Proactive Resource Allocation: Harnessing the Diversity and Multicast Gains

1 Proacti v e Resource Al location: Har nessing the Di ver sity and Multicast Gains John T adrous, Atilla Eryilmaz, and Hesh am El Gamal Abstract —This p aper introduces the nov el concept of proacti ve resour ce allocation through which the predictability of user behavio r is exploited to b alance the wireless trafﬁ c over time, and hence, signiﬁcantly reduce the bandwid th required to achiev e a giv en blocking/outage pro bability . W e start with a simple model in which the smart wireless devices are assumed to predict the arriv al of new requests and submit them to the network T time slots in advance. Usin g tools from large deviation theory , we quantify the r esulting prediction diversity gain to establish that the decay rate of the outage ev ent probabilities increases with the prediction du ration T . This model is then generalized to incorporate the effect of the randomness in the p rediction look- ahead time T . Remarkably , we also show that, in the cognitiv e networking scenario, the appropriate use of proacti ve resource allocation b y the primary users impro ves t he diversity gain of t he secondary network at no cost in the pri mary network div ersity . W e also shed li ghts on multicasting with predictable d emands and show that the proactiv e multicast networks can achieve a signiﬁcantly higher diversity gain that scales su per -linearly wit h T . Fin ally , we conclud e by a discussion of t he new research questions posed un der th e umbrella of the proposed proacti ve (non-causal) wireless netw orking framework. Index T erms —Scheduling, larg e deviations, diver sity gain, multicast alignment, predictiv e trafﬁc. I . I N T R O D U C T I O N I Deally , wireless networks should be op timized to d eliv er the b est Qu ality o f Ser vice (in terms of reliability , delay , and th rough put) to the su bscribers with the minimum exp en- diture in resources. Such re sources include tr ansmitted power , transmitter and rec eiv er complexity , and allocated frequen cy spectrum. Over the last few y ears, we have experienced an ev er increasing deman d fo r wireless spectrum resulting fr om the ado ption of th r oughput hungry ap plications in a v ariety of civilian, military , and scien tiﬁc settings. Since the av ailable spectrum is n on renewable an d limited, this demand mo ti vates the need for efﬁcient wir eless n etworks that maximally utilize the spectrum . I n th is work, we f ocus our attention on the resource allocation aspect of the pr oblem and propose a new paradig m that offers remarkable spectral gains in a variety of relev ant scenarios. Mo re speciﬁcally , our proactiv e resource allocatio n framework exploits the p re- dictability o f our d aily usag e of wireless devices to smooth out the trafﬁc demand in the network, and hence, reduce the required resou rces to achie ve a certain p oint on the Quality o f Service (QoS) curve. This new app roach is m otiv ated by th e following observations. Authors are with the Department of Electrica l and Computer E ngineeri ng at the Ohio State Univ ersity , Columbus, USA. E-mail: { tadrousj,eryilmaz ,helgamal } @ ece.osu.edu • While there is a severe sho rtage in the spectrum , it is well-docum ented that a signiﬁcant fractio n of the av ailable spectrum is under-utilized [1]. In fact, this is the main moti- vation fo r th e co gnitive networking wh ere seco ndary users are allowed to use the spectrum at the off time o f the primary so as to maxim ize the spectral efﬁciency [2]. The cognitive radio approa ch, howe ver , is still facing sig niﬁcant techn ological hurdles [3], [4] an d, will offer only a partial solution to the problem . This limitation is tied to the main reason b ehind the under-utilization of the spectrum; namely th e large disparity between the ave rag e and peak trafﬁc demand in the network . Actually , on e c an see that the trafﬁc d emand at the peak hours is much high er th an that at night. Now , the cognitive radio appro ach assumes that the secondary users will b e able to utilize the spe ctrum at the off-peak times, but at those times one may expect the seconda ry trafﬁ c char acteristics to b e similar to th at of the primary users (e.g., at night most of the pr imary a nd secon dary users a re expe cted to b e idle). Ther eby , the overarchin g go al of the pr oactive resour ce allocation framework is to av oid this limitatio n, and h ence, achieve a signiﬁcant reduction in the peak to a verage dem and ratio without relying on out o f network u sers . • In th e traditiona l app roach, wire less networks are con - structed assuming that th e su bscribers are equipp ed with dumb terminals with very limited com putationa l p ower . It is o bvious that the new generation o f smart devices en joy signiﬁcantly enhanced capabilities in terms o f both processing power and av ailable memo ry .This ob servation sho uld insp ire a similar paradig m shift in the design o f wireless n etworks whereby th e capabilities of th e smar t wireless terminals a re lev eraged to maximize the utility of the freq uency spe ctrum, a non-renew able r esour ce th at does not scale according to Moor e’ s la w . Ou r proactive resource allo cation framew ork is a signiﬁcan t step in this direction. • Th e in troduction o f smar t ph ones h as resulted in a paradig m shift in th e do minant trafﬁc in m obile cellular n etworks. While the primary traf ﬁc source in traditional cellular networks was real-time voice comm unication, o ne can argue that a signiﬁcant fraction of the trafﬁc generated by th e smart ph ones results from non- real-time data r equests (e.g., ﬁle downloads). As d emonstrated in the following, this feature allows for m ore degrees of freedom in the d esign of the sched uling algo rithm. • The ﬁnal piece of our puzzle relates to the observation that the human u sage of the wireless d evices is highly pr edictable . This claim is supp orted by a growing body of evidence that ranges fr om the recent lau nch of Google Instant to the interesting ﬁndin gs on ou r predictab le mobility pattern s [5]. An example would be the fact that o ur preferen ce for a 2 particular news outlet is not expected to change f requen tly . So, if the smart phone ob serves that th e user is downloading CNN, for example, in th e mornin g for a sequenc e of days in a row then it can anticipate that t he user will be intere sted in the CNN again the fo llowing d ay . One c an now see the poten tial for sch eduling early downloads of th e predictable trafﬁc to r educe the peak to average trafﬁc dema nd b y m aximally exploiting the av ailable spectrum in the network idle time . These ob servations m otiv ate us in this work to develop and analyze p roactive resour ce allocation strategies in the presence of user predictab ility under v arious co nditions, d ynamics, and operation al capabilities. In pa rticular, our co ntributions along with their p osition in the rest of the p aper are: • In Section II we state the predictive network mo del and in- troduce the outage probability and the as sociated diversity gain for tw o main scaling regimes, namely , linear and poly nomial scaling. • In Sectio n III, we establish th e diversity gain of non- predictive and pr edictive network s, an d analy ze the effect of the random loo k-ahead wind ow size, T . Our an alysis reveals a minimum improvemen t factor of (1+T ) in the di versity gain for bo th linear and polynomial scaling regimes. • In Section IV, we investigate pr oactive schedu ling in a two- QoS network,typica l of a cogn itiv e rad io network, wh ere we prove the existence o f a proactiv e schedulin g po licy that can maintain the diversity gain level of the primary predictive network while strictly improving it for the seconda ry non- predictive n etwork. • In Section V, we a nalyze the robustness of the proactive resource allocation scheme to the pred iction errors and deter- mine the op timal cho ice of the lo ok-ahe ad win dow size gi ven an imperf ect prediction mechanism to maximize the diversity gain, which is shown to be always strictly greater than tha t o f the no n-pred ictiv e network. • In Section VI, we a nalyze the proactive multicasting with predictable demands, and show the sign iﬁcant gains that can be le veraged throu gh the alig nment pro perty offered by pre- dictable multicast trafﬁc. More speciﬁcally , we sh ow that th e div ersity gain of a p roactive multicasting network is in creasing super-linearly with the window size, T , for the lin ear scaling regime. • In Section VII I, we co nclude th e pap er and hig hlight oth er importan t researc h aspec ts that can b e lev eraged throu gh predictive wire less commu nications. The proactive wireless network can be v iewed as an ordi- nary network with d elay tolerant req uests, that is, when the network pr edicts a requ est a head of time, the actu al arr iv al time of th at request c an be consid ered as a hard deadline that the schedu ler should meet. In [6], sch eduling with deadlines was consider ed for a single packet under th e objective o f minimizing the expected energy consum ed for transmission. In [7], the asym ptotic p erform ance of the error pro bability with the sig nal-to-n oise r atio was analyzed when the b its of each codeword must b e deliv ered under hard deadline con straints. In [8] and [9], schedu ling w ith deadlines was also addressed from queuing the ory point o f vie w under different o bjectives and mu ltiple prior ity classes while op timal schedu ling po licies were inv estigated for different scenarios. Time slots q(n) Prediction time n T q(n) : Prediction duration D q(n) : Deadline Actual arrival time Fig. 1: Prediction Mo del: q ( n ) is a request pr edicted at th e beginning o f tim e slot n , T q ( n ) is the pr ediction duration of request q ( n ) , and D q ( n ) is the actual slo t of arriv al for request q ( n ) which can be considered as the dea dline for such a requ est. In this paper , howe ver , we look at the sched uling pr oblem with deadlines from a different perspec ti ve, wher e we d eﬁne the outage p robab ility as the probability of having a time slot suffering expiring requests, and we analyze the asympto tic decay rate of this outage pr obability w ith the system c apacity , C , when th e in put trafﬁc is increasing in C either linea rly or polynom ially , an d C is appr oaching inﬁn ity . W e call this metric the diversity gain o f the network and show that its behavior can sign iﬁcantly be im proved b y exploitin g the predictable b ehavior of the users. This m etric a nd line of in vestigation are also motiv ated by th e or der-wise difference between the timescale of the pr ediction window length s (typ- ically of the order of tens of minu tes, if not h ours) and the timescale of app lication-based deadline- constraints (of the order of milliseco nds) conside red in other works. I I . S Y S T E M M O D E L W e con sider a simpliﬁed model of a single server , time- slotted wir eless n etwork where the requests a rrive at the beginning of each slot. T he number o f arr iving requests a t time slot n is an integer-valued random variable deno ted by Q ( n ) th at is assumed to be ergodic a nd Poisson distributed with m ean λ . Ea ch request is assumed to consu me one unit of re source and is com pletely served in a single time slot. Moreover , the wireless n etwork has a ﬁ xed capacity C per slot. W e distinguish two types of wireless re source alloca - tion: reactive and proactive . I n reacti ve resour ce allocation , the wireless network respo nds to user requests righ t after they are in itiated by the u ser , wh ereas in proactive resour ce allocation, the n etwork can track, learn an d then predict the user requests ahead of time, and hen ce possesses mo re ﬂexibility in schedulin g th ese requests b efore their actual tim e of arriv al. W e refer