The Spatial Effect of Mobility on the Mean Number of Handoffs: A New Theoretical Result

Transcription

1 The Satial Effect of Mobility on the Mean Number of Handoff: A New Theoretical Reult Wolfgang Bziuk, Said Zaghloul and Admela Jukan Techniche Univerität Carolo-Wilhelmina zu Braunchweig, {bziuk, zaghoul, jukan}@idaingtu-bde Abtract The mean number of handoff i a fundamental erformance meaure in any mobile ytem, a it directly relate to the ignaling load in the network a well a to the delivered QoS A the mobile ervice are evolving from imle cellular voice call toward media and data eion, and a cellular rovider are reinventing their buinee by incororating third arty ervice, the handoff rate will even lay a more ivotal role Thi i becaue future eion are exected to lat longer than voice call and uer are more likely to roam into other network Exiting reult rovide an etimate of the mean number of handoff in network comoed of an infinite number of acce gateway and hence conider neither the toological arrangement of the gateway nor the mobility attern between them In thi aer, we obtain a cloed form olution for the mean number of handoff under generic aumtion of twodimenional Markovian mobility attern, atial arrangement of the acce gateway, a well a generic eion time and gateway reidence time Our olution unveil a new inight into mobility foundation a it how that the conideration of the mobility attern and the acce gateway layout imly tranform the known equation for the mean number of handoff of an infinite network ize from calar to vector rereentation We demontrate that the mean number of handoff i a non-linear function of the gateway atial arrangement and uer mobility I INTRODUCTION Uer mobility ha been and will continue to be a core deign challenge to the wirele mobile ytem In thee ytem, the mean number of handoff i a fundamental erformance meaure, a it directly relate to the ignaling load in the network a well a to the delivered Quality of Service (QoS) Although handoff in radio layer have been tandardized and uccefully imlemented, the imact on higher layer i not well undertood The higher layer ignaling and QoS roviion are triggered when uer move between the o-called IP Acce Gateway (), which tyically erve a grou of bae tation (ee Fig1a) Examle of gateway in the current ytem are GGSN in 3GPP ytem, ASN in WiMAX ytem, and PDSN in 3GPP2 ytem A uer move between gateway, MobileIP ignaling i triggered to maintain IP connectivity, Diameter/RADIUS ignaling i triggered for authentication, and SIP/QoS ignaling [1] i triggered to authorize the eion at the target gateway, a tandardized within the IP Multimedia Sub-ytem (IMS) [2] Therefore, in thee ytem the handoff between gateway, reulting from handoff between border bae tation of each gateway area, are no longer limited to few SS7 exchange, but rather extended to a melange of alication layer ignaling rotocol, with imact on the ignaling load, handoff latency, and eion droing robabilitie Pat etablihed theorie [3], [4], [5] in circuit witched cellular ytem cannot be eaily ued to ae the mean number of handoff generically Firt, they majorly focued on aect of call erformance uch a the etimation of the effective call duration and call blocking in macro and micro cellular ytem In uch tudie, mobility wa only characterized by the mobility factor defined a the ratio of the mean eion duration to the mean time a uer end in a cell (ie, the cellular reidence time) Second, due to the limited voice call duration uch ytem were aumed infinite in ize and hence uer are alway erved by one network for the entire call duration Hence, neither the toology nor the mobility attern were conidered In next generation ytem, at aumtion on the eion characteritic no longer hold a uer are likely to exerience long eion duration and hence are more likely to change their erving network Therefore, etimating the mean number of handoff between gateway deend not only on the mobility factor but alo on the mobility attern among acce gateway and their arrangement in the network Thi require