Handover Rate and Sojourn Time Analysis in Mobile Drone-Assisted Cellular Networks
11 Handover Rate and Sojourn Time Analysis inMobile Drone-Assisted Cellular Networks
Mohammad Salehi and Ekram Hossain
Abstract —To improve capacity and overcome some of thelimitations of cellular wireless networks, drones with aerial basestations can be deployed to assist the terrestrial cellular wirelessnetworks. The mobility of drones allows flexible network recon-figuration to adapt to dynamic traffic and channel conditions.However, this is achieved at the expense of more handovers sinceeven a static user may experience a handover when the dronesare mobile. In this letter, we provide an exact analysis of thehandover rate and sojourn time (time between two subsequenthandovers) for a network of drone base stations. We also showthat among different speed distributions with the same mean, thehandover rate is minimum when all drone base stations movewith same speed.
Index Terms —Drone-assisted cellular networks, drone basestations, handover rate, sojourn time
I. I
NTRODUCTION
Using drones as aerial base stations, to assist terrestrialcommunications, is a promising approach to tackle the 5Gand beyond 5G challenges. Owing to their mobility and highflexibility, drones can provide on-demand communications toground users [1] but the downside is that 5G and beyondwireless networks with drone-assisted communications sufferfrom increased handover rate. In this letter, we derive thehandover rate (mean number of handovers in a unit time) andmean sojourn time (mean service time by each drone basestation) of a network of drone base stations.Existing works study the impact of mobility on handovermostly through handover rate, handover probability, and so-journ time. In this regard, [2] has derived the handover rateand mean sojourn time for a single-tier cellular networkwith Poisson point process (PPP) distributed terrestrial basestations. For multi-tier networks, [3] has studied the handoverrate, and [4] has studies the mean sojourn time. Handoverprobability is also studied in [5] for single-tier networks andin [6] for multi-tier networks. In terrestrial networks, handoveroccurs as the user moves across a cell boundary; however,in drone-assisted communications, even a static user mayexperience a handover due to drones’ mobility. Therefore, itis critical to study the handover rate for these networks.To study the handover rate in the context of drone-assistedcellular communication, most of the existing works onlyconsider the scenario where the drones act as users, i.e., theyfocus on the connection between a drone and terrestrial basestations [7], [8]. In this scenario, handover analysis is similar
The authors are with the Department of Electrical and Computer Engi-neering, University of Manitoba, Canada (Email: [email protected],[email protected]). This work was supported by a DiscoveryGrant from the Natural Sciences and Engineering Research Council of Canada(NSERC). to the previous works except that instead of moving in twodimensions, drones can move in three dimensions. In this case,we need to project drones mobility onto R plane. Recently,[9] has studied the handover probability for a network ofdrone base stations . They have derived the exact result for thescenario where all drones move with the same speed, but, forthe scenario with different speeds, only a bound is provided.However, as the first step toward handover rate analysis, weneed to derive the exact results. For this, we convert drones’mobility to user’s mobility, so that instead of having multiplemobile nodes we only have a single mobile node. Then, fromthe exact results, we can derive the handover rate and meansojourn time following the same steps as in [4].In Section II, we introduce the system model and state themethodology of analysis. Section III provides the analyticalresults. Numerical results are validated in Section IV, wherewe also study the effect of speed distribution for the drones.Finally, Section V concludes the paper.II. S
YSTEM M ODEL AND M ETHODOLOGY OF A NALYSIS
Consider a network of mobile drone base stations (BSs)that serves ground users. Drone BSs are initially (at time0) distributed at height h according to a two-dimensionalhomogeneous Poisson point process (PPP) Φ of density λ .Each drone BS moves in z = h plane in a random direction θ with respect to the positive x -axis with velocity v independentfrom other drone BSs and its location. f Θ ( . ) and f V ( . ) providedistributions of θ and v . Assuming straight line trajectories forthe drones provides performance bound for more complicatedmodels. It also complies with drones’ mobility model in 3GPPsimulations [11]. From the stationarity of the homogeneousPPP, we can assume that a user is located at the origin ofour coordination system. This user is always