Joint Caching and Pricing Strategies for PopularContent in Information Centric Networks

Mohammad Hajimirsadeghi, Student Member, IEEE, Narayan B. Mandayam, Fellow, IEEE,and Alex Reznik, Senior Member, IEEE

Abstract—We develop an analytical framework for distributionof popular content in an Information Centric Network (ICN)that comprises of Access ICNs, a Transit ICN and a ContentProvider. Using a generalized Zipf distribution to model contentpopularity, we devise a game theoretic approach to jointlydetermine caching and pricing strategies in such an ICN. Underthe assumption that the caching cost of the access and transitICNs is inversely proportional to popularity, we show that theNash caching strategies in the ICN are 0-1 (all or nothing)strategies. Further, for the case of symmetric Access ICNs, weshow that the Nash equilibrium is unique and the caching policy(0 or 1) is determined by a threshold on the popularity of thecontent (reflected by the Zipf probability metric), i.e., all contentmore popular than the threshold value is cached. We also showthat the resulting threshold of the Access and Transit ICNs, aswell as all prices can be obtained by a decomposition of the jointcaching and pricing problem into two independent caching onlyand pricing only problems.

Index Terms—Information Centric Networking (ICN), 5G,Content Delivery Networks (CCN), Network Pricing, GameTheory, Content Popularity, Zipfs Law.

I. INTRODUCTION

THE vast majority of Internet traffic relates to contentaccess from the sources such as YouTube, Netflix, Bit

Torrent, Hulu, etc. This rapid increase of content delivery inthe Internet has revealed the need for a different networkingparadigm. Further, as Fig. 1 describes, the emerging trend isthat the users are just interested in the information (content),and not where it is located or perhaps, even how it is delivered.This high increase in demand for video content in the Internetand the need for new approaches to control this large volumeof information have motivated the development of future Inter-net architectures based on named data objects (NDOs) insteadof named hosts [1]. Such architectural proposals are generallyreferred to as Information Centric Networking (ICN) whichis a new communication paradigm to increase the efficiencyof content delivery and also content availability [2]-[4] offuture fifth generation (5G) networks. In this new concept, thenetwork infrastructure actively contributes to content cachingand distribution and every ICN node can cache and serve therequested content. To fulfill that purpose, several architectureshave been proposed for ICN to reflect current and future

Manuscript received April 30, 2016; revised September 29, 2016 andDecember 14, 2016; accepted January 23, 2017.

M. Hajimirsadeghi and N. B. Mandayam are with WINLAB,Rutgers University, Piscataway, NJ 08854-8060 USA (e-mail:narayan@winlab.rutgers.edu; mohammad@winlab.rutgers.edu).

A. Reznik is with Hewlett Packard Enterprise (HPE), New York 10011,USA (e-mail: alex.reznik@hpe.com ).

needs better than the existing Internet architecture [5]-[11].To provide preferable services to the users in ICN, Internetservice providers (ISPs) or access ICNs should be able tomaintain quality of service (QoS) by improving the responsetime for file request. They need to cache the frequentlyrequested or popular content locally and store them near theusers in the network. To provide QoS, in-network caching isintroduced to provide the network components with cachingability. Therefore every node actively contributes in contentcaching and operates as a potential source of content. Thisleads to reduction in network congestion and user accesslatency and increase in the throughput of the network bylocally caching the more popular content [12]-[16]. Severalworks have claimed that web (file) requests in Internet aredistributed according to Zipfs law [17]-[23]. Zipfs law statesthat the relative probability of a request for the ith mostpopular content is proportional to 1

i . However, several otherstudies have found out that the request distribution generallyfollows generalized Zipf distribution where the request rate forthe ith most popular content is proportional to 1

iγ and γ is apositive value less than unity [24]-[25].

Since each ICN requires cooperation in caching from otherICNs to provide a global high performance network, it isnecessary to have pricing policies to incentivize all the ICNsto contribute to the caching process [26]. Several works havebeen done to address the problem of the economics of servicepricing in current Internet and interconnection networks [27]-[31]. Using contemporary pricing policies cannot incentivizethe lower tier ISPs to cooperate in the future Internet archi-tecture [32]; hence, we need to have new models to providethem with monetary incentives to collaborate in caching anddistributing content when content with different popularitiesare available in the network.

In this paper, we investigate joint caching and pricingstrategies of the access ICN, the transit ICN and the contentprovider based on content popularity. We study Nash strategiesfor a non-cooperative game among the above entities using aprobabilistic model by assuming that access requests generallyfollow the generalized Zipf distribution. We then use theinsights gained to simplify the problem by replacing twocaching threshold indices instead of caching parameters forthe symmetric case; where all access ICNs have the sameparameters. In our model, the ICNs caching costs vary inrespect to content popularity while the content provider costper unit data is fixed for all content types.

The rest of the paper is organized as follows. Section IIreviews some related works. In Section III, we describe the

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Content Provider

Transit ICN

Access ICN

Access ICN

Access ICN

Content request in ICN

Content request in IP based Network

Fig. 1. ICN communication model: unlike the current Internet which theusers are interested in the location of the files, the ICN users just look for thecontent regardless of file location [47].

system model. The problem formulation for joint caching andpricing strategies is described in Section IV. We examine thegeneralized scenario for arbitrary number of access ICNs inSection V. The symmetric access ICNs scenario is analyzedin Section VI. Section VII presents numerical results on theresulting caching and pricing strategies, and we conclude inSection VIII.

II. RELATED WORK

The benefits of in-network caching have been investigatedbefore in the setting of distributed file systems in severalrecent works. In [15], the problem of caching is studied froman information-theoretic viewpoint. They propose a codedcaching approach that in addition to the local caching gainis able to achieve a global caching gain. A novel CooperativeHierarchical Caching (CHC) framework is proposed in [33],[34] in the context of Cloud Radio Access Networks (C-RAN).In [35], a collaborative joint caching and processing strategyfor on-demand video streaming in mobile-edge computing net-work is envisioned.Content caching and delivery in device-to-device (D2D) networks have been studied in [36]. The aim ofthis work is to improve the performance of content distributionby the use of caching and content reuse. Several approachessuch as base station assisted D2D network and other schemesbased on caching at the user device are compared to showthe improvement of the network throughput in the presenceof in-network caching. Another recent article [37] studiesthe limitation of current reactive networks and proposes anovel proactive networking paradigm where caching plays acrucial role. It shows that peak data traffic demands can besubstantially reduced by proactively serving users’ demands

via caching. [38] has used a mean field game model to studydistributed caching in ultra dense small cell networks. Zhang etal have proposed an optimal cache placement strategy based oncontent popularity in content centric networks (CCN) in [39].The authors in [40] have proposed a collaborative caching andforwarding design for CCN. The problem of joint caching andpricing for data service for a single Internet service provider(ISP) is studied in [41]. Similar problem but for multiple ISPsin the setting of small cell networks is investigated in [42]using a Stackelberg game. [43] proposes an incentive proac-tive cache mechanism in cache-enabled small cell networks(SCNs) in order to motivate the content providers to participatein the caching procedure.

