Cloud-Aided Interference Management with Cache-Enabled Edge Nodes and Users

Transcription

1 F arxiv: v [cs.it] 0 Jan 09 C F ireless nel Clou-Aie Interference Management with Cache-Enable Ege Noes an Users Seye Pooya Shariatpanahi,, Jingjing Zhang 3, Osvalo Simeone 3, Babak Hossein Khalaj 4, Mohamma-Ali Maah-Ali 5. School of Electrical an Computer Engineering, University of Tehran, Tehran, Iran. School of Computer Science, Institute for Research in Funamental Sciences (IPM, Tehran, Iran 3. Department of Informatics, King s College Lonon, Lonon, UK 4. Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran 5. Nokia Bell Labs, Holmel, NJ, USA Abstract This paper consiers a clou-ran architecture with cache-enable multi-antenna Ege Noes (ENs that eliver content to cache-enable en-users. The ENs are connecte to a central server via limite-capacity fronthaul links, an, base on the information receive from the central server an the cache contents, they transmit on the share wireless meium to satisfy users requests. By leveraging cooperative transmission as enable by ENs caches an fronthaul links, as well as multicasting opportunities provie by users caches, a close-tooptimal caching an elivery scheme is propose. As a result, the minimum Normalize Delivery Time (NDT, a high-snr measure of elivery latency, is characterize to within a multiplicative constant gap of 3/ uner the assumption of uncoe caching an fronthaul transmission, an of one-shot linear precoing. This result emonstrates the interplay among fronthaul links capacity, ENs caches, an en-users caches in minimizing the content elivery time. nt K R I. INTRODUCTION Caching content at the network ege can mitigate the heavy traffic buren at network peak times. Contents are proactively store in caches at the Ege Noes (ENs or at the enusers uring low-traffic perios, relieving network congestion at peak hours [], []. Ege caching at the ENs can enable Clou cooperative wireles transmission in the presence of share cache contents across multiple ENs [4], [9], [4]. In contrast, m(μ, r caching of share content at the users enables the multicasting EN ENof p coe EN q information EN n EN that KT is useful simultaneously for multiple users [5] [8]. m(μ, Inr practice, not all contents can be cache, an requeste uncache contents shoul be fetche from a central server through m(μ, r finite-capacity m(μ, fronthaul r links. This more general set- illustratein βfig., was stuie in [9] [] (see also βup, cache reference therein uncache in the absence of users caches. These references consier as the performance metric of interest the (b μk overall T m elivery min (r latency, incluing both fronthaul an wireless contributions. In particular, in prior works [4] [5], the elivery latency is measure in the high Signal-to-Noise Ratio (SNR regime. While [9] [] allow any form of elivery strategy, incluing interference alignment, in [], the optimal high-snr latency performance is stuie uner the assumption that wireless transmission can only use practical one-shot Library Fronthaul Caches Ege Noes En Users r Clou r Share Wireless Channel K R r Fig.. Clou-RAN system with cache-enable ENs an en-users. linear precoing strategies. Reference [] presents a cachingfronthaul-wireless transmission scheme that is shown to be latency-optimal within a multiplicative factor of 3/. In this paper, we exten the results in [] by allowing caching not only at the ENs but also at the en-users. To this en, we consier the clou-ran Clou scenario in Fig. an evaluate the impact of both cooperative transmission opportunities at the ENs an multicasting opportunities brought by caching at the users. Caching at the users has, in fact, EN EN m(μ,r EN q EN n EN two potentially beneficial effects on the network KT performance. First, since users have alreay cache some parts of the library, m(μ, r m(μ, r they nee not receive from the network the cache portion of the requeste file this isiknown as j the local F C caching gain. Secon, as assume, by a careful cache content placement, a common coe message can cache benefit more than one user, which is known as the global caching gain [5]. Assuming that entire library is cache (a μkacross T m min ENs (r an users an that the fronthaul links are absent, reference [4] prove that the gains accrue form cooperative transmission by the ENs an the global caching gain provie by users caches are aitive. Here, we generalize this conclusion by consiering the role of finite-capacity fronthaul links an by allowing for partial caching of the library of popular files across ENs an users. μ R μ T caches caches EN W n EN W n W n W n3

