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1312 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 10, OCTOBER 2009 Mobility Management Approaches for Mobile IP Networks: Performance Comparison and Use Recommendations Nadjia Kara, Member, IEEE Abstract—In wireless networks, efficient management of mobility is a crucial issue to support mobile users. The Mobile Internet Protocol (MIP) has been proposed to support global mobility in IP networks. Several mobility management strategies have been proposed which aim reducing the signaling traffic related to the Mobile Terminals (MTs) registration with the Home Agents (HAs) whenever their Care-of-Addresses (CoAs) change. They use different Foreign Agents (FAs) and Gateway FAs (GFAs) hierarchies to concentrate the registration processes. For high-mobility MTs, the Hierarchical MIP (HMIP) and Dynamic HMIP (DHMIP) strategies localize the registration in FAs and GFAs, yielding to high-mobility signaling. The Multicast HMIP strategy limits the registration processes in the GFAs. For high-mobility MTs, it provides lowest mobility signaling delay compared to the HMIP and DHMIP approaches. However, it is resource consuming strategy unless for frequent MT mobility. Hence, we propose an analytic model to evaluate the mean signaling delay and the mean bandwidth per call according to the type of MT mobility. In our analysis, the MHMIP outperforms the DHMIP and MIP strategies in almost all the studied cases. The main contribution of this paper is the analytic model that allows the mobility management approaches performance evaluation. Index Terms—Mobile IP, mobility approach, performance evaluation. Ç 1 INTRODUCTION I Pmultimedia applications are becoming popular in the packet-based wireless networks. The integration of these applications in wireless networks requires the support of micromobility is the MT movements through different subnets belonging to a single network domain. For micromobility where the MT movement is frequent, the seamless terminal mobility. Mobile IP (MIP) has been MIP concept is not suitable and needs to be improved [3]. proposed by the Internet Engineering Task Force (IETF) to Indeed, the processing overhead related to location update provide global mobility in IP networks [1]. It allows could be high specifically under high number of MTs and maintaining mobile terminals ongoing communications when MTs are distant from the HAs yielding to high- while moving through IP network [1], [2]. mobility signaling delay [4]. In the MIP protocol, Mobile Terminal (MT) registers with Hierarchical Mobile IP (HMIP) has been proposed to its home network from which it gets a permanent address reduce the number of location updates to HA and the (home address). This address is stored in the Home Agent signaling latency when an MT moves from one subnet to (HA). It is used for identification and routing purpose. If another [5], [6]. In this mobility scheme, FAs and Gateway MT moves outside the home network visiting a foreign FAs (GFAs) are organized into a hierarchy. When an MT network, it maintains its home address and obtains a new changes FA within the same regional network, it updates its one from the Foreign Agent (FA). This Foreign address is CoA by performing a regional registration to the GFA. called Care-of-Address (CoA). To allow continuity of When an MT moves to another regional network, it ongoing communications between the MT and a remote performs a home registration with its HA using a publicly end point, the MT shall inform the HA of its current routable address of GFA. The packets intercepted by the location when it moves outside the home network. The HA HA are tunneled to a new GFA to which the MT is delivers to MT the intercepted packets by tunneling them to belonging (e.g., GF A2 following MT handoff from F A3 to the MT’s current point of attachment. F A5 in Fig. 1). The GFA checks its visitor list and forwards IP mobility in wireless networks can be classified into the packets to the FA of the MT (F A5 in Fig. 1). This macro- and micromobility. The macromobility is the MT regional registration is sensitive to the GFAs failure because mobility through different administration domains. The of the centralized system architecture [7], [8]. Moreover, a high traffic load on GFAs and frequent mobility between regional networks degrade the mobility scheme perfor- . The author is with INRS Energy, Materials et Telecommunications (INRS- `re EMT), Place Bonaventure, 800, de La Gauchetie West, Gate North-West, mance [4]. In order to reduce the signaling load for Suite 6900, Montreal, Quebec, Canada, H5A 1K6. interregional networks, mobility dynamic location manage- E-mail: kara@emt.inrs.ca. ment approaches for MIP have been proposed: A Hier- Manuscript received 21 Sept. 2007; revised 4 Jan. 2009; accepted 21 Jan. 2009; archical Distributed Dynamic Mobile IP (HDDMIP) and published online 6 Feb. 2009. Dynamic Hierarchical Mobile IP (DHMIP). For information on obtaining reprints of this article, please send e-mail to: tmc@computer.org, and