One of the earliest studies of economic incentives in ICNshas been conducted by Rajahalme et al. [32] and has demon-strated that top level providers are not willing to cooperatein the caching process since they cannot get enough revenue.Another recent study by Agyapong et al. [44] has addressedthe economic incentive problem in ICN by building a simpleeconomic model to evaluate the incentive of different types ofnetwork players in a hierarchical network infrastructure. Theyqualitatively showed that without explicit monetary compen-sation from publishers, the network will fail to deploy thesocially optimal number of caches. Few prior works have usedgame theoretic approaches to solve the problem of cachingand pricing in ICN. In [45] the authors have presented a gametheoretic approach using matrix payoff to analyze the processof economic incentives sharing among the major networkcomponents. A pricing model was proposed in [46] to studythe economic incentives for caching and sharing content inICNs which consists of access ICNs, a transit ICN and acontent provider. This work has shown that if each player’scaching (pricing) strategy remains fixed, the utility of eachplayer becomes a concave function of its own pricing (caching)strategy. Therefore a unique Nash Equilibrium exists in a non-cooperative pricing (caching) game among different players.In our earlier work [47], a similar model was adopted toaddress the problem of joint caching and pricing strategies ina network including two access ICNs, one transit ICN andone content provider. However, content popularity was nottaken into account and the ICNs were agnostic to content type.The interaction of the above entities is modeled using gametheoretic approach. It was shown that each player can optimizeits caching and pricing strategies in a non-cooperative game.At the Nash Equilibrium, the caching strategies turn out to be0-1 (all or nothing) strategies, where each access ICN cachesall the requested demand if its caching cost is less than thetransit ICNs caching cost. When the caching cost of the accessICN is higher than the transit ICN’s, all the requested contentwill be served by either the transit ICN or the content provider,whichever has the lower caching cost. It means that the contentwould be cached in the ICN with smallest caching cost. To thebest of our knowledge, none of the earlier works consider theproblem of joint caching and pricing in an ICN in the presenceof content with different popularity. In contrast, in this paper,we study joint caching and pricing strategies with the notion ofcontent popularity for an ICN network with different elementsconsidering asymmetric utility functions.

2

III. SYSTEM MODEL

A. Content Popularity

There are M different types of content in the network thateach user is trying to access. Each type of content has adifferent measure of popularity reflected by the probabilityof requests for it. We consider a model where the popularityof content is uniformly similar in all parts of the network,i.e., all users in the network have the same file popularitydistribution. Analyzing the impact of different per-user filepopularities is an open problem. As in previous works (e.g.[14]-[17], [23]-[25], [36]-[40]), in this paper the distributionof user requests for content is described by a generalized Zipfdistribution function as follows:

qM (m) =Ω

mγ,m = 1, ...,M, (1)

where Ω =

(M∑i=1

1iγ

)−1and 0 ≤ γ ≤ 1 is the exponent

characterizing the Zipf distribution in which γ = 0 makesthe distribution uniform and all the content will be identicalin popularity, whereas the case of γ = 1 corresponds toone where the content popularity distribution is following theclassic Zipf’s law and more popular content is dominant inthe network. The content is ranked in order of their popularitywhere content m is the mth most popular content, i.e., m = 1is the most popular content and m = M is the least popularcontent.

In most of the aforementioned works that have studiedin-network caching, the popularity profile of content wasassumed to be identical and perfectly known by all the networkcomponents. In reality, the demand and popularity are notpredictable and certain [48], [49]. The problem of cachinghas been studied in [50] wherein the users have access todemand history but no knowledge about popularity. Severalother papers have used learning-based approaches to estimatethe popularity profile at the user side [51]-[56]. While contentlearning is more accurate for modeling content popularity,the reason we have used the Zipf model is due to (1)experimental results showing reasonable fit to the Zipf modeland (2) analytical tractability provided by the Zipf model.Our framework can be extended by changing the demandmodel and considering a repeated game with a parametric Zipfdistribution. In each time slot of the game, this parameter canbe estimated in an optimal way using a learning process.

B. Cost Model

Although the prices are fixed for all types of content, sincethe ICNs want to earn more profit by caching the content, theyare more willing to locally store the content which is morepopular. Thus, the access ICNs’ and the transit ICN’s cachestreat the content differently in regard to their popularity. Asthe content gets more popular, the ICNs incur less cachingcosts to locally store the content.

Definition 1. For a finite cache, the caching costs of accessand transit ICNs is defined to be inversely proportional to the

content popularity as follows

cxi =cx0

qM (i), (2)

where, without loss of generality, cx0is a fixed initial caching

cost at ICN x.

Using equations (1) and (2), we see in Fig. 3, for fixedvalues of i, cxi is a decreasing function of γ when i is small(i.e., more popular content). On the other hand, cxi is anincreasing function of γ when i is large (i.e., less popularcontent). Unlike the access and transit ICNs, the contentprovider has no priority for caching the content and cachesall types of content. The content provider incurs the constantcost cO for every unit of data that it serves for the transit ICN.

C. Access and Demand Model

For simplicity in illustration, as shown in Fig. 2, we beginwith a hierarchical network model [46], [57] with two accessICNs (A and B), one transit ICN (C), one content provider(O) and an arbitrary number of users who can switch fromone access ICN to another. The access ICNs connect the end-users to the content network and the transit ICN provideswide-area transport for the access ICNs while the contentprovider provides the content for the users. Fig. 2 also showsthe monetary and data flows among different entities with thevarious prices described in Table I. The network economydepends on two effective factors: caching and pricing. Underthe assumption that each ICN can have access to all content,it can decide to either cache the entire or portion of therequested content, or get it from somewhere else. The cachingstrategy adopted by each entity is denoted by the parameter αthat takes values in the interval [0, 1]. Every ICN decides tocache different types of content independently, therefore, wehave a specific caching variable for each type of content. Wehave denoted αI,Si , where I can either be access or transitICN and S is any cache in the network (possibly anotherICN or content provider), as the fraction of ICN I’s demandfor content type i that comes from cache S. Each ICN alsohas different pricing strategies. These strategies are the pricesthat each player charges others for the provided service. Thepricing is based on the usage, i.e., price per unit data. Eachaccess ICN sets two different prices: (1) the network priceper unit data for transporting the content; and (2) the storageprice per unit data for providing content from its cache forother ICNs. For example, the network and storage prices foraccess ICN A are denoted as P

(n)A and P

(s)A , respectively.

The total price per unit data is the sum of these two pricesand is denoted by PA = P

(n)A + P

(s)A . We will find it useful

to utilize an alternative form of the above. Following thecharging policies of several storage services for the Internetsuch as Amazon S3, it would be practical to assume that thestorage price is typically less than the network price. The linearrelationship between network price and storage price whileempirical, has been used earlier in [46] and we follow thisassumption. It can be represented by P

(n)A = βAP

(s)A where

βA > 1. Thus, the relationship between PA and P (s)A would be

P(s)A =

(1

1+βA

)PA. As a result, each access ICN or transit

3

Fig. 2. Interaction between different entities in a simplified model of an ICN[47].

ICN will have a set of strategies for pricing in the interval[0,∞). The content provider pricing strategies set also consistsof the content price P (c)

O that the users should pay for contentand the storage price P

(s)O which is the price for providing

the content from the content provider cache. The access ICNA and B charge prices PA and PB to their users and P

(s)A

and P(s)A to the transit network if they store or forward the

content that the transit ICN had asked for. The transit ICN Ccharges access ICNs A and B with price PC , if it stores orforwards their requested content. The content provider chargesusers with content price P (c)

O and transit ICN C with storageprice P (s)

O if it stores transit ICN requested content.To model the behavior of users, we have considered content

demand at each access ICN to be a linear function of theprices.