to the networks that perform reactiv e and proactive resou rce allocation, respectively , as non-predictive and predictive networks. The pr edictive wir eless network can anticipate the arrival of requ ests a nu mber of tim e slots ahead. That is, if q ( n ) , q ∈ { 1 , · · · , Q ( n ) } , is the identiﬁer of a req uest predicted at the b eginning of time slot n , the pr edictive network has th e advantage of serving this r equest within th e next T q ( n ) slots. Hence, wh en req uest q ( n ) arrives at a predictive network, it has a d eadline at time slot D q ( n ) = n + T q ( n ) as shown in Fig. 1. Con versely , in a n on-pr edictiv e network, all arriving re- quests at the b eginning of time slot n must be served in 3 the sam e time slot n , i.e., if q ( n ) is an unp redicted req uest, then T q ( n ) = 0 and D q ( n ) = n . At th is point, we wish to stress the fact that the mo del operates as the time scale o f the application la yer at which 1) the curren t parad igm, i.e., non-p redictive networking, treats all the r equests as urgent, 2) each slot duration may b e in the or der of minutes and po ssibly hours, and 3 ) the system capa city is ﬁxed since the chann el ﬂuctuation dyn amics are averaged out at this tim e scale. Deﬁnition 1: Let N 0 ( n ) be the number of requ ests in the system at the beginnin g of time slot n having a deadline of n . Th e outage event O is then deﬁned as O , { N 0 ( n ) > C, n ≫ 1 } , (1) The above deﬁnition states that an outage o ccurs at slot n if and o nly if at least one of the requests in the system expires in this slot. The ter m N 0 ( n ) co incides on Q ( n ) when the network is non-pr edictive, and is different when the network is predic ti ve. Follo wing the deﬁn ition o f the outage event, we denote the probab ility that the wireless network r uns into an outag e at slot n > 0 b y P ( O ) . Throug hout this pape r , we will focus on an alyzing the asym ptotic de cay rate o f th e o utage probab ility with the system capacity C when it approac hes inﬁnity . W e c all this dec ay rate the div ersity gain o f the network. Mor eover , in our ana lysis we assume that the mean input trafﬁc λ scales with the system cap acity in two different regimes as follows. 1) Lin ear Scaling : I n this regime, the arrival p rocess Q ( n ) , n > 0 is Poisson with rate ¯ λ that scales with C as ¯ λ = γ C, 0 < γ < 1 . And with ou tage pro bability deno ted by P ( O ) , the associated diversity gain is deﬁned as d ( γ ) , lim C →∞ − log P ( O ) C . 2) P o lynomial S caling: In this regime, the arriv al proce ss e Q ( n ) , n > 0 , is also Poisson with rate ˜ λ , but th e rate scales with th e system cap acity C poly nomially as ˜ λ = C e γ , 0 < e γ < 1 . And with o utage p robability P ( e O ) , th e associated diver - sity gain is deﬁned as e d ( e γ ) , lim C →∞ − log P ( e O ) C lo g C . W e consider the linear scaling of the in put tra fﬁc with the system resou rces because it is commo nly used in networking literature where the p arameter γ serves a s bandwid th uti- lization factor . As γ a pproach es 1 the average in put tra fﬁc approa ches the cap acity and the sy stem becom es critically stable and m ore subject to o utage events, whereas small values of γ im ply u nderutilized resou rces but small prob ability of outage. The polyn omial scaling regime is also introd uced because under this ty pe of scalin g, the optimal pr ediction div ersity g ain can be fully de termined thr ough the asymp totic analysis of simple sch eduling p olicies like earliest deadlin e ﬁrst. Excep t for Section VI and its associated ap pendices, we consistently use the ac cents ¯ . and ˜ . to denote linear and polyno mial scaling regimes resp ectiv ely , while symb ols without accen ts are used to denote a ge neral case. I I I . D I V E R S I T Y G A I N A N A L Y S I S A. Diversity Gain of Reactive Networks The reac ti ve networks are su pposed to have no pr ediction capabilities so th ey cannot serve any requ est prior to its time of actu al arriv al. Henc e, the r eactive n etwork encoun ters an outage at time slot n if and only if Q ( n ) > C as N 0 ( n ) = Q ( n ) . Theor em 1: Denote the outag e pr obability a nd the diversity gain of the non-p redictive network, r espectively , by P N ( O ) and d N ( γ ) , then d N ( γ ) = γ − 1 − log γ , 0 < γ < 1 , (2) and e d N ( e γ ) = 1 − e γ , 0 < e γ < 1 . (3) Pr oof: Please refer to Appendix A. It can b e no ted th at as γ and e γ appro ach 1 , the corr e- sponding diversity gains d N ( γ ) and e d N ( e γ ) approa ch 0 , as in this case the arriv al rate in both regimes m atches the system capacity , and hence th e system becom es cr itically stable and the logarithm of the outage prob ability d oes not decay with C . Howe ver , the b ehavior of the th e diversity gain is no t the same when both γ a nd e γ approach 0 . As γ → 0 , d N ( γ ) → ∞ because the arri val rate ¯ λ → 0 , th us the resulting outage probab ility app roaches 0 and the d iv ersity gain a pproach es ∞ . Whereas e γ → 0 im plies that e d N ( e γ ) → 1 wh ich is the case when the input trafﬁc is still po siti ve but does not scale with the system cap acity . B. Diversity Gain of Pr oactive Networks Unlike reactive networks, the proactive network has the ﬂexibility to sched ule the predicted requ ests in a window of time slots thro ugh some scheduling policy . Depend ing on the scheduling p olicy employed, th e re sulting ou tage pr obability (and of c ourse the associated diversity gain) varies. By the term optimal prediction div ersity gain, we mean the maximum div ersity gain th at can be achieved by the predictive network, which correspo nds to the minimu m outage probability deno ted P ∗ P ( O ) . In our analysis, we con sider, for simplicity , the earliest deadline ﬁrst ( EDF) schedu ling policy , which has also been called in [13] sho rtest time to extinction (STE). This policy , as pr oved in [13], maximizes the numb er of served requests under a per-request d eadline con straint. Further stud ies o n this policies can be foun d in [8] an d [14]. In the pro posed predictive network, the EDF scheduling policy is deﬁned as follows. Deﬁnition 2 (Earlies Deadline F ir st (E DF)): Let the max- imum prediction interval for a reque st be denoted by T ∗ , i.e., T ∗ = sup q,n  T q ( n )  , and let N i ( n ) , i = 0 , 1 , · · · , T ∗ be the number of requ ests in th e system at the beginn ing o f time slot n and having a deadline of n + i . Th en, at the beginnin g o f 4 slot n , th e EDF p olicy sorts { N i ( n ) } T ∗ i =0 in an ascending or der with respect to i , and serves them in that order u ntil either a total of C r equests get served or the network completes the service of all existing requests in this slot. It can be noted that EDF does not necessarily minim ize the outage p robab ility as it is only co ncerned with m aximizing the number of served requests while the outage event does not take into acc ount the nu mber of droppe d requests. Ho wever , EDF has tw o main cha racteristics that help in analy sis. Nam ely , it always serves req uests as long as th ere are any , i.e., it is a work conserving policy , and it serves requests in the order of their remain ing time to de adline. 1) De terministic Look-ah ead T ime: In this scen ario, T q ( n ) = T for all q ( n ) , n > 0 for some co nstant T ≥ 0 . Hence, assuming that the system employs EDF sch eduling policy , we hav e T ∗ = T and N T ( n ) = Q ( n ) , n > 0 . Th us, the EDF p olicy will reduce to ﬁrst-come-ﬁrst-serve ( FCFS). The outag e proba bility in this case is denoted by P P D ( O ) . Lemma 1: Under EDF , let U D , ( T X i =0 Q ( n − T − i ) > C ( T + 1) , n ≫ 1 ) and L D , { Q ( n − T ) > C ( T + 1) , n ≫ 1 } . Then, the events U D and L D constitute a necessary con dition and a sufﬁcient co ndition on the outage event, r espectively . Hence, P ( L D ) ≤ P P D ( O ) ≤ P ( U D ) . In the ab ove lemma, we assume that n ≫ 1 as we are interested in the stead y state per forman ce. The event U D occurs when the num ber of arr i ving req uests in co nsecutive T + 1 slots is larger than the total capac ity of T + 1 slots, whereas the e vent L D occurs when the number of ar riving requests at any slot is larger than the total ca pacity o f T + 1 slots. Pr oof: Please refer to Appendix B. It is obvio us from the proof that the ev ent U D is related to the outage event O throu gh th e EDF schedu ling policy , whereas the event L D is ind ependen t of th e schedu ling policy employed. Theor em 2: The op timal prediction diversity gain of a proactive network with determ inistic pred iction interval T , denoted d P D ( γ ) , satisﬁes d P D ( γ ) ≥ (1 + T )( γ − 1 − log γ ) , 0 < γ < 1 , (4) e d P D ( e γ ) = (1 + T )(1 − e γ ) , 0 < e γ < 1 . (5) The above resu lt shows that proac ti ve resour ce allo cation offers a multiplicative d iv ersity gain o f at least T + 1 for the linear scaling regime and exactly T + 1 for the p olynom ial scaling regime. Pr oof: Please refer to Appendix C. Note that, an u pper bound on d P D ( γ ) ca n be established using P ( L D ) ≤ P P D ( O ) and f ollowing the same app roach of deriving th e lower bound in the theorem. This up per bou nd will be gi ven by d P D ( γ ) ≤ ( T + 1)  γ T + 1 − 1 + lo g  T + 1 γ  . (6) Comparing the right hand sides of (4), and (6) it can b e seen that they match o nly when T = 0 , and in this case, th ey also match the non- predictive div ersity gain obtaine d in (59). Otherwise, for positi ve v alues of T , the two b ound s dif fer . 2) Ran dom Look- ahead T ime: W e consider a more gener al scenario where T q ( n ) , 0 ≤ q ( n ) ≤ Q ( n ) , n > 0 is a sequence of IID non -negative integer-v alued ra ndom v ariables deﬁned over a ﬁnite support { T ∗ , · · · T ∗ } , wh ere 0 ≤ T ∗ ≤ T ∗ < ∞ . The rand om variable T q ( n ) has the follo wing probab ility m ass function (PMF), P  T q ( n ) = k  ,  p k , T ∗ ≤ k ≤ T ∗ , 0 , otherwise , (7) where P T ∗ k = T ∗ p k = 1 and p k ≥ 0 , ∀ k , the c umulative distribution function (CDF) of T q ( n ) can be written as P ( T q ( n ) ≤ k ) = F k =    1 , k > T ∗ , P k i = T ∗ p i , T ∗ ≤ k ≤ T ∗ , 0 , k < T ∗ . (8) Hence, the overall process Q ( n ) can be d ecompo sed to a superpo sition of independen t Po isson pro cesses, i.e., Q ( n ) = T ∗ X k = T ∗ Q k ( n ) where Q k ( n ) , n > 0 is th e pro cess of requ ests predicted k slots ah ead, k = T ∗ , · · · , T ∗ . The arriv al rate of Q k ( n ) is p k λ . In this scenario, we d enote the outage prob ability under EDF by P P R ( O ) and th e optimal diversity g ain by d P R ( γ ) . Unlike the case of deterministic loo k-ahead time, EDF h ere does no t redu ce to FCFS becau se the arriving r equests at the subsequen t slots c an have earlier d eadlines than some of those who have alread y arri ved. Upper and lower bou nds on P P R ( O ) are introd uced in the f ollowing lem ma. Lemma 2: Let I ,    T ∗ X j =0 T ∗ X i = T ∗ Q i ( n − i − j ) > C ( T ∗ + 1) , n ≫ 1    , J , T ∗ − 1 [ k = T ∗    k X j = T ∗ j X i = T ∗ Q i ( n − j ) > C ( k + 1) , n ≫ 1    , U R , I [ J and L R , T ∗ [ k = T ∗    k X j = T ∗ Q j ( n − j ) > C ( k + 1) , n ≫ 1    , then, the ev ents U R and L R constitute nec essary an d sufﬁcient condition s on the o utage e vent, respectively . Hence P ( L R ) ≤ P P R ( O ) ≤ P ( U R ) . Here also, w e assume the system is at steady state. Pr oof: Please refer to Appendix D. 5 Theor em 3: Let v ∗ , min T ∗ ≤ k ≤ T ∗ − 1 ( ( k + 1) " log k + 1 γ P k − T ∗ i =0 F k − i ! − 1 # + γ k − T ∗ X i =0 F k − i ) , the o ptimal diversity gain of a pro activ e wireless n etwork with random pr ediction interval, d P R ( γ ) , satisﬁes d P R ( γ ) ≥ min { ( T ∗ +1)( γ − 1 − log γ ) , v ∗ } , 0 < γ < 1 (9 ) for the line ar scaling r egime, and e d P R ( e γ ) = ( T ∗ + 1)(1 − e γ ) , 0 < e γ < 1 , (10) for the p olynom ial scaling regime. Pr oof: Please refer to Appendix E. Theorem 3 determ ines a lower bound on