n greater generalty. Among a number of nterestng results, t s shown that the optmal transmsson s probablstc as n our case. Indeed, f there s only one channel, our results are consstent wth that n [15]. Other related work assumng contnuous tme channel occupances can be found n [16], [17], [13], [18]. 1 It was not realzed then that s n fact optmal when the collson constrants are tght due to an error n the calculaton of transmsson probablty. See [6] for the correct expresson. For the characterzaton of effectve bandwdth regons, the only relevant result s [9] where the authors consder the slotted prmary user network wth a sngle cogntve user. Our results n ths paper generalze [9] to the multuser settng nvolvng contnuous tme Markov prmary channels. It s establshed that ZKL s effectve bandwdth optmal when there are two channels wth loose collson constrants. The nner and outer bounds establshed n ths paper are new. II. NETWORK MODEL We frst descrbe the network model and the assumptons. There are N parallel prmary channels ndexed by 1,...,N and K N cogntve users ndexed by 1,..., K. Each prmary user transmts on ts dedcated channel. The transmsson of each prmary user s modeled as a contnuous tme Markov process ndependent of the transmssons of other prmary users. The state space of the th channel s {(dle), 1(busy)} and the holdng tmes are exponentally dstrbuted wth parameters λ 1 for dle and busy states, respectvely. The generator matrx of the th channel s gven by ( ) λ λ Q =, (1) µ µ and µ 1 and the statonary dstrbuton of the th prmary channel for dle and busy states are gven by v () = µ /(µ + λ ) and v (1) = λ /(µ +λ ), respectvely. For the specal case where λ = λ and µ = µ for = 1,...,N, we term the prmary channels homogeneous, and heterogenous otherwse. The cogntve users access the prmary channels followng a slotted randomzed transmsson polcy wth slot length T. In each tme slot, each cogntve user can sense one of the N prmary channels and decde whether to transmt. A cogntve user collects unt reward n slot t f () the cogntve user accesses the channel, () no other cogntve users access the same channel, and () also the channel s dle throughout slot t. We assume perfect sensng for the cogntve users and no collaboraton among the cogntve users. We am to characterze the maxmum regon of throughput and effectve bandwdth of ths multuser cogntve access network. A. Performance measures The two performance measures used n ths paper are throughput and effectve bandwdth. Throughput measures the quantty of servce for the cogntve users. Fx a sensng and access polcy π. Denote by R (k) t the reward that the kth cogntve user collects n slot t under polcy π. The throughput of the kth cogntve user s defned by the nfnte horzon average reward,.e., J (k) π 1 = lm n n E n t=1 R (k) t. (2) Effectve bandwdth characterzes the quantty of servce avalable wth the qualty of servce (QoS) constrant prescrbed by the cogntve users. Specfcally, we consder the QoS constrant beng the buffer overflow probablty below a

3 3 prescrbed parameter ǫ. For a cogntve user wth buffer sze b 1, let θ = log(ǫ)/b and the effectve bandwdth of the cogntve user s defned by (see [19]) B (k) π = lm n log Eexp(θ n t=1 R(k) nθ t ). (3) More detals of effectve bandwdth wll be gven n later secton. B. Collson constrants The transmssons of the cogntve users are subject to collson constrants mposed by the prmary users. For each prmary user, the overall collson caused by the K cogntve users should be lmted below a collson constrant parameter γ. The collson for the th prmary user s defned to be the fracton of the collded slots n the slots fully or partally used by the prmary user. Specfcally, we use the nfnte horzon average collson scaled by the recprocal of the steady state probablty of the th prmary user transmttng n a certan slot. The collson for the th prmary user s gven by 1 E n t=1 C π, = lm 1 {collson wth PU n slot t}. 