Definition 2. The users’ demands are affected by both contentprice and access price and defined to be a linear function asfollows:

σA = 1− ρAPA + ρBPB − ρ0P (c)o

σB = 1 + ρAPA − ρBPB − ρ0P (c)o

(3)

where ρA, ρB and ρ0 are the reflective coefficients of theprices’ influence on users demands and are positive. Thedemands are normalized per unit data. We observe that theusers demands directly depend on the access ICNs pricesand content price. For example, as the content price P

(c)O

increases, the users demands decrease. If one of the accessICNs increases its price, the users will switch to the otherICN. Table I summarizes our notation.

As illustrated in Fig. 2, the interaction between differententities in the ICN model results in a conflict among the ICNs

TABLE ISUMMARY OF NOTATION

Notation DescriptionK Number of access ICNs

P(n)k

Network price of ICN k per unit data

P(s)k

Storage price of access ICN or content provider k per unitdata

PC Transit price charge by transit ICN CPk Access ICN k’s price to its users per unit data

P(c)O Content price of content provider O per unit data

ρkReflective coefficient of prices influence on access ICN k’suser demand

σk Normalized total user demands for ICN k per unit data

σkiNormalized total user demands for ICN k for content typei per unit data

αI,SiFraction of ICN I’s demand for content type i that comesfrom cache S

cki Caching cost of ICN k for content type ick0

Initial caching cost of ICN kcO Content provider O cost

βkScaling parameter between network price and storage price(greater than unity)

γ Zipf popularity law exponentTh Caching threshold index for access ICNsThC Caching threshold index for transit ICN C

qM (m)Popularity (request rate) for content type m over a set ofM content

(players) when they unilaterally try to maximize their revenue.In the following section, we use a game theoretic approach tosolve the joint caching and pricing strategies for each entityin the ICN network.

IV. JOINT CACHING AND PRICING STRATEGY

A. Utility Function

Each ICN can cache the content or just forward the requeststo other ICNs or content provider based on the utility thatit gains. The utility function for each player is defined asthe utility received by providing the services for others. Eachplayer incurs a caching cost with respect to the popularity ofthe content when it stores a unit of data. Therefore, as shownin Fig. 2, the utility functions for the access ICNs A and B, thetransit ICN C and the content provider O can be formulatedas an opportunistic function in terms of the prices, cachingcosts, demands and fraction of content stored and popularity.

The access ICN A incurs a caching cost of cAi for contenttype i if it decides to store a unit of data. Therefore, the utilityfunction of the access ICN A is given as the average utilityfor all different types of content as follows:

UA =

M∑i=1

qM (i)

σAαA,Ai (PA − cAi) +σAαA,Outi (PA − PC) +

σBαB,Ai

((1

1+βA

)PA − cAi

) , (4)

where the first term in (4) is the utility that results when ICNA stores a portion of its users demands in its own cache.qM (i) denotes the popularity (request rate) of content type iover a set of M different types of content, σA is the totaldemand that users have requested to the access ICN A andαA,Ai is the fraction of ICN A’s demand for content type ithat is going to be served by ICN A’s cache. Therefore, σAmultiplied by qM (i) and αA,Ai is the demand of access ICN

4

A for content type i that is served by transit ICN A’s cacheand (PA − cAi) is the revenue of access ICN A by servingthis portion of the requested demand. The second term is theutility that results when ICN A forwards a portion of its usersdemand to the transit ICN C. The third term is the utility thatresults when the transit ICN C forwards a portion of ICN B’susers’ demand to ICN A. The access ICN A can control onlythe caching and pricing parameters αA,Ai , PA and P (s)

A . Notethat αA,Outi = 1−αA,Ai or αA,Outi = αA,Bi+αA,Ci+αA,Oi .The utility function of the access ICN B can be defined in asimilar way as follows:

UB =

M∑i=1

qM (i)

σBαB,Bi (PB − cBi) +σBαB,Outi (PB − PC) +

σAαA,Bi

((1

1+βB

)PB − cBi

) . (5)

The access ICN B can control only the caching and pricingparameters αB,Bi , PB and P

(s)B . Note that αB,Outi = 1 −

αB,Bi or αB,Outi = αB,Ai +αB,Ci +αB,Oi . The transit ICNC gains a profit if the access ICNs A and B request a contentthrough it. Equation (6) consists of two terms. The first termis the utility that results when transit ICN C stores ICN A andB’s content in its own cache and the second term is the utilitythat results when it forwards ICN A and B’s content to otherICN’s or content provider. We model the ICN C’s utility inthe following way:

UC =∑

X∈A,B

M∑i=1

qM (i)σXαX,Ci(PC − cCi)+∑X∈A,B

∑L∈A,B,O,L6=X

L∑i=1

qM (i)σXαX,Li(PC − P(s)L )

,

(6)where the transit ICN C has control over caching variablesαA,Bi , αA,Ci , αA,Oi , αB,Ai , αB,Ci , αB,Oi and PC . Finally, ifthe requests are not served by any access ICNs or transit ICN,they will be forwarded to the content provider that has allthe content in its cache and the costs for all types of contentare identical and equal to cO. The content provider’s utilityfunction can be expressed formally as

UO =M∑i=1

qM (i) [σAαA,Oi + σBαB,Oi ](P

(s)O − cO

)+ (σA + σB)P

(c)O .

(7)

The first term in (7) results when the content provider Ocharges transit ICN C with storage price to deliver the re-quested content to it. The second term comes from the contentprice that content provider charges the users. The contentprovider can control pricing parameters P (c)

O and P (s)O .

Because of the competitive nature of this problem, wecan present a solution in the analytical setting of a gametheoretic framework. Let G = [N, Sj , Uj (.)] denotethe non-cooperative game among players from the set N =A,B,C,O , where Sj = (Pj , αj) is the set of jointcaching (αj) and pricing (Pj) strategies and Uj (.) is theutility function of player j. The strategy space of all the entitiesexcluding the jth player is denoted by S−j . In the joint caching

and pricing game, each player tries to maximize its own utilityby solving the following optimization problem for all j ∈ N ,

maxsj∈Sj

Uj (sj , S−j) ,∀j ∈ N. (8)

It is necessary to characterize a set of caching and pricingstrategies where all the players are satisfied with the utilitythey receive, given the strategy selection of other players.Such an operating point, if it exists, is called equilibrium. Thenotion that is most widely used for game theoretic problems isthe Nash Equilibrium(NE) [58]. A set of pricing and cachingstrategies S∗j =

(P ∗j , α

∗j

)constitutes a NE if for every j ∈ N ,

Uj(s∗j , S−j

)≥ Uj (sj , S−j) for all sj ∈ Sj . The NE of the

game is one where no player benefits by deviating from herstrategy unilaterally.

B. Characterization of Nash Strategies

To find the NE using the best response functions we needto solve the four following optimization problems for ICN A,ICN B, ICN C and the content provider O, respectively. Themaximization problem for ICN A is:

maxαA,Ai ,PA

UA =M∑i=1

qM (i)

σAαA,Ai (PC − cAi) +σA (PA − PC) +

σBαB,Ai

((1

1+βA

)PA − cAi

)

s.t. 0 ≤ αA,Ai ≤ 1 , PA > 0

,

(9)where the access ICN A tries to maximize her utility bychanging its caching (αA,Ai ) and pricing (PA) strategies whileother players’ strategies are unknown to her. Similarly, themaximization problem for ICN B is:

maxαB,Bi ,PB

UB =M∑i=1

qM (i)

σBαB,Bi (PC − cBi) +σB (PB − PC) +

σAαA,Bi

((1

1+βB

)PB − cBi

)

s.t. 0 ≤ αB,Bi ≤ 1 , PB > 0

.

