How durable and stable is File Sharing on the Move in Cellular-assisted D2D Communications? ∗ Department
† School
Yali Wang∗ , Wenye Wang∗ and Mei Song†
of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC 27606 of Electronic Engineering, Beijing University of Posts and Telecommunications, Beijing, China 100876 Email: {ywang116, wwang}@ncsu.edu,
[email protected] Abstract—File sharing is one of the most promising Proximity Service (ProSe) using device-to-device (D2D) communication technology. As the prospective D2D-enabled file sharing rely on mobile peer discovery and connection in physical proximity, the challenges come from the highly dynamic environment as a consequence of the unpredictable user mobility and the temporal-spatial locality of content popularity. A challenging yet open question is how users could move to fetch target files durably and stably. In this paper, we characterize mobile file sharing in temporal and spatial domains by how long time the service can be active (referred as service lifetime 𝜏L ), how far a mobile user can move away while keeping ongoing service (referred as service distance ∥𝐷L ∥), and how fast the ongoing service would be recovered once interruption caused by handover (classified into bundled handover and split handover in this paper) (referred as recover delay 𝜎HO ). Answers to these questions offer a straightforward interpretation of the potentials of D2D communications for file sharing on the move. Both theoretical and simulation results suggest that D2D file sharing is more benefit for popular content among group moving users with longer service lifetime and distance and less delay caused by mobile user handover and peer discovery.
I. I NTRODUCTION D2D communications is regarded as a promising technology to provide low-power, high data rate and low-latency services between end-users. Several D2D communication use cases have been widely investigated as part of the LTE-A networks, in both licensed cellular spectrum and unlicensed bands. 5G networks are expected to continue supporting applications that require devices to set up direct communication links with nearby devices [1]. In this context, Proximity Service (ProSe) open rich collaboration opportunities for user equipments (UEs) in physical proximity. Importantly, ProSe are build on underlying D2D communication technology, which allows for direct data transfer between two proximal devices without the need for expensive cellular network resources. Specifically, file sharing is one of the promising applications based on D2D communications [2]. Previous research has thoroughly characterized file sharing performance aspects from file-sharing mechanisms, such as [3–5], to caching, mode selection, resource allocation, interference management and routing, such as [6–9]. Chen This work is supported by NSF CNS-1423151. Yali Wang is a joint Ph.D student from Beijing University of Posts and Telecommunications.
[3] proposed a minimum the hop method, which can ensure the user obtain required file as soon as possible. Seppala [4] exploited a cluster structure to design an efficient content dissemination approach. Considering the impact of social relationships in file sharing, papers [5] proposed file-sharing mechanisms according to social relationships between users, social contributions of each user and trust degrees of users. Guo [6] investigated the D2D local caching with heterogeneous file preference among different groups. Ye [7] proposed a simple guard-zone based mode selection scheme as an efficient and scalable mechanism to manage the interference in D2D underlaid cellular systems. Ma [8] jointly considered resource allocation and power control with heterogeneous QoS requirements from the applications and developed optimized solutions to coordinate the cellular and D2D communications with the best resource sharing mode. Du [9] propose the peer-to-peer (P2P) share enabled routing schemes over multi-hop interference-constrained D2D networks, where multiple D2D subscribers attempt to download the common data from multiple distributed D2D servers. As introduced above, existing studies of file sharing underlying D2D communication technology mainly focused on sharing mechanisms, file caching, mode selection, resource allocation, power control and routing, which completely ignore the impact of mobility on file sharing. However, prospective D2D-enabled applications and services rely on highly opportunistic device contacts as a consequence of unpredictable human mobility. Mobility’s impact on D2D communication requires careful investigation to understand the practical operation efficiency of future cellular-assisted D2D systems. Therefore, it motivates us to study a challenging yet open question that is how users could move to fetch target files durably and stably. Since handover is frequently triggered by human mobility, which leads to ongoing file sharing interruption. In order to comprehensively understand mobility in D2D file-sharing, we first define two types of handover scenarios, called bundled handover and split handover, according to the relative movement of a D2D pair of UEs. Then, we characterize the file sharing on the move in temporal and spatial domains i) how long time the service can be active (referred as service lifetime 𝜏L ), ii) how far a mobile user can move away while