1 v ()e λt n n (4) where 1 A s the ndcaton functon for event A. The goal of ths paper s to characterze the throughput and effectve bandwdth regons for the multuser cogntve access network under prescrbed collson constrants. Mathematcally we have the followng problem. The set of the admssble polces Π s gven by the set of polces that meet the collson constrants,.e., {π : C π, γ, = 1,...,N}. For a fxed polcy π Π, the throughput s J π = (J π (1),...,J π (K) ) and the effectve bandwdth s B π = (B π (1),...,B π (K) ). We am to characterze the throughput regon π Π J π and the effectve bandwdth regon π Π B π. III. COGNITIVE ACCESS POLICY The polcy for dstrbuted cogntve access s defned by two components: sensng polcy and transmsson polcy. The sensng polcy selects a channel to sense n each slot based on the hstory avalable to the cogntve user whle the transmsson polcy specfes the transmsson probablty upon dle sensng results, also based on the hstory avalable to the cogntve user. In general the sensng polcy and the access polcy would be desgned jontly to acheve the optmal performance. However, n ths paper we wll analyze several cogntve access polces, for whch the transmsson polcy only uses the current sensng result and gnores the prevous hstory avalable. We term such transmsson polces Memoryless Access (MA). In the followng we elaborate two specfc cogntve access polces, as well as the Markov chans they nduce. A. The polcy The cogntve access polcy wth Perodc Sensng and Memoryless Access () s proposed n [2] for the sngle cogntve user network (K = 1). The sensng and transmsson of the cogntve user n the polcy can be descrbed as follows. The cogntve user senses the channels n an ncreasng order at the begnnng of each slot, startng from the channel wth the smallest ndex (say, channel 1). If the th channel s sensed to be dle, the cogntve user transmts n the sensed channel wth fxed probablty β. A sample path of the polcy s llustrated n Fg. 1. nduces N ndependent Markov chans wth state space {, 1}, one for each prmary channel, wth transton matrx exp(nt Q ). Channel 1 Channel 2 Channel 3 Idle sensng Prmary Users Transmssons Busy sensng Cogntve User Transmssons Fg. 1. Illustraton of the polcy. Open crcle: cogntve user decdes not to transmt. Flled crcle: cogntve user decdes to transmt. B. The polcy In [7], the ZKL polcy s proposed for slotted prmary transmssons. The prmary channels are homogeneous and there are no collson constrants. The polcy descrbed below s an extenson of the ZKL polcy for the contnuous tme Markov channels wth collson constrants. In the polcy the cogntve user frst fxes an ordered lst of the N prmary channels and the transmsson probablty β for each channel. To start, the cogntve user senses the frst channel n the lst. If dle the cogntve user accesses the channel wth the correspondng transmsson probablty. The cogntve user then keeps sensng the frst channel untl the frst busy sensng result, after whch the cogntve user swtches to the next channel n the lst and accesses the next channel wth the correspondng transmsson probablty. The cogntve user moves down along the lst as descrbed above untl reachng the last channel. After the frst busy sensng result from the last channel, the cogntve user goes back to the frst channel agan. Equvalently, the cogntve user stays n the same channel wth randomzed transmsson f the channel s sensed to be dle and moves down along the ordered lst otherwse. A sample path of the polcy s llustrated n Fg. 2. nduces a N 2 N state Markov chan havng state space {1,...,N} {, 1} N wth state vector ndcatng the current channel ndex and the state of the N channels. IV. MAXIMUM THROUGHPUT REGION A. Sngle cogntve user network It s establshed n [2] that for the sngle cogntve user network (K = 1), optmal throughput s acheved by under tght collson constrants.