reviiting the baic of at reearch and extending them to account for uch effect In thi article, we derive for the firt time the cloed form exreion for the handoff rate under generic aumtion of two-dimenional arrangement of acce gateway, eion time and acce gateway reidence time ditribution, and arbitrary mobility attern Our framework build on etablihed technique from cellular network[3], [4] and intertwine it with concet of ixel baed mobility behavior ued in realitic imulation tudie [6] uing tranient Markov technique and comlex analyi We obtain a highly intereting reult which reveal the fact that the known equation for the mean number of handoff in the literature imly tranlate from calar to vector rereentation when conidering mobility attern and gateway arrangement In fact, we how that it i no longer ufficient to rereent mobility by the reidence time, a in the current body of literature, but rather a a roduct of a calar quantity rereenting the reidence time and a atial vector comonent correonding to the toology and the mobility attern Thi aer i a major extenion of the reliminary reult we reented in [7], [8], a we here conider generic mobility attern, eion time, and two dimenional network layout The model i yet elegant, a it comutational comlexity i only a function of the number of gateway and doe not deend on the reidence- and eion time tatitical ditribution Thi aer i organized a follow In Section II, we reent /09/$ IEEE Authorized licened ue limited to: TECHNISCHE UNIVERSITAT BRAUNSCHWEIG Downloaded on June 22,2010 at 17:13:40 UTC from IEEE Xlore Retriction aly

2 the analytical model In Section III, we how numerical reult Section IV conclude the aer CallSeion ControlFunction (CSCF) Diam meter II ANALYTICAL MODEL IMSNetwork HomeSubcriber Subytem(HSS) PolicyControl Function(PCF) AAA Home Agent(HA) Alication Server(AS) Diameter IMSNetwork CSCF HSS PCF AAA BaeStation(BS) BS BS BS BS BS BS NetworkUnderConideration Roaming (a) Partner Network S M G d a Handoff b 5 6 c 8 N G S 9 RoamingPartner Network (b) Offnet netarrival OnnetArrival, forclaritynot howninart(b) Fig 1 (a) Network under conideration, and (b) amle toology with n =9 Acce Gateway (); the border of every gateway are marked a, b, c, d correonding to north, eat, outh and wet movement (eg, 5) In thi ection, we derive the mean number of handoff (MNH) for a eion either entirely or artially erved by a network of limited ize (ie, the network under conideration), which i characterized by a quare area erved by M g N g gateway The mobility model which decribe movement between i baed on the concet of ixel baed mobility modeling [6], develoed for realitic mobility imulation Our mobility model i veratile a it can rereent a wide range of movement attern ranging from directed attern (eg, highway) to comletely random movement In the analytical aroach reented here, we tart by jointly conidering eion time, reidence time, and mobility attern uing a tranient Markov chain with an infinite tate ace Baed on the tructure of the tate ace, we then decomoe the Markov chain, which reult in a cloed form olution incororating the atial comonent of the mobility, ie, the gateway layout and network toology A in our at work [8], to imlify the model decrition we ditinguih between the o-called on-net and off-net traffic On-net traffic refer to any eion which tart within the network under conideration Off-net traffic, on the other hand, refer to eion which tart elewhere, and roam into the network under conideration at ome random oint of time To facilitate the dicuion, we will firt dicu the traffic tarting from within the network (on-net traffic) and then briefly dicu the imlicity of incororating offnet traffic at the end of Sec II AS HA A Aumtion The eion duration, S, ha denity f S (t), a rational Lalace tranform f S () and a mean of E The gateway reidence time, R, are indeendent and identically ditributed R i generally ditributed with an exiting Lalace tranform f R () and mean of E R For imlicity, we aume that eion are alway reumed after handoff The incluion