2 The rest of the paper is organize as follows. In Section II we escribe the system moel. Section III presents the main results along with an intuitive iscussion. In Section IV we etail the propose caching an elivery scheme. Then, we erive a converse in Section V, which is prove to be within a multiplicative gap of 3/ as compare to the high- SNR performance achievable by the propose scheme. Finally, Section VI conclues the paper. II. SYSTEM MODEL We consier a content elivery scenario, illustrate in Fig., in which Ege Noes (ENs, each with antennas, eliver requeste contents to K R single-antenna users via a share wireless meium. The contents library inclues N files, each of L bits, which are collecte in set W {W,..., W N }. Furthermore, each file W n is ivie into F packets, collecte in the set W n {W nf } F f, where F is an arbitrary integer an each packet consists of L/F bits. Each EN is connecte to a central server, where the library resies, via a wire fronhaul link of capacity C F bits per symbol of the wireless channel. Moreover, each EN is equippe with a cache of size µ T N files, for µ T. In this paper, in contrast to [], we assume that the users are also cache-enable, each with a cache of size µ R N files, for µ R. Henceforth, for simplicity, we assume that both µ T an µ R K R are integers, with extensions following irectly as in [9], [], [4] The system operation inclues two phases, namely the cache content placement an content elivery phases. In the first phase, each EN an each user caches uncoe fractions of the files in the library at network off-peak traffic hours an without knowing the actual requests of the users in the next phase. In the secon phase, at network peak traffic hours, at any transmission slot, each active user requests access to one of the files in the library, i.e., user k {,..., K R } requests file W k, k {,..., N}. For elivery, first, the clou sens on each fronthaul link some uncoe fractions of the requeste files to the ENs. For more general ways to use the fronthaul links, we refer to [9]. After fronthaul transmission, the ENs collaboratively eliver the requeste contents to the users via the ege wireless ownlink channel base on the cache contents an fronthaul signals. The signal receive by each user k on the ownlink channel is given as y k h H kix i + z k, ( in which h ki C is the complex representation of the faing channel vector from EN i to user k; x i C is the transmitte vector from EN i; z k is unit-power aitive Gaussian noise; an (. H represents the Hermitian transpose. The faing channels are rawn from a continuous istribution an are constant in each transmission slot. The transmission power of each EN is constraine by E [ x i ] SNR. Furthermore, as in [], [4], the ENs transmit using oneshot linear precoing, so that the vector transmitte by each EN at time slot t is given as x i v inf s nf, ( (n,f where s nf is a symbol encoing file fraction W nf, an v inf is the corresponing beamforming vector. Furthermore, we assume that Channel State Information (CSI is available to all the entities in the network. The performance metric of interest is the Normalize Delivery Time (NDT introuce in [9], which measures the high-snr latency ue to fronthaul an wireless transmissions. To this en, we write C F r log SNR, hence allowing the fronthaul capacity to scale with the wireless ege SNR with a scaling constant r 0. Then, enoting the time require to complete the fronthaul an wireless ege transmissions as T F a E (measure in symbol perios of the wireless channel respectively, the total NDT is efine as the following limit over SNR an file size F + E (3 ( lim lim E[TF ] SNR L L/log(SNR + E[T E ]. L/log(SNR In (3, the term L/log(SNR represents the normalizing elivery time on an interference-free channel; the term F lim SNR lim L E[T F ]/(L/log(SNR is efine as the fronthaul NDT; an E lim SNR lim L E[T E ]/(L/log(SNR as the ege NDT. Accoringly, for given clou an caching resources efine by the triple (r, µ T, µ R, the minimal NDT over all achievable policies is efine as (r, µ T, µ R inf{(r, µ T, µ R : (r, µ T, µ R is achievable}, (4 where the infimum is over all uncoe caching, uncoe fronthaul, an one-shot linear ege transmissions policies that ensure reliable elivery for any set of requeste files [9], []. III. MAIN RESULT In this section we state our main result an its implications. We procee by first proposing an achievable scheme an then proving its optimality within a constant multiplicative gap. In the cache content placement phase, the scheme follows the stanar approach of sharing a istinct fraction of a file to all subsets of µ T ENs an µ R K R users, hence satisfying the cache capacity constraints [4]. As a result, each fraction of any requeste file is available at m R µ R K R users, which we efine as receive-sie multiplicity, an at µ T ENs. As we will see, in the content elivery phase, the transmit-sie multiplicity m T, i.e., the number of ENs at which any fraction of a requeste files is available, can be increase beyon µ T by means of fronthaul transmission.