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keeping ongoing service (referred as service distance ∥𝐷L ∥), and iii) how fast the ongoing service would be recovered once interruption caused by handover (referred as recover delay 𝜎HO ). Answers to these questions can provide understanding of the temporal and spatial limits of file sharing on the move in cellular-assisted D2D communication networks. It also sheds light on how the time-sensitive file sharing could be possibly satisfied, which helps network designers design appropriate D2D handover mechanisms according to the various requirements of applications. In this paper, we derive the complementary CDF of the the first time when there are no neighboring UEs that can share files (denoted as service lifetime 𝜏L ), and the upper bounds on the maximum Euclidean distance that UEs can move away from the initial locations while keeping file sharing during service lifetime (denoted as service distance ∥𝐷L ∥) and the average time it takes to recover file sharing after interruptions caused by handover and (or) D2D link disconnection during UEs travelling by time 𝑡 (denoted as recover delay 𝜎HO (𝑡)). Theoretical and simulation results provide a straightforward interpretation of the potentials of D2D communications for file sharing on the move, which indicate that D2D file sharing is more benefit for popular content among group moving users with longer service lifetime and distance and less delay caused by mobile user handover and peer discovery. The remainder of this paper is structured as follows. In Sec.II, we introduce the network models, handover scenarios and formulate research problems. Bounds on service lifetime, service distance and average recover delay are derived in Sec.III. In Sec.IV, we validate our analytic bounds by simulation results. Finally, we conclude this paper in Sec.V. II. N ETWORK M ODELS AND P ROBLEM F ORMULATION In this section, we introduce our network and mobility models for file sharing in cellular-assisted D2D communication networks. Moreover, we formally define the service lifetime, service distance and average recover delay to characterize file sharing on the move in both temporal and spatial domain. Finally, we formulate the problem. A. Network and Mobility Model As shown in Fig. 1, we consider a single-tier cellular network with macro base stations (BSs) (in green squares) and 𝑁 UEs (in black and white circles) 𝒰 = {𝑢1 , ..., 𝑢𝑁 } modeled as an independent homogeneous Poisson point process (PPP) with density 𝜆ℬ and 𝜆𝒰 , respectively. UEs move in an observed network region (a two-dimensional Euclidean plane ℝ2 ) [10]. Network operation is described on a slotted-time basis with the time axis partitioned into equal non-overlapping time slots, with 𝑡 denoting the time slot index. Let 𝒳 (𝑡) = {𝑋1 (𝑡), ...𝑋𝑁 (𝑡)} denote locations of 𝑁 UEs at time 𝑡, which is a set of point in the Cartesian coordinates 𝑋𝑖 (𝑡) = (𝑥𝑖 (𝑡), 𝑦𝑖 (𝑡)), 𝑖 = 1, ..., 𝑁 . For file sharing using D2D communications, we support that each UE can play the roles of File Provider and File Requester.
Assume each UE’s transmission range is 𝑟, we adopt the commonly used protocol model for the simplicity of our analysis. Transmission between two UEs 𝑢𝑖 and 𝑢𝑗 is feasible only if their distance is less or equal to transmission range 𝑟, i.e., ∥𝑋𝑖 − 𝑋𝑗 ∥ ≤ 𝑟. Suppose that UEs move according to a given mobility model ℳ. In other words, the displacement of node 𝑢𝑖 from its position 𝑋𝑖 (𝑡 − 1) to 𝑋𝑖 (𝑡) is distribute according to ℳ. In this paper, we consider a generic mobility model defined as follows. Definition 1. (Generic Mobility) Given initial UEs’ positions 𝒳 (0) at 𝑡 = 0. At each time slot 𝑡, each UE randomly and uniformly chooses a random direction 𝜃𝑖 (𝑡) (𝑖 = 1, ...𝑁 ) from (0, 2𝜋], and travels for a random length 𝑙𝑖 (𝑡) following a uniform distribution from [0, 𝑙𝑚𝑎𝑥 ]. Thus, as shown in Fig. 1, the movement vector of a given UE 𝑢𝑖 at time 𝑡 is 𝑀𝑖 (𝑡) ≜ 𝑙𝑖 (𝑡)𝑒𝑗𝜃𝑖 (𝑡) , 𝑡 = 1, 2, ... (𝑗 is imaginary unit) with origin at 𝑋𝑖 (𝑡) and endpoint 𝑋𝑖 (𝑡 + 1), which is independently and uniformly distributed in the circular region ℛ(𝑋𝑖 (𝑡), 𝑙𝑚𝑎𝑥 ) centered at 𝑋𝑖 (𝑡) with radius 𝑙𝑚𝑎𝑥 . As the tempo-spatial locality of various content popularity analyzed in our previous work [11], we model the count of file provider located in a bounded region as an inhomogeneous Poisson process. As shown in Fig. 1, given a file requester 𝑢𝑖 ∈ 𝒰 located in 𝑋𝑖 (𝑡) at time 𝑡, denote by 𝒱(𝒮(𝑋𝑖 (𝑡), 𝑟)) = {𝑣1 , ..., 𝑣𝑚 }, 𝑚 ≤ 𝑁 − 1 is the neighboring file provider set, in which every UE locates in the circular region 𝒮(𝑋𝑖 (𝑡), 𝑟) centered at 𝑋𝑖 (𝑡) with radius 𝑟 > 0 and caches the requested content at time 𝑡. Hence, the neighboring file provider count is the size of the file provider set, denoted as 𝑉 (𝒮(𝑋𝑖 (𝑡), 𝑟)). Remark 1. The constrained mobility model reflects the speed limit of UEs (usually wireless equipments carried by humans or installed on vehicles) that UEs can travel to adjacent locations with pre-assigned probabilities and each movement step is limited in a circular region around previous location. B. Handover Scenarios In order to detect the average recover delay of a D2D pair, we define two types of handover scenarios according to mobility patterns. Firstly, we measure the relative movement between a pair of D2D UEs (𝑢𝑖 , 𝑢𝑗 ), which is evaluated by the Euclidean distance 𝑑𝑖,𝑗 (𝑡) ≜ ∥𝑋𝑖 (𝑡) − 𝑋𝑗 (𝑡)∥ and the relative velocity 𝑣𝑖∣𝑗 (𝑡) ≜ ∥𝑀𝑖 (𝑡) − 𝑀𝑗 (𝑡)∥. Hence, we define the Relative Movement Similarity 𝑅𝑀 𝑆𝑖,𝑗 (𝑡) of a pair of UEs 𝑢𝑖 and 𝑢𝑗 as follows. Definition 2. (Relative Movement Similarity) 𝑅𝑀 𝑆𝑖,𝑗 (𝑡) := 𝛼 ⋅ (1 −
𝑣𝑖∣𝑗 (𝑡) 𝑑𝑖,𝑗 (𝑡) ) + (1 − 𝛼) ⋅ (1 − ), (1) 𝑟 2𝑙max
where 𝛼 is constant. Intuitively, the relative movement similarity is a dimensionless quantity, which is a linear function of random variables 𝑑𝑖,𝑗 (𝑡) and 𝑣𝑖∣𝑗 (𝑡). The quantity is larger when
V(t)=3
V(τ )=0 L
V(t+1)=2 r
Bounded Cir cular Region S(X(t),r ) r
Cell Boundar y
M(t+1)
X(τ ) L
M(τL)
r
M(t)
X(t+1)
e i,j (t) V(t-3)=7
r
DL
X(t-3)
r
M(t-1) r
X(t-1)
Xj (t) Mj (t+1) v j (1)
M(t-2)
M(t-3)
r
X(0)
Content Requester
Content Pr ovider
Macr o BS
Non-Content Pr ovider
Xj (t+1)
Xi (t+2) r
Mi (t) v i (2)
r
Xi (t+1)
Mj (t) v j (2)
r
Xi (t+1) Mi (t+1) v i (1)
e i,j (t) r
e i,o (t) Xo (t+2)
Cell Boundar y
Fig. 1. User movement trajectory illustration.
