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On the Interaction between Content Caching and RequestAssignment in Cellular Cache Networks
K.P. NaveenINRIA Saclay, Campus de
l’Ecole PolytechniquePalaiseau 91120, [email protected]
Laurent MassoulieMSR-INRIA, Campus de
l’Ecole PolytechniquePalaiseau 91120, France
Emmanuel BaccelliINRIA Saclay, Campus de
l’Ecole PolytechniquePalaiseau 91120, France
[email protected] Carneiro Viana∗INRIA Saclay, Campus de
l’Ecole PolytechniquePalaiseau 91120, [email protected]
Don TowsleyDept. of Computer ScienceUniversity of Massachusetts
Amherst MA 01003, [email protected]
ABSTRACTThe potential availability of storage space at cellular andfemtocell base-stations (BSs) raises the following question:How should one optimize performance through both loadbalancing and content replication when requests can be sentto several such BSs? We formally introduce an optimizationmodel to address this question and propose an online algo-rithm for dynamic caching and request assignment. Cru-cially our request assignment scheme is based on a serverprice signal that jointly reflects content and bandwidth avail-ability. We prove that our algorithm is optimal and sta-ble in a limiting regime that is obtained by scaling the ar-rival rates and content chunking. From an implementationstandpoint, guided by the online algorithm we design a light-weight scheme for request assignments that is based on loadand cache-miss cost signals; for cache replacements, we pro-pose to use the popular LRU (Least Recently Used) strat-egy. Through simulations, we exhibit the efficacy of ourjoint-price based request assignment strategy in comparisonto the common practices of assigning requests purely basedon either bandwidth availability or content availability.
CCS Concepts•Networks → Network performance modeling; Net-work simulations; •Theory of computation→Designand analysis of algorithms;
KeywordsCellular cache networks; online algorithm; fluid approxima-tions; Lyapunov stability; LRU caches.
∗This work was supported by the EU FP7 ERANET pro-gram under grant CHIST-ERA-2012 MACACO.
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AllThingsCellular’15, August 17-21, 2015, London, United Kingdomc© 2015 ACM. ISBN 978-1-4503-3538-6/15/08. . . $15.00
DOI: http://dx.doi.org/10.1145/2785971.2785975
1. INTRODUCTIONMobile data traffic is expected to increase by more than
57% annually to 24.3 exabytes per month by 2019 [1]. Thisincrease will likely result in durably congested wireless accessat the edge despite advances in radio technologies. Conges-tion is also anticipated at the core, driven not only by mobileInternet access but also by the emergence of new cloud ser-vices and Machine-to-Machine (M2M) communications.
Figure 1: A cellular network with a dense deploy-ment of heterogeneous base-stations (BS).
To alleviate congestion at the Internet edge, one promisingapproach is to target denser deployments of wireless base-stations (BSs), such as cache enabled femtocells in additionto the existing cellular BSs [12, 18]. Such a scenario is de-picted in Figure 1, where the alternating black and gray hor-izontal sections represent regions that are within the rangeof distinct subset of BSs. Thus, mobile users, dependingon their respective locations, will be potentially connectedto several BSs from which content may be directly down-loaded. In this context, BSs can be heterogeneous in termsof bandwidth and memory capacities. For instance, a nearbyfemtocell BS will provide more bandwidth than the heavilyloaded cellular BS; however, the chance of finding a desiredcontent at the femtocell is lower due to its lower cache ca-pacity.
Thus, both bandwidth and content availability criticallyaffect “by which BS” and “with which performance” incom-ing requests for contents can be served. In turn, contentrequest assignment strategies impact targeted BSs throughboth their bandwidth loads and cache contents as they trig-ger cache update decisions. The goal of this paper is tounderstand this intricate interplay between content requestassignment strategies, bandwidth load, and cache manage-ment in cellular cache networks. Specifically, we strive to
37
determine routing mechanisms and associated cache updaterules that together achieve a target trade-off between cache-miss probability and bandwidth loads.