3 NDT As prove in [4], an briefly reviewe below, the content multiplicities m T an m R can be leverage in orer to erive a elivery scheme that serves simultaneously u(m T, m R min(k R, m T + m R (5 users at the maximum high-snr rate of log(snr. Unlike [4], however, here the transmit-sie multiplicity m T is not fixe, since any uncache fraction of a file can be elivere to an EN by the clou on the fronthaul. The multiplicity m T can be hence increase at the cost of a larger fronthaul elay F. Therefore, the multiplicity m T shoul be chosen carefully, by accounting for the fronthaul latency F as well as for the wireless NDT E, which ecreases with the size of the number u(m T, m R of users that can be serve simultaneously. Our main result below obtains an approximately optimal solution in terms of minimum NDT. Before etailing the main result, we briefly present how the scheme in [4] serves u(m T, m R users simultaneously at rate log(snr by leveraging both multicasting an cooperative Zero-Forcing (ZF precoing. Assume that m T +m R K R (the complement case follows in a similar way. At any given time, m T + m R ENs transmit simultaneously to eliver fractions of the requeste files to m T + m R users. To this en, the active ENs are groupe into all subsets of m T active ENs. Note that there are ( m T +m R m T such groups, an that each EN generally belongs to multiple groups. All groups transmit at the same time, with each group elivering collaboratively a share fraction of a file to the requesting user. Transmission by a group is one within the null space of the channel of other m T active users by means of Zero-Forcing (ZF one-shot linear precoing. The interference create by this transmission to the remaining m R active users is remove by leveraging the information in the receive-sie caches. This is possible since the caching strategy ensures that the message transmitte by a group of m T ENs is also available to m R users. Note that the scheme in [4] assumes, but the extension escribe above is straightforwar. Base on the above mentione achievable scheme, along with an optimize transmit-sie multiplicity, the following theorem characterizes the minimum NDT (4 to within a multiplicative constant equal to 3/. Theorem (Multiplicative gap on minimum NDT. The NDT up (r, µ T, µ R K R (m(r, µ T, µ R µ T + K R ( µ R + min {K R, m(r, µ T, µ R + K R µ R } (6 is achievable, where m(r, µ R if µ T < m(r, µ R, m(r, µ T, µ R µ T if m(r, µ R µ T m max, m max if µ T > m max, (7 [ ] an ( µ R r n m(r, µ R T K Rµ R if r < r th, (8 m max if r r th, Achievable NDT Lower boun Fig.. Achievable NDT up(r, µ R, µ T an lower boun (r, µ R, µ T versus µ T for ifferent values of µ T an, with, K R 4 an r 4. with an r th ( m max + K Rµ R, (9 ( µ R { m max min, KR ( µ R Moreover, the minimum NDT satisfies the inequalities }. (0 3 up(r, µ T, µ R (r, µ T, µ R up (r, µ T, µ R. ( Theorem implies that the multiplicity m T m(r, µ R, µ T in (7 is optimal in terms of NDT, up to a constant multiplicative gap. Importantly, in contrast to [], the choice of m T in (7 epens also on the caching capacity µ R at the users, an it reuces to selection in [] when µ R 0. The first term in (6 is the fronthaul NDT F require to convey the uncache portions of files to achieve the esire multiplicity m T m(r, µ T, µ R in (7. The secon term is the ege transmission NDT E, which accounts for the local caching gain (i.e., ( µ R, an for the combine global caching gain ue to the users caches an for the cooperation gain ue to the ENs caches an to fronthaul transmission (i.e., m+k R µ R. The result hence generalizes the main conclusion from [3] an [4] that the gains from coe caching multicasting opportunities at the receive sie an cooperation at the transmit sie are aitive. Example. The achievable NDT up (r, µ T, µ R in (6, along with the lower boun (r, µ T, µ R erive in Lemma in Section V, are plotte in Fig. as a function of the users cache capacity µ R for ifferent values of the parameters, r an µ T. We set the number of ENs an users to an K R 4 an the fronthaul rate to r 4. Note that for non-integer values of µ R K R, the achievable NDT is obtaine by memory sharing between the receive-sie multiplicities µ R K R an µ R K R [6]. It is observe that caching at the en-users is more effective when the number of EN transmit antennas an/or the transmit-sie caches are small. Furthermore, when the transmit-sie multiplicity is sufficient to serve all K R users at the same time, en-user caching only provies local caching gains. In particular, this happens when µ T 6 an 4, in which case the NDT is seen to ecrease linearly with µ R K R.