4 4 Channel 1 Channel 2 Channel 3 Idle sensng Prmary Users Transmssons Busy sensng Cogntve User Transmssons Fg. 2. Illustraton of polcy. Open crcle: cogntve user decdes not to transmt. Flled crcle: cogntve user decdes to transmt. of for homogeneous channels wth equal collson constrants. Theorem 1. For homogeneous channels wth equal collson constrants γ, s throughput optmal n the set of all admssble polces Π for all γ [, 1]. Proof: Omtted due to the space lmt. See [2]. In Fg. 4 the throughput s shown as a functon of the collson constrant parameter γ. For the full sensng (FO) upper bound we refer to [2]. J Lemma 1. [2] Let φ 1 v()exp( λt) 1 exp( λ T) and γ v () Nφ. Gven N ndependent contnuous tme Markov channels wth parameters (λ, µ ) and statonary dstrbutons v () for dle states, the throughput of for general γ s can be characterzed as J = N =1 φ exp( λ T){γ 1 {γ γ } +γ 1 {γ>γ } }. Under tght collson constrants,.e., γ γ, the PS- MA polcy s throughput optmal for the sngle cogntve user network. We extend the tght collson constrants regme larger by for homogeneous channels. We have the followng characterzaton of sngle user throughput of. Proposton 1. Denote by ω(, x) the statonary dstrbuton of the Markov chan nduced by where s the channel ndex the cogntve user currently senses and x s the current state for the N channels. Let γ 1 φ x = ω(, x). The throughput of can be characterzed as J = N =1 φ exp( λ T){γ 1 {γ γ } +γ 1 {γ>γ } }. For homogeneous channels, γ > γ. Proof: Omtted due to the space lmt. See [2]. Proposton 1 gves the tght collson constrants regme {γ γ } for the polcy, n whch the throughput s lnear n γ. For homogeneous channels ZKL- MA has strctly larger tght collson constrants regme than and the throughput performance of s superor to that of. We remark that for heterogenous channels we may have for certan s γ γ. The possble stuatons of tght collson constrants regmes for and are llustrated n Fg. 3 for heterogenous channels. Wth the extended tght collson constrants regme f we assume equal collson constrants for the prmary channels we can further show the sngle user throughput optmalty γ γ γ FO Fg. 4. Throughput versus collson constrant parameter. B. Multuser network: tght collson constrants For the multuser cogntve access network we state the followng theorem for maxmum throughput regon for tght collson constrants. Theorem 2. Under tght collson constrants,.e., γ for = 1,...,N, the throughput regon s gven by γ K N {(y 1,..., y K ) y k exp( λ T)φ γ, y k }. (5) k=1 =1 Proof: Omtted due to the space lmt. See [2]. The throughput regon under tght collson constrants s a polytope. Specfcally, t s the convex hull of the orgn and the K ponts correspondng to exclusvely servng one sngle cogntve user. A pont n the postve orthant s n the throughput regon f and only f the total throughput of the K cogntve users s below an upper bound gven by a lnear combnaton of the collson constrant parameters. In the next subsecton we gve the mult-access scheme whch acheves the throughput regon gven n Theorem 2. C. Optmal mult-access scheme under tght collson constrants We use the O polcy (see [2]) to acheve the throughput regon n Theorem 2. Snce n the network there are fewer cogntve users than prmary channels (K N), we can ft the K cogntve users n K orthogonal sensng phases such γ

5 5 γ 2 γ 2 γ 2 γ 2 J γ 1 γ 1 γ 1 γ 2 γ 2 J γ 1 γ 1 γ 1 γ 1 γ 1 (a) Followng the lne. γ γ 1 γ 1 (b) Followng the dashed lne. γ Fg. 3. Tght collson constrants regme: heterogeneous channels, no defntve orderng between and. that each cogntve user performs wth ts own sensng phase and no collson between cogntve users would occur. Therefore the collson suffered by a prmary user s the sum of collsons caused by each ndvdual cogntve user. In O, the kth cogntve user transmts wth probablty β (k) on the th channel upon dle sensng result n channel. Under tght collson constrants, the transmsson probablty on the th channel upon dle sensng results for sngle user network s β = γnφ v (). Let β(k) = β α (k) where α (k) for all and k, and K k=1 α(k) 1 for all. The α (k) s are back-off coeffcents to guarantee the collson constrants to be met. Each cogntve user transmts less aggressvely to accommodate other cogntve users. Dfferent α n the O polcy would yeld dfferent ponts n the throughput regon. O wth all possble α s acheves the throughput regon n Theorem 2. A sample path of the OPS- MA polcy s llustrated n Fg. 5. Channel 1 Channel 2 Channel 3 Idle sensng for cogntve user 1 Busy sensng for cogntve user 1 Idle sensng for cogntve user 2 Busy sensng for cogntve user 2 Prmary Users Transmssons Cogntve User 1 Transmssons Cogntve User 2 Transmssons Fg. 5. Illustraton of O polcy. Open crcle: cogntve users decde not to transmt. Flled crcle: cogntve users decde to transmt. We remark that n order to use O the exstence of a base staton who knows the currently n-use sensng phases s needed. Ths s reasonably practcal and the base staton does not ntroduce collaboraton among cogntve users concernng ther sensng results. D. Multuser network: homogeneous channels wth looser collson constrants Theorem 2 s vald under tght collson constrants. In ths subsecton we loose the collson constrants and consder multuser network wth homogeneous channels. Specfcally, we have the followng theorem for throughput regon. Theorem 3. For multuser network wth homogeneous channels 1) If γ γ for = 1,...,N, the throughput regon s gven by {(y 1,...,y K ) K y k k=1 N exp( λ T)φ γ, y k }. =1 (6) 2) If there exsts such that γ > γ, an clarvoyant settng (see [2]) and mxtures of and gve outer and nner bounds for the throughput regon, respectvely. In the followng specal cases the bounds get better. a) Wth equal collson constrants the throughput of any ndvdual cogntve user s bounded below by the throughput of sngle user. b) If N = K, acheves optmal sumthroughput n a regon near the drecton of vector (1, 1,...,1).