Related Work: The problem of caching at femtocell basestations or heterogeneous caches has been recently studiedin [10, 12, 18]. Authors in [12] address content placement atthe femtocell base stations to maximize the probability ofcache hit. A similar setup, but with mobile users, is studiedin [18]. In both these works, request routing is purely basedon content availability (so that cache hits are maximized),while ignoring the bandwidth costs at the base stations. Fur-ther, the authors focus on static content placement algo-rithms. For instance, in [12] the content placement problemis formulated as that of maximizing a monotone submodu-lar function under matroid constraints; a static greedy al-gorithm that is within a constant factor of the optimal isproposed.
Borst et al. in [7] address the problem of content place-ment and routing to optimize the total bandwidth cost, buttheir formulation is limited to linear cost functions, in con-trast to our work where we allow for general convex costs.Moreover, the focus in [7] is again on designing static al-gorithms. Similarly, there are other works in the literatureaddressing the problem of static content placement so as tooptimize some linear system cost [4, 8, 13].
There is an extensive literature on online cache manage-ment algorithms and in particular on LRU (Least RecentlyUsed) caches [5, 9, 11]. There is also an abundant literatureon dynamic request routing [6, 15], but very few work ad-dress the design of online algorithms for both routing andcache management, and the interaction between the two.Notable exceptions are the works on cache networks [2, 3],which focus on stabilizing queues of pending requests byjoint management of bandwidth and caches. In contrast weinstead focus on loss-based models and on the objective ofcost minimization rather than queue stabilization.
With a similar focus as ours, the authors in [14] studythe problem of cache replication and request forwarding inCDNs comprising distributed caches, e.g., household set-top boxes. However, the objective is again limited to lin-ear traffic costs. Moreover, they focus on scenarios withmany servers with identical characteristics. This is in con-trast with our work as we consider a system with few poten-tially heterogeneous caches; our limiting regime is insteadobtained by scaling the arrival rates and the chunk size.There are other work in the literature, e.g., [16, 19], whereproportional placement strategies and matching algorithmsfor routing requests to content available caches are studied.These differ again from our present work by focusing on sce-narios with many identical servers.
2. SYSTEM MODELConsider a system comprising a finite collection of base-
stations (BSs) S and locations L where a location ` ∈ Lcan be thought of as representing the set of all physical lo-cations within the range of a subset of BSs. For instance,in Figure 1 the alternating black and grey shaded horizontalsegments represents different locations. Thus, each locationis connected to a subset of the BSs. These interconnectionscan be effectively depicted as in Figure 2, where location `is connected to BSs s and s′ while `′ is connected to s′ ands′′. Let A denote the adjacency matrix representing the in-terconnections, i.e., A`s = 1 implies that location ` is within
{λℓ′c}
Location ℓ′
{rs′ℓc}
{rs′ℓ′c}
{rs′′ℓ′c}
{λℓc}{rsℓc}
BS s
BS s′
BS s′′storing {θs′′c}
storing {θs′c}
storing {θsc}
Location ℓ
Figure 2: Illustration of the interconnections be-tween BSs and locations.
the range of BS s; A`s = 0 otherwise. We assume a finiteset C of contents in the system with the size of each contentbeing one unit. We allow content chunking whereby a con-tent can be completely reconstructed using k independentlycoded packets of the content. Each BS s ∈ S includes acache that can store up to Ms units of contents (memoryconstraints). We allow the BSs to store chunks of each con-tent instead of complete copies. We assume that the chunksstored at different servers are always independent so that itonly suffices to download k chunks from any subset of BSsto completely reconstruct the content.
Requests for content c arrive at location ` according to a
Poisson process with rate λ(k)`c = kλ`c, which can be thought
as the aggregate rate for c from of all users stationed at `.The number of chunks of content c, available at server s,restricts the rate at which s can serve incoming requestsfor c (content-availability constraints). Content downloadrequests, if accepted, result in service at unit rate. In thissense we consider a loss model (in contrast to a queuingmodel).