4 IV. ACHIEVABLE SCHEME The achievable scheme generalizes the strategies propose in [] an [4] by accounting for fronthaul transmission an for the caches available at the users. As iscusse in Section III, the cache content placement phase uses the same approach propose in [4], which guarantees content replication of µ T an m R µ R K R at the transmit an receive sies, respectively. In the content elivery phase, fronthaul transmission provies packets from the requeste files to the ENs in orer to increase the transmit-sie multiplicity m T to the esire value m. This is at the cost of the fronthaul elay F (m K R (m µ T +, ( given that (m µ T + L bits nee to be elivere for each requeste file (see also []. Base on the multiplicities m R an m T, the number of users that can be serve at the same time is (5. Since each user has cache a ( µ R -fraction of its requeste file, the ege NDT is given by [] E (m K R( µ R. (3 u(m The transmit-sie multiplicity m T m shoul be tune such that the total elivery latency E (m + F (m is minimize. First, we etermine the maximum possible multiplicity m max from the following necessary conitions m, m + K R µ R K R, (4 which result in m max given in (0. To procee, we first focus on the case of µ T 0, an fin a close-to-optimal multiplicity m(r, µ R. Then, base on the expression for m(r, µ R, we propose a specific choice for the multiplicity for the general case where µ T 0. To start, in the case when µ T 0, from ( an (3, the total NDT is (m K Rm + K R( µ R. (5 u(m In orer to optimize over m, we fin the (only stationary point for function (5 as ( µ R r m 0 K Rµ R. (6 We then approximate the integer solution of the original problem to be the nearest positive integer smaller than m max, yieling (8. For the general case µ T 0, we propose the choice (7 for the transmit-sie multiplicity. Accoringly, when µ T < m(r, µ R, an hence the transmit-sie caches are small, packets are sent over the fronthaul links so that the aggregate multiplicity is equal to the value m(r, µ R selecte above when µ T 0. For the case µ T m(r, µ R, instea, the transmit-sie multiplicity (7 only relies on EN caching, an fronthaul transmission is not carrie out. In particular, Fig. 3. Transmit-sie multiplicity m(r, µ T, µ R in (7 versus K R µ R for, K R 4, µ T 4 an ifferent values of r. when µ T m max, the maximum multiplicity m max can be guarantee irectly by EN caching. Theorem emonstrates the near-optimality of this choice. As illustration for how the user cache capacity µ R K R affects transmit-sie multiplicity m(r, µ T, µ R is shown in Fig. 3 for, K R 4, µ T 4. As µ R K R increases, user-sie caching becomes more effective, an less EN-sie cooperation is neee to null out interference. Accoringly, the transmit-sie multiplicity ecreases with µ R K R, an it epens on r only when µ R K R is sufficiently small. V. MULTIPLICATIVE OPTIMALITY In this section we emonstrate that the achievable NDT iheorem is within a multiplicative constant gap of the minimum NDT by proving (. To this en, we exten the converse proof evelope in [] in orer to account for the presence of users caches. First, without loss of generality, consier a split of each file W n into ( K R subsets of packets W n {W nτt τ R }, such that each part W nτt τ R is inexe by the subsets of inices τ T [ ]\{ } an τ R [K R ]. The subset W nτt τ R inclues the packets of W n that are present at all the ENs i τ T after fronthaul transmissions an at all the users j τ R. We also efine τ R as the number of packets of file W n that are cache at all the ENs in τ T, an at all the users in τ R ; f nτt τ R ( to be number of packets from file W n that are transmitte to all the users in τ T via the fronthaul for a given eman vector an cache at all the users in τ R. Note that these quantities are well-efine for every policy. With these efinitions, NDT of any achievable policy can be lower boune by the solution to the following optimization problem: (a min { τ R } {f nτt τ R (} subject (b (c N n to i τ T max K R K R E ({ τ R (}, + F ( + f nτt τ R ( F, n, (7 τ R µ T F N, i [ ]