6 6 Proof: Omtted due to the space lmt. See [2]. The throughput regon n (6) s acheved by a mxed polcy, whch mxes the polces correspondng to the K vertces of the polytope. The nner and outer bounds of the throughput regon are llustrated n Fg. 6(a) and Fg. 6(b). E. Numercal results In the numercal results subsecton we only show the throughput regons for the cogntve network wth two homogeneous channels and one cogntve user or two cogntve users. We pont out that the optmalty result for only holds for homogeneous channels. The results for hold for both homogeneous and heterogeneous channels and the plots obtaned from the two cases have smlar qualtatve features. Therefore the plots obtaned from homogeneous channels suffce n valdatng our results. The channel parameters we use are as follows. µ = 1/2, λ = 1/3, slot length T =.25. We use three dfferent collson constrant parameters γ =.2, γ =.7 and γ =.9 for three regmes, γ γ, γ γ γ, and γ γ, separately. Fg. 7(a) depcts the throughput versus the collson constrant parameter γ for, and the upper bound obtaned by the clarvoyant settng (see [2]) for sngle cogntve user network. We term the clarvoyant settng full sensng coordnated (FO-Coordnated) from now on. The plot shows that for the two breakponts, γ < γ and valdates the throughput characterzaton for n Proposton 1. Fg. 7(b), Fg. 7(c) and Fg. 7(d) depct the throughput regons of, and the upper bound obtaned by the clarvoyant settng (see [2]) under the three values of γ, separately. In Fg. 7(b) the throughput regon of matches wth the upper bound, valdatng that acheves the throughput regon, and the corner ponts A and B obtaned by match wth the corner ponts obtaned by and the upper bound. In Fg. 7(c) we observe that for at pont A and B lowerng the throughput of one cogntve user does not ncrease the throughput of the other snce the transmsson probablty s saturated by 1 for. In contrast we observe no saturaton n the throughput regon for the upper bound snce under the clarvoyant settng the cogntve users are able to see more channels and therefore able to place the transmsson probablty on all the current dle channels. Also we observe at pont C and D matches wth the upper bound, ndcatng the optmalty of for the sngle cogntve user network. Also note that does not acheve the maxmum throughput regon. However, the throughput regon of matches wth the upper bound near the drecton of vector (1, 1), ndcatng that acheves sum-throughput optmalty. It can be seen from Fg. 7(d) that both and the upper bound get more saturated as γ gets looser and the regon near the drecton of vector (1, 1) n whch the throughput of PS- MA matches wth the upper bound shrnks. The corner ponts A and B obtaned by le n between and the upper bound. A. Effectve bandwdth V. EFFECTIVE BANDWIDTH REGION We gve some background for effectve bandwdth n ths subsecton. Effectve bandwdth measures the quantty of servce wth a requred QoS. We consder the queueng process at the kth cogntve user. For ease of notaton we drop the superscrpt k. Assume that the ncomng traffc of the kth cogntve user s a constant arrval process wth an ntensty of a unts of data per slot and the arrved bts are stored n a buffer of sze b 1 before beng transmtted. For a fxed sensng and access polcy π, denote by Q π t the queue sze at the end of slot t. Then (Q π t ) t s gven by the followng recurson Q π t = max{q π t 1 + a R π t, }, t 1, Q π =. (7) The reward process R π t s also the output process of the cogntve user queue. From now on, we shall omt the superscrpt π. We frst state a well known result n effectve bandwdth theory (see for example [21] and [19]). The followng lemma characterzes the decay rate of the steady state tal dstrbuton of the queue. Lemma 2. Assume that the queue s stable and that the output process (R t ) t 1 satsfes the Gärtner-Ells lmt,.e., there exts a dfferentable functon Ψ R (θ) such that