reference IEEECS Log Number TMC-2007-09-0290. In the HDDMIP approach, each FA can act either as an Digital Object Identifier no. 10.1109/TMC.2009.36. FA or GFA according to the user mobility. The traffic load 1536-1233/09/$25.00 ß 2009 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS KARA: MOBILITY MANAGEMENT APPROACHES FOR MOBILE IP NETWORKS: PERFORMANCE COMPARISON AND USE... 1313 Fig. 2. DHMIP mobility approaches. Fig. 1. MIP and DHMIP mobility approaches. in a regional network is distributed among the FAs. The location update and packet route processing in FAs number of FAs attached to a GFA is adjusted for each MT. belonging to the hierarchy increasing the mobility signaling Thus, the regional network boundary varies for each MT. and packet delivery delay. Moreover, the path extension through the FAs hierarchy increases the network resources This number is computed according to the MT mobility used for packet delivery and location update signaling for characteristics and the incoming packet arrival rate. This an ongoing communication. number is adjustable from time to time according to the In [12], another inter-FAs tunneling approach has been variation of the mobility and the packet arrival rate for each proposed to optimize the route between the remote end MT. In [9] and [10], analytic models are proposed to point and the MT. This approach enables remote end point compute this number such as the total signaling traffic for to get the CoA associated to the MT and to use it to reach location update and packet delivery is transferred with the MT through the foreigner network without passing minimal network resource and low delay, respectively. through the home network. When the MT moves from one Nevertheless, this approach requires that each FA is able to foreigner network to another, it communicates its new CoA act as an FA and a GFA. Moreover, it adds processing load to its previous FA through its new FA. The previous FA on the MT to estimate the average packet arrival rate and tunnels the received traffic from the remote end point to the the subnet residence time. Hence, the main advantage of MT’s new location. At the same time, it sends a message to this approach is the system robustness enhancement since the HA requesting that the remote end point be notified of the GFA failure affects only the packets routing to MTs the MT’s new CoA. Upon receiving this new CoA, the belonging to this GFA. The disadvantages are the system remote end point uses it to reach the MT through the new infrastructure and MTs costs which could be high. foreigner network without passing through its previous The DHMIP approach has been proposed to reduce the foreigner network. This approach requires to restore an location update messages to the HA by registering the new optimized route after each CoA change. It aims to transfer CoA to the previous FA and building a hierarchy of FAs . packets through the resulting route with smaller delay than Hence, the user’s packets are intercepted and tunneled that experienced when these packets transit through the along the FAs hierarchy to the MT. The hierarchy level home network. However, this may not be always the case, numbers are dynamically adjusted based on mobile user’s and such performance will depend on the route optimiza- mobility and traffic load information. Fig. 2 illustrates an tion mechanism used and a set of influencing factors such example of DHMIP approach with a maximum of hierarchy as remote end point to FAs distance, the loads of the level number equal to 3. When MT is attached to F A2 , F A3 , networks the optimized route should pass through, and the F A5 , or F A6 , the CoA update is sent to the previous FAs. If MT inter-FAs mobility frequency. Such analysis is needed the MT becomes attached to F A4 the level number reach the to compare this approach with the existing ones, but it is out threshold and the MT will set up a new hierarchy. The MT of the scope of this paper. registers its new CoA directly to the HA. In this approach, Another alternative that reduce the signaling load in the location update to the new FA, which is close to the Mobile IP network is to use a multicast-based mobility previous FAs, could be less expensive than that to the HA. approaches. These approaches have been proposed to In [11], authors propose an analytic performance model to reduce the mobility signaling delay by setting a multicast evaluate the signaling transmission, the packet delivery, and group (see Section 2). The MTs address update processes the total costs of HMIP, HDDMIP, and DHMIP mobility are concentrated into the multicast network nodes (e.g., approaches using a one-dimensional random walk model. routers). They are reachable under these multicast group The performance analysis shows that the DHMIP scheme addresses. However, these approaches could be resource outperforms compared to the HMIP and HDDMIP ones. consuming except for next-generation IP-based radio access Despite that, the DHMIP approach still requires the new technologies such as 