of blocking i traightforward and can be carried out a in [3] B Handoff Probability Preliminarie Since a new eion tart inide an coverage, the reidence time in the firt, R 1, i different from the ubequent one Auming that R 1 i given by the reidual of the reidence time, then it denity and Lalace tranform are given a f R1 (t) = 1 f R (t) E R reectively Let R (k) = R 1 + k, () = 1 f R () E R R denote the um j=2 of reidence time incurred ince the eion tarting event until the k th handoff event R(k) ha Lalace tranform of fk () = f R 1 ()(fr ())k 1, k 1 It follow that the robability that a call ha already incurred k-1 handoff and that it eion duration i large enough to include at leat one more handoff i [4], P H (k) =P {R (k) S R (k 1) S } = G (k) G (k 1) where G (0) = 1 G (k) i the robability that the eion contain at leat k handoff (k 1) (ie, P {R (k) S}) and i calculated a [4], [9], G (k) = 1 σ+j (1) ()(fr ())k 1 fs ( ) d (2) Furthermore, if fs () i a rational function and if the et of ole, Ξ S,offS ( ) only contain ole in the right half ide of the comlex lane (ie R{ } > 0), the reidue theorem can be alied to olve the integral (2) Let Re = denote the reidue at ole =, then we have [4], G (k) = fr Re 1 () (fr ()) k 1 fs ( ) (3) = P Ξ S Uing (1)-(3), it can be hown that the mean number of handoff during the whole eion i given a [9], [12], E {HO} = C Mobility Model P Ξ S Re = () fs ( ) (1 fr ()) During a eion, a mobile node may travere multile area To imlify the dicuion, let u firt conider an area coniting of rectangularly arranged (ie, M g N g number of gateway within the network under conideration (ee Fig1b)) In the end of thi ection we how how to relax thi aumtion When a mobile enter thi area it future movement i decribed by a et of tranition robabilitie which deend on the entering and exit border [6] Thu each (4) Authorized licened ue limited to: TECHNISCHE UNIVERSITAT BRAUNSCHWEIG Downloaded on June 22,2010 at 17:13:40 UTC from IEEE Xlore Retriction aly

3 i decribed by 4 4 tranition robabilitie Uing the border labeling hown in Fig1b for 5, the tranition robabilitie are denoted a jxy, where j denote the, x and y define the entering and the exit border reectively uch that (x, y) {a, b, c, d} The tranition robabilitie can be arranged into a et of two different one te tranition matrice, one that define the initial tranition robabilitie P MI (ie, when the eion i firt erved by the network) and another, P M that define the tranition robabilitie afterward (ie, after the initial handoff) a, P MI = ( ) Q MI A MI, PM = ( ) Q M A M (5) Auming a network of n = M g N g acce gateway, the matrix Q M decribe the movement of a handoff eion between and ha 4n 4n element decribing movement between neighboring For examle, a eion leaving 5 in Fig1b at border a enter 2 at border c Thu the exit border a of j i linked to the entry border c of (j N g ), where N g i the number of in a row Let u number the column and row of Q M by the entry border of the a 1a, 1b, 1c, 1d, 2a, 2b, 2c, 2d,, na, nb, nc, nd For a given entry border there are u to four oible tranition For intance, entering from border a in the j th (ie (ja) th row of Q M ), the tranition robabilitie (which um to one) are jaa, jab, jac, jad They have the column number (j N g ) c, (j +1)d, (j + N g ) a and (j 1) b, reectively, [6] Additionally for an at the network boundary, all tranition to the roaming artner (V) are lited in the 1 4n matrix A M An examle for 2 (row (2a) and (2b)) i hown for the matrixe Q M and A M below, where the tate numbering i added on to and to the right ide for clarity Q M 1a 1b 3d 5a V 0 0 2ad 2bd 2ab 2bb 2ac 2bc 2a, A 2b m 2 2 The matrix Q MI i of dimenion n 4n and contain the tranition robabilitie for a new eion A eion tarting in j and leaving at border y correond to the jth row of Q MI and ha tranition robabilitie ˆ ja, ˆ jb, ˆ jc, ˆ jd Again, all tranition to the roaming artner created by boundary are combined in the matrix A MI Finally, it hould be noted that our analyi i indeendent of the grid like arrangement of For