lm n log E[exp(θ n t=1 R t)] = Ψ R (θ). (8) n Assume also that there exsts a unque soluton θ (a) > of the equaton aθ + Ψ R ( θ) =. (9) Then Q t converges n dstrbuton to steady state dstrbuton Q and log Pr(Q > x) lm = θ (a). (1) x x Lemma 2 mples that for a buffer sze b large enough, the buffer overflow probablty P o Pr[Q > b] can be approxmated by P o γ exp( θ b). The constant γ s n general dffcult to obtan, but t has been suggested [22], [23] that γ 1 s n general a good approxmaton. The QoS constrant requres that the buffer overflow probablty for the queue of the kth cogntve user s lmted below a prescrbed parameter ǫ and the effectve bandwdth s defned to be the maxmum constant arrval rate a that can be supported provded that the buffer overflow probablty satsfes P o ǫ. Adoptng the large buffer approxmaton for the buffer overflow probablty, the maxmum sustanable arrval rate (effectve bandwdth) a (ǫ) can be defned as a (ǫ) max{a : exp( bθ (a)) ǫ}. (11)

7 7 J 2 J 2 J 1 J 1 (a) Throughput regon: loose collson constrants, N = (b) Throughput regon: loose collson constrants, K < K, equal collson constrants. N, equal collson constrants. Fg. 6. Bounds for throughput regon: loose collson constrants. for a buffer of sze b 1. Under the assumptons of Lemma 2, the effectve bandwdth a (ǫ) of a cogntve user wth buffer sze b 1 and QoS parameter ǫ s gven by a (ǫ) = Ψ R(θ) log Eexp(θ n t=1 = lm R t) (12) θ n nθ where θ = log(ǫ)/b. Recall that and nduce Markov chans n the cogntve access network. Snce and use memoryless access, the output processes of the cogntve user queue,.e., the reward processes, for the two polces form Markov modulated processes. Before presentng the results of the effectve bandwdth regon, we ntroduce a lemma whch computes the effectve bandwdth for the case when the output process of the queue forms a Markov modulated process (see for nstance [24, pp ]). Lemma 3. Let X(t) be a dscrete tme Markov chan on the state space {1,..., M} wth transton matrx P. Let {Y (t), t = 1,...}, = 1,...,M be M sequences of..d. random varables wth moment generatng functons G (θ) = Eexp(θY (1)). The process Z(t) = Y X(t) (t) s then a Markov modulated process. The effectve bandwdth for the output process Z(t) s log(ρ(g(θ)p)) (13) θ where ρ( ) s the spectral radus of a matrx, θ = log(ǫ)/b and G(θ) = dag{g 1 (θ),..., G M (θ)}. B. Sngle cogntve user network: homogeneous channels In ths subsecton we consder the cogntve network wth two homogeneous channels and equal collson constrants gven by γ. Followng the structure of the ZKL polcy [7] we can show that Theorem 4. For the cogntve access network wth two homogeneous channels, equal collson constrants and sngle cogntve user, acheves strctly larger effectve bandwdth than. X(t) Transmsson opportunty assgnment (,) No transmsson (,1) CU 1 on Ch 2 w.p. β 1, CU 2 on Ch 2 w.p. β 2 (1,) CU 1 on Ch 1 w.p. β 3, CU 2 on Ch 1 w.p. β 4 (1,1) CU 1 on Ch 1 and CU 2 on Ch 2 w.p. β 5, CU 2 on Ch 1 and CU 1 on Ch 2 w.p. β 6 CU 1 on Ch 1 only w.p. β 7, CU 1 on Ch 2 only w.p. β 8 CU 2 on Ch 1 only w.p. β 9, CU 2 on Ch 2 only w.p. β 1 TABLE I. Coordnated transmsson opportunty assgnment. Furthermore, f the collson constrant parameter satsfes γ γ, s effectve bandwdth optmal. Proof: Omtted due to the space lmt. See [2]. Theorem 4 parallels Proposton 1 and Theorem 1, mplyng that has superor effectve bandwdth performance to and partally characterzes the effectve bandwdth optmalty of. C. Multuser network: outer bound The exact effectve bandwdth regon s dffcult to obtan. We derve outer and nner bounds for the effectve bandwdth regon. We derve outer bound for effectve bandwdth regon va a clarvoyant settng wth coordnaton among the cogntve users and dvsble transmsson opportunty (see [2] for more detal). We assume there exsts a coordnator who observes n slot t the current state of the N channels X(t) = x {, 1} N and make a coordnated assgnment of the transmsson opportunty. The coordnator can dvde the overall transmsson opportunty arbtrarly and assgn to the cogntve users. For ease of presentaton we examne the case for K = N = 2 for whch the decson varables β s are gven n Table V-C. We fx the effectve bandwdth of cogntve user 1 and optmze the effectve bandwdth of cogntve user 2. Defne the functon ψ k ( ) of β, k = 1, 2: ψ k (β) = log(ρ(pλk )) θ k where P s the 4 4 transton matrx of the state vector X(t), and Λ k s a 4 4 dagonal matrx wth dagonal entres gven

