In this section we present an extension of Algorithm 5.2 to accommodate any values L1, L2 ∈[1, L−1], such that L1+L2 =L. The main premise is to increase the per-type subpacketization by valued∈N, such thatd·L1, d·L2 ∈N, thus the total subpacketization becomes
S =d2(L1+K1γ1) K1
K1γ1
(L2 +K2γ2) K2
K2γ2
. (5.59)
The new scheme works by repeating d2 times Alg. 5.2, with the difference that some transmissions will allocate ⌈L1⌉ streams to users of set K1 and at the same time it will allocate ⌊L2⌋ streams to set K2 and in some transmis-sions will allocate ⌊L1⌋ streams to users of set K1 and at the same time it will allocate ⌈L2⌉ streams to set K2.
This way, it allows the average allocation of L1 and L2 streams to each user type, which leads to the DoF DL(K1, γ1, K2, γ2) = L+K1γ1+K2γ2.
Channel Unevenness Bottleneck
In this chapter, our focus is concentrated on the worst-user effect that is caused by the uneven channel nature of wireless channels, which is inter-twined with the multicasting transmission structure of Coded Caching. This performance bottleneck has its roots in the observation that in a wireless setting users will experience different channel capacities thus, a multicast message intended to convey information to many users at a time will have to be delivered with the worst user’s rate so as to be decodable by all users.
The uneven channels bottleneck does not only affect a single user, but instead the whole system performance, thus even a single user’s low-capacity channel can reduce the performance of all users. This performance degradation can be seen as an important drawback in the implementation of Coded Caching in wireless communications.
The model of interest is the wireless Broadcast Channel (BC) with a single antenna transmitter that serves K cache-aided, single antenna receivers with potentially different channel rates.
Related Work
Many previous works have considered user channel qualities not being equal, in the context of Coded Caching. Specifically, a lot of effort has been re-cently made in understanding the impact of coded caching under the more realistic scenario where the transmit Signal-to-Noise Ratio (SNR) is finite.
In this direction many interesting results, showed that the performance of single-stream Coded Caching can outperform that of multiple streams in the low SNR region [41], while the works in [43,96] (see also [97]) revealed the importance in designing the multicast beamformers in order to take into account the fact that users are cache-aided. Further, the work in [74] stud-ied the SNR performance of multi-antenna algorithms depending on their subpacketization requirements.
Moreover, the works in [98,99] used feedback to select the user XORs that can maximize the sum-rate and a fairness function that ensures the that users with lower rates can also receive their requested content in a timely manner.
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The works in [100,101] (see also [102–104] studied broadcast channels with channel erasures and showed how an erasure to one user can still be useful in the other users as side information.
Works [77,78] studied the setting with K transmitters each storing either part or the whole library, and which transmitters where tasked with serving K cache-aided receivers. In this setting, the direct links are “strong” thus are able to facilitate a high rate, while the cross links are “weak”. The proposed setting considered the case where no Channel State Information at the Transmitters (CSIT) was available to the transmitters and it was shown that the lack of CSIT can be ameliorated by exploiting the topological factor of the channel and the multicast transmissions.
Further, the work in [79] studied a multiple-antenna setting where the antennas were used as a means to increase diversity and achieve higher rates. The work showed that in order to achieve a scalable transmission rate by transmitting a single XORed message (as in the algorithm of [1]) but from a total ofL antennas would require the number of antennas to scale as O(lnK) in order for a scaling, with the number of users rate to be achieved.
A similar setting to the one proposed in this work was studied in works [105,106]. Specifically, the work in [105] studied the single antenna channel where each user has a - potentially - different channel strength in the finite Signal-to-Noise-Ratio (SNR) region. The authors proposed algorithms that could outperform the naive implementation of the algorithm of [1]. Further, the work in [106] studied the single antenna channel where the K users are split into two categories, those with full-rate channels and those with reduced rate channels. The authors designed a superposition code-based algorithm and showed its order optimal performance.
Another similar setting to the one presented in this chapter, was explored in [107] where the authors divided each file into 2K−1 subfiles (each for a different user subset) and further used opportunistic scheduling of users and superposition coding to maximize the rate.