(10)The transit ICN C maximization problem is given as thefollowing equation:

maxαA,Ci ,αB,Ci ,αA,Bi ,αA,Oi ,αB,Ai ,αB,Oi ,PC

UC =M∑i=1

qM (i)σA

×

αA,Ci (PC − cCi) +

αA,Bi

(PC − P (s)

B

)+ αA,Oi

(PC − P (s)

O

) +

M∑i=1

qM (i)σB

αB,Ci (PC − cCi) +

αB,Ai

(PC − P (s)

A

)+ αB,Oi

(PC − P (s)

O

) ,

s.t.

αA,Ai + αA,Bi + αA,Ci + αA,Oi = 1αB,Bi + αB,Ai + αB,Ci + αB,Oi = 10 ≤ αA,Bi ≤ 1, 0 ≤ αA,Ci ≤ 1, 0 ≤ αA,Oi ≤ 10 ≤ αB,Ai ≤ 1, 0 ≤ αB,Ci ≤ 1, 0 ≤ αB,Oi ≤ 1PC ≥ 0

(11)

5

and the content provider O maximizes its utility using equation(12)

maxP

(c)O ,P

(s)O

UO = (σA + σB)P(c)O

+M∑i=1

qM (i) [σAαA,Oi + σBαB,Oi ](P

(s)O − cO

)s.t. P

(C)O > 0, P

(s)O > cO

(12)

Theorem 1. The caching variables αI,Si take on values ofeither 0 or 1 at the equilibrium of the caching and pricingstrategies game.

Proof: The solution of a maximization (minimization)problem with an objective function that has a linear re-lationship with the variable is the boundary point of thefeasible interval. Therefore, since the relationship betweenutility function and caching parameters are linear to maximizethe utility functions, they just take on the boundary values.Since αI,Si ∈ [0, 1], therefore, they can be either 0 or 1.

Given Theorem 1, it follows that all caching variables adoptbinary values. For example, according to equation (9), if PC >cAi , the caching variable αA,Ai should be 1 to maximize theaccess ICN A’s utility function. Whenever PC > cAi it meansthat the transit price for delivering the requested content typei to users under the access ICN A is smaller than the cachingcost of access ICN A for locally storing the requested fileitself. Therefore the access ICN decides to store the contentlocally in its cache rather than transferring the request to transitICN C. That is the reason that caching variable αA,Ai getsthe value 1. The caching strategies at equilibrium for differentconditions are summarized in Table II.

Unlike the caching parameters which only get binary values;the pricing parameters can be continuous. Therefore, thisoptimization problem is a mixed integer program with multipleobjective functions, and in general, the uniqueness of theNE in terms of pricing and caching strategies cannot becharacterized.

V. GENERALIZATION TO K ACCESS ICNS

We can extend the case of two access ICNs, one transit ICNand one content provider to a generalized scenario of K accessICNs, one transit ICN and one content provider. We considerK = A1, A2, ..., AK as the set of access ICNs which areconnected to transit ICN C and α is the set of all cachingvariables. The demand function of each access ICN is definedin Definition 3.

Definition 3. The received demands at access ICN Aj isdefined as

σAj = 1− ρAjPAj + 1K−1

[K∑

k=1,k 6=jρAkPAk

]−ρ0P (c)

o ,∀j = 1, ...,K.

(13)

The received demand of content type i by access ICN Ajcan be shown as σAji = σAjqM (i). Following the previous

section, the maximization problem of each access ICN Aj ∈K can be defined as follows:

maxαAj,Aji

,PAj

UAj =M∑i=1

qM (i)

σAjα(Aj,Aj)i

(PC − cAji

)+σAj

(PAj − PC

)+((

11+βAj

)PAj − cAji

)×∑

k∈K,k 6=Ajσkα(k,Aj)i

s.t. 0 ≤ αAj ,Aji ≤ 1 , PAj > 0

(14)The transit ICN C maximization problem is given by thefollowing equation:

maxα,PC

UC =∑k∈K

M∑i=1

qM (i)σkαk,Ci(PC − cCi)

+∑k∈K

∑L∈K∪O,L 6=k

M∑i=1

qM (i)σkαk,Li(PC − P(s)L )

s.t.

α(Aj ,Aj)i+

∑L∈K∪C,O

L 6=Aj

α(Aj ,L)i= 1

0 ≤ α ≤ 1PC ≥ 0

(15)and the content provider O maximizes its utility using equation(16).

maxP

(c)O ,P

(s)O

UO =M∑i=1

qM (i)

[ ∑k∈K

σkαk,Oi

](P

(s)O − cO

)+∑k∈K

σkP(c)O

s.t. P(C)O > 0, P

(s)O > cO

(16)Theorem 1 can be extended for the generalized K Access

ICN case with the same reasoning and all the caching variablestake binary values (i.e., all or nothing 0-1 strategies). So,the joint caching and pricing strategy game in the generalform is also a mixed integer program. In the next section wewill simplify the problem with the assumption of symmetricaccess ICNs with similar characteristics and try to give someanalytical and intuitive results.

VI. SYMMETRIC ACCESS ICNS SCENARIO ANALYSIS

In the previous section, the general form of the caching andpricing strategies for ICNs was formulated through equations(14)-(16) as a set of mixed integer programs. In this section,in order to analytically study the equilibrium of our proposedmodel, we consider the symmetric scenario, where all accessICNs have the same specifications. For the symmetric scenario,where all the access ICNs are exactly the same, we considerρk = ρ, βk = β, ck0 = c0∀k ∈ K.

Theorem 2. In the symmetric case, for each content type i,αAk,Aji = 0, ∀k 6= j.

Proof: According to equations (14), when access ICNAj receives a request for content type i and transit price

6

TABLE IICACHING TABLE FOR EACH CONTENT TYPE i

Condition αA,Ai αA,Bi αA,Ci αA,Oi αB,Bi αB,Ai αB,Ci αB,Oi1 PC >

(cAi&cBi

)1 0 0 0 1 0 0 0

PC > cAi&PC < cBi2 cCi = min

cCi , P

(s)O , P

(s)A , P

(s)B

1 0 0 0 0 0 1 0

3 P(s)O = min

cCi , P

(s)O , P

(s)A , P

(s)B

1 0 0 0 0 0 0 1

4 P(s)A = min

cCi , P

(s)O , P

(s)A , P

(s)B

1 0 0 0 0 1 0 0

PC < cAi&PC > cBi5 cCi = min

cCi , P

(s)O , P

(s)A , P

(s)B

0 0 1 0 1 0 0 0

6 P(s)O = min

cCi , P

(s)O , P

(s)A , P

(s)B

0 0 0 1 1 0 0 0

7 P(s)B = min

cCi , P

(s)O , P

(s)A , P

(s)B

0 1 0 0 1 0 0 0

PC <(cAi&cBi

)8 cCi = min

cCi , P

(s)O , P

(s)A , P

(s)B

0 0 1 0 0 0 1 0

9 P(s)O = min

cCi , P

(s)O , P

(s)A , P

(s)B

0 0 0 1 0 0 0 1

is greater than its caching cost for that particular type ofcontent (PC ≥ cAji ), access ICN Aj decides to serve therequested content itself by adopting value 1 for cachingparameter αAj ,Aji . Since αAj ,Aji = 1, then all the othercaching parameters for content type i would be equal to zero.On the other hand, if the transit price is less than the accessICN’s caching cost for content type i (PC < cAji ), theaccess ICN Aj will forward the request to the transit ICNC to be served by choosing αAj ,Aji = 0. When the transitICN C receives the request, it should decide to either cachethe content or forward it to the content provider or otheraccess ICNs based on the payoff that it gains according toequations (15). Considering the Theorem 1 and the constraintα(Aj ,Aj)i