other arrangement (eg, irregular arrangement of area with more than 4 neighbor), we imly add row and column with the correonding tranition robabilitie for each edge in the mobility matrice in (5) D Mean Number of Handoff In thi ubection, we derive the mean number of handoff, E {N H }, for eion artially erved by a network comried of M g N g by majorly extending the tranient Markov aa ba V T 1a 1 k 1 P H 1 P H ˆ ˆ 1d 0,1 3,2c 3,4b 3,6d 3,8a 2,1c tranient tate 3 th handoff, matrix D 3 P H P H 3 5 dc 3 5 da 2,5d 2 4 aa 1,2d 1,4a P H 1 ˆ 1b P H 1 ˆ 1c no handoff, matrix D 0 2,7a P H 2 4 ab P 2 ac P H H 4 P H P H 3 5 db 3 5 dd P H 2 4 P H ad 1 ˆ ˆ Fig 2 Model diagram for the network of Fig1b with elected tate and tranition only (all tranient tate can reach tate T, tate V i drawn twice for clarity, initial tate haded, tranient tate dahed) model in our reviou work [8] to incororate the mobility model decribed in Section IIC and to account for two dimenional arrangement In our context, a tranient Markov chain contain tranient and aborbing tate For a tranient tate, there i a chance of never being reviited after the initial viit Thi i due to the exitence of aborbing tate that are never abandoned once entered In our model, tranient tate rereent the gateway inide the network, while the aborbing tate rereent dearture from the network area to roaming artner network or eion termination Since it i ractically infeaible to track eion in roaming artner network, dearting eion return only a off-net arrival (ee [8] for more detail) Let the tranition robabilitie P be, P = ( Q A 0 I ), where Q contain only tranition between tranient tate, A contain only tranition to the aborbing tate Λ i, and I i the identity matrix with roer dimenion Let the row vector P S denote the initial robabilitie for the tranient chain Uing the fundamental matrix M, given a M =(I Q) 1, the mean number of viit to tranient tate, excluding the firt and before abortion i [8], [10], 0,9 E {N} = P S Me T 1, (6) where e = {1, 1,, 1} and e T denote the tranoe The 1 in (6) i ubtracted becaue P S Me T include the initial viit (ie, the eion tart) The robability of being aborbed into tate Λ i i [10], β i = P S M A i where A i i the i th column of A (7) Since we deal with generally ditributed eion time when calculating the likelihood of making future handoff, we mut track the number of comleted handoff Thu our tranient chain conit of 2-tule tranient tate definition, 9b 9c V Authorized licened ue limited to: TECHNISCHE UNIVERSITAT BRAUNSCHWEIG Downloaded on June 22,2010 at 17:13:40 UTC from IEEE Xlore Retriction aly

4 where (0,j) rereent a eion tarting inide the j th, while the tate (k, jx) i aigned to a eion entering the j th through a border x {a, b, c, d} after comleting k handoff We alo define two aborbing tate: V rereenting a eion leaving the domain to the roaming artner and T rereenting the eion termination Fig2 how a hort ummary of the model, where the initial tate are haded and the tranition to the aborbing tate are dahed For a eion tarting in j (ie, tate (0,j)), the tranition robabilitie are given by the joint event comried of the tranition robabilitie ˆ jy for the initial movement (ummarized in matrix Q MI ), and the robability that the eion contain at leat one more handoff given that it made no handoff, P H (1) For examle in Fig2, a eion tarting in 1 (ie, tate (0, 1)), leaving through border c and handing over to border a of 4 (tate (1, 4a)) i decribed by the tranition robability P H (1) ˆ 1c With (1) and (5) all tranient tranition from (0,j) to (1,ix) are written in matrix form a D 0 = P H (1) Q MI, where D 0 i a n 4n matrix Similarly, the initial tranition from tranient tate (0,j) to the aborbing tate V are decribed by A 0 = P H (1) A MI Otherwie, if the eion ha terminated, the chain goe to the aborbing tate T with a robability of 1 P H (1) Now conider the examle, that a eion enter 5 through border d after the econd handoff (ie, tate (2, 5d) in Fig2) If the eion leave the through border b, the tranition robability to neighboring