8 PS MA.25.2 FO Coordnated Throughput γ (a) Sngle cogntve user Throughput for CU Throughput for CU 1 (b) Two cogntve users, tght constrants Throughput for CU C A Throughput for CU A.2 B Throughput for CU 1 D Throughput for CU 1 (c) Two cogntve users, medum constrants. Ponts A and B are the (d) Two cogntve users, loose constrants. Ponts A and B are corner end ponts of the regon where O s sum-throughput optmal. ponts obtaned by ZKL. Ponts C and D are corner ponts obtaned by ZKL..1 Fg. 7. Throughput B by Eexp(θ k R (k) x ),.e., the moment generatng functon of the reward cogntve user k collects at state x evaluated at θ k. Snce the clarvoyant settng FO-Coordnated nduces a Markov chan wth state vector X(t) {, 1} N, by Lemma 3 ψ k (β) gves the effectve bandwdth for the kth cogntve user under the setup of FO-Coordnated. The functons ψ k (β) are concave n β (see [25]). Denote by f(x) the statonary dstrbuton of the state vector X(t). Gven the effectve bandwdth requrement of cogntve user 1 that ψ 1 t 1, the maxmum effectve bandwdth for cogntve user 2 and the correspondng β s can be determned by solvng the followng convex optmzaton problem P(t 1 ): subject to max β ψ2 (β) (14) f(, 1)(β 3 + β 4 ) + f(, )(β 5 + β 6 + β 7 + β 9 ) γ 1 φ 1 (15) f(1, )(β 1 +β 2 )+f(, )(β 5 +β 6 +β 8 +β 1 ) γ 2 φ 2 (16) β 1 +β 2 1, β 3 +β 4 1, β 5 +β 6 +β 7 +β 8 +β 9 +β 1 1 (17) β (18) ψ 1 (β) t 1 (19) Denote the optmal value of P(t 1 ) by g(t 1 ). The curve (t 1, g(t 1 )) gves the outer bound for the effectve bandwdth regon for the multuser cogntve access network. D. Multuser network: nner bound We derve nner bound for effectve bandwdth regon va O. The decson varables for the kth cogntve user are α (k) = (α (k) ) N =1, the back-off coeffcents for the transmsson probablty to accommodate the other cogntve users. Defne the functon ψ (k) ( ) of α (k), k = 1,...,K: ψ (k) (α (k) ) = log(ρ(ent Q Φ (k) )) Nθ k

9 9 where Φ (k) s a 2 2 dagonal matrx gven by Φ (k) = dag{e, α (k) exp( λ T)e θk +(1 α (k) exp( λ T))e θk }. The dagonal entres have smlar nterpretaton to those of Λ k. By Lemma 3 ψ (k) (α (k) ) gves the effectve bandwdth the kth cogntve user gets from the th prmary channel under O and s concave n α (k) (see [25]). The effectve bandwdth for the kth cogntve user under O s gven by the sum of ψ (k) (α (k) ) over channel ndex due to the structure of O and the assumpton that the prmary channels are ndependent. Gven the effectve bandwdth requrements of K 1 cogntve users, say cogntve user 2 to cogntve user K, that N =1 ψ(k) (α (k) ) t k for k = 2,...,K, the maxmum effectve bandwdth for cogntve user 1 and the correspondng α s can be determned by solvng the followng convex optmzaton problem P(t 2,..., t K ): subject to N =1 max α K k=1 N =1 ψ (1) (α (1) ) (2) α (k) 1, (21) α (k), k (22) ψ (k) (α (k) ) t k, 2 k K (23) Denote the optmal value of P(t 2,...,t K ) by h(t 2,..., t K ). A pont (y 1,..., y K ) s n the regon gven by the nner bound f and only f y 1 h(y 2,..., y K ). Due to Proposton 4 we can extend the achevable effectve bandwdth regon obtaned by O on the axes by ZKL- MA. E. Numercal results In the numercal results subsecton we also just show the effectve bandwdth regons for the cogntve network wth two homogeneous channels and one cogntve user or two cogntve users. The same channel parameters wth the subsecton for