The bandwidth availability at a server s ∈ S is reflected
using a bandwidth-cost function, C(k)s (r), which represents
the cost incurred by s for serving requests at a total rate
of r. We assume that C(k)s , k ≥ 1, can be expressed as
C(k)s (r) = Cs( r
k), where Cs, s ∈ S, is strictly convex and
increasing. We refer to the above system as the k-th system.To obtain a benchmark for performance comparison, we
consider a system where the caching and routing variablesare relaxed to be reals. Let θsc ∈ [0, 1] denote the fraction ofcontent c stored at BS s, and krs`c the rate at which s servesrequests for c from `. Then, the memory constraint at BSs ∈ S is given by
∑c θsc ≤ Ms. Next, since only a fraction
θsc of a request for c can be served by s, we have the followingcontent availability (CA) constraints: krs`c ≤ A`skλ`cθsc,s ∈ S, ` ∈ L, c ∈ C, where we have approximated the numberof ongoing requests (of content c at `) by the steady statearrival rate, kλ`c (assuming an average service time of one).
In order to obtain an unconstrained optimization formula-tion, we replace the above hard memory and CA constraints
with strictly convex and increasing penalty functions, Cs
and Cs`c, in the cost to be minimized1. Cs(θ) denotes thecost of storing θ units of contents at BS s. For the CA con-straint, denoting x := krs`c − kλ`cθsc, Cs`c(
xk
) representsthe cost of respecting (x negative) the CA constraint, or
1Formulation directly involving the hard constraints is avail-able in our technical report [17].
38
exceeding the constraint (x positive) by x units. Since theBSs are physically constrained to always satisfy the CA con-straints, a positive x represents the content cache-miss ratefrom BS s for requests from ` for c. Thus, Cs`c(
xk
) representsthe cost incurred for fetching the missed content rate fromthe main data center at the backend. Another possibility isto let Cs`c(
xk
) = bmax(xk, 0) for some b > 0, which can be
interpreted as a cost of b per cache miss.We now define the total cost incurred as,
C({θsc}, {rs`c}) =∑
s Cs
(∑`cA`srs`c
)+∑
s Cs
(∑c θsc
)+∑
s`cA`sCs`c
(rs`c − λ`cθsc
)(1)
and propose the following optimization problem:Total Cost (TC):
Minimize: C({θsc}, {rs`c}) (2a)
Over: θsc ∈ [0, 1], s ∈ S, c ∈ C (2b)
rs`c ≥ 0, s ∈ S, ` ∈ L, c ∈ C (2c)
Subject to:∑s
A`srs`c = λ`c, ` ∈ L, c ∈ C. (2d)
The equality constraint (2d) is used to ensure that all in-coming requests are handled by the BSs. In Section 3, wepropose an online algorithm to solve this optimization prob-lem. We show that our algorithm converges to the optimalsolution of (2) as the scale of the system in terms of arrivalrate and number of chunks per content (i.e., k) increases.
3. ONLINE ALGORITHMThe main objective of this section is to propose an al-
gorithm for assigning incoming requests to BSs based ontheir current “prices”; the key idea is that these prices re-flect both the load as well as the content availability at therespective BSs (which is in contrast to the common practiceof assigning requests purely based on either load or con-tent availability). Due to space constraints, we only brieflymention about the optimality and stability properties of ouralgorithm; theoretical details and proofs are available in ourtechnical report [17].
Let Θsc(t) denote the number of chunks of content c storedat BS s at time t, and Gslc(t) the total number of chunksof content c served (possibly for more than one request)from s to ` at time t. Thus, Θsc(t)/k is the fraction ofcontent c stored at s at time t, and Rslc(t) := Gslc(t)/kis the rate at which s serves requests for c from `. Thechunk service times are i.i.d (independent and identicallydistributed) exponential random variables with unit rate.Cache Updates: Updating a BS’s cache for a content
involves either adding new chunks or removing some existingchunks. In the k-th system, let kνsc denote the rate at whichBS s updates the cache for content c. Suppose BS s initiatesa cache update for c at time t. Then, the number of chunksthat are added or removed is proportional to
∆Θsc(t) = −Qs(t) +∑`
A`sλ`cPs`c(t), (3)
where (using F ′ to denote the derivative of F ),
Qs(t) := C′s
(∑c
Θsc(t)
k
)(4)
Ps`c(t) := C′s`c
(Rs`c(t)
k− λ`c
Θsc(t)
k
). (5)
Formally, Θsc(t) is updated as follows: Θsc(t+) = Θsc(t)
+[ε∆Θsc(t)], where t+ denotes the time instant just after t,ε > 0 is a constant, and the notation [x] represents the inte-ger closest to x. We refer to Qs(t) and Ps`c(t), respectively,as the memory price and the content availability (CA) price(w.r.t (`, c)) of BS s at time t.