border d of 6 i given by P H (3) 5db Uing (1) and (5) all tranient tranition out of tate (k, jx) to tate (k +1,iy) can be written in matrix form a D k = P H (k +1)Q M, where D k i a 4n 4n matrix The tranition from tranient tate to the aborbing tate V are imilarly decribed by A k = P H (k +1)A M Ordering tate lexograhically a (0, 1),, (0,n),(0, 1a),, (0,nd),(1, 1a),, (1,nd),, (k, 1a),, (k, nd), the tranient Markov chain i given a, Q= 0 D D D 2 0, A= A 0 A 1 A 2 (8) where Q i a matrix of unlimited ize ince the number of handoff k can go to infinity The element of (8) are, D 0 = P H (1) Q MI, A 0 = P H (1) A MI (9) D k = P H (k +1)Q M, A k = P H (k +1)A M, k 1 Let the initial tate robabilitie for new eion be defined a P S = [P I, 0, 0, ] where P I = [ε 1,ε 2,, ε n ] and ε j rereent the robability of tarting a eion from j, then uing (6) and (7), the mean number of handoff before leaving the network, E {N}, to the roaming artner, E {V H }, and in total, E {N H },are, E {N}=P S Me T 1,E{V H }=β V,E{N H }=E {N}+β V (10) E From Scalar to Vector Rereentation Although (10) give the olution for the MNH, the matrix M =(I Q) 1 may have coniderable ize reulting in comutational and numerical iue In the following rooition we avoid the inverion oeration by tudying (8) a, Prooition 21: Let the matrix Q of (8) be truncated to an arbitrary finite number of handoff k le than K (ie Q contain ub-matrice u to D K ), then the matrix M = (I Q) 1 ha the tructure M = I D 0 D 0 D 1 D 0 D 1 D 2 D 0 D 1 D 2 D 3 0 I D 1 D 1 D 2 D 1 D 2 D I D 2 D 2 D 3 (11) Eq(11) i imle to obtain by olving the linear ytem M (I Q) = I and alying rincile of induction The detail of the roof are omitted due to ace limitation Uing (6) and (11), the mean number of handoff before leaving the network E {N} in (10) i given a, K k E {N} = P S Me T 1= P I I + j=0 D j e T 1 (12) Let u define (Q M ) 0 = I, then utilizing (9) we get, K k E {N} = P I Q MI (Q M ) k P H (j +1) e T j=0 ( K ) = P I Q MI G (k +1)(Q M ) k e T (13) where we ued the obervation that P H (k)=g(k)/g (k 1) k and G (0) = 1, (ie, j=0 P H (j +1) = k j=0 G(j+1) G(j) = G (k +1)) Uing the comlex integral rereentation (2) for G (k) and letting the limit K, we get, E {N} = P I Q MI ( K k Q σ+j M σ+j = 1 = 1 σ+j () fs ( ) ) (fr ()) k d e T () P I Q MI fs ( ) K (fr () Q M ) k e T d () P I Q MI fs ( ) M R () e T d where we have defined M R ()=(I fr () Q M) 1 Uingthe reidue theorem we obtain the final cloed form olution a, E {N}= fr Re 1 () P I Q MI fs ( ) M R ()e T (14) = P Ξ S Comaring the mean number of handoff in (4) and our reult in (14), we oberve that the calar reidence time term () and fr () reflecting the eed of the uer and the ize are now multilied by a atial comonent which Authorized licened ue limited to: TECHNISCHE UNIVERSITAT BRAUNSCHWEIG Downloaded on June 22,2010 at 17:13:40 UTC from IEEE Xlore Retriction aly

5 correond to the mobility model () i multilied by the initial movement matrix and the initial robabilitie (ie, P I Q MI ) and fr () i multilied by the movement matrix Q M Such elegant reult allow u to ay that the conideration of the atial aect due to mobility attern and arrangement imly tranform the known olution for the mean number of handoff from the calar form in (4) to the vector rereentation in (14) Finally, uing (11) and auming that A K+1 = 0, the roaming robability in (10) i given a, K k 1 β V =P S MA = P I A 0 + P I D j A k (15) k=1 j=0 ( K = G (1) P I A MI + P I Q MI G (k+1)(q M ) )A k 1 M k=1 Uing (2), the econd term in (15) denoted a ˆβ i given a, ˆβ = P [ σ+j IQ MI f R1 () fr () f S ( ) ] K (fr () Q M ) k 1 A M d k=1 Taking the limit K and alying the Reidue theorem a hown in (3), we get a cloed form exreion a, β V = G (1) P I A MI (16) fr Re 1 () P I Q MI fs ( ) M R () fr () A M = P Ξ S Examle A an alication we conider the cae, where the reidence time i generally ditributed and the eion time follow a hyer-erlang ditribution with Lalace tranform, J ( ) mj fs μj () = α j + μ j j=1 The ole of f S ( ) are located at Pj = μ j > 0 and hence atifie