Macr o BS
Movement Vector
D2D UEs
D2D Link
Fig. 2. Bundled handover scenario.
two D2D UEs locating within smaller distance, moving with more similar speed and directions, in other words, the D2D pair of UEs are more likely keep durable and stable connections. Therefore, according to the relative movement similarity, we classify the D2D pair handover to two scenarios: Bundled Handover and Split Handover. Definition 3. (Bundled Handover and Split Handover) Initially a pair of D2D UEs, UE𝑖 (0) and UE𝑗 (0), locate at 𝑋𝑖 (0) and 𝑋𝑗 (0) within the same cell and move following the defined mobility model. Based on the different results of movement patterns, we define bundled handover and split handover, respectively. (i) At the end of movement interval, if D2D links are interrupted by the handover and re-established with the original UEs, we call this Bundled Handover, such as movement within interval (0, 𝑡 − 1] and (𝑡 − 1, 𝑡] shown in Fig. 2. (ii) At the end of interval, if D2D links are interrupted and re-established with a new neighbor, we call this Split Handover, which generates the peer discovery delay 𝑑(D) (or both delay 𝑑(H) and 𝑑(D) ), such as movement within interval (0, 𝑡 − 1] and (𝑡 − 1, 𝑡] shown in Fig. 3. C. Problem Formulation Definition 4. (Service Lifetime 𝜏L ): is defined as the first time that the file provider set becomes empty after time 0, i.e., the file provider count equals to 0. 𝜏L := inf{𝑡 > 0 : 𝒱(𝑋(𝑡)) = ∅}.
(2)
𝜏L is a r.v., which takes value in (0, ∞). Definition 5. (Service Distance ∥𝐷L ∥): 𝐷L denotes the service vector of UEs from origin at 𝑋(0) to the farthest location on the moving trajectory in the time window [1, 𝜏L ]. service distance is denoted as ∥𝐷L ∥ := max {∥𝑋(𝑡) − 𝑋(0)∥}.
(3)
Xi (t+2)
r
Xj (t+2)
Obser ved Networ k Region D2D UEs
Mi (t+2) v i (2)
Xi (t)
Inter -cell link
Obser ved Networ k Region
Movement Vector Ser vice Vector Mobility
Xj (t)
e i,j (t+2)
e i,j (t+1)
V(t-1)=12
X(t-2)
V(t-2)=6
V(0)=10
Mj (t+1) v j (1)
Mi (t+1) v i (1)
Mz(t+2) v z(2) e i,z(t+1)
Mz(t+1) Vz(1)
Xj (t+1)
Xi (t) r
Xz(t+2) Vz(t+2)
Xz(t+1)
Xz(t)
X(t)
Macr o BS
Xo (t+1)
Mo (t+2) v o (t+2)
Obser ved Networ k Region
Movement Vector
D2D Link
Fig. 3. Split handover scenario.
D2D discovery, respectively. During UEs travelling by time ˜HO is 1 ≤ 𝑡 ≤ 𝜏L , the recover delay per time slot 𝜎 { 𝑑H 𝐻𝑂∥𝑀 (𝑡)∥ 𝑝 = ℙB (𝑡), 𝜎 ˜HO (𝑡) := (4) (𝑑H + 𝑑D )𝐻𝑂∥𝑀 (𝑡)∥ 𝑝 = ℙS (𝑡). where ∥𝑀 (𝑡)∥ = ∥𝑋(𝑡) − 𝑋(𝑡 − 1)∥ is the length of movement vector, ℙB (𝑡) ≜ ℙ[𝑅𝑀 𝑆(𝑡) ≥ 𝑎] is the bundled handover probability, where 𝑎 is a constant and ℙS (𝑡) = 1 − ℙB (𝑡) is the split handover probability. Following √ [12], the handover rate 𝐻𝑂 per unit length is equal to 4 𝜆ℬ /𝜋. From Eq. (4), we have the average recover delay as 𝑡 ∑ 𝔼[𝜎HO (𝑡)] = 𝔼[˜ 𝜎HO (𝑘)] = 𝑘=1
𝐻𝑂
𝑡 ∑
(5)
[𝑑H ℙB (𝑘) + (𝑑H + 𝑑D )ℙS (𝑘)]𝔼(∥𝑀 (𝑘)∥).