5 ( (e N n F r j τ R n i τ T K R + f nτt τ R ( µ R F N, j [K R ], f nτt τ R ( F (, i [ ], (f{ τ R, f nτt τ R (} 0, 0 F ( Fmax, (8 where function E ({c nτ T τ R (}, is implicitly efine as the minimum ege NDT in (3 for given cache an fronthaul policies when the request vector is, while F ( is a function for that satisfies conitions (e an (f. We have also efine Fmax K R(m max µ T +. (9 In (7, the equality (b guarantees the availability of all the requeste files; inequalities (c are ue to the fact that the size of the cache content of each EN i [ ] is limite by the cache capacity µ T F N; similarly, inequalities ( enforce the cache capacity constraint at each user j [K R ]; inequalities (e follow from the efinition of fronthaul NDT in (3, since the left-han sie is the number of packets sent to EN i via the fronthaul link; in (f, F ( is upper boune by F max since the maximum multiplicity is m max an hence the total number of bits require via fronthaul is K R (m max µ T +. In (7, the expression of function E ({c nτ T τ R (}, is generally unknown. Notwithstaning this complication, the following Lemma gives a lower boun to the solution of the above optimization problem. Proof can be foun in Appenix A. Lemma. The minimum value of the optimization problem in (7 is lower boune by f(x K R (x µ T + K R( µ R x + µ R K R, (0 where we have efine x K R ib ij, fnτt τ R an b ij :n f nτ T τ R ( K R π(n, K R, N. ( n Finally, the following lemma analyzes the gap between the lower boun erive in Lemma an the upper boun (6, completing the proof of Theorem. A proof can be foun in Appenix B. Lemma. Function f(x in (0 is lower boune by 3 up(r, µ T, µ R, where up (r, µ T, µ R is given iheorem. VI. CONCLUSIONS For a cache-enable clou-ran architecture where both the ENs an the en-users have caches, this paper has characterize the minimum elivery latency in the high-snr to within a multiplicative gap of 3/. Uner the practical constraint that the ENs can only transmit using one-shot linear precoing, the main result shows that the cooperation gains accrue by EN cooperation via EN caching an fronthaul transmission are aitive with respect to the multicasting gains offere by en-user caching. ACKNOWLEDGEMENTS Jingjing Zhang an Osvalo Simeone have receive funing from the European Research Council (ERC uner the European Union s Horizon 00 Research an Innovation Programme (Grant Agreement No REFERENCES [] J. Kangasharju, J. Roberts, an K. Ross, Object Replication Strategies in Content Distribution Networks, Computer Communications, vol. 38, no. 4, pp , 00. [] E. Nygren, R.K. Sitaraman, an J. Sun. The Akamai Network: A platform for high-performance Internet application, ACM SIGOPS Operating Systems Review, vol. 44, no. 3, pp. 9, 00. [3] K. Shanmugam, N. Golrezaei, A. G. Dimakis, A. F. Molisch an G. Caire, FemtoCaching: Wireless Content Delivery Through Distribute Caching Helpers, IEEE Trans. Inf. Theory, vol. 59, no., pp , Dec. 03. [4] A. Liu an V. K. N. Lau, Exploiting Base Station Caching in MIMO Cellular Networks: Opportunistic Cooperation for Vieo Streaming, IEEE Trans. Signal Process., vol. 63, no., pp , Jan. 05. [5] M. A. Maah-Ali an U. Niesen, Funamental limits of caching, IEEE Trans. Inf. Theory, vol. 60, no. 5, pp , 