the Reidue theorem in (3) Thu uing (14) we get, E {N}= J j=1 where we have et, α j ( μ j ) mj P I Q MI (m j 1)! E {R} lim d mj 1 μ j d mj 1V N () e T V N ()= 1 f R () 2 M R ()= 1 f R () 2 (I fr () Q M ) 1 For the ractically intereting cae of m j = 2, where firt and econd moment matching i traightforward to calculate, the required derivative dv N () /d can be eaily obtained, becaue in thi cae we have dm R () /d μj = dfr () /d μ j M R (μ j ) Q M M R (μ j ) [11] Finally, let u dicu how to alter the analyi for off-net traffic Since off-net eion tart outide the network and enter the network at an arbitrary time intant, the eion duration S i now given by it reidual, S, (ie f S () = 1 f S () E S ) In addition, the reidence time of the firt, R 1, i now equal to the full reidence time R (ie () = fr ()) becaue a roaming eion can only enter a boundary [8] Thu, the initialization matrix Q MI ha only row correonding to entry border of lying at the network edge (eg in Fig1b row 1a and 1d for 1) Accordingly P I now contain the initial roaming robabilitie at the exterior gateway border III NUMERICAL RESULTS We now how the main reult which can be obtained from our model Let u take an examle of the mean ignaling rate toward any of the ytem in Fig 1, uch a Policy Control Function (PCF) erver The mean ignaling rate at PCF due to the handoff can be written a φ = λe(n H ), where λ i the eion arrival rate In thi cae, we only need to conider the mean number of handoff for a eion, in articular a a function of the mobility ratio, defined a ρ = ES E R and the uer mobility attern The eion duration ha a mean of E S = 40 min and i modeled by an Erlang ditribution with a coefficient of variation of c S = 05 or, alternatively, by an Hyerexonential ditribution with c S =3 Since the reult mainly deend on the mobility ratio and not on the abolute value of E S, we do not conider other mean duration For the gateway reidence time a Gamma ditribution i aumed, a it i wa hown in [3] that it can realitically aroximate meaured field data The gateway reidence time ha it mean value defined by the mobility ratio and it coefficient of variation i et to c R =3 The mobility behavior i conidered in two different toologie, linear and rectangular For the linear toology, defined a M g =1 N g =9we conider two different mobility attern, referred to a Random Route and Directed Route The mobility attern Random Route allow to change the direction at each Thu the matrix Q M ha nonzero entrie 1bb = 9dd =05, jbb = jbd = jdb = jdd = 05,j = 2,(n 1) and A M i given by [0, 05, 0, 0, 05] T In the mobility attern Directed Route, on the other hand, the uer cannot change the initial direction, thu Q M ha element jbd = jdb =1,j =2,(n 1) and A M i given by [0, 1, 0, 0, 1] T For the quare toology, defined a M g =3 N g =3 baed on the layout hown in Fig1b, we conider the o-called Random 3 3 and Hotot 3 3 mobility attern In Random 3 3 mobility, all tranition robabilitie are et to 025 The initial robability P I for a new eion i equally ditributed The mobility model Hotot 3 3 i defined by a combined random and directed movement attern, a hown in the to art of Fig 3, where a handoff eion can chooe a random direction only in 6 and 7 A new eion tart with robability of 08 in 5 and with equal likelihood in the remaining gateway In Fig 3, we how the effect of different toologie and movement attern A exected, the mean number of handoff E[N H ], i alway maller than the mean number of handoff for the whole eion, ie for a network of unlimited ize The latter increae with increaing the mobility ratio a Authorized licened ue limited to: TECHNISCHE UNIVERSITAT BRAUNSCHWEIG Downloaded on June 22,2010 at 17:13:40 UTC from IEEE Xlore Retriction aly