𝑘=1
III. A NALYSIS ON 𝜏L , ∥𝐷L ∥ AND 𝜎HO (𝑡) In this section, we drive complementary CDF of service lifetime ℙ(𝜏L > 𝑡), and upper bounds on average service distance 𝔼(∥𝐷L ∥) and average recover delay 𝔼[𝜎HO (𝑡)] according to the temporal and spatial locality of content popularity, the defined generic mobility and the handover classification. Initially, a given D2D pair of UE𝑖 (0) (the file requester) and UE𝑗 (0) (the file provider) locate in 𝑋𝑖 (0) and 𝑋𝑗 (0) and move with initial velocity 𝑣𝑖 (0) and 𝑣𝑗 (0) Let in the initial direction 𝜃𝑖 (0) and 𝜃𝑗 (0), respectively. ∑𝑡 ≈ 0, 𝑋𝑖 (𝑡) − 𝑋(0) = 𝑘=1 𝑀𝑖 (𝑘) ∥𝑋𝑖 (0)∥ ≈ 0, ∥𝑋𝑗 (0)∥ ∑ 𝑡 and 𝑋𝑗 (𝑡) − 𝑋(0) = 𝑘=1 𝑀𝑗 (𝑘). In the following, the analysis object is the file requester UE𝑖 . For simplify, we omit the index 𝑖 if we have no special statement. A. Service Lifetime
We find that the vector of a UE from origin at 𝑋(0) to ∑𝑡 the location 𝑋(𝑡) at time 𝑡 is 𝑋(𝑡) − 𝑋(0) = (𝑖). The service distance can be denoted as ∥𝐷L ∥ = 𝑖=1 𝑀∑ 𝑡 max ∥ 𝑖=1 𝑀 (𝑖)∥.
1≤𝑡≤𝜏L
Lemma 1. In the time slot 𝑘 (𝑘 ≥ 1), the count of file provider 𝑉 (𝒮(𝑋(𝑘), 𝑟)) located in the bounded circular region 𝒮(𝑋(𝑘), 𝑟) (simplified as 𝒮𝑘 ) within the ℝ2 plane is an inhomogeneous Poisson process with the intensity function 𝜆(𝑥, 𝑦, 𝑡) and the intensity measure Λ(𝒮𝑘 , [𝑘−1, 𝑘]) shown in Eq. (6) and (7), respectively.
Definition 6. (Recover Delay 𝜎 ˜HO (𝑡)): Denote by 𝑑H and 𝑑D are the time taken by per-user handover and peer discovery. The subscripts H and D denote events, the handover and
Proof: Let a file requester’s location at time 𝑘 is denoted by a point 𝑋(𝑘) = (𝑥(𝑘), 𝑦(𝑘)) in Cartesian Coordinates and the origin at 𝑋(0) = (0, 0), we have
1≤𝑡≤𝜏L
vector 𝑋(𝑘) − 𝑋(0) = (𝑥(𝑘), 𝑦(𝑘)) = (
𝑘 ∑ 𝑧=1
𝑙(𝑧) cos 𝜃(𝑧),
𝑘 ∑ 𝑧=1
𝑘 ∑ 𝑧=1
𝑀 (𝑧) =
𝑙(𝑧) sin 𝜃(𝑧)). Following the temporal
and spatial locality of content request analyzed in [11], ℎ−1 −𝑡/𝜃 𝑒 is the probability density function 𝑓 (𝑡; ℎ, 𝜃) = 𝑡 𝜃ℎ Γ(ℎ) (pdf) of the file request time 𝑇R following Gamma distribution with shape ℎ and scale 𝜃, where Γ(ℎ) the gamma function evaluated at ℎ. Inter-arrival time 𝑇 of a user’s requests follows general distribution with expectation 𝜇. Let ∑∣𝒞∣ 2 2 1/2 𝑔(𝑥, 𝑦∣𝑡) = 𝑐=1 {𝑒−[(𝑥(𝑡)+𝑥−𝑥𝑐 ) +(𝑦(𝑡)+𝑦−𝑦𝑐 ) ] } denote the intensity function of the inhomogeneous-spatial PPP, which is a function of Cartesian coordinates 𝑥 and 𝑦, where 𝒞 is a set of community composed by UEs who have common interests. 𝑥𝑐 and 𝑦𝑐 are the Cartesian coordinates (𝑐 ∈ 𝒞). Then, the intensity function 𝜆(𝑥, 𝑦, 𝑡) is defined as 𝜆𝒰 𝑔(𝑥, 𝑦∣𝑡)𝑓 (𝑡; ℎ, 𝜃) = 𝜆(𝑥, 𝑦, 𝑡) = 𝜇 (6) ∣𝒞∣ 𝜆𝒰 𝑡ℎ−1 𝑒−𝑡/𝜃 ∑ −[(𝑥(𝑡)+𝑥−𝑥𝑐 )2 +(𝑦(𝑡)+𝑦−𝑦𝑐 )2 ]1/2 {𝑒 }. 𝜇 𝜃ℎ Γ(ℎ) 𝑐=1 Then, the density measure is ∫ ∫ 𝜆𝒰 𝑘 Λ(𝒮𝑘 , [𝑘 − 1, 𝑘]) = 𝑓 (𝑡; ℎ, 𝛿)𝑑𝑡 𝑔(𝑥, 𝑦∣𝑡)𝑑𝑥𝑑𝑦 𝜇 𝑘−1 𝒮𝑘 ∫𝑟 ∫2𝜋 𝛾(ℎ, 𝑘𝜃 ) − 𝛾(ℎ, 𝑘−1 𝜃 ) = 2𝜋𝑠 ⋅ 𝑔(𝑠 cos 𝜑, 𝑠 sin 𝜑∣𝑡)𝑑𝜑𝑑𝑠. 𝜇Γ(ℎ)/𝜆𝒰 0
0
(7)
𝑡 ∏
1 − 𝑒−Λ(𝒮𝑘 ,[𝑘−1,𝑘]) .