the o ptimal pr e- diction d iv ersity gain of the linear scaling regime and fully characterizes the optimal predic tion diversity . It is obvious that the lower boun d o n d P R ( γ ) d epends on the distribution of T q ( n ) , howev er , th is lower bound is alw ays larger than d N ( γ ) as long as T ∗ > 0 an d p T ∗ > 0 . This can be viewed by co nsidering th e term ( T ∗ + 1)( γ − 1 − log γ ) wh ich is strictly larger th an d N ( γ ) an d v ∗ where fo r any k such that T ∗ ≤ k ≤ T ∗ − 1 , ( k + 1) " γ P k − T ∗ i =0 F k − i k + 1 ! − log P k − T ∗ i =0 F k − i k + 1 − 1 − lo g γ # ( a ) > ( k + 1)( γ − 1 − lo g γ ) ( b ) ≥ d N ( γ ) . Inequa lity (a) follows as 0 < P k − T ∗ i =0 F k − i k + 1 < 1 and γ x − log x > γ , ∀ x ∈ (0 , 1) , while inequality (b) follo ws because k ≥ T ∗ ≥ 0 . Hen ce, the proactive network in linea r scaling regime with T ∗ > 0 and p T ∗ > 0 always impr oves the diversity gain. For the polyn omial scaling regime, Theorem 3 shows that the p rediction d i versity gain of a p roactive wireless network with ra ndom lo ok-ah ead interval is dominated b y ar riv als with T q ( n ) = T ∗ . Hence, t he main drawback of this is that, if T ∗ = 0 the pred iction di versity becomes tantamount to that of the non - predictive scenar io. Howe ver , even thou gh T ∗ = 0 , the outage probab ility of the predictive n etwork is ev aluated numerically in Section VII and shown to outpe rform the no n-pred icti ve case. I V . H E T E RO G E N O U S Q O S R E Q U I R E M E N T S W e consider two types of u sers with different QoS re- quiremen ts, the ﬁrst is a primary user who has the prior ity to access th e network, whereas the second is a secondary user that is allowed to access the pr imary network resources oppor tunistically . Th at is, it can use the pr imary reso urces at any time slot only when there is sufﬁcient capa city to serve all primary requests at that slot with the remainin g capacity assigned to the second ary user . This type of opportun istic access to the pr imary network adds more utilization to its resources while it may get paid by the seco ndary user for the offered service. The primary and secondar y requ ests arrive to the net- work following two Poisson pro cesses Q p ( n ) , n > 0 and Q s ( n ) , n > 0 with arriv al rates λ p and λ s respectively . W e also assume that the network is stable and domina ted by p rimary arriv als as follows. Assumption 1: λ s + λ p < C, (11) λ s < λ p . (12) The network is reactive to the secondar y req uests and hence each seconda ry request will expire if it is not ser ved in the same slot of arrival. In the following subsection, we analyze the perfo rmance o f the secon dary outage prob ability and div ersity gain when the p rimary n etwork is also reactive, then we pro ceed to the p roactive case. A. Non -pr edictive Primary Network At the beginning of time slot n the network has Q p ( n ) + Q s ( n ) arriv als that sho uld be served within the same slot, i.e., all have a deadline of n . The network ty pically serves the primary requ ests befor e the secon dary . Hence, the diversity gain of the p rimary network in th is scheme, denoted d p N ( γ p ) , follows the same expressions obtained in Theorem 1, i.e., d p N ( γ p ) = γ p − 1 − lo g γ p , 0 < γ p < 1 ( 13) e d p N ( e γ p ) = 1 − e γ p , 0 < e γ p < 1 , (14) where ¯ λ p = γ p C and ˜ λ p = C e γ p . The secondary user , the refore, suffers an outage at time slot n if and only if Q p ( n ) + Q s ( n ) > C, Q s ( n ) > 0 . Theor em 4: The diversity gain of th e seco ndary ne twork, d s N ( γ p , γ s ) , when th e primary network is non-pr edictiv e, sat- isﬁes d s N ( γ p , γ s ) ≤ γ p − 1 − lo g γ p , (15) d s N ( γ p , γ s ) ≥ γ p + γ s − 1 − lo g( γ p + γ s ) , (16) e d s N ( e γ p , e γ s ) = 1 − e γ p , (17) where ¯ λ s = γ s C , ˜ λ s = C e γ s and 0 < γ s < γ p < 1 , γ p + γ s < 1 and 0 < e γ s < e γ p < 1 . Pr oof: Please refer to Appendix F. Theorem 4 reveals that the diversity gain o f the secondar y user , un der non -predic ti ve network, is at most e qual to the div ersity gain of the p rimary network in the linear scaling regime a nd is exactly equal to it in the polynom ial scaling regime althoug h the seco ndary u ser ha s strictly less trafﬁc rate than the primary . It can also be noted that e d s N ( e γ p , e γ s ) is independen t of e γ s , that is, regard less of how small e γ s is, the diversity gain o f the secondar y user is kept ﬁxed at e d p N ( e γ p ) 6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.5 1 1.5 2 2.5 3 ¯ γ p Upper and lower b ounds on ¯ d s N (¯ γ p , ¯ γ s ) Lower bound Upper bound ¯ γ p − 1 − lo g ¯ γ p (¯ γ p + ¯ γ s ) − 1 − lo g(¯ γ p + ¯ γ s ) Fig. 2: The ga p between th e upper an d lo wer bounds o n d s N ( γ p , γ s ) declines wh en γ s ≪ γ p . In this ﬁg ure, γ s = 0 . 02 and γ p ∈ ( γ s , 1 − γ s ) . as long as e γ s > 0 . The lower bo und in (16), althoug h do es not match the upp er bound in (15), it is alw ays positive and approa ches the u pper b ound when γ s is much smaller than γ p as shown in Fig. 2 . B. Predictive Primary Network When the primar y network is p redictive, the arriving pri- mary requ ests Q p ( n ) , n > 0 ar e assumed to be pred ictable with a deterministic look- ahead time T . The secondary r e- quests, Q s ( n ) , co n versely , ar e all u rgent. Let N p i ( n ) b e the number of all primary r equests awaiting in the network at the beginning o f time slot n with deadline n + i , i = 0 , · · · , T and let N p ( n ) = P T i =0 N p i ( n ) . 1) S elﬁsh Primary S cheduling: By a selﬁsh prim ary beh av- ior we mean the primary network has a dedicated capacity C per slot a nd no second ary request is served at th e beginnin g of time slot n un less all prima ry r equests N p ( n ) are served at this slot and C − N p ( n ) > 0 . The op timal p rediction diversity gain and the o utage pr obability of the p rimary network in this case are no t affected by the presence of the seco ndary user . On the oth er h and, the selﬁsh behavior o f the primary predictive network c annot im prove the outag e prob ability of the secondary user . T o sho w this, let P P ( O s ) d enote the outage probab ility of the secondar y user when the primary n etwork is predic ti ve. Then P P ( O s ) = P ( N p ( n ) + Q s ( n ) > C, Q s ( n ) > 0) ≥ P ( Q p ( n ) + Q s ( n ) > C, Q s ( n ) > 0) (18) = P N ( O s ) , (19) where in equality (1 8) follows since N p T ( n ) = Q p ( n ) and N p ( n ) ≥ N p T ( n ) . Her e we note that the above result ho lds for any scheduling policy that serves all p rimary requests in the ne twork at any slot bef ore the secondar y requests. 2) Co operative Primary User: The p redictive p rimary n et- work, h owe ver , can act in a less-selﬁsh man ner witho ut losin g perfor mance an d, at the same time, enh ance the diversity g ain of the secondar y user . This can be don e by limitin g the per- slot capacity dedicated to serve the primary requests in the system. One po ssible way to do so is to d ecide the cap acity for th e prim ary n etwork dynam ically at th e beginn ing o f each slot. W e sugg est the fo llowing less -selﬁsh po licy . Deﬁnition 3: The number of primary req uests to b e served at slot n is denoted by C p ( n ) and g iv en by C p ( n ) , min ( C, N p 0 ( n ) + f × T X i =1 N p i ( n ) ) , (20 ) where 0 ≤ f ≤ 1 , and the prim ary requests are served accordin g to EDF . This scheme determines the maximum num ber of primary requests that th e primary network can serve at th e beginning of each slot de pending on the n umber of primary req uests with deadline at this slot as well as some factor of the number of other primary req uests in the system. Hence, a t the b eginning of time slot n , ar riving secondary requests will have the chance to get service if C − C p ( n ) > 0 , while the primary network has the capab ility to schedule the C p ( n ) r equests a ccording to a service p olicy that minimizes the prima ry o utage proba bility (we ad dress the EDF scheduling, howev er , for simplicity). I n the above schem e, if f is c hosen to be 1 , th e primary network will act selﬁshly , whereas f = 0 implies a perfor mance of primary non-predictive network. In the following theor em we show that f or some range of f , the diversity gain expressions for the prim ary network satisfy th e same bo unds of th e selﬁsh scenario. Theor em 5: Under the dy namic capacity assignme nt policy in Def. 3 with f ∈ [0 . 5 , 1 ] , the di versity g ain of the primary network satisﬁes d p P ( γ p ) ≥ ( T + 1 )( γ p − 1 − lo g γ p ) , 0 < γ p < 1 , (21) e d s P ( e γ p ) = ( T + 1 )(1 − e γ p ) , 0 < e γ p < 1 . (22) Pr oof: Please refer to Appendix G. The above theorem thus shows that the pred ictiv e pr imary network satisﬁes the same di versity g ain boun ds of the selﬁsh behavior un der the p roposed dynamic capacity assignment policy as lon g as f ∈ [0 . 5 , 1] . Moreover , it g iv es a po tential for improvement in the outage per forman ce o f the second ary users by limiting the n umber of p rimary req uests served per slot. The outag e pro bability of the secondary network in th is case is g i ven by P P ( O s ) = P ( Q s ( n ) + C p ( n ) > C, Q s ( n ) > 0) = P Q s ( n ) + min n C, N p 0 ( n ) + f T X i =1 N p i ( n ) o > C, Q s ( n ) > 0 ! . (23) T o show that even the diversity gain of the second ary network is imp roved u nder such policy , we c onsider the ca se when f = 0 . 5 , and T = 1 for simplicity . In this case, the per-slot capacity of the p rimary network turns out to be C p ( n ) = min { C , N p 0 ( n ) + 0 . 5 Q p ( n ) } (24) 7 with N p 0 ( n + 1) =              Q p ( n ) , if N p 0 ( n ) = C, 0 . 5 Q p ( n ) + N p 0 ( n ) − C , if N p 0 ( n ) < C , N p 0 ( n ) +0 . 5 Q p ( n ) ≥ C, 0 . 5 Q p ( n ) , if N p 0 ( n ) +0 . 5 Q p ( n ) < C. (25) It is clea r fro m (25) that P ( N p 0 ( n + 1 ) = l | N p 0 ( n ) = i, · · · , N p 0 (1) = k ) = P ( N p 0 ( n + 1 ) = l | N p 0 ( n ) = i ) . That is, the discrete-tim e rand om pr ocess N p 0 ( n ) , n > 0 satisﬁes the Mar kov proper ty , and hen ce, it is a Markov chain . Moreover , it can be easily veriﬁed that N p 0 ( n ) , n > 0 is irreducib le and aperiodic as P ( Q p ( n ) = q ) > 0 for all q ≥ 0 . The drift o f the ch ain can thu s be ob tained as E [ N 0 ( n +1) − N 0 ( n ) | N 0 ( n ) = i ] ( ≤ − (1 − γ p ) C, if i ≥ C, ≤ γ p 2 C, if i < C. (26) Then, by Foster’ s th eorem [1 5], the Markov chain is positiv e recurren t, and hence has a stationary state distribution. Theor em 6: Suppose that the system is operating at the stationary d istribution of N p 0 ( n ) , n > 0 , the diversity gain of the secondar y n etwork, d s P ( γ p , γ s ) , unde r the dyn amic capacity allocatio n for the primary satisﬁes d s P ( γ p , γ s ) ≥ − γ s ( y 2 − 1) − 2 γ p ( y − 1) + 2 log( y ) , (27) where y = − γ p 2 γ s + q (4 γ s + γ p 2 ) 2 γ s and e d s P ( e γ p , e γ s ) ≥ ( (1 − e γ p ) , 1 + e γ s ≥ 2 e γ p , 1 2 (1 − e γ s ) , 1 + e γ s < 2 e γ p . (28) Pr oof: Please refer to Appendix H. The right hand side of ineq uality ( 27) will be shown in Section VII to be strictly larger than the righ t hand side of (15) for a range of γ s , which implies a strict improvement in the diversity gain of th e secondary network witho ut any loss in the di versity g ain o f the primary . Howe ver , the right hand side of inequality (28) shows that if 1 + e γ s < 2 e γ p , then the div ersity g ain of the secondary network is at least equal to its non-p redictive co unterp art. V . R O B U S T N E S S T O P R E D I C T I O N E R RO R S In the previous s ections we have assumed that the prediction mechanism is e rror free , that is, all predicted r equests are tr ue and will arrive in f uture after exactly the same look -ahead period of pred iction. Under th is assumptio n, we managed to treat the predicted ar riv al pr ocess with d eterministic look- ahead time as a delay ed version of the actual arriv al process. Howe ver , in practical scenarios, this is not necessarily the case. In this section we pr ovide a model f or the im perfect pred iction process and in vestigate its effect on th e p rediction div ersity gain with ﬁx ed look-ahead in terval T assuming a single c lass of QoS. Let Q ( n ) , n > 0 be the actual arrival p rocess that the n et- work should pr edict T slots a head. Th is pro cess, as intr oduced in Sectio n II, is Po isson with rate λ . Because the pr ediction mechanism emp loyed by th e network may cause errors, the predicted arriv al process differs from the a ctual arriv al process. The p rediction mecha nism is su pposed to cause two typ es of errors: 1) I t pr edicts false requ ests, those will not arrive actually in future, and serves them, resulting in a waste of resources. 