+∑

L∈K∪C,O,L 6=Ajα(Aj ,L)i

= 1, one of the caching

parameters should be 1 and others should be 0. If cCi or P (s)O

are the minimums amongcCi , P

(s)O , P

(s)k ∀k ∈ K, k 6= Aj

,

the transit ICN C (αAj ,Ci = 1) or the content providerO (αAj ,Oi = 1) will serve the request, respectively. Nowassume that one of the access ICN’s storage price P

(s)k

is the minimum. If P(s)k is the minimum, it means that

P(s)k <

(cCi&P

(s)O

). On the other hand, P (s)

k > cKi inorder to the access ICN K accepts the request; otherwise itdoes not accept the content to prevent from losing payoff.Therefore, cKi < P

(s)k < cCi . Note that cKi = cAji in

symmetric scenario. Thus P (s)k should adopt a value greater

than access ICN Aj’s caching cost and less than transit ICNC’s caching cost for content type i (cAji < P

(s)k < cCi ).

On the other hand, we know that PC < cAji , therefore

PC < cAji < P(s)k <

(cCi&P

(s)O

). It means that (PC − cCi),(

PC − P (s)k

)and

(PC − P (s)

O

)are negative and that is a

contradiction since the transit ICN C is trying to choose PCin a way to get at least zero payoff. Thus P (s)

k can neverbe the minimum value among the others and accordinglyαAk,Aji = 0, ∀k 6= j,∀i, in the symmetric scenario case.

To better understand the implication of Theorem 2, we

can refer to Table II that shows the caching strategies at theequilibrium for asymmetric case with two access ICNs whichare not identical. By assuming two identical access ICNs thathave the same characteristics, the cases 2-7 can be removedand the table will reduce to just three cases and in all of thesecases αA,Bi = αB,Ai = 0. This table can also be extendedfor the generalized scenario with K access ICNs.

What the above theorem reveals is that in the symmetriccase, access ICNs have no motivation to serve each other’susers. This is not against the philosophy of the content centricnetwork paradigm, since in this setup the access ICNs and alsothe transit ICN are capable of caching the requested content.Besides, this theorem is just for the Symmetric Scenario andin the asymmetric setup the access ICNs are able to cacherequests for users of other access ICNs.

Moreover, according to Theorem 2, when the system issymmetric, we can add the following facts to our models.

αAj ,Ci = αAk,Ci , ∀iαAj ,Oi = αAk,Oi , ∀i (17)

So using Theorem 2 and (17), we can reformulate ourmaximization problem described in (14)-(16) for symmetriccase. The maximization problems for access ICN Aj can beexpressed in equation (18) as follows:

maxPAj ,αAj,Aji

UAj = σAjM∑i=1

qM (i)

αAj ,Aji

(PC − cAji

)+(PAj − PC

) s.t. αAj ,Aji ∈ 0, 1 , PAj > 0

(18)The maximization problem for transit ICN C can be definedin the following equation:

maxαAj,CiαAj,OiPC

UC =K∑k=1

σAk .M∑i=1

qM (i)×

αAj ,Ci (PC − cCi) +

αAj ,Oi

(PC − P (s)

O

)

s.t.αAj ,Aji + αAj ,Ci + αAj ,Oi = 1,

αAj ,Ci ∈ 0, 1 , αAj ,Oi ∈ 0, 1 , PC > 0,(19)

7

and finally, the content provider maximization problem can bereformulated as the following

maxP

(c)O

P(s)O

UO =K∑k=1

σAk

[P

(c)O +

M∑i=1

qM (i)αAj ,Oi

(P

(s)O − cO

)]

s.t. P(C)O > 0, P

(s)O > cO

(20)As mentioned in Theorem 1, the caching parameters still

take on binary values. Moreover, since the content popularityprobability function qM (i) is monotonically decreasing; ac-cording to (2), the access and transit ICNs’ caching costs aremonotonically increasing depending on the content type. Foraccess ICN Aj , the caching parameter αAj ,Aji adopts value1 when the transit price PC is greater than the access ICNcaching cost cAji . Hence, if αAj ,Aji is 1 for content type i,it would also be 1 for content type i − 1. It means that ifaccess ICN decides to cache the content type i, it will cacheall the other content that are more popular than it. So, therewould be a caching threshold index for the number of contenttype that access ICN is willing to locally store. We denote theoptimum caching threshold index by ThAj for access ICN Aj .In the symmetric scenario, the caching threshold indices ThAjare identical for all access ICNs, so we consider Th as thecaching threshold index for all the access ICNs. Since cachingcontent types Th+ 1 to M is not beneficial for access ICNs,they will forward these to the transit ICN C to be served.The transit ICN C should decide to either serve the contentitself or forward it to somewhere else. In the symmetric case,as mentioned in Theorem 2, in case it decides not to servethe content itself, it can forward it to the content provider O.According to (19), if the content provider storage price P (s)

O

is greater than the transit ICN C caching cost cCi , the cachingparameter αAj ,Ci adopts value 1 and the transit ICN C cachesthe content type i. On the other hand, if P (s)

O is less than cCi ,the caching parameter αAj ,Ci will be 0 and αAj ,Oi adoptsvalue 1. In this case, the content provider O will take care ofthe request for content type i. As discussed, for access ICNs,the transit ICN also can have a caching threshold index. Itmeans that if it caches content type i, it would also be ableto cache the content type i− 1. So there would be a cachingthreshold index for the number of content type that the transitICN C is willing to locally store. We denote this cachingthreshold index by ThC . So the transit ICN C will serve thecontent with popularity index Th + 1 to ThC and contentwith popularity index greater than ThC will be served by thecontent provider O. We can summarize these new parametersin (21) and (22).

αAj ,Aji =

1 i ≤ Th0 i > Th

,∀j ∈ 1, ...,K (21)

αAj ,Cji =

1 Th+ 1 ≤ i ≤ ThC0 i ≤ Th ,∀j ∈ 1, ...,K

αAj ,Oji =

1 ThC + 1 ≤ i ≤M0 i ≤ ThC

,∀j ∈ 1, ...,K(22)

Thus, all the caching variables α will be replaced by twocaching threshold indices Th and ThC . By (21) and (22),the problem set of (18)-(20) can be rearranged using newparameters Th and ThC . The new maximization problem foraccess ICN Aj is given as the following:

maxPAj ,Th

UAj = σAj

[PAj − Th.c0 − PC

M+1∑i=Th+1

qM (i)

]s.t. 0 ≤ Th ≤M, PAj > 0

(23)The transit ICN C maximizes its utility function as thefollowing:

maxPC ,ThC

UC =(K −Kρ0P (c)

O

)

PCM+1∑

i=Th+1

qM (i)

− (ThC − Th) cC0

−P (s)O

M+1∑i=ThC+1

qM (i)

s.t. Th ≤ ThC ≤M, PC > 0

(24)And the content provider O maximization problem is formu-lated using new parameters in (25)

maxP

(c)O ,P

(s)O

UO =(K −Kρ0P (c)

O

)P

(c)O +

(P

(s)O − cO

).