Moreover, the work in [41] (see also [108]) exploited multiple antennas to combat the worst user effect, by transmitting a XOR message, as in the scheme of [1] and multiplied it by a beamforming vector designed to allocate power in such a way that will “raise” the worst user’s channel.
Further, the work in [109] has considered the multiple-antenna channel with asymmetric channel qualities and created the content placement and subsequent delivery phases in such a way to allow for a scaling rate.
Results Overview
Here we will propose a new algorithm that is designed to elevate the single-antenna performance when users are experiencing uneven channels. The algorithm borrows the placement strategy and generation of XORs from [1]
but delivers these XORs in a more efficient manner, based on statistical knowledge of the user channel strengths and through the use of superposi-tion coding. For this proposed setting we prove a performance bound and
further continue to show that our proposed algorithm performs within a multiplicative gap of 4 from this bound.
A very interesting result that is derived from this work is that, under the assumption of users being sorted with respect to their channel strength, i.e.
αk ≤αk+1,∀k ∈[K], when the channel qualities satisfy
αk ≥1−e−k·γ, ∀k∈[K] (6.1) then the system can perform exactly as if there was no channel degradation, thus showing how the “strong” users can elevate the “weaker” ones, and further showing that systems with smaller cache sizes are more immune to channel unevenness.
6.1 System Model
We consider the single-input-single-output (SISO), wireless, broadcast chan-nel (BC) with K single-antenna receivers.
The objective is to design the pre-fetching at the caches and the delivery of requests in such a way to minimize the delivery time T(K, γ,α), where α denotes the vector containing the channel strength of each of the K users.
The transmission/reception model follows the Generalized Degrees of Free-dom (GDoF) framework of [45] (see also [46,49]), thus a message received at some user k∈[K] takes the form
yk=√
Pαkhkx+wk, (6.2)
where P represents the transmitting power, x∈C is the output signal of the transmitter satisfying the power constraint E{|x|2} ≤ 1 and hk ∈ C corre-sponds to the channel coefficient of user k. Further, αk∈(0,1] represents the normalized, by factor logP, link strength of user k. Finally, wk ∼ CN(0,1) represents the Gaussian noise at user k. From the above we can deduce the average signal-to-noise-ratio (SNR) at user k∈[K] as
E{|yk|2}=Pαk (6.3)
which amounts to a (normalized) user rate of rk=αk. It should be noted that without loss of generality α = 1 corresponds to the highest possible channel strength, i.e. αK = 1.
For the model under study, we consider the most generic case, where each user has – potentially – a distinct link strength, namely, any two channel strength parameters αk, k ∈[K] may be different from each other. Without loss of generality, we assume an ascending ordering of the strength set α≜{αk}Kk=1, i.e., αk+1 ≥αk, for any k ∈[K].
Notation We will use Xσ to denote a XOR generated as in the algorithm of [1] i.e.,
Xσ =M
k∈σ
Wσd\{kk}. (6.4)
Further, we will use sets Xk, k ∈ [K] to denote the sets of XORs whose minimum numbered user is user k, i.e.
Xk =
where (⋆) comes directly from Pascal’s triangle and further,
In this section we present¹ the achievable performance of the algorithm of [1]
under a naive implementation and further, in Theorem 6.1 we describe the achievable performance of our scheme along with its order optimality.
Proposition 6.1. In the SISO-BC with user channel strength αk ≤ 1 and caching of fraction γ at each user, a naive implementation of the algorithm of [1] would amount to the delivery time of
Tuc(K, γ,α) = 1
Proof. The proof comes directly by allocating normalized time Tσ to XOR Xσ, according to XOR Xσ, to decode almost surely the transmitted message.
Theorem 6.1. In the K-user SISO Broadcast Channel with single antenna receivers, receiver channel strengths
αk K
k=1, αk ≤ αk+1, ∀k ∈ [K] and each receiver equipped with a cache of normalized sizeγ, the achieved worst-case delivery time takes the form
Tsc(K, γ,α) = max and has a multiplicative gap from the optimal performance of at most4.
¹We note that the derivation of the expression in Proposition6.1has also been calculated in previous works which focused on a finite SNR region analysis of Coded Caching (see [41,105]).