associated withthe drone BS that provides the maximum averaged receivedpower (i.e., nearest drone). Since all drone BSs are at thesame height, for the purpose of handover rate and sojourntime analysis, we can ignore h and focus on the R plane.Therefore, in the following section, by location we mean theprojected location of the drone BS onto the x - y plane.The methodology of analysis of handover rate and sojourntime is as follows: • Step 1: We derive the conditional distribution of thesojourn time for the initially serving drone base station, Definition of handover probability in [5], [6] is different from [9]. Toanalyze the handover probability [5], [6] only consider two time instants; forexample, time 0 and t . On the other hand, [9] considers the entire time intervalbetween 0 and t , i.e., [0 , t ] . To understand the difference and relation betweenthese two, refer to [4], [10]. a r X i v : . [ c s . N I] J un given that this drone moves with velocity v in direction θ (Section III.A). • Step 2: We calculate the handover rate from a drone basestation with velocity v and movement direction θ toany other drone base station (Section III.B). • Step 3: Finally, the handover rate (inverse of mean so-journ time) can be obtained by integrating over different v , and θ (Section III.B).III. H ANDOVER R ATE AND S OJOURN T IME A NALYSIS
The initial serving drone BS moves with velocity v indirection θ with probability P ( v , θ ) = f V ( v )d v f Θ ( θ )d θ . (1)Let us denote the distance between the (projected) location ofthe initial serving drone BS and the origin at time by r .Conditional probability density function (PDF) of r given v and θ is f R ( r | v , θ ) = 2 λπr e − λπr . (2)Since the model is isotropic (invariant under rotation), wecan assume that (projected) location of the serving drone BSat time 0 is at [ r , T . In this section, we first derive thedistribution of the time until the first handover, denoted by ˜ S ,given that the initial serving drone BS moves with velocity v in direction θ . Then following the same steps as in [4], wederive the handover rate and sojourn time. A. Conditional Distribution of ˜ S The complementary cumulative distribution function(CCDF) of the time until the first handover ˜ S given r , v ,and θ can be obtained by P (cid:16) ˜ S > s | r , v , θ (cid:17) = P (cid:18) (cid:92) v (cid:92) θ Φ ( t ) v,θ (cid:0) b (cid:0) [0 , T , r ( t ) (cid:1)(cid:1) = 0 ∀ t ∈ (0 , s ] | r , v , θ (cid:19) , (3)where r ( t ) = (cid:112) r + v t + 2 r v t cos( θ ) is the distancebetween the projected location of the initial serving droneBS and the origin at time t . b (cid:0) [0 , T , r (cid:1) denotes a ball withradius r centered at the origin. Φ (0) v,θ ⊂ Φ \ { [ r , T } denotesthe initial location of the non-serving drone BSs that movewith velocity v in direction θ . According to the thinningproperty, their spatial distribution follows a PPP with density λf V ( v )d vf Θ ( θ )d θ in R \ b (cid:0) [0 , T , r (cid:1) . At time t , we have Φ ( t ) v,θ = Φ (0) v,θ + vt [cos( θ ) , sin( θ )] T . Since Φ (0) v,θ s are independentfor different v and θ , (3) can be further simplified as P (cid:16) ˜ S > s | r , v , θ (cid:17) = (cid:89) v (cid:89) θ P (cid:18) Φ (0) v,θ ( b ( x v,θ ( t ) , r ( t ))) = 0 ∀ t ∈ (0 , s ] | r , v , θ (cid:19) , (4)where x v,θ ( t ) = vt [cos( π + θ ) , sin( π + θ )] T . From (4) wecan understand that sojourn time analysis for a static user in anetwork of moving drone BSs with velocity v and direction θ is similar to the sojourn time analysis for a mobile user withvelocity v and direction π + θ in a network of static droneBSs. To calculate the right hand side of (4), let us define A ( v, θ, r , v , θ , s ) (cid:44) (cid:40)(cid:91) t b ( x v,θ ( t ) , r ( t )) : t ∈ [0 , s ] (cid:41) . (5)Using the above definition, we can write P (cid:16) ˜ S > s | r , v , θ (cid:17) = (cid:89) v (cid:89) θ P (cid:16) Φ (0) v,θ (cid:0) A ( v, θ, r , v , θ , s ) \ b (cid:0) [0 , T , r (cid:1)(cid:1) = 0 | r , v , θ (cid:17) . (6)The above probability can be calculated by using the voidprobability of the PPP. P (cid:16) ˜ S > s | r , v , θ (cid:17) (a) = (cid:89) v (cid:89) θ e − λ ( |A ( v,θ,r ,v ,θ ,s ) |− πr ) f V ( v )d vf Θ ( θ )d θ = e − λ (cid:82) v (cid:82) θ |A ( v,θ,r ,v ,θ ,s ) | f V ( v ) f Θ ( θ )d θ d v + λπr . (7)where |A| denotes area of the region A , and (a) fol-lows from b (cid:0) [0 , T , r (cid:1) ⊂ A ( v, θ, r , v , θ , s ) . Note that |A ( v, θ, r , v , θ , s ) | does not depend on θ ; thus, we can write P (cid:16) ˜ S > s | r , v , θ (cid:17) = e − λ (cid:82) v |A ( v,r ,v ,θ ,s ) | f V ( v )d v + λπr , (8)where A ( v, r , v , θ , s ) = (cid:40)(cid:91) t b (cid:0) [ vt, T , r ( t ) (cid:1) : t ∈ [0 , s ] (cid:41) . (9)We can calculate |A ( v, r , v , θ , s ) | from Theorem 1 in [4]by changing r → vv r , β kj → vv , θ → π + θ , and T → s .Using (2) yields P (cid:16) ˜ S > s | v , θ (cid:17) = (cid:90) ∞ λπr e − λ (cid:82) v |A ( v,r ,v ,θ ,s ) | f V ( v )d v d r . (10) Remark : To derive the distribution of ˜ S for a static user ina hybrid network with terrestrial BSs and mobile drones , letus use subscript 1 for the tier of drone BSs and subscript 2for the tier of terrestrial BSs. B i , h i , λ i , and α i denote thebias factor , height, density, and path-loss exponent of tier i ,where i ∈ { , } . Let us define f i,j ( x ) = (cid:118)(cid:117)(cid:117)(cid:116)(cid:34)(cid:18) B i B j (cid:19) /α i (cid:0) r j + h j (cid:1) α j /α i − h i (cid:35) + , where i, j ∈ { , } and [ y ] + = max(0 , y ) . With maximumbiased averaged receive power association, the user is initiallyserved by a tier j BS if r i > f i,j ( r j ) , i (cid:54) = j , where r i is For this hybrid network, we only derive the distribution of ˜ S since this isthe fundamental step. Other steps are straightforward as will be discussed inthe next subsection. Bias factor also incorporates the effect of transmission power and meanpower of small scale fading. the projected distance of the nearest tier i BS to the origin attime 0. Due to independence of drone BSs and terrestrial BSs,when the user is initially served by a drone BS, we have P (cid:16) ˜ S > s | r , v , θ , tier = 1 (cid:17) = e − λ (cid:82) v |A ( v,r ,v ,θ ,s ) | f V ( v )d v + λ πr × e − λ π [ f , (max( r ,r ( s )) − f , ( r ) ] . When the serving BS at time 0 is a terrestrial BS, we have P (cid:16) ˜ S > s | r , tier = 2 (cid:17) = e − λ E [ v ] sf , ( r ) . B. Main Results
To derive the handover rate and mean sojourn time, we firstneed to calculate [4] E [ L | v , θ ] = lim z → z − P (cid:16) ˜ S > zv | v , θ (cid:17) . (11)For a drone BS with velocity v and movement direction θ , E [ L | v , θ ] is the average length of its trajectory duringwhich the drone BS serves the user at the origin. Thus, meansojourn time for drone BSs that move with velocity v indirection θ is E [ S | v , θ ] = E [ L | v , θ ] v (a) = 1 v × − dd z P (cid:16) ˜ S > zv | v , θ (cid:17) (cid:12)(cid:12)(cid:12) z =0 , (12)where (a) follows from the L’Hospital’s Rule. From (10), wehave dd z P (cid:18) ˜ S > zv | v , θ (cid:19) = − (cid:90) ∞ λπr e − λ (cid:82) v |A ( v,r ,v ,θ , zv ) | f V ( v )d v × λ (cid:90) v dd z |A ( v, r , v , θ , zv ) | f V ( v )d v d r . (13)Denominator of (12) can be calculated by substituting |A ( v, r , v , θ , | = πr , and dd z |A ( v, r , v , θ , zv ) | (cid:12)(cid:12)(cid:12) z =0 = 2 r × (cid:114)(cid:16) vv (cid:17) − cos θ + cos θ cos − (cid:16) − cos θ vv (cid:17) , if | cos θ | ≤ vv , , if vv < − cos θ ,π cos θ , if vv < cos θ , . in (13). Therefore, E [ S | v , θ ] = 1 √ λ E v [ F ( v, v , θ )] , (14)where E v denotes the expectation with respect to v , and F ( v, v , θ ) = πv cos θ (cid:18) vv < cos θ (cid:19) + v (cid:32)(cid:115)(cid:18) vv (cid:19) − cos θ + cos θ cos − (cid:32) − cos θ vv (cid:33) (cid:33) × (cid:18) | cos θ | ≤ vv (cid:19) . (15) ( . ) , in (15), is the indicator function. From (14), we cancalculate the mean number of handovers from (to) a droneBS with velocity v and movement direction θ to (from) anyother drone BS as [4] H v ,θ = P ( v , θ ) E [ S | v , θ ]= √ λ E v [ F ( v, v , θ )] f V ( v )d v f Θ ( θ )d θ . Finally, handover rate and mean sojourn time are obtained by H = 1 E [ S ] = (cid:90) v (cid:90) θ H v ,θ = √ λ E [ F ( v, v , θ )] , (16)where the expectation is over all the random variables (i.e., v , v , and θ ). Special Case I : When all drone BSs move with samevelocity v , E [ F ( v, v , θ )] = π v . Thus, H = π √ λv whichis equal to the handover rate of a mobile user with velocity v in a single-tier network of terrestrial BSs [2]. Special Case II : Consider a scenario where each drone BSeither moves with velocity v > or remains static . Let us de-note the probability that a drone BS moves with p m . The han-dover rate for this case is H = 2 √ λvp m (cid:0) − (cid:0) − π (cid:1) p m (cid:1) , where √ λvp m (1 − p m ) is the handover rate from a movingdrone BS to a static drone BS which is equal to the handoverrate from a static drone BS to a moving drone BS. π √ λvp m is also the handover rate from a mobile drone BS to anothermobile drone BS.Next, we solve the following optimization problem: min f V H = √ λ E v,v ,θ [ F ( v, v , θ )] subject to v, v ∼ f V , (17) E [ v ] = c, (18)supp ( f V ) ∈ R + , (19)i.e., we want to find the speed distribution for which thehandover rate is minimum. Condition (17) indicates that v and v are two independent realizations of the distribution f V .According to (18) and (19), f V is a distribution with mean c and positive support (it only outputs positive real numbers).In the following corollary, we provide the solution of thisoptimization problem. Corollary 1.
The handover rate is minimum when all dronebase stations move with speed c , compared to any other speeddistribution with mean c .Proof: Since F ( v, v , θ ) is convex with respect to v and v , from Jensen’s inequality, we have E v,v ,θ [ F ( v, v , θ )] ≥ E v ,θ [ F ( E [ v ] , v , θ )] ≥ E θ [ F ( E [ v ] , E [ v ] , θ )] . This scenario can be further extended to study the handover rate for ahybrid network of terrestrial and drone base stations.
SimulationUpper BoundAnalysis
Fig. 1. Conditional CCDF of time until first handover given r = 12 , v =10 , and θ = π/ . λ = 0 . . v is uniformly distributed in the interval [5 , . Drone BS Density ( ) -4 H ando v e r R a t e ( H ) v~Unif[8,12] v~Unif[10,20]v~Unif[5,15] Simulation - V~Unif[5,15]Analysis - V~Unif[5,15]Simulation - V~Unif[8,12]Analysis - V~Unif[8,12]Simulation - V~Unif[10,20]Analysis - V~Unif[10,20]Analysis - V=10Analysis - V=15 (a) Different means and variances.
Drone BS Density ( ) -4 H ando v e r R a t e ( H ) v~Unif[5,15]v~Exp(1/10)v=10v~Unif[10,20]v~Exp(1/15)v=15 (b) Different mobility models.Fig. 2. Handover rate with respect to drone density for different speeddistributions. The equality holds only when v and v are constants. Since, v and v are two realizations of the same distribution with mean c , we have the equality only when v = v = c .IV. N UMERICAL R ESULTS
In this section, we validate the analytical results by com-paring them with simulation results. We also study the effectof mean and variance of the speed distribution of drones onthe handover rate.We compare (8) with simulation in Fig. 1. ˜ S denotes thetime until the first handover and is greater than s when thereis no handover in the time interval [0 , s ] . For the scenariowhere different drone BSs move with the same speed, whenthere is a handover, the initially serving drone BS will notserve the user again (always exist a closer drone base stationto the user than the old drone base station after handover).This is also proved in Lemma 2 in [9]. Therefore, in thiscase, handover does not occur in the time interval [0 , s ] if theserving base station at time s is the same as the initial servingbase station. However, in a scenario where drone BSs movewith different speeds, a drone BS can serve a user at t < t and t < t (for t < t ), while in the time interval [ t , t ] another drone BS serves the user. In other words, ˜ S may beless than s even when the serving base stations at time and s are the same. Therefore, in this case, checking for ahandover event by comparing serving base stations at time and time s (instead of the whole interval [0 , s ] ) provides anupper bound. In Fig. 2, we show the effect of speed distributionof drones on the handover rate for different density values ofdrone BSs. According to Fig. 2(a), with increasing the meanspeed of drones, the handover rate increases. Also, increasingthe variance of the speed distribution (while keeping its mean the same) increases the handover rate. Therefore, when alldrone base stations move with same speed v , the handoverrate is minimum, compared to the any other distributionwith mean v . In Fig. 2(a), the handover rate is illustrated fordifferent mobility models. Specifically, we compare uniformdistribution with exponential distribution and deterministicdistribution. These distributions are related to random walkand modified random waypoint mobility models [10].V. C ONCLUSION
We have derived the handover rate and mean sojourn timefor a network of drone base stations. We have also shown that,handover rate is minimum when all drone base stations movewith same speed (compared to any other distribution with thesame mean). Although we have considered a simple network,our results can be easily extended for more complicatedscenarios. Specifically, the handover rate in a hybrid networkof terrestrial and aerial base stations can be derived followingthe same approach. R
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