M+1∑i=ThC+1

qM (i)

s.t. P

(C)O > 0, P

(s)O > cO

(25)Note that in the above equations qM (0) = qM (M + 1) = 0.

The problem of joint caching and pricing strategies forthe case of symmetric ICNs can be decomposed into twoindependent caching and pricing optimization problems. Thecaching problem is dealing with the parameters that affect thecaching process and is stated as follows:• Caching Problem:

minTh

Th.c0 + PCM+1∑

i=Th+1

qM (i)

s.t. 0 ≤ Th ≤M(26)

maxPC ,ThC

PCM+1∑

i=Th+1

qM (i) + (Th− ThC) cC0

− P (s)O

M+1∑i=ThC+1

qM (i)

s.t. Th ≤ ThC ≤M,PC > 0

(27)

maxP

(s)O

(P

(s)O − cO

).

M+1∑i=ThC+1

qM (i)

s.t. P(s)O > cO

(28)

The outcome of this problem is a 4-tuples(Th∗, ThC

∗, PC∗, P

(s)O

∗). The pricing problem is defined in

(29) and (30) by substituting the 4-tuple resulting from thecaching problem.• Pricing Problem:

8

maxPAj

UAj = σAj

[PAj − Th∗.c0 − PC∗

M+1∑i=Th∗+1

qM (i)

]s.t. PAj > 0

(29)

maxP

(c)O

UO =(K −Kρ0P (c)

O

)P

(c)O +

M+1∑i=ThC∗+1

qM (i)

×(P

(s)O

∗− cO

)

s.t. P(c)O > 0

(30)The K+1-tuple

(PA∗1, ..., PA

∗K , P

(c)O

∗)is the outcome of the

pricing problem. The NE of the joint caching and pricingproblem is

(Th∗, ThC

∗, PC∗, P

(s)O

∗, PA

∗1, ..., PA

∗K , P

(c)O

∗).

Theorem 3. The caching problem introduced above is atwo player matrix game between transit ICN C and contentprovider O.

Proof: According to (14), when the transit price for ICNC is greater than access ICN’s caching cost for content typei(PC ≥ cAji

), the access ICN caches all the content which

are more popular than content type i. Therefore, when cAji ≤PC < cAji+1

, the optimum caching threshold index chosenby access ICN will be Th∗ = i. On the other hand, since theutility function of the transit ICN C has a linear relationshipwith the transit price PC , the transit ICN will choose themaximum value possible that is cAji+1

− ε (ε is a very smallvalue). Thus, the transit ICN C also can adopt its actionsfrom a discrete set. There is a caching threshold index Thcorresponding with each transit price chosen by ICN C. Itshows that the transit ICN is the leader in its relationshipwith the access ICNs and its action is (PC , Th) from a set ofM + 1 feasible choices. The relationship between the transitICN C and the content provider is also a leader follower game.Depending on the storage price P (s)

O , the transit ICN C mightforward some part of demands to the content provider O to beserved. If the access ICN decides to cache the content morepopular than content type Th, the rest of the content shouldbe forwarded to transit ICN. Therefore, the content type withindex Th+ 1 to M is going to be served in either the transitICN or the content provider. When cCj ≤ P

(s)O < cCj+1 , the

optimum caching threshold index chosen by the transit ICNC will be ThC∗ = j. Since, the utility of the content providerO has a linear relationship with the storage price P

(s)O ; the

content provider will pick the maximum value possible for thestorage price that is cCj+1−ε. Thus, for every content providerstorage price, there is a corresponding caching threshold indexchosen by the transit ICN C. Since both the transit ICN C andthe content provider O have a limited set of discrete actions,the problem introduced in (26)-(28) is a matrix game betweenthe transit ICN C and the content provider O when the transitICN action is the transit price PC and content provider actionis the storage price P (s)

O . The access ICNs cannot change theresults and they just follow the transit ICN and their actions.

By Theorem 3, we can discard (26) and solve equations(27) and (28) jointly to find the integer thresholds Th andThC . Note that PC and P

(s)O are functions of Th and ThC ,

respectively as follows:

PC (Th) =

cAjTh+1

− ε 0 ≤ Th ≤M − 1

cAjTh + ε Th = M(31)

P(s)O (ThC) =

cCTh+1

− ε 0 ≤ ThC ≤M − 1cCTh + ε ThC = M

(32)

Proposition 1. f (Th) = PC (Th)M+1∑

i=Th+1

qM (i)+ThcC0 and

g (ThC) =(P

(s)O (ThC)− cO

).

M+1∑i=ThC+1

qM (i) are concave

sequences and have a unique maximum.

The proof of the above proposition can be found in Ap-pendix A.

Theorem 4. The symmetric joint caching and pricing gamehas a unique NE.

Proof: The solution of the caching and pricing problemsin (26)-(30) is the NE. On the one hand, the caching problemset is like a leader-follower game and the transit ICN C(leader) maximizes (27) and the content provider O (follower)maximizes (28). Since both of them are concave sequencesbased on Proposition 1, they have only one maximum intheir feasible sets. By (27), Th∗ can be defined as theunique optimum value for the access ICN caching thresholdindex and by (28), ThC∗ can be defined as the optimumcaching threshold index of the transit ICN C that shouldalways be greater or equal than Th∗. Assume that ThCmax

is the value that maximizes (28). If ThC∗ < ThCmax thenThC

∗ = ThCmax and if ThC∗ ≥ ThCmax then ThC∗ = Th∗.By finding the first set of parameters and substituting themin (29) and (30), we can find the second set of parameters.Since these are concave quadratic functions the problem hasthe following unique solution

P(c)O

∗= max

0 ,1−ρ0

(P

(s)O

∗−cO

) M+1∑i=ThC

∗+1

qM (i)

2ρ0

,

PA∗ = PB

∗ = 1ρ

ρ

(Th∗.c0 + PC

∗M+1∑

i=Th∗+1

qM (i)

)+1− ρ0P (c)

O

∗

.(33)

Note that PA∗j ’s are always greater than zero. Hence, as wehave unique set of results for Th∗ and ThC

∗, so PA∗j ’s and

P(c)O

∗are also unique. Therefore the NE exists and is unique.

As the analytical results show, the NE for the symmetriccase is independent of number of access ICNs. So, for thenumerical results section, we consider the scenario with onlytwo access ICNs.

9

0 10 20 30 40 50 60 70 80 90 1000

100

200

300

400

500

600Access ICN caching cost

Index of content (m)

Cac

hin

g c

ost

per

unit

dat

a

γ=0.3γ=0.6γ=0.8γ=1

Fig. 3. Access ICN caching costs vs index of content for different Zipfsfactor γ.

VII. NUMERICAL RESULTS

We consider the interaction among two symmetric accessICNs, one transit ICN and a content provider who are compet-ing to maximize their utilities. In this scenario, the reflectivecoefficients of price’s influence on users’ demands ρA, ρBand ρ0 are set identically to 0.1. These parameters modelthe sensitivity of the demands to an increase in the prices.The scaling parameter between network price and storageprice is set to βA = βB = 10, i.e., the storage price is anorder of magnitude less than the network price. There areM = 100 different types of content in this network whichare randomly requested by users according to a generalizedZipf distribution. Fig. 3 shows how the access ICN cachingcost varies for different types of content and different valuesof Zipfs factor γ when initial caching cost c0 = 1. As Zipf’sfactor γ is increasing, the distribution of content requestedis getting skewed and according to (2), the caching costs ofmore popular content decrease while the caching costs of lesspopular content increase. Note that the transit ICN C possiblyhas (on average) access to cheaper caches. We denote the ratioof transit ICN caching cost to access ICN caching cost by R.We assume that the content provider cost (cO) for all types ofcontent is identical.