Remark: For the sake of analysis, we assume that thecaches are updated instantaneously at the update instants,although in practice a small time is required to downloadnew chunks from a backend data center. We further assumethat the backend data center always supplies a new batch ofindependently coded chunks, so that it is possible to recon-struct a content by downloading k chunks from any subsetof BSs, without worrying about chunks being dependent.
Note that ∆Θsc(t) is simply the negative of the gradient(w.r.t θsc) of the cost function in (1). Thus, our cache up-date strategy is essentially the gradient descent algorithm.The novelty of our work appears next, where we proposea request assignment strategy that couples the update in-stants of {Rs`c(t)} variables with the request arrival times.Further, incoming requests are assigned to download chunksfrom a (state dependent) selected subset of BSs, thus alwaysincreasing the current Rs`c(t) values of the selected BSs.This differs from the gradient descent algorithm where, de-pending on the gradient, an update could also recommenddecreasing the value of some rate variables. We formallydiscuss our request assignment strategy next.
Request Assignments: Suppose a request for contentc arrives at location ` at time t. Then, the decision as tohow many chunks should be downloaded from different BSsis based on the “prices” associated with the BSs connectedto `. Formally, for each (`, c), we define the price of BS s(such that A`s = 1) at time t as,
Πs`c(t) = Ps(t) + Ps`c(t) (6)
where, Ps`c(t) is the CA price (recall (5)) and
Ps(t) := C′s
(∑`′c′
A`′sRs`′c′(t)
k
)represents the bandwidth price of BS s.
Remark: Although the price depends on various cachingand routing variables, its calculation in practice can be basedon, for instance, an estimate of the current load at a BS tocompute its bandwidth price; The content availability pricecan be simplified to reflect the cost of fetching the unavail-able chunks from the backend data center. We will studythe performance of such a light-weight scheme in Section 4.
Now, given the BS prices, let S`c(t) denote the set of allminimum price BSs reachable from `, i.e.,
S`c(t) = argmins
{Πs`c(t) : A`s = 1
}. (7)
Our request assignment strategy centers on downloading suf-ficient chunks of c from the BSs in S`c(t) to construct thecontent at `. However, if there are an insufficient numberof chunks at these BSs, signals in the form of dummy chunkassignments are issued to increase their CA price and thustrigger future replication of content c chunks at these BSs.Formal details follow, where we have chosen to distributethe dummy chunks equally among all the BSs in S`c(t) (see(8a) and (8a)); however, note that any other dummy chunk
39
assignment would work equally well, as long as the totalnumber of chunks allocated (actual+dummy) is k.
Let n denote the number of BSs in S`c(t) (i.e., n = |S`c(t)|).Now, arrange the BSs in S`c(t) in any arbitrary order, forinstance, e.g., in increasing order of the BS IDs (assum-ing that each BS is given a unique integer ID). Suppose(s1, s2, · · · , sn) is such an arrangement, then BS si is as-
signed to serve Γsi`c(t) = Γ(1)si`c
(t) + Γ(2)si`c
(t) chunks (ac-
tual+dummy, resp.) of content c, where (using bxc to de-note the largest integer smaller than x), for i = 1, · · · , n,
Γ(1)si`c
(t) = min{k −
i−1∑j=1
Γ(1)sj`c
(t),Θsic(t)},
Γ(2)si`c
(t) =
k −∑nj=1 Γ
(1)sj`c
(t)
n
, i = 1, · · · , n− 1
Γ(2)sn`c(t) = k −
n∑j=1
Γ(1)sj`c
(t)−n−1∑j=1
Γ(2)sj`c
(t).