6 Mean Number of Handoff (MNH) g MNH 8 9 Hotot 3x3 V Whole Seion Handoff Random Route Directed Route Random 3x3 Random Route Mobility Ratio Fig 3 Mean number of handoff (MNH) and roaming robability β V v mobility ratio for different mobility attern (E S =40 min, c S =05,c R =2) wa alo hown in [9] Firt, we oberve that the Random Route mobility attern reult in a much larger number of handoff than the Directed Route Comared to the directed movement, the random movement incur everal direction change inide the network and ince the uer can only leave the network at 1 or 9, a higher mean number of handoff i oberved due to the low roaming robability (ee Fig 3) However, thi behavior highly deend on the toology and the mobility attern Thi i evident by comaring the Random 3 3 with the Hotot 3 3 mobility attern In thi cae, the random mobility behavior reult in a much higher robability of roaming, β V and thu the more directed Hotot attern achieve a higher mean number of handoff Fig 3 clearly how that only by joint conideration of the network toology and the mobility attern can the mean number of handoff for a eion be accurately etimated In Fig 4, we invetigate the effect of different eion time ditribution For all mobility attern, a higher coefficient of variation c S dratically reduce the mean number of handoff For the Random Route mobility attern, the the mean number of handoff for an exonentially ditributed eion duration, widely ued in literature, i alo hown Fig 4 trongly ugget that the ditribution of the eion time ha a great imact on the mean number of handoff and deerve a earate and a more in-deth tudy IV CONCLUSIONS In thi aer, we obtained a new reult for the mean number of handoff under generic aumtion of two-dimenional Markovian mobility attern, atial arrangement of the acce gateway, a well a generic eion time and gateway reidence time Our olution unveiled a new inight into mobility foundation a it how that the conideration of the mobility 2 1,5 1 0,5 0 ility( v ) Roaming Probabi attern and the acce gateway layout imly tranform the known equation for the mean number of handoff of an infinite network ize from calar to vector rereentation We demontrated that the mean number of handoff i a non-linear function of the gateway atial arrangement and uer mobility and that it aement trongly deend on the eion time ditribution Mean Number of Handoff (MNH) c S 05 c S 3 1 c S Random Route Hotot 3x Mobility Ratio Directed Route Fig 4 Mean number of handoff (MNH) v mobility ratio for different mobility attern and eion time ditribution (E S =40min,c R =2) REFERENCES [1] 3GPP2 X , All-IP Core Network Multimedia Domain - Service Baed Bearer Control Stage 2, Draft V 0210, July, 2006 [2] G Camarillo, M Garcia-Martin, The 3G IP Multimedia Subytem (IMS), John Wiley and Son, 2004 [3] Y Fang and I Chlamtac, Call erformance of a PCS network, IEEE J Selec Area Commun, vol 15, no 8, , 1997 [4] Y Fang and I Chlamtac, Analytical Generalized Reult for Handoff Probabilitie in Wirele Network, IEEE Tran Communication, vol 50, no 3, , 2002 [5] Rodriguez-Dagnino et al, Counting Handover in a Cellular Mobile Communication Network: Equilibrium Renewal Proce Aroach, Performance Evaluation, vol 52, no 2, 2003, [6] R Perera, ete al, Pixel oriented mobility modeling for UMTS network imulation, in Proc of IST Mobile and Wirele Telecommunication Summit 2002, Thealoniki, Greece, , 2002 [7] S Zaghloul, W Bziuk, A Jukan, Relating the Number of Acce Gateway Handoff to Mobility Management: A Fundamental Aroach, ubmitted to IEEE INFOCOM 2009 Conference [8] S Zaghloul, W Bziuk, A Jukan, Signaling and Handoff Rate at the Policy Control Function (PCF) in IP Multimedia Subytem (IMS), IEEE Commun Letter Journal, vol 12, no 7, , 2008 [9] W Bziuk, S Zaghloul, A Jukan, A New Framework for Characterizingthe Number of Handoff in Cellular Network, acceted for ublication in PGTS 08, Berlin, Germany, 2008 [10] U Bhat, G Miller, Element of Alied Stochatic Procee, Wiley,2002, 3rd ed [11] W Ficher, K Meier-Helltern, The Markov-modulated Poionroce cookbook, Performance Evaluation, vol 18, 1992 [12] Y Fang, Modeling and Performance Analyi for Wirele MobileNetwork: A New Analytical Aroach, IEEE Tran Networking,vol 13, no5, , 2005 Authorized licened ue limited to: TECHNISCHE UNIVERSITAT BRAUNSCHWEIG Downloaded on June 22,2010 at 17:13:40 UTC from IEEE Xlore Retriction aly