(8)
𝑘=1
Proof: Based on the definition ∩of service lifetime in Eq. (2), we have ℙ{𝜏L > 𝑡} = ℙ{ 𝒱(𝒮(𝑋(𝑘), 𝑟)) ∕= ∅}. 𝑘=1,...,𝑡
As 𝑉 (𝒮(𝑋(𝑘), 𝑟)) are i.i.d. random variables, we have ℙ{𝜏L > 𝑡} = =
𝑡 ∏ 𝑘=1 𝑡 ∏
Let 𝑒𝑘 the unit vector is in the same direction as the vector 𝑀 (𝑘) (𝑘 = 1, ..., 𝑡), then ∥𝑀 (𝑘)∥ = 𝑀 (𝑘) ⋅ 𝑒𝑘 and ∥𝑋(𝑡) − 𝑡 ∑ ∥𝑀 (𝑘)∥. Therefore, based on the results of 𝑋(0)∥ ≤ 𝑘=1
Lemma 2, we have 𝔼(∥𝑋(𝑡) − 𝑋(0)∥) ≤
𝑡 ∑
𝔼(∥𝑀 (𝑘)∥) =
𝑘=1
2𝑡𝑙max . 3
(11)
Based on the Cauchy-Schwarz inequality [13], we have 2
𝔼(∥𝑋(𝑡) − 𝑋(0)∥ ) ≤
𝑡 ∑
𝔼(∥𝑀 (𝑘)∥2 ) =
𝑘=1
2 𝑡𝑙max . 2
(12)
Then, we derive the upper bound on 𝔼(∥𝐷L ∥) in the service lifetime window 𝑡 ∈ [1, 𝜏L ].
Theorem 1. Service lifetime 𝜏L satisfies that ℙ{𝜏L > 𝑡} =
movement vector 𝑀 (𝑡) starts from origin at 𝑋(𝑡 − 1) to the endpoint at 𝑋(𝑡) that uniformly locates in the circular region ℛ(𝑋(𝑡), 𝑙𝑚𝑎𝑥 ) centered at 𝑋(𝑡) with radius 𝑙𝑚𝑎𝑥 > 0. The length of a movement step ∥𝑀 (𝑡)∥ = ∥𝑋(𝑡) − 𝑋(𝑡 − 1)∥. To drive the probability ℙ(∥𝑀 (𝑡)∥ = 𝑥), we separate the circular region ℛ(𝑋(𝑡), 𝑙𝑚𝑎𝑥 ) into 𝑂 concentric rings indexed by 𝑜, 1 ≤ 𝑜 ≤ 𝑂. Each of rings is with equal ring width 𝜖, which is assumed much smaller than 𝑙max . Hence, 2 . we have ℙ(∥𝑀 (𝑡)∥ = 𝑥) ≈ 2𝑥/𝑙max Then, we have the 𝔼(∥𝑀 (𝑡)∥) and 𝔼(∥𝑀 (𝑡)∥2 ) as ∫ 𝑙max 2𝑥2 2 (9) 𝔼(∥𝑀 (𝑡)∥) = 𝑑𝑥 = 𝑙max , 2 𝑙 3 0 max ∫ 𝑙max 2𝑥3 𝑙2 (10) 𝔼(∥𝑀 (𝑡)∥2 ) = 𝑑𝑥 = max . 2 𝑙max 2 0
ℙ{𝑉 (𝒮(𝑋(𝑘), 𝑟)) ≥ 1} 1 − 𝑒−Λ(𝒮𝑘 ,[𝑘−1,𝑘]) .
𝑘=1
Theorem 2. During service lifetime, average service distance 𝔼(∥𝐷L ∥) is upper bounded by 23 𝑙max 𝜏L + √ 𝑙max
(𝜏L2 − 1)[ 14 −
2 27 (2𝜏L
+ 1)].