2) I t fails to predict requests and, as a con sequence, the network encoun ters urgent arrivals (unpredicted requests that should b e served in the same slot of a rriv al). So, we mo del the pr edicted process as Q E ( n ) = Q ′ ( n ) + Q ′′ ( n ) (29) where Q ′ ( n ) , n > 0 is the arriv al pr ocess o f th e predicted requests. It represents the nu mber of arr iving req uests at the beginning of tim e slot n with d eadline n + T . The pro cess Q ′′ ( n ) , n > 0 represents the nu mber of unp redicted req uests that arri ve at the beginning of time slot n and must be served in the same slot because the network has f ailed to predict them. W e assum e further that Q ′ ( n ) and Q ′′ ( n ) are independ ently Poisson distributed with arrival ra tes λ ′ and λ ′′ , respec ti vely . Since Q ′′ ( n ) is a part of the requests Q ( n ) , th en 0 ≤ λ ′′ < λ (30) where the secon d ineq uality is strict because we assume that Q ′ ( n ) contain s truly predicted requests as well as mistakenly predicted requ ests, which also implies λ ′ + λ ′′ ≥ λ (31) Moreover , the network is stable as lo ng as λ ′ + λ ′′ < C. (32) For the linear scalin g regime, the ar riv al p rocesses Q ′ ( n ) and Q ′′ ( n ) , n > 0 ha ve ar riv al rates α ′ γ C and α ′′ γ C r espec- ti vely . Applying c ondition s (30)-(3 2) to α ′ γ C and α ′′ γ C we obtain α ′′ < 1 (33) and 1 ≤ α ′ + α ′′ < 1 γ (34) So, if th e pred iction mechan ism is perfe ct, then α ′ = 1 whereas α ′′ = 0 . The arriv al process Q E ( n ) , n > 0 , can b e considered as a pre dicted process with random look-ahea d in terval that takes on values 0 and T . Hence, using the event U R deﬁned in Lemma 2, we obtain the following lower bound on the prediction diversity gain, d E P ( γ ) , d E P ( γ ) ≥ min { ( T + 1) [( α ′ + α ′′ ) γ − 1 − log ( γ ( α ′ + α ′′ ))] , α ′′ γ − 1 − lo g ( α ′′ γ ) } (35 ) The best o perating point (p rediction window) that maximizes the rig ht hand side of (3 5) is whe n bo th terms in the min { . } are equal, wh ich implies ¯ T cr it = α ′′ γ − 1 − lo g ( α ′′ γ ) ( α ′ + α ′′ ) γ − 1 − log ( γ ( α ′ + α ′′ )) . (36) 8 Since α ′′ < 1 , then for ¯ T cr it derived in ( 36), we obtain d E P ( γ ) > d N ( γ ) . For th e po lynomia l scaling regime, the processes e Q ′ ( n ) a nd e Q ′′ ( n ) , n > 0 have a rriv al rates C e α ′ e γ and C e α ′′ e γ respectively . Applying condition s (30)-(3 2) to th e arriv al ra tes C e α ′ e γ and C e α ′′ e γ , we ob tain, e α ′′ < 1 , (37) C e α ′ e γ + C e α ′′ e γ ≥ C e γ , (38) and C e α ′ e γ + C e α ′′ e γ < C . (39) If the pr ediction mechanism is per fect, then e α ′ = 1 whe reas e α ′′ = − ∞ . W e also use events U R and L R from Lemm a 2 to deter- mine the p rediction d i versity gain with im perfect pre diction mechanism, e d E P ( e γ ) , as e d E P ( e γ ) = min { ( T + 1 ) [1 − ma x { e α ′ , e α ′′ } e γ ] , 1 − e α ′′ e γ } . (40 ) Nev ertheless, since at e d E P ( e γ ) is at C → ∞ , then from (38), (39), as C → ∞ , we ob tain, 1 ≤ e α ′ < 1 e γ . And from (37), max { e α ′ , e α ′′ } = e α ′ . Hence , e d E P ( e γ ) = min { ( T + 1 )(1 − e α ′ e γ ) , 1 − e α ′′ e γ } . (41 ) So, to obtain th e m aximum diversity gain, the best p redic- tion window ˜ T cr it should satisfy ˜ T cr it = ( e α ′ − e α ′′ ) e γ 1 − e α ′′ e γ , ( 42) and at th is poin t, since e α ′′ < 1 , we have e d E P ( e γ ) > e d N ( e γ ) . This section hence has shown theoretically that e ven u nder imperfect pred iction mechanisms, the pred iction window size is jud iciously chosen to strike the best balan ce between th e predicted trafﬁc and the urgent on e. V I . P RO A C T I V E S C H E D U L I N G I N M U LT I C A S T N E T W O R K S This section sheds light on the pr edictive m ulticast networks and inv estigates th e diversity gain s that can b e leveraged fro m efﬁcient schedu ling o f multicast trafﬁc. T ypically , multicast trafﬁc minimizes the usage of th e network resources becau se the same d ata is sent to a group of u sers co nsuming th e same amount of resources that serve on ly a single user whic h is taken to be u nity [16]. So , ev en in the n on-pr edictiv e case, the multicast trafﬁc is expected to resu lt in a n improved div ersity gain per forman ce over its unicast co unterpar t, discussed in th e previous sections. Furthermo re, when the m ulticast tr afﬁc is predictable, there is an ad ditional gain that can be o btained from the ability to alig n the trafﬁc in time. That is, th e ne twork can keep on receiving pred ictable requ ests that target th e same data over time then serve them altog ether as th e earliest dea dline approa ches. I n this case, th e network will end up servin g all the gather ed requests in a window o f tim e slots with th e same resources requir ed to serve one req uest, and hence will signiﬁcantly imp rove the diversity gain of the network. W e assume that there are L data sources av ailable (e.g. ﬁles, packets, movies, podcasts, etc.) for multicast tr ansmission. The number of multicast requ ests a rriving at th e beginning of time slot n > 0 is a rando m variable Q m ( n ) wh ich is assumed to be Poisson distrib uted with mea n λ m . Assuming that the data sources are demand ed independen tly across time and requ ests, the pr ocess Q m ( n ) , n > 0 can be decomp osed into Q m ( n ) = L X l =1 Q m, [ l ] ( n ) , f or all n > 0 , where Q m, [ l ] ( n ) denotes th e number of multicast requests for data source l ∈ { 1 , · · · , L } arr i ving in slot n , and is Poisson distributed with mean λ m, [ l ] , p [ l ] λ m , wh ere p , ( p [ l ] ) L l =1 is a valid pro bability distribution 1 capturing the potentially asymmetric multicast deman ds over the pool of L data sources. In this section we focus only on the an alysis of the lin- ear scaling regime wher e the potential improvement in the div ersity gain is tangible 2 . T he mean number of ar riving multicast reque sts scales with C as λ m = γ m C , γ m ∈ (0 , 1) . The n umber of data so urces L scales also linearly with C as L = θC , θ > 0 . The binary para meter X m, [ l ] ( n ) for each multicast data source l ∈ { 1 , · · · , L } is deﬁned as X m, [ l ] ( n ) ,  1 , if Q m, [ l ] ( n ) > 0 , 0 , if Q m, [ l ] ( n ) = 0 , l = 1 , · · · , L, (43) which gives th e indicator of at least one multicast request for data sou rce l arrives at slot n. And, und er the aforementione d Poisson assumptions, X m, [ l ] ( n ) is a simple Bernoulli random variable with parameter A m, [ l ] = 1 − e − p m, [ l ] λ m , l ∈ { 1 , · · · , L } . (44) W e d enote the total number o f distinct multicast data requests arriving in slot n as S m ( n ) , d eﬁned as S m ( n ) , L X l =1 X m, [ l ] ( n ) . (45) Deﬁnition 4: Let N m, [ l ] 0 ( n ) denote the in dicator that there is at least one aw aiting mu lticast r equest f or data sou rce l ∈ { 1 , · · · , L } that expir es in slot n. T hen, letting N m 0 ( n ) , P L l =1 N m, [ l ] 0 ( n ) , th e multicast ou tage ev ent is deﬁn ed as O m , { N m 0 ( n ) > C, n ≫ 1 } . The pure multicast ne twork will b e in vestigated in the following subsection wher e th e diversity gain o f its no n- predictive side will be shown to be larger th an its unicast coun- terpart, further more, the alignment proper ty of th e predictive multicast will b e proven to result in a sign iﬁcantly im proved div ersity gain, that scales super-linearly with the prediction interval T . Th en, the subsequen t subsection will add ress a composite n etwork co nsisting of unicast an d m ulticast trafﬁcs; the p otential div ersity g ain will b e investigated under different prediction scenarios. 1 p is a v alid distributio n if 0 ≤ p [ l ] ≤ 1 and P L l =1 p [ l ] = 1 . 2 The additional multic ast gain s do not appear in the pol ynomial scali ng regi me because the traf ﬁc to each data source vanishe s asymptotically , as C → ∞ , when the number of data sources L scales with C , implying that at m ost one reque st can ta rget a data sourc e at each slot , i.e., the multic ast traf ﬁc will approach the unica st as C → ∞ . 9 0 1 0 θ Div er s ity gain Multicast Unicast − log( 1 − e − γ ) γ − 1 − lo g γ Fig. 3: Di versity gain of the non-pre dictiv e multicast network monoto nically dec reases with θ . Howev er , it is lower b ound ed by the di versity gain of non -predictive unicast networks. A. S ymmetric Multicast Deman ds Suppose that the number of d ata sources scales w ith C a s L = θC , θ > 0 . Then, θ ≤ 1 implies zero o utage probability and inﬁnite di versity ga in r egardless o f the value of γ m . This is the ﬁrst gain improvement that can be leveraged f rom the nature of the multicast tr afﬁc. W e now conﬁn e the analysis to the case wh en θ > 1 . Assume that the multicast demand s are equally distributed on the a vailable data sources, i.e. p [ l ] = p = 1 θ C , A m, [ l ] = A m = 1 − e − γ m θ , ∀ l ∈ { 1 , · · · , L } . 1) No n-pr edictive Multicast Network: Under the above symmetric setup (and assumin g θ > 1 ), the ran dom variable S m ( n ) tu rns out to ha ve a bin omial distribution with parameter A m and th e outag e probab ility in this case, denoted by P N ( O m ) , is equ al to P ( S m ( n ) > C ) . I n other words, th e multicast outage occu rs in slot n if and only if the numbe r of distinct data sour ces requested at this slot is larger than the network capacity . Theor em 7: The di versity ga in of non -predictive m ulticast- ing, den oted by d N ( γ m , θ ) , is giv en by d N ( γ m , θ ) = ( θ − 1) lo g( θ − 1) − θ log θ + γ m  θ − 1 θ  − log  1 − e − γ m θ  , 0 < γ m < 1 , θ > 1 . (46) Pr oof: Please refer to Appendix I. Theorem 7 and Fig. 3, wh ich depicts the div ersity gains of non-p redictive multicast (4 6) and unicast (2 ) networks with γ m = γ , show that d N ( γ m , θ ) is monotonically d ecreasing in θ . As θ increases, the nu mber of data sources in th e network grows faster with C , and hence, fr om (46), lim θ →∞ d N ( γ m , θ ) = γ m − log γ m − 1 = d N ( γ m ) . (47) That is, multica st div ersity gain d N ( γ m , θ ) is strictly gre ater than its unicast co unterpar t d N ( γ m ) , and conv erges to it in the limit as θ → ∞ . In fact, a much stron ger result is th at, when γ m = γ , lim θ →∞ LA m = lim θ →∞ θC  1 − e − γ m θ  = γ m C, 0 < γ m < 1 , (48) we have also A m → 0 and L = θC → ∞ as θ → ∞ . Therefo re, S m ( n ) conver ges in distribution to Q ( n ) , and consequen tly , P N ( O m ) → P N ( O ) , θ → ∞ . In this subsection , we h av e high lighted the extra diversity gain a chieved through one of the multicast properties, that is all the requests arriving to the network at time slot n an d demand ing a certain data sou rce are all served with one u nit resources exactly as if o nly one request de mands th at data source. 