throughput are used. In addton, we use buffer sze b = 4 and QoS parameter ǫ = 1e 7. Snce we use dentcal buffer sze and QoS parameters among the cogntve users, we would lke to comment that unlke the throughput regon the maxmum avalable effectve bandwdth n a multuser cogntve access network may vary among cogntve users due to dfferent buffer szes and QoS parameters. Fg. 8(a) depcts the effectve bandwdth versus the collson constrant parameter γ for, and the upper bound obtaned by the clarvoyant settng (see [2]) for the sngle cogntve user network. The effectve bandwdth has the same saturatng feature as the throughput wth the same breakpont. However, before saturatng the effectve bandwdth s nonlnear n γ, as opposed to the lnear dependency n γ for the throughput. The plot valdates that acheves strctly larger effectve bandwdth than. Fg. 8(b), Fg. 8(c) and Fg. 8(d) depct the effectve bandwdth regons of, and FO-Coordnated under the three values of γ, separately. The three plots show the same trend of saturaton as those of throughput. Nether the regon of O nor the corner ponts obtaned by ZKL-Ma match the regon of FO-Coordnated for effectve bandwdth. The nner and outer bounds are closer near the lne y = x than near the axes. VI. CONCLUSIONS In ths paper we have nvestgated the throughput and effectve bandwdth regons of multuser cogntve access network wth K N cogntve user sharng N Markov channels wth prescrbed collson constrants. We characterze the throughput regon under tght collson constrants by analyzng two transmsson polces O and FO-Coordnated. We also analyze the transmsson polcy and obtan ts sngle cogntve user throughput optmalty for dentcal prmary channels wth equal collson constrants. We derve nner and outer bounds for the effectve bandwdth regon from O and FO-Coordnated va a par of convex optmzatons. ZKL- MA also enables us to extend the achevable throughput and effectve bandwdth regons for general collson constrants on the axes. There are several future drectons that we wsh to pursue such as the determnaton of the sngle cogntve user effectve bandwdth optmal polcy for dentcal prmary channels under tght collson constrants and possble extenson of to the multuser case. REFERENCES [1] Q. Zhao and B. M. Sadler, A survey of dynamc spectrum access, IEEE Sgnal Processng Magazne, vol. 24, pp , May 27. [2] X. L, Q. Zhao, X. Guan, and L. Tong, Optmal cogntve access of Markovan channels under tght collson constrants, IEEE J. Selected Areas on Communcatons, 29. submtted n December 29. [3] Q. Zhao, S. Gerhofer, L. Tong, and B. Sadler, Optmal dynamc spectrum access va perodc channel sensng, n Proc. Wreless Communcatons and Networkng Conference (WCNC), (Hong Kong), March 27. [4] Q. Zhao, S. Gerhofer, L. Tong, and B. M. Sadler, Opportunstc spectrum access va perodc channel sensng, IEEE Trans. Sgnal Processng, vol. 36, pp , Feb 28. [5] I. A. Akbar and W. H. Tranter, Dynamc spectrum allocaton n cogntve rado usng hdden Markov models: Posson dstrbuted case, n IEEE Proc. SoutheastCon, pp , March 27. [6] X. L, Q. Zhao, X. Guan, and L. Tong, On the optmalty of memoryless cogntve access wth perodc spectrum sensng, to appear n IEEE Internatonal Conference on Communcatons, 21. [7] Q. Zhao, B. Krshnamachar, and K. Lu, On myopc sensng for multchannel opportunstc access: structure, optmalty, and performance, IEEE Trans. Wreless Comm., vol. 7, pp , Dec 28. [8] S. Ahmad, M. Lu, T. Javd, Q. Zhao, and B. Krshnamachar, Optmalty of myopc sensng n mult-channel opportunstc access, IEEE Transactons on Informaton Theory, vol. 55, no. 9, pp , 29. [9] A. Laourne, S. Chen, and L. Tong, Queueng analyss n multchannel cogntve access: A large devaton approach, n Proc. 21 INFOCOM, (San Dego, CA), March 21. [1] S. Wang, J. Zhang, and L. Tong, Delay analyss for cogntve rado networks wth random access: a flud queue vew, n Proc. 21 IEEE INFOCOM, (San Dego, CA), March 21.