The caching strategies Th∗ and ThC∗ at the equilibrium areshown in Fig. 4, where Zipfs factor γ is varying in the intervalof 0.01 to 1. In this scenario, the caching cost of transit ICNC is set to 70% of the caching cost of access ICNs for everytype of content i, i.e., R = 0.7. The results are compared forcO = 40, 60, 100. As seen, for small amount of γ the contentare less skewed and the caching cost for them is very similar.Since the caching cost of transit ICN is less than access ICNs’and content provider’s, it decides to cache most of the content.But as γ increases, some of the content is getting relativelymore popular. In this case, the access ICNs prefer to cache themore popular content locally and smaller amount of contentwill be cached by transit ICN and content provider. For the

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

90

100

Caching strategies for various content provider cost cO

, R=0.7

Zipfs factor γ

Cac

hin

g t

hre

shold

index

Th, co=100

ThC

, co=100

Th, co=60

ThC

, co=60

Th, co=40

ThC

, co=40

Fig. 4. Access ICN/transit ICN Caching threshold vs Zipfs factor γ forvarious content provider cost cO when R = 0.7.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100

Caching strategies for various content provider cost cO

, R=0.5

Zipfs factor γ

Cac

hin

g t

hre

shold

index

Th, co=100

ThC

, co=100

Th, co=60

ThC

, co=60

Th, co=40

ThC

, co=40

Fig. 5. Access ICN/transit ICN Caching threshold vs Zipfs factor γ forvarious content provider cost cO when R = 0.5.

higher γ, the caching cost of access ICNs and transit ICN isgetting higher and they do not have an incentive to locallystore them. Therefore, at this point, the content provider startsto serve more content than before and the transit ICN doesnot cache content and just forwards requests to the contentprovider. For higher content provider cost cO = 100, sincethe cost is so high the content provider just serves the lesspopular requested content but as the cost cO increases, it startsto serve more content. For the case which cO is relativelysmall (cO = 40), the transit ICN does not have the incentiveto spend its resources for caching the requested content andjust forwards all the requests for content from access ICNsto the content provider. The same scenario with R = 0.5 isexamined in Fig. 5. In this case, the transit ICN caching costis half of the access ICN caching cost for each type of content.

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 192

94

96

98

100

102

104

106

108

Access price PA

for for various content provider cost cO

Zipfs factor γ

Pri

ce p

er u

nit

dat

a

co=100

co=80

co=60

co=40

Fig. 6. Access price PA vs Zipfs factor γ for various content provider costcO and R = 0.7.

So, the transit ICN caches more requested content in its cache.Fig. 6 shows the access ICN price PA as function of γ

for different content provider costs cO. PA decreases as theγ gets higher when the content provider cost cO is relativelylow. The reason is that as γ increases, the caching costs ofmore popular content at access ICNs caches are getting lower.Therefore, the access ICNs need to decrease their price PA inorder to compete with other access ICNs. But as the relativecontent provider cost cO increases, the access price is gettinghigher since both the access ICN and the transit ICN have agreater incentive to locally cache the content. However, theprice for greater value of cO displays a bimodal behavior as afunction of γ. According to Fig. 3, for relatively small valueof γ, the different types of content have similar popularity andthe access and transit ICN should incur more or less similarcaching costs for each type of content. But as γ increases,some of the content is getting more popular and the costof caching them at the access ICN is decreasing. Therefore,the access ICN decreases its access price in order to induceincreased demand from the users (see relationship between PAand demand in equation (3)). On the other hand, when γ keepsincreasing, the transit price and the content provider storageprice increase. Thus, the access ICN needs to slightly increaseits price to compensate the increase in the transit price. Afterthe slight increase, as γ increases further, the access pricePA decreases again. This is because, as seen from Fig. 4, thecaching strategies Th∗ and ThC∗ are the same for large γ. Atthis point, the access ICNs decide to cache the content that aremore popular to get higher payoff. As a result, the access ICNdecreases the price to further incentivize greater user demandfor popular content.

The transit price PC and the content provider storage priceP

(s)O are shown in Fig. 7. As γ increases, the caching costs of

less popular content that are going to be cached in transit ICNor content provider caches are increasing. Therefore, both thetransit ICN and content provider should increase their prices

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 150

100

150

200

250

300

350

Transit price and content provider storage price for various content provider cost cO

, R=0.7

Zipfs factor γ

Pri

ce p

er u

nit

dat

a

Transit price, co=100

content provider storage price, co=100

Transit price, co=80

content provider storage price, co=80

Transit price, co=60

content provider storage price, co=60

Transit price, co=40

content provider storage price, co=40

Fig. 7. Transit price / content provider storage price vs Zipfs factor γ forvarious content provider cost cO and R = 0.7.

as shown in the figure.

VIII. CONCLUSION

In this paper, we developed an analytical framework fordistribution of popular content in an Information CentricNetwork (ICN) that comprises of Access ICNs, a Transit ICNand a Content Provider. By modeling the interaction of theabove entities using game theory and under the assumptionthat the caching cost of the access ICNs and transit ICNsis inversely proportional to popularity, which follows a gen-eralized Zipf distribution, we first showed that at the NE,the caching strategies turn out to be 0-1 (all or nothing).Further, for the case of symmetric Access ICNs, we showedthat a unique NE exists and the caching policy (0 or 1) isdetermined by a threshold on the popularity of the content(reflected by the Zipf probability metric), i.e., all contentmore popular than the threshold value is cached. Moreover,we showed that the resulting threshold indices and pricescan be obtained by a decomposition of the joint cachingand pricing problem into two independent caching only andpricing only problems. Finally, using numerical results weshowed that as the Zipf’s factor γ increases and the relativepopularity of the content gets skewed, the access ICN justcaches the more popular content and the content providerserves only requests for less popular content while the transitICN just forwards the demands to the content provider withoutlocally caching any content itself. The insights obtained fromthe analysis here warrants further investigation into the caseof asymmetric access ICNs. In this paper, we discussed ahierarchical scenario with K access ICNs, one transit ICNand one content provider. In the special case that the transitICNs are not ”interconnected’ to each other, the model in thispaper is readily extendable for multiple transit ICNs. However,a generalized scenario with either multiple ”interconnected”transit ICNs or multiple content providers is a direction forfuture work. Extension of the above model to scenarios where

11

the content popularity may need to be learned is also ofinterest.

APPENDIX APROOF OF PROPOSITION 1

Two sequences f (Th) and g (ThC) defined in Proposition1 are concave sequences.

A sequence S is strictly concave if the below inequalityholds for every n.