2) Predictive Multica st Network: No w suppose that the symmetric multicast network has p redictable dema nds with a p rediction window of T > 0 slots. The tr afﬁc alignment in this case a ppears in the f ollowing sense, the resour ce serving a g roup of r equests arriving at slot n also serves all other requests in th e system (that ha ve arrived withing the previous T slots) requesting the same data source. So, the resource value is extendab le ac ross time. T he pred iction capability o f the network is thus equal to inﬁnity as long as θ ≤ ( T + 1) , which implies a multiplicative gain of T + 1 in the number of da ta sou rces that the network can suppor t with z ero outage probab ility , as comp ared to the non -pred icti ve case. Consider then the o ther rang e of θ , that is θ > ( T + 1) . The n etwork now is sub ject to o utage events a nd e fﬁcient scheduler has to b e employed . Because of the symme tric demand s, we focu s the analysis on th e EDF schedu ling. Let the optimal div ersity gain in this predictive scenario be deno ted by d P ( γ m , θ ) , in [1 7], we have sh own th at d P ( γ m , θ ) ≥ ( T + 1) d N ( γ m , θ ) which is consistent with the results of Subsection III -B as the p redictability multiplies th e diversity gain by a factor of at least T + 1 . Howe ver , we sho w now that the alignmen t p roperty can even imp rove the diversity gain and result in a super-linear scaling of d P ( γ m , θ ) with T . Theor em 8: The op timal diversity g ain of the p redictive multicast network with symmetric de mands, d P ( γ m , θ ) , satis- ﬁes d P ( γ m , θ ) ≥ ( T + 1) log  (1 − ξ m T )( T + 1) ξ m T ( θ − ( T + 1))  − θ log  1 − ξ m T + (1 − ξ m T )( T + 1) θ − ( T + 1 )  , , L sym . (49) where ξ m T = 1 − exp  − ( T + 1 ) γ m θ  . Pr oof: Please refer to Appendix J. The new lo wer bound L sym takes into account the align ment proper ty o f the predictable mu lticast trafﬁc, and thus shows signiﬁcant in crease in the diversity gain with T as compared to the o lder bou nd ( T + 1 ) d N ( γ m , θ ) in Fig. 4. 10 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Prediction interval (T) Lower bounds on the diversity gain (T+1)d N ( γ m , θ ) L sym Fig. 4: Super linear increase in the diversity gain of the multicast network with the p rediction in terval T beca use of the align ment proper ty . I n this ﬁgure γ m = 0 . 9 and θ = 15 . B. Multica st and Unica st T rafﬁc Generally , wireless networks suppo rt both ty pes of trafﬁ c: multicast and unicast. For instance, a smart phon e user my receive unicast data su ch as e-mail or electronic bank statement as well a s multicast data such as m ovies or podcasts. In this subsection we investigate the potential diversity gain of wireless networks enco mpassing both ty pes of trafﬁc un der different predictability conditio ns. The mu lticast trafﬁc model ad opted here is exactly as de- ﬁned in th e beginnin g o f this sectio n, with th e only d ifference is we assume that L = θ C , wh ere θ ∈ (0 , 1) . The multicast data sources are also equally d emanded , each w ith pro bability A m = 1 − exp  − γ m θ  . The unicast trafﬁc arrives at the beginn ing of each slot n accordin g to Q u ( n ) which is Poisson distributed with mean λ u = γ u C , γ u ∈ (0 , 1) . Each of the unicast requests consumes one unit o f th e sy stem capacity . The stability condition of th e non-p redictive n etwork necessitates that A m θ + γ u < 1 . (50) Deﬁnition 5: Letting N u 0 ( n ) denote th e number of unicast requests in th e system at the beginning of time slo t n , the combined o utage event o f th e wire less n etwork with unicast and multicast trafﬁc is d eﬁned as O A = { N m 0 ( n ) + N u 0 ( n ) > C, n ≫ 1 } . In [17], we have addr essed the case whe n o nly on multicast data sou rce exists in the ne twork an consumes µC , µ ∈ (0 , 1) of the available resour ces to supply data. T his data source shares the network with un icast tra fﬁc. W e have shown the impact of the m ulticast trafﬁc alignm ent on the d i versity gain where mor e gains can be leveraged b y ga thering more of the predictable multicast tra fﬁc and serving them altogeth er in a single slot. Alternatively , in this sub section we address the scenario o f mu ltiple data sources e ach con sumes on e unit of th e av ailable re sources. W e will inves tigate the div ersity gain of th e ne twork in the following fo ur scen arios of d emand predictability : 1) Bo th unicast a nd multicast trafﬁcs are no n-pred ictiv e. 2) U nicast is n on-pr edictive but multicast is predictiv e. 3) Bo th unicast a nd multicast trafﬁcs are pr edictive. 4) U nicast is p redictive but multicast is non-predictive. 1) Scen ario 1: Both Unicast and Multicast T rafﬁcs a r e Non- pr edictive: In this scena rio, all of the arriving requests are urgent and hence, N m 0 ( n ) = S m ( n ) and N u 0 ( n ) = Q u ( n ) . Theor em 9: Let the outage prob ability in Scenario 1 be denoted by P 1 ( O A ) a nd the associated diversity g ain b e denoted by d 1 ( γ u , γ m , θ ) , the n d 1 ( γ u , γ m , θ ) = log( y 1 ) + γ u (1 − y 1 ) − θ log  e − γ m θ + y 1  1 − e − γ m θ  , (51) where y 1 = 1 2 γ u  e γ m θ − 1  "  ( θ 2 − 2 θ + 1) e 2 γ m θ +  − 2 θ 2 + 2 θ ( γ u + 2) + 2 ( γ u − 2)  e γ m θ + θ 2 − 2 θ ( γ u + 1) + γ u 2 − 2 γ u + 1  1 2 + (1 − θ ) e γ m θ + θ − γ u − 1 # . Pr oof: Please refer to Appendix K. Theorem 9 th us tightly characterizes the di versity g ain of the n etwork in Scenario 1. The expression of d 1 ( γ u , γ m , θ ) , howe ver , is not insigh tful, so it will be compared graphica lly to the r esults of the o ther scenar ios. 2) Scen ario 2: Unicast is Non-pr edictive but Multica st is Pr edictive: In th is scenario, the network can pred ict the multicast req uests T slots ahead, wh ereas the u nicast trafﬁc is urgent. W e consider a scheduling policy π 2 to establish a lo wer bound o n th e op timal diversity gain, deno ted d 2 ( γ u , γ m , θ ) , o f this scenario . Deﬁnition 6 (Scheduling P o licy π 2 ): At each slot n , the scheduling policy π 2 serves as much as possible of the e xisting requests in the system in the following o rder: 1) M ulticast data sources demande d by urgent r equests, N u 0 ( n ) . 2) U nicast requests, Q u ( n ) . 3) T he r est of th e mu lticast d ata sou rces acc ording to EDF . The p olicy π 2 is a sligh tly mo diﬁed versio n of EDF with priority given to urgent multicast requests. Theor em 10: L et the o utage prob ability in Scenario 2 un der the scheduling policy π 2 be denoted P 2 ( O A ) and the optim al div ersity gain be de noted by d 2 ( γ u , γ m , θ ) , then d 2 ( γ u , γ m , θ ) ≤ min n d N ( γ u ) , ( T + 1) log y 2 − ( T + 1) γ u ( y 2 − 1) − θ log (1 − ξ m T + ξ m T y 2 ) o , , L 2 . (52) 11 1 2 3 4 5 6 7 8 9 10 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Look ahead interval (T) Bounds on d 2 ( γ u , γ m , θ ) L 2 (T) U 2 (T) Fig. 5: As T in creases, the system attains the same diversity gain o f the non -predictive unicast network. In this ﬁgure, θ = 0 . 7 , γ m = 0 . 9 an d γ u = 0 . 4 . and d 2 ( γ u , γ m , θ ) ≤ d N ( γ u ) , U 2 , (53) where d N ( γ u ) is as derived in (2) with γ = γ u , and y 2 = 1 2 ξ m T γ u ( T + 1 ) "  (1 − ξ m T ) 2 γ u 2 + 2 ξ m T γ u (1 − ξ m T ) + ξ m T 2  2 T 2 +  [2 ξ m T γ u (1 − ξ m T ) − 2 ξ m T 2 ] θ + 2 ξ m T 2 (1 − ξ m T ) 2 + 4 ξ m T γ u (1 − ξ m T ) + 2 ξ m T 2  T + [2 ξ m T θ (1 − ξ m T ) − 2 ξ m T 2 ] θ + γ u 2 (1 − ξ m T ) 2 + 2 ξ m T θ (1 − ξ m T ) + ξ m T 2 (1 + θ ) 2 ! 1 2 +  ( ξ m T − 1) γ u  T − ξ m T θ + γ u ( ξ m T − 1) + ξ m T # . Pr oof: Please refer to Appendix L. The up per and lo wer bo unds on d 2 ( γ u , γ m , θ ) e stablished in Theorem 10 match each oth er as T incr eases. In fact, the second term in min { ., . } of expression (52) is m onoton ically increasing in T , and hen ce ∃ t such that T ≥ t implies d 2 ( γ u , γ m , θ ) = d N ( γ u ) . This result means that, efﬁcient scheduling of the pr edictable mu lticast trafﬁc results in the same diversity gain tha t will be o btained if the system sees only the unicast tr afﬁc. This resu lt is clar iﬁed in Fig. 5 where the lower bo und L 2 increases in T until it becomes dominated by d N ( γ u ) at T = 2 , an d from this p oint on, L 2 and U 2 coincide and the diversity gain of th e network is only determined by the non-pre dictiv e un icast trafﬁc. 3) S cenario 3: Bo th Unica st and Multicast T raf ﬁcs a r e Pr edictive: In this scenario we a ssume th at b oth trafﬁcs are predictable with the same look-ahead interval of T slots. The scheduling policy we consider is EDF wh ere requests are served in the o rder of their arri val. Theor em 11: L et the outage probab ility o f the network in Scenario 2 under EDF scheduling policy b e denoted by P 3 ( O A ) and the optimal diversity gain of this scenario be 0 0.1 0.2 0.3 0.4 0.5 0 5 10 15 γ u Bounds on the diversity gain d 1 L 2 U 2 L 3 U 4 Fig. 6 : Boun ds on th e op timal diversity gain versus the unicast trafﬁc factor γ u . In this ﬁgure, γ m = 0 . 9 , θ = 0 . 7 and T = 4 for any predictive n etwork. denoted by d 3 ( γ u , γ m , θ ) , the n d 3 ( γ u , γ m , θ ) ≥ ( T + 1) log y 2 − ( T + 1) γ u ( y 2 − 1) − θ log(1 − ξ m T + ξ m T y 2 ) , L 3 . (54) Pr oof: Please refer to Appendix M. In Scenar io 3 one shou ld expect that the op timal diversity gain sh ould be the largest amon gst the othe r three scenarios. T o high light this intuition, an up per bou nd will be established on the diversity gain of Scenario 4 . 4) Scen ario 4: Unicast is Predictive but Multicast is Non- pr edictive: Assumin g that the unicast trafﬁc is predictab le with a look-ah ead window of T slots, and the multicast trafﬁ c is urgent. Theor em 12: L et the optimal diversity ga in of Scenar io 4 be denoted by d 4 ( γ u , γ m , θ ) and the minimu m possible outage probab ility be denoted by P ∗ 4 ( O A ) , then d 4 ( γ u , γ m , θ ) ≤ d 1 ( γ u , γ m , θ ) + T h 2 log y 4 − γ u ( y 4 − 1) − 2 θ log(1 − A m + A m y 4 ) i , U 4 , (55) where y 4 = 1 2 γ u A m "  (4 θ 2 − 4 θ ( γ u + 2) + (2 − γ u ) 2 ) A m 2 + γ u 2 + (4 γ u θ − 2 γ u 2 + 4 γ u ) A m  1 2 + ( − 2 θ + γ u + 2) A m − γ u # . Pr oof: Please refer to Appendix N. T o co llecti vely co mpare th e ob tained b ounds o n th e op timal div ersity gain of the discu ssed scenarios, Fig.6 plo ts th e different bound s o btained in the last four theorems versus γ u , where the range of γ u ensures that (50) is satisﬁed, an d h ence the non -predictive n etwork always sees a positive diversity gain. It is clear from the ﬁg ure tha t the to tally pred ictiv e 12 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Prediction look−ahead time (T) Bounds on the diversity gain d 1 (no prediction) L 2 U 2 L 3 U 4 Fig. 7: Bounds on the optimal di versity gain versus the p re- diction look -ahead time T . In this ﬁgure , γ u = 0 . 4 , γ m = 0 . 9 and θ = 0 . 7 . network (of Scenario 3 ) h as the highest p ossible d i versity g ain as the lo wer bound L 3 ev en exceed s th e u pper bound U 4 on the entire range of plotted γ u . Also, it shows that L 2 and U 2 are coincidin g a t d N ( γ u ) , and of co urse this is the be st div ersity gain that the netw ork can achieve with u npred ictable unicast trafﬁc. Also, Fig. 7 demonstrates the ef fect o f the prediction look- ahead period T o n the derived b ounds. It shows that both L 3 and U 4 are both incr easing in T , and th at as T in creases L 3 exceeds U 4 and L 2 matches U 2 . V I I . S I M U L A T I O N R E S U LT S The analy tical results obtained in this p aper are demon - strated throu gh nu merical simulation s in this section. T he o ut- age p robab ility is qu antiﬁed as the ratio of the n umber o f slots that su ffer e xpired r equests to th e to tal numb er of simulate d slots. Each simulation result is ob tained by a veraging a 1 00 sample paths each contains a 10 0 0 slots. A. Diversity Gain of Deterministic and Random T Scen arios Fig. 8 compares the ou tage probab ility of proa cti ve net- works with different look -ahead schemes to the n on-pr edictive network. Th e results ob tained fo r the lin ear scalin g regime a re plotted versus C in Fig. 8a and for the poly nomial scaling regime are p lotted versus C log C in Fig . 