Proof: As the definition of service distance in Eq. (3), ∥𝐷L ∥ equals to the maximum of several random variables ∥𝑋(𝑡) − 𝑋(0)∥, 𝑡 = 1, .., 𝜏L − 1. To analyze this, we apply Aven’s upper bound on the mean of the maximum of a number of random variables with general distributions (not necessarily i.i.d) [14]. Hence, we have 𝔼(∥𝐷L ∥) = 𝔼{ max ∥𝑋(𝑡) − 𝑋(0)∥} 1≤𝑡≤𝜏L
B. Service Distance To find upper bound on ∥𝐷L ∥, we first examine movement vector 𝑀 (𝑡). Lemma 2. Under constrained i.i.d. mobility, the length of the movement vector satisfies that 𝔼(∥𝑀 (𝑡)∥) = 2𝑙max /3 2 and 𝔼(∥𝑀 (𝑡)∥2 ) = 𝑙max /2, where 𝑙max is the maximum moving distance per time slot. Proof: Let a certain UE locate at 𝑋(𝑡 − 1) and 𝑋(𝑡) at time 𝑡 − 1 and 𝑡, respectively. As shown in Fig. 1, the
≤ max {𝔼(∥𝑋(𝑡) − 𝑋(0)∥)}+ 1≤𝑡≤𝜏L √ 𝜏L 𝜏L − 1 ∑ [ Var(∥𝑋(𝑡) − 𝑋(0)∥)]1/2 . 𝜏L 𝑡=1
(13)
Based on Eq. (11) and (12), we have √ 2 2 1 𝔼(∥𝐷L ∥) ≤ 𝑙max 𝜏L + 𝑙max (𝜏L2 − 1)[ − (2𝜏L + 1)]. 3 4 27
C. Average Recover Delay According to the definition in Eq. (4), to find the upper bound on 𝔼[𝜎HO (𝑡)], we first analyze the average relative movement similarity 𝔼[𝑅𝑀 𝑆(𝑡)]. 2
Theorem 3. 𝔼[𝑅𝑀 𝑆(𝑡)] is lower bounded by 𝛼𝑒−𝜆𝒰 𝜋𝑟 + (1 − 𝛼)/3. Proof: Based on the Eq. (1), we have 𝔼(𝑅𝑀 𝑆𝑖,𝑗 (𝑡)) = 𝔼[𝑣 (𝑡)] 𝔼[𝑑 (𝑡)] ) + (1 − 𝛼) ⋅ (1 − 2𝑙𝑖∣𝑗max ). 𝛼 ⋅ (1 − 𝑖,𝑗 𝑟 Assume UE𝑖 locates at 𝑋𝑖 (𝑡) at time 𝑡 and 𝒱(𝒮(𝑋𝑖 (𝑡), 𝑟)) is the set of UE𝑖 ’s neighboring file providers. Denote by {∥𝑋𝑖 (𝑡) − 𝑋𝑗 (𝑡)∥} the maximum 𝐿(𝑡) = max 𝑢𝑗 ∈𝒱(𝒮(𝑋𝑖 (𝑡),𝑟))
distance between UE𝑖 and its neighboring file providers at time 𝑡. Apparently, the Euclidean distance 𝑑𝑖,𝑗 (𝑡) is equal to or less than 𝐿(𝑡) and the equality holds if and only if UE𝑖 connects its farthest provider to fetch content. Hence, for 0 ≤ 𝑥 ≤ 𝑟, we have ℙ(𝑑𝑖,𝑗 (𝑡) ≥ 𝑥) ≤ ℙ(𝐿(𝑡) ≥ 𝑥). Let 𝐴(𝑚, 𝑎, 𝑏) denote the event that there exist 𝑚 neighboring file provider in region {𝑥 : 𝑎 ≤ ∥𝑥 − 𝑋𝑖 (𝑡)∥ ≤ 𝑏}, (0 ≤ 𝑎 < 𝑏 ≤ 𝑟). As 𝑁 UEs are Poisson distributed in the observed network region with density 𝜆𝒰 at all times. Hence, we have (𝜆𝒰 𝜋(𝑏2 − 𝑎2 ))𝑚 −𝜆𝒰 𝜋(𝑏2 −𝑎2 ) 𝑒 . 𝑚! For 0 ≤ 𝑥 ≤ 𝑟, ℙ(𝐿(𝑡) ≥ 𝑥) = 1 − ℙ(𝐴(0, 𝑥, 𝑟)) = 2 2 1 − 𝑒−𝜆𝒰 𝜋(𝑟 −𝑥 ) . Hence, we have ℙ[𝐴(𝑚, 𝑎, 𝑏)] =
𝔼[𝑑𝑖,𝑗 (𝑡)] ≤ 𝔼[𝐿(𝑡)] ∫ 𝑟 2 2 2 𝑒𝜆𝒰 𝜋𝑥 𝑑𝑥 ≤ 𝑟(1 − 𝑒𝜆𝒰 𝜋𝑟 ). = 𝑟 − 𝑒𝜆𝒰 𝜋𝑟
(14)
0
As 𝑣𝑖∣𝑗 (𝑡) ≜ ∥𝑀𝑖 (𝑡) − 𝑀𝑗 (𝑡)∥ ≤ ∣∥𝑀𝑖 (𝑡)∥ + ∥𝑀𝑗 (𝑡)∥∣ and i.i.d. mobility, we have 𝔼[𝑣𝑖∣𝑗 (𝑡)] ≤ 𝔼(∣∥𝑀𝑖 (𝑡)∥ + ∥𝑀𝑗 (𝑡)∥∣) = 4𝑙max /3. Hence, 𝔼[𝑅𝑀 𝑆(𝑡)] satisfies that 2
𝔼[𝑅𝑀 𝑆(𝑡)] ≥ 𝛼𝑒−𝜆𝒰 𝜋𝑟 + (1 − 𝛼)/3.