No chunks are downloaded from the BSs not in S`c(t), i.e.,we have Γs`c(t) = 0 for s /∈ S`c(t).
Thus, Γ(1)si`c
(t) is the actual number of chunks that si can
serve, while∑n
j=1 Γ(2)sj`c
(t) represents the total number of
chunks that are not present in the caches, i.e., results incache misses. Although, physically it is not possible for sito serve Γ
(2)si`c
(t) number of additional chunks, we proceedto allocate these as dummy chunks served by si. As men-tioned earlier, the idea is to increase the CA price Psi`c(t)(recall (5)), so that more chunks of c will be replicated by sduring its subsequent cache updates (see (3)). Practically,the dummy chunks are downloaded from the backend datacenter.
Finally, the associated rate variables are updated accord-ing to the following expression: Rs`c(t
+) = Rs`c(t)+Γs`c(t).Theoretical Guarantees: Using mean field approxima-
tion and Lyapunov function techniques, we show that ouronline algorithm is optimal and stable in a limiting fluidregime that is obtained by scaling k, i.e., the arrival ratesand the number of chunks per content; formal details areavailable in our technical report [17]). From a practicalstandpoint, our theoretical results imply that for a largesystem2 the performance of the online algorithm is expectedto converge in time to a close-to-optimal performance.
4. SIMULATION EXPERIMENTSWe first consider a simple setting comprising 2 locations
and 3 BSs; the content catalog size is |C| = 10. After demon-strating the efficacy of the joint-price based request assign-ment schemes for this setting, we present results for a largersystem.
We assume that BSs 1 and 3 are femtocells serving loca-tions 1 and 2, respectively, while BS 2 is a common cellularBS connected to both the locations (recall Figure 2). Thus,the requests arriving at a location can be assigned eitherto the respective femtocell or to the cellular BS. The total
2Note that large here refers to a system with large arrivalrates and content chunking. It does not imply a large net-work in terms of number of BSs and locations; all our anal-ysis holds for a network of any finite size.
4 6 8 10 12 14 16 18 20 220
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Average Bandwidth Cost
Cach
e−
mis
s P
rob
ab
ilit
y
Optimal
Online Algorithm
Load−only Signaling
Content−only Signaling
Joint Signaling
b
Figure 3: Performance curves.
(unscaled) content request rate from each location is 1, i.e.,∑c λ`c = 1 for ` = 1, 2. The content popularity at each
location follows a Zipf distribution with parameter 0.8, i.e.,the probability that an incoming request is for the i-th mostpopular content is proportional to 1/i0.8. In order to ob-tain a scenario where the same content can have differentpopularities at different locations, we randomly shuffle thedistributions for the two locations. The (unscaled) cacheupdate rate of each BS for each content is 1, i.e., νsc = 1.
The various costs (recall (1)) used in the simulations areas follows:• Bandwidth cost: Cs(r) = exp(2(r − Rs)), where R1 =R3 = 0.5 and R2 = 0.1; thus, the cellular BS is costlier thanthe femtocell BSs.• Memory cost: Cs(θ) = exp(5(θ−Ms)) where M1 = M3 =2 while M2 = 8; thus, the cellular BS has more memorycompared with the femtocells.• Content availability (CA) cost: Cs`c(x) = exp(bx); we areparticularly interested in studying the effect of b on perfor-mance. A large value of b, which yields a steep cost func-tion, results in a system where cache misses are costlier thanbandwidth. Thus, b can be used to trade off bandwidth costand cache-miss probability. We refer to b as the cache-missexponent.
Performance of the online algorithm: In Figure 3we first plot the optimal performance curve, obtained bydirectly solving the problem in (2) for different values of b(curve labeled ’Optimal’). As highlighted in the figure, thecache-miss exponent b increases from left to right along allthe curves. Also shown in the figure is the performance of theonline algorithm, described in Section 3, for a system withk = 10 chunks per content. Although the online algorithm isproven to converge to the optimal performance as k →∞, itis interesting to observe that its performance is comparablewith the optimal already for k = 10.