3rd Generation Partnership Program 1314 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 10, OCTOBER 2009 (3GPP) and 3GPP2 future cellular communication system Different Mobile IP multicast protocols have been called Long Term Evolution (LTE) [13], [14]. In LTE proposed. In [20], Mobility Supporting Agents (MSA)-based systems, where small cells deployment is expected, MT architecture has been proposed using IGMPv2 and PIM SM with high mobility will be able to access different wireless IP multicast protocols. In [21], an Core Based Trees (CBT)- networks frequently yielding to increase traffic overhead based multicast mobile IP approach has been proposed for due to MIP signaling and tunneling. This signaling includes micromobility. In [22], authors propose a set of multicast not only location update signaling but also security mobility protocols called Candidate Access Router set (CAR- set). The performance of multicast mobility approaches has association signaling required for MIP support [14], [15]. been evaluated through simulation or through analytic HAs could be signaling traffic bottleneck for such future models [22], [23]. In [22], a set of performance metrics (such mobile networks with high-mobility MTs. Hence, MHMIP as handoff delay, packet loss, and bandwidth overhead due mobility approach is proposed to reduce the signaling delay to handoff) have been identified and evaluated for multicast using multicast groups. The MT with high mobility could mobility approaches that have been simulated using NS2 reuse the multicast resources for signaling and packet network simulator. In [23], a software platform, set up delivery for several handoff events that occur during its call testbeds, has been used to analyze multicast mobility holding time. From that we expect that the resource usage is protocols in terms of handoff delay, packets losses and no greater than that of the DHMIP mobility approach. duplications, and relative TCP throughput. There is a large Hence, we propose to compute the mean bandwidth per number of multicast approaches that could be used to call and the mean handoff delay per call used for signaling implement mobility into MIP networks. The analysis of these and packet delivery according to the MT mobility and call approaches and their design is not the focus of this paper. We holding time duration, and to compare the performance of a refer the reader to [23], where four case studies for multicast- Multicast Hierarchical Mobile IP approach (MHMIP) with based mobility are presented based on different multicast those of the DHMIP and MIP mobility strategies. We derive service models and protocols. In this paper, we focus on a set of recommendations for the usage of these mobility usage of the multicast hierarchical architecture for IP management approaches according to the MTs mobility. mobility support and its performance in terms of bandwidth The main contribution of this paper is the analytic model usage and handoff delay. The example used in this paper of such architecture is given in Section 2.2. that allows performance evaluation of three mobility management approaches. 2.2 Multicast Hierarchical Mobile IP This paper is organized as follows: Section 2 discusses In this approach, we propose to build hierarchical multicast the multicast-based mobility approaches. Sections 3 and 4 groups. In each group, FAs are connected to each other present the analytic model and the numerical results, through a GFA. A set of GFAs are connected to an HA. respectively. Section 5 gives the conclusions. When an MT moves through FAs belonging to the same group, the GFA of this group multicasts the received packet 2 MULTICAST-BASED MOBILITY APPROACHES (coming from the HA) to the MT. When the MT moves outside a group, the new CoA is registered to the GFA of 2.1 Overview the new group to which the MT is currently belonging. This The multicast has been proposed to be used for mobility GFA sends this CoA to the HA. This latest tunnels the support and specifically in wireless networks with small packet to the new GFA which will multicast the received radio cells and high mobility of MTs. Several multicast- packets within the new FAs group. This approach reduces based mobility approaches have been proposed. They can the frequency of the location update to the HA. This update be classified into multicast-based mobility in connection- is performed every inter-GFAs mobility rather than every oriented and connection-less networks. For connection- inter-FAs mobility limiting the location update processing oriented networks, Acampora and Naghshineh propose a only at the GFA. In this example, the group creation is static virtual tree concept, where a multicast connection tree is in the sense that the numbers of groups and FAs do not preestablished. This tree is a collection of radio base stations change and remain fix. and ATM network switches connected to the tree’s root. The In Fig. 3, when the MT moves from F A2 to F A5 , the signaling delay is limited to the activation and deactivation location