(15)
Corollary 1. During UEs travelling by time 0 < 𝑡 < 𝜏L , the average recover delay 𝔼[𝜎HO (𝑡)] is upper bounded by √ 8 𝜆ℬ 3𝜋 𝑡𝑙max [𝑑H
constant.
+ 𝑑D (1 −
2
𝛼𝑒−𝜆𝒰 𝜋𝑟 +(1−𝛼)/3 )], 𝑎
where 𝑎 is
Proof: As ℙB (𝑡) ≜ ℙ[𝑅𝑀 𝑆(𝑡) ≥ 𝑎] ≤ 𝔼[𝑅𝑀𝑎𝑆(𝑡)] and 2 𝔼[𝑅𝑀 𝑆(𝑡)] ≥ 𝛼𝑒−𝜆𝒰 𝜋𝑟 + (1 − 𝛼)/3, we have √ 𝑡 4 𝜆ℬ ∑ 𝔼[𝜎HO (𝑡)] = [𝑑H + 𝑑D (1 − ℙB (𝑡))]𝔼(∥𝑀 (𝑘)∥) 𝜋 𝑘=1 √ 2 8 𝜆ℬ 𝛼𝑒−𝜆𝒰 𝜋𝑟 + (1 − 𝛼)/3 𝑡𝑙max [𝑑H + 𝑑D (1 − )]. ≤ 3𝜋 𝑎 IV. S IMULATIONS In this section, our theoretic analysis is validated via simulations. The network is realized in a 1 km × 1 km area via two independent PPPs of macro BSs with density 𝜆ℬ = 52.6 BSs/km2 and mobile users with density
𝜆𝒰 = 10000 users/km2 (a city with large population, e.g., New York City). 10000 pedestrians move according to the constrained mobility model. Each time slot is 1 minute and the maximum movement length 𝑙max = 80 meters per time slot, which means the speed limits is about 1.4 m/s [15]. The transmission range of UE is 𝑟 = 200 meters. Following [16], let handover time 𝑑(H) = 0.085 sec and peer discovery time 𝑑(D) = 0.7 sec. Let simulation period 𝑇 = 60 (minutes). During the simulation period, the content request time 𝑇R follows Gamma distribution with shape ℎ = 2 and scale 𝜃 = 5. The expectation of inter-arrival time of users’ repetitive content requests is 𝜇 = 2 minutes. Fig. 4 and 5 illustrates the analysis accuracy of ℙ(𝜏L > 𝑡) shown in Theorem 1 with different mobile user density and transmission range. It shows that with the increases of mobile user density and transmission range, the service lifetime is extended. From Fig. 6, we find that when the content popularity is higher, the probability of achieving longer service lifetime is larger. Moreover, Fig. 7 shows that simulation results of 𝔼(∥𝐷L (𝑡)∥) is well bounded by corresponding analytic bounds in Theorem 2. The average service distance exhibits approximately linear increase with service lifetime and is larger with higher velocity. Remark 2. Theoretical and simulation results in Fig. 4, 5, 6 and 7 indicate that D2D file sharing is benefit for popular content among high density users with large transmission range, which can realize a durable file-sharing service with long service lifetime and distance. Fig. 8 shows that simulation results of 𝔼[𝜎HO (𝑡)] is well bounded by analytic bounds in Corollary 1. In the figure, the average recover delay exhibits linear increase with time slot, and is longer for higher speed users. Moreover, simulation results in Fig. 9 shows that the average recover delay per time slot decreases linearly with the increases of RMS when 𝑅𝑀 𝑆 ≤ 0.5. Especially, the recover delay drops significantly at the threshold 𝑅𝑀 𝑆 = 0.5. Similarly, the recover delay per time slot is longer for higher speed users, while is shorter for UEs with smaller transmission range. Remark 3. Theoretical and simulation results in Fig. 8 and 9 demonstrate that movement velocity and relative movement between D2D pair of users determines the handover patterns and recover delay. In other word, D2D file sharing is more benefit for low-speed group moving users in low-density networks (e.g., macro networks). As the complex of users mobility, D2D handover mechanisms designing is also a key problem for achieving a reliable D2D file sharing to satisfy the quality of experience. V. C ONCLUSION In this paper, we study the temporal and spatial limits of file sharing on the move in cellular-assisted D2D communication network. To characterize the D2D file sharing on the move, based on relative movement, we first define two D2D handover scenarios, i.e., bundled handover and split handover. For constrained mobility model, we derive
1
1
0.25
Simulation result (λ U=104 users/km 2) 4
Analytic result (λ U=10 users/km )
0.8
Simulation result (r=200 m) Analytic result (r=200 m) Simulation result (r=100 m) Analytic result (r=100 m)
2
0.8
Simulation result (λ U=5*103 users/km 2)
Request time Service lifetime
0.2
0.15 PDF
0.6
L
P(τ >t)
L
0.4
0.4
0.1
0.2
0.2
0.05
0
0
10
20
30 Time slot
40
50
0
60
0
Fig. 4. ℙ(𝜏L > 𝑡) for various 𝜆U .