04. [6] N. Karamchanani, U. Niesen, M. A. Maah-Ali, an S. N. Diggavi, Hierarchical Coe Caching, IEEE Trans. Inf. Theory, vol. 6, no. 6, pp. 3 39, 06. [7] R. Pearsani, M. A. Maah-Ali an U. Niesen, Online Coe Caching, IEEE/ACM Trans. Netw., vol. 4, no., pp , April 06. [8] J. Zhang an P. Elia, Funamental limits of cache-aie wireless BC: Interplay of coe-caching an CSIT feeback, IEEE Trans. Inf. Theory, vol. 63, no. 5, pp , 07. [9] A. Sengupta, R. Tanon, an O. Simeone, Fog-aie wireless networks for content elivery: Funamental latency traeoffs, IEEE Trans. Inf. Theory, vol. 63, no. 0, pp , Oct. 07. [0] R. Tanon an O. Simeone, Clou-aie wireless networks with ege caching: Funamental latency trae-offs in fog raio access networks, in Proc. IEEE Int. Symp. on Inf. Theory (ISIT, pp , 06. [] S. M. Azimi, O. Simeone, A. Sengupta, an R. Tanon, Online Ege Caching in Fog-Aie Wireless Network, in Proc. IEEE Int. Symp. on Inf. Theory (ISIT, pp. 7, 07. [] J. Zhang, O. Simeone, Funamental Limits of Clou an Cache-Aie Interference Management with Multi-Antenna Base Stations in Proc. IEEE Int. Symp. on Inf. Theory (ISIT, pp , 08. [3] S. P. Shariatpanahi, S. A. Motahari an B. H. Khalaj, Multi-Server Coe Caching, IEEE Trans. Inf. Theory, vol. 6, no., pp , Dec. 06. [4] N. Naerializaeh, M. A. Maah-Ali an A. S. Avestimehr, Funamental Limits of Cache-Aie Interference Management, IEEE Trans. Inf. Theory, vol. 63, no. 5, pp , May 07. [5] M. A. Maah-Ali an U. Niesen, Cache-aie interference channels, in Proc. IEEE Int. Symp. on Inf. Theory (ISIT, pp , 05. [6] H. Ghasemi an A.Ramamoorthy, Improve Lower Bouns for Coe Caching, IEEE Trans. Inf. Theory, vol. 63, no. 7, pp , July, 07.

6 APPENDIX A PROOF OF LEMMA To prove Lemma, we substitute the maximum over all the possible request vectors in the objective of (7 with an average over them. The solution of the resulting problem yiels a lower boun to the solution of the orginal problem. Mathematically, the objective (a in (7 is substitute with min { τ R } {f nτt τ R (} π(n, K R ( [ E ({ τ R (}, + F (], where we have efine π(n, K R N! /(N K R!. In orer to eal with the unknown function E ({c nτ T τ R (},, we nee the following lemma. Lemma 3. Define τ T l as the subset of ege noes that have access to the packet W nl f l after the fronhaul transmission, an τ Rl as the subset of users that have cache the packet W nl f l. Then, the number u of users that can be serve at the same time is upper boune by u min l [u] τ T l + τ Rl. (3 Proof. This lemma is generalization of Lemma in [] to the case where we have cache at the users. The main proof is consiering the role of receivers caches in a similar way as in Lemma 3 of [4], an the rest of the proof is the same. Using the above lemma we will have the following lower boun on the minimum ege NDT: E ({ τ R (}, ( µ R F K R K R k c k τ T τ R + f k τ T τ R (, i + j (4 respectively. Now we lower boun the first term in ( as π(n, K R K R k E ({ τ R (}, (a µ R F π(n, K R K R k K R (b µ R F π(n, K R K R( µ R K R K R K R (c K R( µ R ( K R( µ R (e K R( µ R K R i + j µ R F π(n, K R c k τ T τ R + f k τ T τ R ( i + j (c k τ T τ R + f k τ T τ R ( K R π(n, K R i + j N n i + j N n ( KT KT KT b ij i + j K R ( µ R KT KR ib ij + KR b ij KR (i + jb ij ( KR (i + jb ij KT KR jb, ij where (a hols because of Lemma 3; in (b we have efine :n f nτt τ R f nτ T τ R ( K R π(n, K R ; (5 in (c we have efine b ij N ; (6 n in ( we have use the inequality i,j u ij v ij i,j u ij i,j v ij ; (7 since at most i + j users can be serve simultaneously when the multiplicities at the ENs an the users are i an j, with u ij b ij /(i + j an v ij b ij (i + j; an finally (e results from the equality KR b ij.