registration is performed between HA and GF A2 . of preestablished branch in the tree [16]. GF A2 multicasts packets to F A4 , F A5 , and F A6 . Thus, For Connection-less network, Seshan, in [17], proposes to when MT moves to F A6 or F A4 there is no need for the MT apply a multicast to Mobile IP to reduce the handoff delay. location registration. Hence, this approach allows reducing The HA encapsulates the intercepted packets into multicast the mobility signaling delay compared to the HMIP and packets and sends them to the targeted MT over multiple DHMIP mobility approaches specifically for high-mobility FAs. In [18], Ghai and Singh propose to divide the wireless MTs. However, it is network resources consuming ap- network into regions controlled by a supervisor host. Each proach due to multicast protocol use. Consequently, it is region includes groups of cells such as each cell may be part required for comparison purpose to evaluate the perfor- of several of these groups. A unique IP multicast ID is mance not only in term of handoff signaling delay but also assigned to each of these groups. In [19], authors extend this in term of bandwidth use. This latest is the bandwidth used work by considering multiple wireless networks and cases for signaling transfer and packet delivery. where mobile device is not able to use channel character- If we take the same MIP network architecture for the istics to trigger handoffs due to the frequent network three mobility management approaches, the bandwidth interface change. used by MHMIP signaling is smaller than that of MIP or KARA: MOBILITY MANAGEMENT APPROACHES FOR MOBILE IP NETWORKS: PERFORMANCE COMPARISON AND USE... 1315 The DHMIP uses also path extension which requires additional signaling messages to establish the path part that extends the mobile connection from the previous FA to the new one when the mobile move and becomes attached to this latest. Each connection is subjected to a certain number of handoffs through its life duration (call holding time). This latest is divided into n time intervals enough small to allow the occurrence and the end of only one handoff during this interval. In each time interval, we define . qa as the probability that an FAs handoff (handoff between two FAs) occurs and ends in this interval and . qf as the probability that the call ends in this interval. The number of handoffs that could occur during a call Fig. 3. Hierarchical handoff scheme. holding time depends on the MT dwelling time in a radio cell and the traffic type: voice or data. Several voice traffic DHMIP approaches because the path reestablishment is researches have supposed that the dwelling time in a radio performed only between HA and GFAs. However, the cell is an exponential distribution [24], [25]. In fact, this bandwidth used by an MT for packet delivery is high assumption depends on the shape of the radio cell and the because several connections are used for packets’ transfer to specific distributions of the mobile’s speed and direction the MT. It is clear that the total bandwidth used for which are difficult to characterize. In [26], [27], [28], [29], signaling and packet delivery in MHMIP approach is higher [30], authors have demonstrated that the exponential than that used by the other approaches. Nevertheless, in distribution for the dwelling time in radio cell is not case of MTs with high mobility (high handoff requests), the appropriated. They propose to replace it with complex multicast resource in the GFA groups are reused by the MT distributions such as Phase-Type, Lognormal, Hyperexpo- every handoff event that occurs during its call holding time. nential, and HyperErlang requiring the identification of Consequently, we expect that the MHMIP mean bandwidth several parameters related to the selected traffic model. In per call for MTs with high mobility is no greater than that of order to simplify the computation of the mean bandwidth the DHMIP and MIP mobility approaches. We also expect and mean delay per call, we consider that the time between that the MHMIP mean handoff delay (including signaling the handoff events and the call duration is a geometric and packet delivery delays) is smaller than that of the distribution of mean 1=qa 1 and 1=qf , respectively. DHMIP and MIP mobility approaches. Hence, we propose to derive an analytic model that For data traffic, researches have addressed the problem allows computation of mean bandwidth and mean handoff of the persistent congestion periods with non-negligible delay per call for MIP, DHMIP, and MHMIP mobility packet losses [31], [32], [33], [34]. They show that these losses approaches. These performance measurements are com- do not allow the usage of Poisson model to model the TCP puted according to the MTs mobility type (high or low) and traffic. In [33], [34], authors have demonstrated that the Self- the call holding time duration. The model description and Similar processes are better models for TCP traffic mod- the performance comparison of the three mobility ap- elization than