8 b. It is obviou s from bo th ﬁgures that being proactive signiﬁcan tly enhance s the ou tage p robability perf ormance at a given capacity , or it c onsiderab ly redu ces the required capa city to satisfy a giv en level of outage perfo rmance. This ascribes to the m ore ﬂexibility given to the predictive network th at allows it to schedule the arriving re quests over a lon ger time ho rizon compare d to the urgent deman d of th e no n-pre dictiv e network. The effect o f the distribution o f rando m lo ok-ahea d pred iction interval is demonstrated in Fig. 9 f or both the linear and polyno mial scaling regimes. The predictive network in each regime is assum ed to an tici- pate requ ests by a rando m period which varies between T ∗ and T ∗ where T ∗ = 0 an d T ∗ = 5 . W e con sider a gen eral binomial distribution with parame ter p , 0 ≤ p ≤ 1 to repr esent the 2 4 6 8 10 12 14 16 18 20 10 −4 10 −3 10 −2 10 −1 10 0 System capacity (C) Outage probability Deterministic T=4 Uniformly distributed T from 1 to10 Non−predictive network T=0 (a) Linear scaling regime: γ p = 0 . 8 . 0 5 10 15 20 25 10 −4 10 −3 10 −2 10 −1 10 0 C log C Outage probability Deterministic T=4 Uniformly distributed T from 1 to 10 Non−predictive network: T=0 (b) Polynomial scaling regime: e γ p = 0 . 8 . Fig. 8: Ou tage pr obability is signiﬁcan tly impr oved b y proac- ti ve networks. PDF of th e look-ah ead inter val. He nce, the pro bability that an arriving req uest at the beginning of time slot n has a deadline at slot n + T , T ∗ ≤ T ≤ T ∗ , is gi ven by P ( T q ( n ) = T ) = p T =  T ∗ T  p T (1 − p ) T ∗ − T . (56) W e consider different values of p in each r egime in a ddition to the no n-pred icti ve network scenario. The obtained results for the linear scaling regime are shown in Fig. 9a wh ere at p = 0 . 1 , d P R ( γ ) ≥ γ p 0 − log( γ p 0 ) − 1 , a nd d P R ( γ ) = ( T ∗ +1)( γ − 1 − log γ ) at p = 0 . 9 . The results of the polynomial scaling regime are shown in Fig. 9b. Although the di versity gain is tantamou nt to that of the n on-pr edictiv e network, it is clear fro m the ﬁgure th at the ou tage probab ility is signiﬁcantly improved. Her e, we want to po int out tha t diversity ga in represents the asym ptotic d ecay ra te of the outage probab ility with the system capacity (or C log C ) , b ut it does no t capture the relative difference b etween th e ou tage prob ability curves themselves. This is why the curves show different trends a t small values of C . After all, the ﬁgure shows that even if T ∗ = 0 the network ach iev es a signiﬁcantly better outage perfor mance wh en it f ollows a proactive resour ce allo cation technique . Finally , f rom Figs. 9a, 9b, we can rou ghly infer that as p in creases, it is more likely to ha ve a rriving re quests with larger prediction interval and hen ce the network ge ts more d egrees 13 2 4 6 8 10 12 14 16 18 20 10 −4 10 −3 10 −2 10 −1 10 0 System Capacity Outage probability T~ binomial, p=0.1 T~ binomial, p=0.9 Non−predictive network (a) Linear scaling regime: γ p = 0 . 6 . 0 5 10 15 20 25 10 −4 10 −3 10 −2 10 −1 10 0 C log C Outage probability Binomial with p=0.1 Binomial with p=0.9 Non−predictive (b) Polynomial scaling regi me: e γ p = 0 . 9 . Fig. 9: Ou tage probability is signiﬁcantly improved b y proac- ti ve networks. of freed om in sched uling such req uests in an e fﬁcient way that reduces the n umber of o utage ev ents. B. T wo -QoS Network Fig. 10 demo nstrates the result (19) for bo th the linear scaling an d poly nomial scaling regimes. T he simulation is run assum ing 10 3 time slo ts and averaged over 10 2 sample paths. For the selﬁsh p redictive primary network, we assume that T = 4 and the primar y req uests a re served according to EDF . T he results of the lin ear scaling regime are de picted in Fig. 10a, whereas that of the polynom ial scaling regime ar e depicted in Fig . 10 b. Figure 11 sh ows the potential improvement in the di versity gain of th e secondary n etwork b y efﬁcient u se of p rediction at the primary side only . Also, simulation results and analytical results are plotted toge ther on the sam e ﬁgure to show the relativ e differences. The perform ance of the dy namic-pr imary-cap acity scheme, has been evaluated num erically and plotted in Fig. 12 f or different values of f an d und er the two scaling regimes, namely , the lin ear scaling in Fig. 12a and the p olynom ial scaling in Fig. 1 2b. The p rediction interval is chosen to be T = 4 and at each slot n , the primary network is assumed to serve the C p ( n ) p rimary requests according to EDF policy . For the two schem es, the selﬁsh pr imary network, at f = 1 , results 2 4 6 8 10 12 14 16 18 20 10 −4 10 −3 10 −2 10 −1 10 0 System Capacity (C) Secondary outage probability Primary predictive network Primary non−predictive network (a) Linear scaling regime: γ p = 0 . 6 , γ s = 0 . 1 . 0 5 10 15 20 25 10 −3 10 −2 10 −1 10 0 C log C Secondary outage probability Primary predictive network Primary non−predictive network (b) Polynomial scaling regime: e γ p = 0 . 75 , e γ s = 0 . 05 . Fig. 10 : Selﬁsh p rimary pr edictive network canno t improve the outage probability of the secon dary . 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.05 0.1 0.15 0.2 0.25 γ s Diversity gain bounds Non−predictive upper bound Predictive lower bound Non−predictive lower bound Numerical non−predictive Numerical predictive Fig. 1 1: Improvement in the div ersity gain of the secondary network unde r p redictive prim ary with T = 1 and dy namic capacity assignment. Con sidered in th e ﬁgure is the line ar scaling r egime with γ p = 0 . 6 . The lower bou nd on d P ( γ p , γ s ) is shown in red, and obviou sly it strictly exceeds the upper bound on d N ( γ p , γ s ) determined in Theorem 4 plotted in blue. in the smallest primary outage probability , while a t f = 0 . 5 , the p rimary outag e prob ability is slightly increased beyo nd the selﬁsh case, but th e seco ndary outag e pro bability outper forms its co unterpa rt of the non -predictive primar y network o btained at f = 0 . It is clear from the ﬁgur es that at f = 0 . 5 the secondar y outage probability achieves the prim ary outage probab ility of the primary no n-pred ictiv e ne twork a t f = 0 in the linear scalin g regime, and is even better in the polyno mial 14 2 4 6 8 10 12 14 16 18 20 10 −4 10 −3 10 −2 10 −1 10 0 System Capacity (C) Outage Probability Primary, f=0.5 Primary, f=0 Secondary, f=0.5 Secondary, f=0 Primary, f=1 (a) Linear scaling regime: γ p = 0 . 6 , γ s = 0 . 1 . 0 5 10 15 20 25 10 −3 10 −2 10 −1 10 0 C log C Outage probability Primary, f=0.5 Primary, f=0 Secondary, f=0.5 Secondary, f=0 Primary, f=1 (b) Polynomial scaling regi me: e γ p = 0 . 75 , e γ s = 0 . 05 . Fig. 12: Prim ary predictive network can to lerate a trivial loss in outage prob ability at a sign iﬁcant improvement in the secondary ou tage probab ility . 2 4 6 8 10 12 14 16 18 20 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 System capacity (C) Outage probability Multicast T=2, θ =6 Multicast T=2, θ =10 Unicast T=2 Multicast T=0, θ =6 Multicast T=0, θ =10 Unicast T=0 Fig. 1 3: Ou tage p robab ility versus C . In this simula tion, γ u = 0 . 6 . scaling regime. The simulation is for 10 3 time slots averaged over 10 2 sample p aths. C. P r oactive Multicasting with S ymmetric Demands The outage prob ability o f th e predictive multicast an d unicast network s of the symm etric input trafﬁc is compa red numerically to that of non-p redictive network and is p lotted in Fig. 13. T he ﬁgure shows the signiﬁcant enhance ment to the outage prob ability of the multicast network when prediction is employed. Mo reover , we can see tha t the ou tage probab ility of the unicast p redictive n etwork is better than that of the multicast non-p redictive network . The impact of θ also appea rs clearly , as it c an easily be no ticed that as the θ d ecreases, the outage perfo rmance is enh anced ev en for the same value of T . When θ → ∞ the multicast curves co incide o n the unicast as shown in Section VI. V I I I . C O N C L U S I O N A N D D I S C U S S I O N W e have pr oposed a novel paradig m for wireless resou rce allocation which exploits the pr edictability of u ser b ehavior to m inimize the spectral resources (e.g., bandwidth ) needed to achie ve cer tain QoS metrics. Unlike the traditional reactive resource allocation approach in which the network can only start serv ing a p articular user request up on its initiatio n, our propo sed scheme anticipates future requests. This grants the network mo re ﬂexibility in sch eduling tho se potential req uests over an extended period of time. By adopting the outag e (blockin g) pro bability as our Qo S metric, we ha ve e stablished the po tential of the proposed framework to ac hieve signiﬁcan t spectral efﬁciency gains in sev eral interesting scenar ios. More sp eciﬁcally , we have introd uced the notion of pre- diction diversity gain and used it to quantify the g ain offered by the prop osed resource allocation algorithm under different assumption o n the perfo rmance of the trafﬁc prediction tech- nique. Moreover , we have sho wn that, in the cog nitiv e network scenario, pre diction at one side only do es not only enhance its diversity gain, but it also impr oves the diversity gain perfor mance of the other u ser class. On the multicasting fron t, we have shown that th e diversity gain of predictive mu lticast network scales super-linearly with the pr ediction win dow . Ou r theoretical claims were supported by num erical results that demonstra te the remark able gains that can be le veraged fro m the prop osed techniques. W e believe that this work has only scratched the surface of a very interesting research area which spans several disciplines and could potentially have a signiﬁcan t impact on the design of future wireless network s. In fact, on e can imm ediately identify a mu ltitude of interesting resear ch p roblems at the inter section of inf ormation theor y , machin e learning, b ehavioral science, and networking. For example, the analy sis have focu sed on the case of ﬁxed su pply an d variable d emand. Clearly , the same appro ach can be used to match deman d with sup ply under mor e general assumption s on the two pr ocesses. A P P E N D I X A P R O O F O F T H E O R E M 1 Let Λ Q ( r ) den ote the log moment gen erating fu nction [ 12] of a Po isson rando m v ariable Q ( n ) , n > 0 with mean λ , i.e., Λ Q ( r ) = λ ( e r − 1) , r ∈ R . For the linear scaling regime, let X i , i = 1 , 2 , · · · be a sequence of inde penden t and identically distributed (II D) random v ariables, each with a Poisson d istribution with mean γ , and deﬁne S C , C X i =1 X i . The outag e probability , P N ( O ) , can then be written a s P N ( O ) = P ( Q ( n ) > C ) = P  S C C > 1  (57) 15 Applying Cramer ’ s the orem [12] to (57), we get lim C →∞ 1 C log P  S C C > 1  = inf r > 0 { Λ X ( r ) − r } , (58) where Λ X ( r ) = γ ( e r − 1) . By the conv exity of the log moment generating functio n, we obtain inf r > 0 { Λ X ( r ) − r } = 1 − γ + lo g γ . Then, it follows tha t d N ( γ ) = − lim C →∞ log( P ( O )) C = γ − 1 − log γ , 0 < γ < 1 . (59) For the p olynom ial scaling regime, we determine th e diver - sity gain using tight lower and u pper b ound s. First, th e outage probab ility is giv en by P N ( e O ) = P ( e Q ( n ) > C ) (60) = ∞ X k = C +1 ( C e γ ) k k ! e − C e γ ≥ ( C e γ ) ( C +1) ( C + 1)! e − C e γ . Using Stirling’ s for mula to approximate th e factorial function , we have ( C + 1 )! . = p 2 π ( C + 1 )  C + 1 e  ( C +1) , where . = means that the lef t hand side ap proache s the right hand side in the limit as C → ∞ . Hence, lim C →∞ − log P N ( e O ) C lo g C ≤ lim C →∞ − 1 C lo g C log e − C e γ p 2 π ( C + 1)  C e γ e C + 1  C +1 ! Therefo re, e d N ( e γ ) ≤ 1 − e γ . (61) Second, ap plying tightest Chernoff bound [12] on (6 0), we have P ( e Q ( n ) > C ) ≤ inf r > 0 e Λ e Q ( r ) − r C (62) where Λ e Q ( r ) = C e γ ( e r − 1 ) . And since Λ e Q ( r ) − r is conv ex in r , b y simple d ifferentiation, we get P N ( e O ) ≤ e C − C e