10 1 Effectve bandwdth PS MA Effectve bandwdth for CU γ (a) Sngle cogntve user Effectve bandwdth for CU 1 (b) Two cogntve users, tght constrants Effectve bandwdth for CU Effectve bandwdth for CU Effectve bandwdth for CU 1 (c) Two cogntve users, medum constrants Effectve bandwdth for CU 1 (d) Two cogntve users, loose constrants Fg. 8. Effectve bandwdth [11] Q. Zhao, L. Tong, and A. Swam, Decentralzed cogntve MAC for dynamc spectrum access, n Proc. of the frst IEEE Symposum on New Fronters n Dynamc Spectrum Access Networks (DySPAN), (Baltmore, MD), Nov 25. [12] Q. Zhao, L. Tong, A. Swam, and Y. Chen, Decentralzed cogntve MAC for opportunstc spectrum access n ad hoc networks: a POMDP framework, IEEE J. Select. Comm., vol. 25, pp , Aprl 27. [13] S. Huang, X. Lu, and Z. Dng, Opportunstc spectrum access n cogntve rado networks, n Proc. 28 IEEE INFOCOM, (Phoenx, AZ), 28. [14] S. Huang, X. Lu, and Z. Dng, Optmal sensng-transmsson structure for dynamc spectrum access, n Proc. 29 IEEE INFOCOM, 29. [15] S. Huang, X. Lu, and Z. Dng, Optmal transmsson strateges for dynamc spectrum access n cogntve rado networks, IEEE Trans. Moble Computng, vol. 8, pp , Dec 29. [16] S. Gerhofer, L. Tong, and B. M. Sadler, Cogntve medum access: constranng nterference based on expermental models, IEEE J. Select. Areas Communcatons, Specal Issues on Cogntve Rado: Theory and Applcatons, vol. 36, pp , Feb 28. [17] S. Gerhofer, L. Tong, and B. M. Sadler, Dynamc spectrum access n the tme doman: modelng and explotng whtespace, IEEE Communcatons Magazne, vol. 45, pp , May 27. [18] S. Gerhofer, L. Tong, and B. Sadler, Interference-aware ofdma resource allocaton: A predctve approach, n IEEE MILCOM 28, pp. 1 7, November 28. [19] C.-S.Chang, Stablty, queue length, and delay of determnstc and stochastc queueng networks, IEEE Transactons on Automatc Control, vol. 39, pp , May [2] S. Chen and L. Tong, Maxmum throughput regon of multuser cogntve access of markovan channels, to be submtted. [21] P. Glynn and W. Whtt, Logarthmc asymptotcs for steady-state tal probabltes n a sngle-server queue, Journal of Appled Probablty, vol. 31, pp , [22] J. Abate, G. Choudhury, and W. Whtt, Exponental approxmatons for tal probabltes n queues, : watng tmes, Operatons Research, vol. 43, no. 5, pp , [23] J. Abate, G. Choudhury, and W. Whtt, Exponental approxmatons for tal probabltes n queues : sojourn tme and workload, Operatons Research, vol. 44, no. 5, pp , [24] C. Chang, Performance guarantees n communcaton networks. Sprnger Verlag, 2. [25] J. C. Kngman, A convexty property of postve matrces, Quart. J. Math., vol. 12, pp , 1961.

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