S (n+ 1) + S (n− 1)− 2S (n) < 0

As stated before, qM (i) = 1(M∑j=1

1jγ

)iγ

for = 1, ...,M and

qM (i) = 0 for i > M . Using (31) for Th = 1, ...,M − 1, wehave

PC (Th) = c0(Th+ 1)γM∑i=1

(1i

)γ− ε

Since ε is so small and tends to zero, it can be consideredas zero. Therefore, for Th = 1, ...,M − 1, the sequence f isdefined as follows

f (Th) =

[c0(Th+ 1)

γM∑i=1

(1i

)γ] M+1∑i=Th+1

qM (i) + ThcC0

⇒ f (Th) = [c0(Th+ 1)γ]M+1∑

i=Th+1

(1i

)γ+ ThcC0

To prove concavity, we need to have

f (Th+ 1) + f (Th− 1)− 2f (Th) < 0⇒

[(Th+ 2)γ − 2(Th+ 1)

γ+ Thγ ]

M+1∑i=Th+2

(1i

)γ< 1−

(ThTh+1

)γ.

If this inequality holds for all M and Th = 1, ...,M − 1,then the sequence f is concave. We introduce ϕγ (Th) =[(Th+ 2)

γ − 2(Th+ 1)γ − Thγ ]. Two functions ϕ0 (Th)

and ϕ1 (Th) are zero for all Th that satisfy the aboveinequality. It means that for γ = 0 and γ = 1, f is a concavesequence. Moreover, ϕγ (Th) can be written as

ϕγ (Th) = [(Th+ 2)γ − (Th+ 1)

γ]− [(Th+ 1)

γ+ Thγ ]

⇒ ϕγ (Th) =(

ThTh+1

)γ+(Th+2Th+1

)γ− 2.

Since(

ThTh+1

)γ+(Th+2Th+1

)γ< 2 for 0 < γ < 1, ϕγ (Th)

is negative for all Th = 1, ...,M − 1 and M which causethat above inequality be satisfied. Therefore, f is a concavesequence for all M and all 0 ≤ γ ≤ 1 .

Using (32) for ThC = 1, ...,M − 1, we have

P(s)O (ThC) = cC0

(ThC + 1)γM∑i=1

(1i

)γ− ε

Since ε is so small and tends to zero, it can be considered aszero. Therefore, for ThC = 1, ...,M − 1, the sequence g canbe defined as

g = cC0(ThC + 1)

γM∑

i=ThC+1

(1i

)γ− cO

M+1∑i=ThC+1

qM (i)

On the other hand, we can show that if two sequences areconcave the sum of them is also concave. Assume that g =Ψ + ∆. If Ψ and ∆ are concave, we have

Ψ (n+ 1) + Ψ (n− 1)− 2Ψ (n) < 0

and

∆ (n+ 1) + ∆ (n− 1)− 2∆ (n) < 0.

Therefore,

Ψ (n+ 1) + ∆ (n+ 1) + Ψ (n− 1) + ∆ (n− 1)−2 [Ψ (n) + ∆ (n)] < 0

That means

g (n+ 1) + g (n− 1)− 2g (n) < 0

We already showed that the first term of sequence g isconcave, so we just need to show that the second term

∆ (ThC) = −cOM+1∑

i=ThC+1

qM (i) is also concave. Then, we

have to show that the following inequality holds for everyThC = 1, ...,M − 1

∆ (ThC + 1) + ∆ (ThC − 1)− 2∆ (ThC) < 0

By doing some algebra, we will get qM (ThC) >qM (ThC + 1) that holds for ThC = 1, ...,M−1, since qM (i)is a strictly decreasing function. That completes the proof.

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PLACEPHOTOHERE

Mohammad Hajimirsadeghi (S15) received theB.S. degree in electrical engineering (with honors,first rank) from Shahed University, Tehran, Iran,in 2008 and the M.S. degree in telecommunica-tions from Sharif University of Technology (SUT),Tehran, Iran, in 2011. Currently, he is pursuing aPh.D. degree in electrical and computer engineeringat WINLAB at Rutgers-The State University of NewJersey, Piscataway, NJ under the guidance of Dr.Narayan Mandayam. His research interests are in thegeneral area of communications and networking, in-

cluding wireless resource allocation, Network economic and network security.Before coming to the United States, he was a Communication Engineer at IranTelecommunication Research Center (ITRC), Tehran, Iran.

PLACEPHOTOHERE

Narayan. B. Mandayam (S89-M94- SM99-F09)received the B.Tech (Hons.) degree in 1989 fromthe Indian Institute of Technology, Kharagpur, andthe M.S. and Ph.D. degrees in 1991 and 1994 fromRice University, all in electrical engineering. Since1994 he has been at Rutgers University where he iscurrently a Distinguished Professor and Chair of theElectrical and Computer Engineering department.He also serves as Associate Director at WINLAB.He was a visiting faculty fellow in the Departmentof Electrical Engineering, Princeton University, in

2002 and a visiting faculty at the Indian Institute of Science, Bangalore,India in 2003. Using constructs from game theory, communications andnetworking, his work has focused on system modeling, information processingas well as resource management for enabling cognitive wireless technologiesto support various applications. He has been working recently on the useof prospect theory in understanding the psychophysics of data pricing forwireless networks as well as the smart grid. His recent interests also includeprivacy in IoT as well as modeling and analysis of trustworthy knowledgecreation on the Internet.

Dr. Mandayam is a co-recipient of the 2015 IEEE Communications SocietyAdvances in Communications Award for his seminal work on power controland pricing, the 2014 IEEE Donald G. Fink Award for his IEEE Proceedingspaper titled Frontiers of Wireless and Mobile Communications and the 2009Fred W. Ellersick Prize from the IEEE Communications Society for his workon dynamic spectrum access models and spectrum policy. He is also a recipientof the Peter D. Cherasia Faculty Scholar Award from Rutgers University(2010), the National Science Foundation CAREER Award (1998) and theInstitute Silver Medal from the Indian Institute of Technology (1989). He is acoauthor of the books: Principles of Cognitive Radio (Cambridge UniversityPress, 2012) and Wireless Networks: Multiuser Detection in Cross-LayerDesign (Springer, 2004). He has served as an Editor for the journals IEEECommunication Letters and IEEE Transactions on Wireless Communications.He has also served as a guest editor of the IEEE JSAC Special Issues onAdaptive, Spectrum Agile and Cognitive Radio Networks (2007) and GameTheory in Communication Systems (2008). He is a Fellow and DistinguishedLecturer of the IEEE.

PLACEPHOTOHERE

Alex Reznik [M94, SM10] is currently a Solu-tion Architect on Hewlett Packard Enterprises telcostrategic account team. In this role he is involvedin various aspects of helping a Tier 1 telco evolvetowards a flexible infrastructure capable of deliver-ing on the full promises of 5G. Prior to May 2016Alex was a Senior Principal Engineer at InterDigital,leading the companys research and development ac-tivities in the area wireless internet evolution. Sincejoining InterDigital in 1999, he has been involvedin a wide range of projects, including leadership

of 3G modem ASIC architecture, design of advanced wireless securitysystems, coordination of standards strategy in the cognitive networks space,development of advanced IP mobility and heterogeneous access technologiesand development of new content management techniques for the mobile edge.Alex earned his B.S.E.E. Summa Cum Laude from The Cooper Union, S.M. inElectrical Engineering and Computer Science from the Massachusetts Instituteof Technology, and Ph.D. in Electrical Engineering from Princeton University.He held a visiting faculty appointment at WINLAB, Rutgers University, wherehe collaborated on research in cognitive radio, wireless security, and futuremobile Internet. He served as the Vice-Chair of the Services Working Groupat the Small Cells Forum. Alex is an inventor of over 130 granted U.S. patents,and has been awarded numerous awards for Innovation at InterDigital.

14