7 This results from summing up (8b for all π(n, K R request vectors an for all K R files in each request vector as follows: π(n, K R K R F (8 n K R K R π(n, K R N K R n K R π(n, K R K R + f nτt τ R ( N n K R K R π(n, K R b ij. Now we lower boun for the term relate to the fronthaul elay as follows: π(n, K R F ( (9 π(n, K R F r n i π(n, K R F r i π(n, K R F r K R i N F r n i τ T K R π(n, K R N n K R K R i N F r b ij K R (a K R K R f nτt ib ij N n i K R ib ij µ T, f nτt ( f nτt ( N n f nτt N n in which we have efine f nτt f nτt K R K R K R τ R f nτt τ R f nτt τ R, an in (a we have use the following inequality ue to the ege noes cache size constraint in (8c µ T F N N n N n N i n i (30 i τ T N. n Finally, we can a up the terms corresponing to the lower bouns for the elay an ege NDT to arrive at the NDT lower boun K R K R ib ij µ T + (3 K R ( µ R KT KR ib ij + KT KR jb ij K R (x µ T + K R( µ R, x + µ R K R in which we have efine x K R ib ij, (3 an have use the following inequality ue to cache size constraint at the users state in (8 K R jb ij (33 K R j j µ R K R. N n

8 APPENDIX B PROOF OF LEMMA To prove the inequality f(x up (r, µ T, µ R /3, we first focus on the minimum f min of f(x. To this en, we first erive the omain of function f(x. From (7 an (9, we can have the inequalities K R (x µ T /( F ( F max. Hence, the maximum value of x is given as x max max{m max, µ T }. We also have the inequality KT KR ib ij ue to the equality KR b ij in (8. This yiels the necessary conition x. Also because x 0, the minimum value is given as x min max{µ T, }. Hence, x lies in the interval [x min, x max ]. Since function f(x is convex for x > 0, an the only stationary point is x m 0, i.e., f (m 0 0, where we have m 0 ( µ R r/ K R µ R /. Therefore, the esire minimum f min is given as { f(m0, if x f min min m 0 x max min{f(x min, f(x max }, otherwise, (34 which can be rewritten as K R (m (r,µ R µ T + K f min R ( µ R m (µ R,r +µ R K R, : µ < m (r, µ R K R ( µ R µ T +µ R K R, : µ m (µ R, r (35 where we have efine m (r, µ R { max{m 0, } : r < r th, m max : r r th. (36 As a result, if we can prove the inequality f min up (r, µ T, µ R /3, then Lemma hols immeiately. Since f min is quite intractable, we turn to choose a simpler function, enote as lb (r, µ T, µ R that satisfies lb (r, µ T, µ R lb (r, µ T, µ R, where we have efine lb (r, µ T, µ R f min. The lower boun lb (r, µ T, µ R is given as lb(r, µ T, µ R ( (i + µt K R ( µ R + (µ T i (i + + µ R K R (i + + µ R K R (37 for µ T [i, i +, with m(r, µ R i m max ; an lb(r, µ T, µ R { KR (m (r,µ R µ T + K R ( µ R m (r,µ R +µ R K R, K R (m(r,µ R µ T + lb (r, m(r,µ R, µ R, (38 for µ T m(r, µ R, an the first expression is for the regime m (r [(m(r, µ R 0.5, m(r, µ R ], while the secon expression is for the regime m (r [m(r, µ R, (m(r, µ R + 0.5], where m (r, µ R is given in (36. To procee, we now prove the inequality lb (r, µ T, µ R lb (r, µ T, µ R. Proof. Since m(r, µ R is the nearest integer point of m 0 when r < r th, we have the inequality m(r, µ R + > m 0, yieling m(r, µ R + > m (r, µ R for this range of r. Furthermore, we also have m(r, µ R m (r, µ R when r r th. Hence, the inequality m(r, µ R + m (r, µ R hols for any value of r. As a result, for any sub-interval µ [i, i +, with m(r, µ R + i m max, lb (r, µ T, µ R, i.e., f min, is given as K R ( µ R /(µ T +µ R K R from (35 since µ T m (r, µ R. By simple comparison between the above lb (r, µ T, µ R an lb (r, µ T, µ R in (37, we have the inequality lb (r, µ T, µ R lb (r, µ T, µ R. For the remaining interval µ istinguish the following two cases. m(r, µ R +, we Case : m (r, µ R [(m(r, µ R 0.5, m(r, µ R ]. Hence, we have the inequality m (r, µ R m(r, µ R. For interval µ T [m(r, µ R, m(r, µ R + ], the inequality lb (r, µ T, µ R lb (r, µ T, µ R hols with the same reason as above. Moveover, by comparison, we have that lb (r, µ T, µ R in (38 is equal to lb (r, µ T, µ R in (35 for µ m (r, µ R. Instea, for µ [m (r, µ R, m(r, µ R ], since both lb (r, µ T, µ R in (38 an lb (r, µ T, µ R K R ( µ R /(µ T + µ R K R are ecreasing functions of µ T, they are equal for µ T m (r, µ R, an the former has a smaller graient for the whole range of value of µ T at han, we have lb (r, µ T, µ R lb (r, µ T, µ R. Case : m (r, µ R [m(r, µ R, (m(r, µ R + 0.5]. In this range, the inequality m (r, µ R m(r, µ R hols immeiately. For interval µ T [m(r, µ R, m(r, µ R + ], since both lb (r, µ T, µ R in (38 an lb (r, µ T, µ R are ecreasing functions of µ T, they are equal for µ T m(r, µ R +, an the former has a smaller graient, we have lb (r, µ T, µ R lb (r, µ T, µ R. This hols also for the value µ T m(r, µ R. Combining this with the fact that lb (r, µ T, µ R in (38 an lb (r, µ T, µ R in (35 are linear an parallel for µ T m(r, µ R, we have lb (r, m(r, µ R/, µ R lb (r, µ T, µ R in this range. To complete the proof, we procee to prove the multiplicative gap between the upper boun up (r, µ T, µ R (6 an the lower boun lb (r, µ T, µ R. For µ T [i, i +, with m(r, µ R i m max, we have up (r, µ T, µ R lb (r, µ T, µ R + (a up(r, µ T i/, µ R lb (r, µ T i/, µ R (i + K R µ R ((i K R µ R 3, (39 where inequality (a hols because up (r, µ T, µ R an lb (r, µ T, µ R are both linearly ecreasing an they coincie at the enpoint µ T i +. For µ T m(r, µ R in

9 Case, the gap is given as up (r, µ T, µ R lb (r, µ T, µ R (b (a up(r, µ T m(r, µ R /, µ R lb (r, µ T m(r, µ R /, µ R m(r,µ R +µ R K R m (r,µ R m(r,µ R + m (r,µ R +µ R K R /p(m(r, µ R /p(m (r, µ R p(m(r, µ R /p(m (r, µ R (c /4 + (m(r, µ R + µ R K R (m(r, µ R + µ R K R ( 3, where inequality (a hols because up (r, µ T, µ R an lb (r, µ T, µ R ecrease with the same slope an the maximum ratio is at the enpoint µ T m(r, µ R ; equality (b hols ue to the efinition of p(x x + µ R K R an m (r, µ R ; inequality (c hols ue to the constraints m (r, µ R [(m(r 0.5, m(r]; an inequality ( hols for any m(r, while for m(r, we have / [0, ] an µ [0, ]. With simple comparison, we can get up (r, µ T, µ R lb (r, µ T, µ R. Finally, for µ m(r in Case, the gap is given as up (r, µ T, µ R lb (r, µ T, µ R + (a up(r, µ T m(r, µ R /, µ R lb (r, µ T m(r, µ R /, µ R (m(r, µ R + K R µ R ((m(r, µ R K R µ R 3, (4 where inequality (a hols as inequality (a in (39. This completes the proof.