Before proceeding further, let us briefly discuss the trade-off curves. Note that, as b increases the cache-miss prob-ability improves, however at the expense of an increasedbandwidth cost. This is because, for a larger b, the cellularBS offers a lower price for the incoming requests as it holdsmore contents. Thus, most of the requests are forwarded tothe cellular BS, increasing its (and the overall) bandwidth
40
10−2
100
102
104
106
108
1010
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Average Bandwidth Cost
Cach
e−
mis
s P
rob
ab
ilit
y
Load−only Signaling
Content−only Signaling
Joint Signalingτ=0
τ=10
τ=1 b
τ=100
τ=∞
Figure 4: Performance of the light-weight schemesfor a large network.
cost. In contrast, when b is small, bandwidth cost is thekey component in deciding the BSs’ prices. Since femtocellsare less expensive in this regard, more requests are now for-warded to the femtocells, resulting in more cache misses asthey only hold few contents.
Drawbacks of the online algorithm: The implemen-tation of the online algorithm however requires the BSs tomaintain routing variables in the form of number of chunksof each content downloaded by requests from each location.This would require identifying each chunk with its request’slocation, which is not possible in practice when the numberof ongoing requests are large. Further, estimates of the re-quest arrival rates, λ`c, are required by the BSs to computetheir content availability prices. Finally, the cache updatein the online algorithm is server-driven requiring the BSs toupdate their caches regularly. This would result in unneces-sary updates whenever the request rate is less.
A light-weight signaling scheme: In view of the abovedrawbacks, we modify the online algorithm to obtain a light-weight signaling scheme for request assignments. We refer toit as the joint-price scheme since, as in the online algorithm,the request assignment here is based jointly on the contentand bandwidth availability. The bandwidth price, as before,remains to be the same convex increase function of the cur-rent load on the BS; thus, by using the congestion signalsfrom the BS, it is possible to estimate the bandwidth costat a location. However, for the content availability price,we now simply use the cost of fetching unavailable chunks ifan insufficient number of chunks of the requested content isavailable at the BS. Formally, the content availability priceof BS s for request c (from any location) at time t is givenby b(k −Θsc(t))/k, where Θsc(t) is the number of chunks ofcontent c at BS s at time t and b now actually representsthe cost of fetching the unavailable fraction of content c.Thus, as with the online algorithm, a larger b implies thatthe cache misses are costly.
We use the above joint price signaling scheme in conjunc-tion with the LRU (Least Recently Used) cache eviction al-gorithm [5, 9], which is a request driven cache managementstrategy. Under LRU, cache replacements are initiated byincoming requests that do not find the desired content in
the cache (a cache-miss event). As a consequence, the leastrecently used content is removed from the cache to free mem-ory for storing the new content. Since we deal with contentsat the chunk level, we have implemented a variation of LRU,referred to as the CLRU (Chunk LRU) strategy. In CLRU,the incoming request at a cache for m chunks of a contenttriggers the relocation of such m chunks to the head of thecache queue to be served. The placement of extra cachedchunks at the cache (i.e., if more than m chunks are cached)remain unchanged. In the opposite case, if chunks are miss-ing (i.e., if less than the requested m chunks are cached)and the cache is full, new chunks of the content are down-loaded and placed at the head of the queue by removingleast recently used chunks at the tail of the queue.
Performance of joint-price scheme: In Figure 3 wehave depicted the performance of the joint-price signalingscheme. Also shown in the figure are the performance oftwo heuristic policies (commonly used in practice) where therequest assignment is purely based on bandwidth availabilityor content availability signals (points ’Load-only Signaling’and ’Content-only Signaling’, respectively). The joint-pricescheme provides a favorable trade off between bandwidthcost and cache-miss probability, encompassing the heuristicpolicies as its extreme end points.
Although the joint-price scheme suffers some degradationin performance when compared with the online algorithm,it is useful to emphasize that the simplicity of the formerscheme would render it more favorite than the latter when itcomes to considerations for practical implementation. More-over, the performance degradation is less for low to moder-ate values of cache-miss cost b, which is typically the case inpractice.