30 Time slot
40
50
1600
Simulation (l max=100 m/min, λ B=52 BSs/km ) Simulation (l max=80 m/min, λ B=520 BSs/km s )
(t)] (s)
25
HO
L
1200
E[δ
800
Bound (l max=80 m/min, λ B=52 BSs/kms ) Bound (l max=100 m/min, λ B=52 BSs/kms )
20
Bound (l max=80 m/min, λ B=520 BSs/km s )
15 10
400 5 10
15 τL (min)
20
30 Time slot
40
50
60
1.5
30
Analytic bound (l max=100 m/min)
5
10
Fig. 6. PDF of request time 𝒯R and service lifetime ℙ(𝜏L = 𝑡).
s
Analytic bound (l max=80 m/min)
0
0
Simulation (l max=80 m/min, λ B=52 BSs/kms )
Simulation result (l max=100 m/min)
0
0
60
35 Simulation result (l max=80 m/min)
(m)
20
Fig. 5. ℙ(𝜏L > 𝑡) for various 𝑟.
2000
E(|D |)
10
Average recover delay per time slot (s)
P(τ >t)
Analytic result (λ U=5*103 users/km 2)
0.6
20
25
30
Fig. 7. Average of service distance 𝔼(∥𝐷L ∥).
0
0
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15 Time slot
20
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30
r=200 m, l max=80 m/min, λ B=52 BSs/km2
1
r=100 m, l max=80 m/min, λ B=52 BSs/km2 r=200 m, l max=100 m/min, λ B=52 BSs/km2 r=200 m, l max=80 m/min), λ B=520 BSs/km 2
0.5
0
0.1
0.3
0.5 RMS
0.7
0.9
1
Fig. 8. Average recover delay by time slot 𝑡 Fig. 9. Average recover delay per time slot 𝑡 𝔼[𝜎HO (𝑡)]. 𝔼[˜ 𝜎HO (𝑡)].
complementary CDF of service lifetime, and theoretical upper bounds on service distance and recover delay, and validate the results with simulations. R EFERENCES [1] M. N. Tehrani, M. Uysal, and H. Yanikomeroglu, “Deviceto-device communication in 5g cellular networks: challenges, solutions, and future directions,” IEEE Communications Magazine, vol. 52, no. 5, pp. 86–92, 2014. [2] B. Bangerter, S. Talwar, R. Arefi, and K. Stewart, “Networks and devices for the 5g era,” IEEE Communications Magazine, vol. 52, no. 2, pp. 90–96, 2014. [3] M. Chen, L. Wang, and J. Chen, “Users’ media cloud assisted d2d communications for distributed caching underlaying cellular network,” China Communications, vol. 13, no. 8, pp. 13–23, 2016. [4] J. Sepp¨al¨a, T. Koskela, T. Chen, and S. Hakola, “Network controlled device-to-device (d2d) and cluster multicast concept for lte and lte-a networks,” in Proc. WCNC 2011. IEEE, 2011, pp. 986–991. [5] L. Wang, L. Gao, A. Zhang, and M. Chen, “Social-aware filesharing mechanism for device-to-device communications,” in Proc. WCSP 2015. IEEE, 2015, pp. 1–5. [6] Y. Guo, L. Duan, and R. Zhang, “Cooperative local caching and file sharing under heterogeneous file preferences,” arXiv preprint arXiv:1510.04516, 2015. [7] J. Ye and Y. J. Zhang, “A guard zone based scalable mode selection scheme in d2d underlaid cellular networks,” in Proc. ICC 2015. IEEE, 2015, pp. 2110–2116. [8] X. Ma, J. Liu, and H. Jiang, “Resource allocation for heterogeneous applications with device-to-device communication underlaying cellular networks,” IEEE Journal on Selected Areas in Communications, vol. 34, no. 1, pp. 15–26, 2016.
[9] Q. Du, P. Ren, H. Song, Y. Wang, and L. Sun, “On p2p-share oriented routing over interference-constrained d2d networks,” in Mobile Ad-hoc and Sensor Networks (MSN), 2014 10th International Conference on. IEEE, 2014, pp. 138–143. [10] J. G. Andrews, F. Baccelli, and R. K. Ganti, “A tractable approach to coverage and rate in cellular networks,” IEEE Transactions on Communications, vol. 59, no. 11, pp. 3122– 3134, 2011. [11] Y. Wang, Y. Li, W. Wang, and M. Song, “A locality-based mobile caching policy for d2d-based content sharing network,” in Proc. GLOBECOM 2016. IEEE, 2016, pp. 1–6. [12] W. Bao and B. Liang, “Stochastic geometric analysis of user mobility in heterogeneous wireless networks,” IEEE Journal on Selected Areas in Communications, vol. 33, no. 10, pp. 2212–2225, 2015. [13] D. Callebaut, “Generalization of the cauchy-schwarz inequality,” Journal of mathematical analysis and applications, vol. 12, no. 3, pp. 491–494, 1965. [14] T. Aven, “Upper (lower) bounds on the mean of the maximum (minimum) of a number of random variables,” Journal of applied probability, pp. 723–728, 1985. [15] R. C. Browning, E. A. Baker, J. A. Herron, and R. Kram, “Effects of obesity and sex on the energetic cost and preferred speed of walking,” Journal of Applied Physiology, vol. 100, no. 2, pp. 390–398, 2006. [16] D. Han, S. Shin, H. Cho, J.-m. Chung, D. Ok, and I. Hwang, “Measurement and stochastic modeling of handover delay and interruption time of smartphone real-time applications on lte networks,” IEEE Communications Magazine, vol. 53, no. 3, pp. 173–181, 2015.