γ − (1 − e γ ) C log C . (63) Now , taking the logarithm of both sides of (63), dividing by − C log C , and letting C → ∞ , it follows that e d N ( e γ ) ≥ 1 − e γ . (64) By (61), ( 64), 1 − e γ ≤ e d N ( e γ ) ≤ 1 − e γ , then e d N ( e γ ) = 1 − e γ , 0 < e γ < 1 . (65) A P P E N D I X B P R O O F O F L E M M A 1 For U D , we need to show that the outage occ urring at time slot n implies P T i =0 Q ( n − T − i ) > C ( T + 1) . T o see this, assume th ere is an outag e at slot n . Since in our scenario EDF reduces to FCFS, then: 1) the outag e a t slot n occur s only on the arrivals of slot n − T and 2) du ring the in terval of slots n − T , n − T + 1 , · · · , n , the system does not serve any of the arriving re quests at slots beyon d n − T . Let N ( m ) , m > 0 denote the num ber of requests in the system a t the beginning of slot m , then having an outage at slot n implies N ( n − T ) > C ( T + 1) . And since at any slot m > 0 , there ar e n o requests in the system arriving at slots prior to m − T , it follows th at P T i =0 Q ( n − T − i ) ≥ N ( n − T ) > C ( T + 1) . For L D , we need to show that Q ( n − T ) > C ( T + 1 ) implies an outage at slot n . This is straightfo rward as the arriv als at slot n − T ca n no t remain in the system at any slot beyond n , further more, since Q ( n − T ) > C ( T + 1) , the cap acity of the system at the slot of arrival in addition to the next T slots is not sufﬁcient to ser ve the Q ( n − T ) requ ests, hence the system encoun ters an outage at slot n . A P P E N D I X C P R O O F O F T H E O R E M 2 For th e linear scaling regime, we have from Lem ma 1, P ( O ) P D ≤ P ( U D ) , hen ce, P P D ( O ) ≤ P T X i =0 Q ( n − T − i ) > C ( T + 1) ! . (66) Using the same deﬁnition of the seq uence of IID random variables X i , i > 0 as in th e pro of of T heorem 1, we have S C ( T +1) = P C ( T +1) i =1 X i and P T X i =0 Q ( n − T − i ) > C ( T + 1) ! = P S C ( T +1) C ( T + 1) > 1 ! . (67) Using Cramer’ s theorem , lim C →∞ − log P ( U D ) C ( T + 1) = γ − 1 − lo g γ . (68) Since P ∗ P D ( O ) ≤ P P D ( O ) ≤ P ( U D ) , we ha ve lim C →∞ − log P ∗ P D ( O ) C ≥ lim C →∞ − log P ( U D ) C = ( T + 1)( γ − 1 − lo g γ ) , (69) for which (4) follows . For the polyn omial scaling regime , ﬁrst we u se the up per bound P P D ( e O ) ≤ P ( f U D ) to establish a lower b ound on the optimal d iv ersity gain e d P D ( e γ ) as follows. Using Chernoff bound on P ( f U D ) , P P D ( e O ) ≤ P T X i =0 e Q ( n − T − i ) > C ( T + 1) ! ≤ inf r > 0 e ( T +1)Λ e Q ( r ) − C ( T +1) r , (70) 16 where Λ e Q ( r ) = C e γ ( e r − 1) . T hen, using differentiation , P P D ( e O ) ≤ e ( T +1)( C − C e γ ) − ( T +1)(1 − e γ ) C log C . (71) And since P ∗ P D ( e O ) ≤ P P D ( e O ) , we get e d P D ( e γ ) ≥ (1 + T )(1 − e γ ) . (72) Second, we use the lower boun d P P D ( e O ) ≥ P ( f L D ) to establish an u pper boun d on e d P D ( e γ ) . P ( f L D ) = P ( e Q ( n − T ) > C ( T + 1)) = ∞ X k = C ( T +1)+1 ( C e γ ) k k ! e − C e γ ≥ ( C e γ ) ( C ( T +1)+1) ( C ( T + 1) + 1)! e − C e γ . = e − C e γ p 2 π ( C ( T + 1) + 1)  C e γ e C ( T + 1 ) + 1  C ( T +1)+1 And since P ∗ P D ( e O ) ≤ P P D ( e O ) , we obtain e d P D ( e γ ) ≤ (1 + T )(1 − e γ ) . (73) By (72), ( 73), it fo llows th at e d P D ( e γ ) = (1 + T )(1 − e γ ) , 0 < e γ < 1 . A P P E N D I X D P R O O F O F L E M M A 2 First, we show tha t U R is a n ecessary con dition for the outage event, that is, if an ou tage occurs at slot n , th en U R = I ∪ J o ccurs. Sup pose there is an o utage at slot n . This outage o ccurs o n the arriv als, Q k ( n − k ) , k = T ∗ , · · · , T ∗ , hence, P T ∗ i =0 N i ( n − T ∗ ) > C ( T ∗ + 1) , i.e., in the interval n − T ∗ , · · · , n th e system is serving r equests with dead lines not exceeding n . Event I rep resents the case when at slot n − T ∗ , the number of r equests in the system in addition to the req uests that will arrive with deadline s not lar ger than n is larger than C ( T ∗ + 1) , i.e., larger than the maximum number of requests that the system can serve in the subsequ ent T ∗ +1 sl ots (Fig. 14a shows the reque sts co nsidered in e vent I as blue circles for T ∗ = 1 , T ∗ = 3 .). Howev er , e vent I alot is not a necessary conditio n for an outage as, f or instance, we may hav e Q T ∗ ( n − T ∗ ) > C ( T ∗ + 1) but P T ∗ j =0 P T ∗ i = T ∗ Q i ( n − i − j ) < C ( T ∗ + 1) . Now , suppose that I d id not occur becau se of the outage at slot n , then th ere exists at least one slot n − l , T ∗ < l ≤ T ∗ such that P l i =0 N i ( n − l ) ≤ C (Otherwise, the system will be serving requ ests w ith deadline of at mo st n in slots n − T ∗ , · · · , n − T ∗ which imp lies n ∈ I .). In o ther words, at slot l , the system will be empty of all r equests that hav e deadlines no t beyond slot n . L et l ∗ = min ( l : l X i =0 N i ( n − l ) ≤ C, T ∗ < l ≤ T ∗ ) , then P l ∗ − 1 j = T ∗ P j i = T ∗ Q i ( n − j ) > C l ∗ , he nce J occurs. Tha t is, all of the arriving r equests in slots n − l ∗ + 1 , · · · , n − T ∗ with n 1 n 3 n ) ( 1 m Q ) ( 2 m Q ) ( 3 m Q (a) Blue ci rcles represent an upper bound on the requests that must be served by slot n in the interva l of slots n − T ∗ , · · · , n . Red circl es represent requests that are no longer in the system at slot n − T ∗ whereas white circles represent requests wit h deadli ne larger than n . n 1 n 3 n ) ( 1 m Q ) ( 2 m Q ) ( 3 m Q (b) Here, e v ent I is not satisﬁe d. At slot n − 3 the system has managed to serve all req uests with deadli nes not exce eding n . Howe v er , l ∗ = 3 , meaning that all of the next arri v als with deadlines not e xceedi ng n will co nsume the whol e system capac ity till slot n inclusi ve. Fig. 14: An outag e o ccurs at slot n whe re T ∗ = 1 , T ∗ = 3 . At the beginning of any time slot, arriving re quests with the same deadline a re represen ted by a circle. deadlines not beyond n are more than C l ∗ (Fig. 14b shows the case wh en event I is not occu rring while l ∗ = 3 .). Second, we sho w that L R is a sufﬁcient conditio n on the outage ev ent. The proo f is straigh tforward as for e very k , T ∗ ≤ k ≤ T ∗ , the ev ent that P k i = T ∗ Q i ( n − i ) > C ( k + 1) means the number of requests th at must be served in the interval n − k , · · · , n is larger than C ( k + 1) which is sufﬁcient to cause an outage at slot n . Then, taking the un ion over all k ∈ { T ∗ , · · · , T ∗ } is a lso a sufﬁcient condition for an outage at slot n . A P P E N D I X E P R O O F O F T H E O R E M 3 P P R ( O ) ≤ P ( U R ) ≤ P   T ∗ X j =0 T ∗ X i = T ∗ Q i ( n − j − i ) > C ( T ∗ + 1)   + T ∗ − 1 X k = T ∗ P   k X j = T ∗ j X i = T ∗ Q i ( n − j ) > C ( k + 1)   . ≤ inf r I > 0 e Λ Q I ( r I ) − r I C ( T ∗ +1) + T ∗ − 1 X k = T ∗ inf r k > 0 e Λ Q k ( r k ) − r k C ( k +1) where Λ Q I ( r I ) = λ ( T ∗ + 1 )( e r I − 1 ) and Λ Q k ( r k ) = λ P k − T ∗ i =0 F k − i , T ∗ ≤ k ≤ T ∗ − 1 . For the linear scalin g regime, P P R ( O ) ≤ e (1 − γ ) C ( T ∗ +1)+ C ( T ∗ +1) log γ + T ∗ − 1 X k = T ∗ e C ( k +1)  1 − γ P k − T ∗ i =0 F k − i k +1 +log γ P k − T ∗ i =0 F k − i k +1  . 17 Let v ( C ) , max T ∗ ≤ k ≤ T ∗ − 1 ( C ( k + 1 ) " 1 − γ P k − T ∗ i =0 F k − i k + 1 + log γ P k − T ∗ i =0 F k − i k + 1 #) and m ( C ) = max { C ( T ∗ + 1)(1 − γ + log γ ) , v ( C ) } , then d P R ( γ ) ≥ lim C →∞ − 1 C log e m ( C ) = lim C →∞ − m ( C ) C = min { ( T ∗ + 1)( γ − 1 − lo g γ ) , v ∗ } (74) which proves (9). For the poly nomial scaling regime, P P R ( e O ) ≤ e ( T ∗ +1)( C − C e γ − C log C 1 − e γ ) + T ∗ − 1 X k = T ∗ e C ( k +1)  1 − P k − T ∗ i =0 F k − i C 1 − e γ ( k +1) +log P k − T ∗ i =0 F k − i C 1 − e γ ( k +1)  . Let e v ( C ) = max T ∗ ≤ k ≤ T ∗ − 1 ( C ( k + 1 ) " 1 − P k − T ∗ i =0 F k − i C 1 − e γ ( k + 1) + log P k − T ∗ i =0 F k − i C 1 − e γ ( k + 1) #) (75) and e m ( C ) , lim C →∞ − e m ( C ) C lo g C , for large values of C , the terms in the max { . } of (75) are decreasing in k , h ence e d P R ( e γ ) ≥ ( T ∗ + 1)(1 − e γ ) . (76) Then, we u se the event L R with the p olynom ial scalin g as follows. P P R ( e O ) ≥ P ( f L R ) ≥ max T ∗ ≤ k ≤ T ∗ ( P k X i = T ∗ e Q i ( n − i ) > C ( k + 1) !) ≥ max T ∗ ≤ k ≤ T ∗  F k C e γ  C ( k +1)+1 C ( k + 1 ) + 1! e − F k C e γ . = max T ∗ ≤ k ≤ T ∗  F k C e γ e C ( k + 1 ) + 1  C ( k +1)+1 × e − F k C e γ p 2 π ( C ( k + 1) + 1) =  F k C e γ e C ( T ∗ + 1) + 1  C ( T ∗ +1)+1 × e − p T ∗ C e γ p 2 π ( C ( T ∗ + 1) + 1 ) . Hence, e d P R ( e γ ) ≤ ( T ∗ + 1)(1 − e γ ) . (77) From (76) an d (77), result (10) follows. A P P E N D I X F P R O O F O F T H E O R E M 4 Let the outage probability o f the secondary user while the primary network is n on-pr edictive b e den oted by P N ( O s ) , then P N ( O s ) = P ( Q p ( n ) + Q s ( n ) > C, Q s ( n ) > 0) . (78) Since Q p ( n ) + Q s ( n ) and Q s ( n ) are two depende nt rando m variables, we use upp er and lower bou nds on P s N ( O ) to characterize d s N ( γ p , γ s ) as follows. P N ( O s ) = P ( Q p ( n ) + Q s ( n ) > C | Q s ( n ) > 0) P ( Q s ( n ) > 0) ( a ) ≥ P ( Q p ( n ) > C | Q s ( n ) > 0) P ( Q s ( n ) > 0) ( b ) = P ( Q p ( n ) > C ) P ( Q s ( n ) > 0) , (79) where ( a) follows f rom the fact that Q s ( n ) ≥ 0 and (b) follows as Q p ( n ) and Q s ( n ) are in depend ent. Moreover , since P ( A , B ) ≤ P ( A ) , th en, from (7 8), we can wr ite P N ( O s ) ≤ P ( Q p ( n ) + Q s ( n ) > C ) . (80) For the linear scaling regime, we have ¯ λ p = γ p C an d ¯ λ s = γ s C . From (11), (12) we obtain 0 < γ s < γ p < 1 and γ s + γ p < 1 . From (79), P N ( O s ) ≥ P ( Q p ( n ) > C ) P ( Q s ( n ) > 0) = P ( Q p ( n ) > C )  1 − e − γ s C  . Hence d s N ( γ p , γ s ) ≤ lim C →∞ − log P ( Q p ( n ) > 0) C − log  1 − e − γ s C  C ( c ) = γ p − 1 − lo g( γ p ) , (81) where (c) fo llows by Cramer’ s th eorem. This pr oves (15). Since Q p ( n ) , Q s ( n ) are indep endent Poisson ran dom vari- ables, th en Q p ( n ) + Q s ( n ) is a Poisson p rocess with rate ( γ p + γ s ) C . Applying Cramer’ s theorem to (80), we obtain d s N ( γ p , γ s ) ≥ ( γ p + γ s ) − 1 − log ( γ p + γ s ) which proves (16). For the polynomia l scaling r egime, ˜ λ p = C e γ p , ˜ λ s = C e γ s . From (11), ( 12), we g et 0 < e γ s < e γ p < 1 . From (80), P N ( e O s ) ≥ P ( e Q p ( n ) > C ) P ( e Q s ( n ) > 0) = P ( e Q p ( n ) > C )(1 − e − C e γ s ) ≥ C e γ p ( C +1) ( C + 1)! e − C e γ p  1 − e − C e γ s  . =  C e γ p e C + 1  C +1 e − C e γ p p (2 π ( C + 1 ))  1 − e − C e γ s  . 18 Hence, e d s N ( e γ p , e γ s ) ≤ lim C →∞ − log P ( e Q p ( n ) > C ) C lo g C −  1 − e − C e γ s  C lo g C ≤ 1 − e γ p . (82) From (80), we obtain, using tighte st Chernoff bound , P N ( e O s ) ≤ inf r > 0 e Λ e Q s + e Q p ( r ) − r C , ( 83) where Λ e Q p + e Q r ( r ) = ( C e γ p + C e γ s )( e r − 1) . Then it follows that, e d s N ( e γ p , e γ s ) ≥ 1 − max { e γ p , e γ s } = 1 − e γ p . (84) From (82) an d (84), the result (17) fo llows. A P P E N D I X G P R O O F O F T H E O R E M 5 Let the o utage probab ility of the primar y network under the dynamic sch eduling p olicy be denoted by P P ( O p ) . T o upp er bound this outage probability , it sufﬁces to show th at f ∈ [0 . 5 , 1 ] im plies P P ( O p ) ≤ P ( U D ) , where U D is as deﬁn ed in Lemma 1. So, suppose tha t there is an ou tage a t slot n , h ence, accordin g to the dynam ic policy , C p ( n ) = C as N p 0 ( n ) > C . Moreover , that outage is occ urring on Q p ( n − T ) . Now , at time slot n − 1 , assume towards contradiction that C p ( n − 1 ) < C , th en f N 1 ( n − 1) < C . Th is must lead to N 0 ( n ) ≤ (1 − f ) N 1 ( n − 1) < C as 1 − f ≤ f , f ∈ [0 . 5 , 1] , which is a contradiction . T herefor e, C p ( n − 1 ) = C . Since th e EDF n ature of the d ynamic p olicy implies th at the network resources are only dedicated to serve primary requests that arrived pr ior to slot n − T + 1 , then C p ( n − 1) an d C p ( n ) represent the served requests tha t arr iv ed at slots n − T − 1 and n − T . But, C p ( k ) ≤ min { C , f ( C p ( n − 1) + C p ( n )) } , k = n − T , · · · , n . Hen ce, C p ( k ) = C for all k = n − T , · · · , n as f ∈ [0 . 