Large network: Motivated by the observations made inthe previous section, we proceed to study the performance ofthe joint-price scheme for a large network. We now considera content catalog of size 1000. The number of locationsare increased to 10. It will be useful to think of a location` ∈ {1, 2, · · · , 10} as the segment [0, `] on the real line (recallFigure 1 for illustration). A cellular base station is locatedat the center (i.e., at 5) so that, as in the earlier case, itcan cover all the locations. However, we now increase thenumber of (small-size) femtocells to 9, which are located atpositions 1, 2, · · · , 9; each small-size femtocell has a range of1. Thus, each location (except the end locations) is servedby two adjacent femtocells. To model the heterogeneity inthe femtocell base stations, we introduce 5 moderate-sizefemtocells. The moderate femtocells have a larger range of 2and are located at 3, 4, · · · , 7. Note that, the inner locationsare covered by more femtocells (small+moderate) than theouter ones.
Small-size femtocells have smaller caches (of size 10) com-pared to moderate-size femtocells (whose cache capacity is100), and the latter’s capacity is smaller than the cellularBS which can store up to Ms = 800 contents. Bandwidthcost increases as we move from the small to moderate fem-tocell and to cellular BS; recalling the cost functions fromthe previous section, the respective values of Rs are 5, 2.5and 1.
In Figure 4 we first plot the performance achieved by thejoint-price signaling scheme along with the performance ofthe two heuristic schemes. Again, we observe that the joint-price scheme can achieve a large range of performance values,which is in contrast to the heuristic schemes whose perfor-
41
mance is fixed at either optimizing bandwidth cost or cache-miss probability, without regarding for the other metric.
In the thus far implemented joint-price schemes, we havebeen assigning requests to download chunks strictly from theset of min-priced servers. We now relax this strict consid-eration by regarding any BS, whose price is within a toler-ance value of τ from the minimum price, as eligible to servechunks for an incoming request (provided as before the el-igible BSs are connected to the request’s location); chunksare however downloaded by sorting the eligible BSs in theincreasing order of their prices. In Figure 4 we have depictedthe performance of the joint-price scheme for different val-ues of τ , where the earlier strict min-price approach nowcorresponds to τ = 0; the case where all connected BSs (ir-respective of their prices) are eligible corresponds to τ =∞.
We observe that as τ increases the trade-off curves shiftdownwards, implying that it is possible to achieve a lowercache-miss probability for a given target bandwidth cost.However, the range of trade-off that is possible reduces as τincreases. For instance, using τ = 100 it is not possible toreduce the bandwidth cost to less than 40; In fact, τ = ∞case achieves the lowest cache-miss probability, however atthe expense of being constrained to operate at a higher rangeof bandwidth cost. This shift in performance is because, asτ increases, more BSs become eligible to serve a requestso that the cache-miss probability decreases; however, moreBSs are now loaded, resulting in a bandwidth scarce system.Thus, further trade-off in performance can be achieved bysuitably choosing the value of τ .
In summary, the joint-price scheme, in conjunction withthe CLRU policy for cache management, can be a practicalcandidate for request assignments in futuristic cellular net-works where a dense deployment of heterogeneous femtocellbase-stations are expected. The cache-miss cost b, alongwith the tolerance value τ , can be used to serve as “tunableknobs” to trade-off one metric for the other.
5. CONCLUSIONWe proposed an optimization framework to study the prob-
lem of cost minimization in cellular cache networks. Towardsthis direction, we proposed an online algorithm for cachingand request assignments which is shown to be optimal andstable in a limiting regime that is obtained by scaling thearrival rates and the content chunking. Based on the onlinealgorithm, we proposed a light-weight joint-price scheme forrequest assignment that can be used in conjunction with theLRU cache management strategy. Through simulations wefound that our joint-price based request assignment strategyoutperforms the common practices of routing purely basedon either load or content availability. Our proposed joint-price routing mechanisms are thus an appealing candidatecombining sound theoretical guarantees with good experi-mental performance while being simple to implement.
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