5 , 1 ] . Therefo re, an outage at slot n implies P T i =0 Q p ( n − i − T ) > C ( T + 1) , and c onsequen tly , we obtain the lower bound s on d p P ( γ p ) and e d s P ( e γ p ) in th e same ma nner as in Theorem 2. Also, it is straigh tforward to see that the event L D of Lemma 1 satisﬁes P ( L D ) ≤ P P ( O p ) . So the d i versity g ain of the polynomial scaling regime is fully determined. A P P E N D I X H P R O O F O F T H E O R E M 6 W e will show the result f or the linear scalin g regime wh ile its polynom ial scaling regime counterpart is obta ined through the same appr oach b y tak ing into accou nt the difference in the div ersity gain deﬁnitio ns. From (23) an d (24), we can upp er bound P P ( O s ) by P P ( O s ) ≤ P ( Q s ( n ) + N p 0 ( n ) + 0 . 5 Q p ( n ) > C, C p ( n − 1) < C ) + P ( Q s ( n ) + N p 0 ( n ) + 0 . 5 Q p ( n ) > C, C p ( n − 1) = C ) . But C p ( n − 1) < C implies N p 0 ( n ) = 0 . 5 Q p ( n − 1) and h ence the joint event Q s ( n ) + N p 0 ( n ) + 0 . 5 Q p ( n ) > C, C p ( n − 1) < C implies Q s ( n ) + 0 . 5 Q p ( n − 1) + 0 . 5 Q p ( n ) > C . Therefore, P P ( O s ) ≤ P ( Q s ( n ) + 0 . 5 Q p ( n − 1 ) + 0 . 5 Q p ( n ) > C ) + P ( C p ( n − 1 ) = C ) . (85) Now , we show that the decay rate of th e second term on the right hand side of (8 5) with C is larger tha n the ﬁrst. W e start with the seco nd term P ( C p ( n − 1) = C ) which can b e upper bound ed by P ( C p ( n − 1) = C ) ≤ P ( N p 0 ( η ) + 0 . 5 Q p ( η ) > C, C p ( η − 1) < C for some η ≤ n − 1) + P ( N p 0 ( m ) + 0 . 5 Q p ( m ) > C, C p ( m − 1) = C for all m ≤ n − 1) ≤ P (0 . 5 Q p ( η − 1) + 0 . 5 Q p ( η ) > C ) + P ( C p ( m ) = C, for all m ≤ n − 1) . (86) Fix 0 ≤ M ≤ n − 1 . The last term on the right hand side of (86) satisﬁes P ( C p ( m ) = C, for all m ≤ n − 1) ≤ P ( C p (1) = · · · = C p ( M ) = C ) , where P ( C p (1) = · · · = C p ( M ) = C ) ≤ P ( C p (1) = · · · = C p ( M ) = C, No outages in 1 , · · · , M ) + M X l =1 P ( C p (1) = · · · = C p ( M ) = C , l outages in 1 , · · · , M ) implying P ( C p (1) = · · · = C p ( M ) = C ) ≤ P ( C p (1) = · · · = C p ( M ) = C, No outages in 1 , · · · , M ) + (2 M − 1) P p P ( O p ) . Since M is constant, the term (2 M − 1) P p P ( O p ) d ecays with the system capacity as d p P ( γ p ) . The joint event C p (1) = · · · = C p ( M ) = C and no ou tage in 1 , · · · , M im plies N p 0 ( M ) = N p 0 (1) − ( M − 1) C + M − 1 X i =1 Q p ( i ) ≤ − ( M − 1) C + M − 1 X i =0 Q p ( i ) . and hen ce, P ( C p (1) = · · · = C p ( M ) = C , N o outage in 1 , · · · , M ) ≤ P − ( M − 1) C + M − 1 X i =0 Q p ( i ) + 0 . 5 Q p ( M ) > C ! ≤ P M X i =0 Q p ( i ) > M C ! ≤ inf r > 0 n e Λ( r ) − r M C o , where, for the linear scaling regime, Λ( r ) = ( M + 1) γ p C ( e r − 1) . 19 Hence, lim C →∞ − 1 C log P  C p (1) = · · · = C p ( M ) = C, No ou tage in 1 , · · · , M  ≥ ( M + 1) γ p − M + M log  M ( M + 1) γ p  (87) with th e right hand side of ( 87) mon otonically increasing in M a s lon g as M M +1 > γ p . Then , M can be cho sen sufﬁciently large 3 so that lim C →∞ − 1 C log P  C p ( m ) = C for all m ≤ n − 1  ≥ d p P ( γ p ) = 2( γ p − 1 − log γ p ) . Also, the ﬁrst term o n the right hand side of (86) ca n be written as P (0 . 5 Q p ( η − 1) + 0 . 5 Q p ( η ) > C ) = P ( Q p ( η − 1) + Q p ( η ) > 2 C ) ≥ P p P ( O p ) , where T = 1 . Hence , lim C →∞ − log P  C p ( n − 1) = C  C ≥ d p P ( γ p ) = 2( γ p − 1 − lo g γ p ) . (88) Now , com paring the two terms P ( Q s ( n ) + 0 . 5 Q p ( n − 1) + 0 . 5 Q p ( n ) > C ) in (85) and P (0 . 5 Q p ( η − 1 ) + 0 . 5 Q p ( η ) > C ) in (86), we have by the station arity of Q p ( n ) , n > 0 and the non-n egati vity o f Q s ( n ) , n > 0 , P ( Q s ( n ) + 0 . 5 Q p ( n − 1) + 0 . 5 Q p ( n ) > C ) ≥ P (0 . 5 Q p ( η − 1) + 0 . 5 Q p ( η ) > C ) . This implies that th e asymptotic decay rate of log P P ( O s ) with C is lo wer bou nded by the decay rate of P ( Q s ( n ) + 0 . 5 Q p ( n − 1 ) + 0 . 5 Q p ( n ) > C ) with C . Now , we can use Chernoff boun d to lower bound d s P ( γ p , γ s ) as follows P ( Q s ( n )+0 . 5 Q p ( n − 1)+0 . 5 Q p ( n ) > C ) ≤ inf r > 0 n e Λ tot ( r ) − r C o , (89) where Λ tot ( r ) = γ s C ( e r − 1) + 2 γ p C ( e 0 . 5 r − 1) . By differentiation, the optimal value of r , denoted r ∗ , satisﬁes γ s e r ∗ + γ p 0 . 5 r ∗ − 1 = 0 . Let y , e 0 . 5 r ∗ , we obtain y = − γ p 2 γ s + p 4 γ s + γ p 2 2 γ s and r ∗ = 2 log y . Substituting with r ∗ in (8 9), tak ing − log of both sides, dividing by C and sen ding C → ∞ , the div ersity g ain of the secon dary network in the linear scaling regime satisﬁes d s P ( γ p , γ s ) ≥ − γ s ( y 2 − 1) − 2 γ p ( y − 1) + 2 log( y ) . 3 The s ystem is assumed to operate in the steady state, i.e., n ≫ 1 . A P P E N D I X I P R O O F O F T H E O R E M 7 P N ( O m ) = P ( S m ( n ) > C ) = P  S m ( n ) θC > 1 θ  = P P θ C l =1 X [ l ] θC > 1 θ ! . Applying Cramer’ s Theorem [12], d N ( γ m , θ ) = − inf r > 0 { θ Λ X [ l ] ( r ) − θr } , (90) but Λ X [ l ] ( r ) = log(1 − A m + A m e r ) = lo g  e − γ m θ +  1 − e − γ m θ  e r  , Then, r ∗ = lo g     e − γ m θ ( θ − 1)  1 − e − γ m θ      . The cond itions 0 < γ < 1 , θ > 1 ensur e that r ∗ > 0 . Substituting with r ∗ in (9 0), we ob tain (46). A P P E N D I X J P R O O F O F T H E O R E M 8 Under EDF scheduling, an outage occurs in slot n ≫ 1 if and only if N m ( n − T ) > C ( T + 1) , where N m ( n − T ) is the number of distinct multicast data sources targeted by existing requests in the system at slot n − T . Hence P ∗ P ( O m ) ≤ P P ( O m ) = P ( N m ( n − T ) > C ( T + 1)) . Let Z m T ( n − T ) be the n umber of distinct data sources that were requested in the window o f slots [ n − 2 T , · · · , n − T ] , then accor ding to EDF , N m ( n − T ) ≤ Z m T ( n − T ) . (91) Therefo re P ( N m ( n − T ) ≤ C ( T + 1)) ≤ P ( Z m T ( n − T ) > C ( T + 1)) . Since each data sou rce is requ ested independ ently of the others at each slot and from slot to another, then the probability that a data so urce is requested at least once in a window of T + 1 slots, den oted ξ m T , is eq ual to ξ m T = 1 − (1 − A m ) T +1 = 1 − exp  − ( T + 1 ) γ m θ  , hence P ( Z m T ( n − T ) = k ) = (  θC k  ξ m T k (1 − ξ m T ) θC − k , k = 0 , · · · , θ C 0 , otherwise . 20 Now we can upper-boun d P ∗ P ( O m ) using Che rnoff b ound as P ∗ P ( O m ) ≤ P P ( O m ) ≤ P ( Z m T ( n − T ) > C ( T + 1)) ≤ inf r > 0 { e Λ Z ( r ) − r C ( T +1) } , where Λ Z ( r ) = θC log (1 − ξ m T + ξ m T e r ) . Solving for r ∗ > 0 that min imizes e Λ Z ( r ) − r C ( T +1) , we ob tain r ∗ = log  (1 − ξ m T )( T + 1) ξ m T ( θ − ( T + 1))  . Now , tak ing − log P ∗ P ( γ m , θ ) , dividing b y C and taking the limit as C → ∞ , we obtain (49). A P P E N D I X K P R O O F O F T H E O R E M 9 W e have by the deﬁnition of O A in Scena rio 1 that P ( O A ) = P ( S m ( n ) + Q u ( n ) > C ) . By Cramer’ s theorem, we h av e d 1 ( γ u , γ m , θ ) = inf r > 0 { r − Λ m + u ( r ) } , ( 92) where Λ m + u ( r ) = γ u ( e r − 1) + θ log  e − γ m θ + e r − e r − γ m θ  . Differentiating r − Λ m + u ( r ) with respect to r an d equating with 0 , we obtain γ u  e γ m θ − 1  e 2 r ∗ +  ( θ − 1) e γ m θ − θ + γ u + 1  e r ∗ − 1 = 0 . (93) Set y 1 = e r ∗ , th en (93) is a quadratic eq uation in y 1 , that can be solve ana lytically f or two po ssible roots. Choosing the r oot y 1 > 1 for r ∗ > 0 , we get y 1 = 1 2 γ u  e γ m θ − 1  "  ( θ 2 − 2 θ + 1) e 2 γ m θ +  − 2 θ 2 + 2 θ ( γ u + 2) + 2( γ u − 2)  e γ m θ + θ 2 − 2 θ ( γ u + 1) + γ u 2 − 2 γ u + 1  1 2 + (1 − θ ) e γ m θ + θ − γ u − 1 # . Substitution with y 1 = e r ∗ into ( 92), we obtain ( 51). A P P E N D I X L P R O O F O F T H E O R E M 1 0 Under the policy π 2 , suppo se th at an outage event h as occurre d in slot n ≫ 1 , th en N m 0 ( n ) + Q u ( n ) > C , wh ich can be decomp osed to either of the fo llowing to events: 1) Q u ( n ) > C o r 2) Q u ( n ) ≤ C but N m 0 ( n ) > 0 so th at N m 0 ( n ) + Q u ( n ) > C . Now , focus on the second e vent, speciﬁcally , N m 0 ( n ) > 0 . T o each data source of the N m 0 ( n ) , at least one c orrespon ding request has already arriv ed at slot n − T . Since N m 0 ( n ) > 0 an d N m 0 ( n ) + Q u ( n ) > C , then the system is o perating at full capacity in the slots [ n − T , · · · , n ] . That is, N m ( n − T ) + T X i =0 Q u ( n − i ) > C ( T + 1) , where N m ( n − T ) is the nu mber of d istinct m ulticast data sources deman ded by at least one request existing in the system at slot n − T . From (91), N m ( n − T ) ≤ Z m T ( n − T ) , where Z m T ( n − T ) is as deﬁn ed in Ap pendix J, then we can now write P 2 ( O A ) ≤ P ( Q u ( n ) > C ) + P T X i =0 Q u ( n − i ) + Z m T ( n − T ) > C ( T + 1) , Q u ( n ) < C ! ≤ P ( Q u ( n ) > C ) + P T X i =0 Q u ( n − i ) + Z m T ( n − T ) > C ( T + 1) ! . W e ha ve from Th eorem 1 that lim C →∞ − log P ( Q u ( n ) > C ) C = γ u − 1 − lo g γ u . (94) Also, Cramer’ s theo rem can be used in the same way of Append ix K to show that lim C →∞ − 1 C log P T X i =0 Q u ( n − i ) + Z m T ( n − T ) > C ( T + 1 ) ! = ( T + 1) log y 2 − ( T + 1) γ u ( y 2 − 1) − θ log (1 − ξ m T + ξ m T y 2 ) , (95) where y 2 = 1 2 ξ m T γ u ( T + 1 ) "  (1 − ξ m T ) 2 γ u 2 + 2 ξ m T γ u (1 − ξ m T ) + ξ m T 2  2 T 2 +  [2 ξ m T γ u (1 − ξ m T ) − 2 ξ m T 2 ] θ + 2 ξ m T 2 (1 − ξ m T ) 2 + 4 ξ m T γ u (1 − ξ m T ) + 2 ξ m T 2  T + [2 ξ m T θ (1 − ξ m T ) − 2 ξ m T 2 ] θ + γ u 2 (1 − ξ m T ) 2 + 2 ξ m T θ (1 − ξ m T ) + ξ m T 2 (1 + θ ) 2 ! 1 2 +  ( ξ m T − 1) γ u  T − ξ m T θ + γ u ( ξ m T − 1) + ξ m T # . Therefo re, from (94) and ( 95), (52) follows. T o see (53), it sufﬁces to note that Q u ( n ) > C is a suf ﬁcient condition for an o utage at slot n in depend ently of the service policy u sed. Hen ce, P 2 ( O A ) ≥ P ( Q u ( n ) > C ) , there fore, d 2 ( γ u , γ m , θ ) ≤ d N ( γ u ) . 21 A P P E N D I X M P R O O F O F T H E O R E M 1 1 An outage e vent a t slot n implies N u ( n − T ) + N m ( n − T ) > C ( T + 1) wh ere N u ( n − T ) is the number o f un icast requests existing in the n etwork at time slot n − T . Hence P 3 ( O A ) ≤ P ( N u ( n − T ) + N m ( n − T ) > C ( T + 1)) , but N u ( n − T ) ≤ T X i =0 Q u ( n − i − T ) , and N m ( n − T ) ≤ Z m T ( n − T ) . Therefo re P 3 ( O A ) ≤ P T X i =0 Q u ( n − i − T ) + Z m T ( n − T ) > C ( T + 1) ! . Since { Q u ( i ) } i are IID random variables, then from (95), we ob tain (54). A P P E N D I X N P R O O F O F T H E O R E M 1 2 Regardless of the scheduling policy used, the following ev ent is sufﬁcient for an o utage at slot n . Q u ( n − i − T ) > 2 C − S m ( n − 2 i ) − S m ( n − 2 i + 1) , i = 1 , · · · , T , and Q u ( n − T ) > C − S m ( n ) . The above ev ent ensures that the number of delayed unicast re- quests is increasing over the window o f slots [ n − 2 T , · · · , n − T ] wh ere at slot n − T , the n etwork will end up h aving T X i =0 Q u ( n − i − T ) + S m ( n − i ) > C ( T + 1) , implying th at the total n umber o f resou rces that h av e to be consumed by slot n inclusive is greater th an th e ag gregate av ailable capacity C ( T + 1) which would cau se an ou tage. Noting that { S m ( i ) } i are IID, we can write P ∗ 4 ( O A ) ≥ P ( Q u ( n − T ) + S m ( n ) > C ) × P  Q u ( n − T + 1) + S m ( n − 2 ) + S m ( n − 1 ) > 2 C  T , which, using Chernoff bou nd, leads to ( 55). R E F E R E N C E S [1] FCC. Spectrum policy task force report, FCC 02-155. Nov . 2002. [2] J. Mitola III,“Cogniti v e Radio: An Integ rated Agent Architecture for Softwa re Deﬁned Radio” Doctor of T ec hnology Dissertati on, Roya l Institut e of T echn ology (KTH), Sweden, May , 2000 [3] I. Akyildi z, W . Lee, M. V ura n, and S. 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Proactive Resource Allocation: Harnessing the Diversity and Multicast Gains

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