the exponential ones. However, in this study, proaches are discussed in the following sections. we are interested by the data session arrivals rather than the data packet generation in the sessions. Hence, we propose that the assumption made for the voice traffic remains valid 3 ANALYTIC MODEL for the data traffic. This section describes the analytic model and the set of The proposed discrete time model is a generalization of established assumptions. the one proposed in [35]. The novelty of this model consists in the definition of generic analytical model that applies to 3.1 Assumptions more than one handoff approach and that allows to Generally, during each handoff, a path reestablishment is compute not only mean bandwidth due to handoff but also required to maintain or to improve call quality. This mean handoff delay of the analyzed handoff approaches. reestablishment uses signaling messages and involves a The temporal diagram given in Fig. 4 is used to compute change in the number of links of the mobile connection. these means. First, we compute the bandwidth and the Note that the three mobility approaches described here delay within each interval and their means over the handoff are based on a mobile connection path reestablishment events. Then, we compute the bandwidth and the delay which leads to perform the following operations: sums over the total call holding time. Finally, we evaluate . CoA update with the HA, their means over all the call durations. In order to . new path establishment from HA to FA for DHMIP understand the modelization mechanism, we illustrate by and MIP, and from HA to GFA for MHMIP, taking as an example the mean bandwidth computation. In . user data traffic transfer from the previous path to this figure, the holding time of ongoing call is divided into the new one, 1. The respective temporal means are obtained while multiplying by the . previous path discard. interval duration. 1316 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 10, OCTOBER 2009 represents the mean number of intervals during a call. Then, we obtain l s XÂ 1 Ã B¼B þB ¼ Bl ðnÞ þ Bs ðnÞ P ðnÞ n¼1 ð5Þ X 1 X 1 ¼ Bl ðnÞP ðnÞ þ Bs ðnÞP ðnÞ: n¼1 n¼1 Fig. 4. Discrete diagram of a call holding time. Let E½Bx ; x 2 fs; lg, a variable that designates the E½Bl or i i E½Bs entity. In the bandwidth computations given later, i time intervals small enough that we may assume that in E½Bx could be variable or not during a call. If variable, then i each time interval i; i þ 1, at most one handoff may occur. we obtain In each interval, let X 1 XX Â Ã 1 nÀ1 . n be the number of intervals for a call, Bx ¼ Bx ðnÞP ðnÞ ¼ E Bx P ðnÞ i n¼1 n¼1 i¼0 . Bl be the bandwidth used by a call during the time i ð6Þ interval i; i þ 1, X 1 ðnÀ1Þ X Â Ã nÀ1 ¼ qf ð1 À qf Þ E Bx ; . Bs be the signaling bandwidth used by a call during i n¼1 i¼1 i handoff that occurred in the time interval i; i þ 1, and . Bi be the total bandwidth used by a call during the otherwise, we have time interval i; i þ 1; X 1 XX Â Ã 1 nÀ1 Bl and Bs are random variables with values that depend on i i Bx ¼ Bx ðnÞP ðnÞ ¼ E Bx P ðnÞ i the occurrence or not of a handoff during the interval i; i þ 1 n¼1 n¼1 i¼0 and on the possible path reestablishment once the handoff X Â ÃX 1 nÀ1 Â ÃX1 ¼ E Bx P ðnÞ ¼ E Bx nP ðnÞ ð7Þ occurs. The variable Bl can take two values. When a handoff i n¼1 i i i¼0 n¼1 occurs for a call in the interval i; i þ 1, Bl represents the sum Â Ã i Â ÃX 1 ðnÀ1Þ E Bxi of the allocated bandwidth over the original path and the ¼ qf E Bx i nð1 À qf Þ ¼ : n¼1 qf one allocated over the links of the new established path. Otherwise, it represents the bandwidth used on the link of The same procedure applies for the mean handoff delay the ongoing connection. Bi represents the sum of the computation by substituting to variable B the variable D bandwidth used by the ongoing call (Bl ) and the bandwidth i which represents the delay. Note that B and D are random used for signaling (Bs ). Otherwise, it represents the allocated variables due to handoff. i bandwidth to the ongoing call (Bl ). Then, we obtain In the following sections, the mean bandwidth per call is i the network bandwidth needed to support a mobile & l s Bi þ Bi ; if a handoff occurs in i; i þ 1; connection over its total duration, and it is given by the Bi ¼ ð1Þ sum of the bandwidth used on the paths’ links of the Bl ; i otherwise: ongoing connection and the signaling bandwidth due to The mean of Bi over the handoff events is given by handoffs. Likewise, the mean handoff delay per call is the Â Ã Â Ã Â Ã network handoffs’ durations to support a mobile connec- E Bi ¼ E Bl þ E Bs : ð2Þ i i tion which is given by the sum of the duration of the For fixed value of n, the total mean bandwidth BðnÞ used resource establishment on the paths’ links and the signaling by an ongoing call during the n time intervals is duration due to handoff. The paths’ links are the total network links of all paths used by the ongoing connection X Â Ã X Â Ã nÀ1 nÀ1 during its holding time. Let BðnÞ ¼ Bl ðnÞ þ Bs ðnÞ ¼ E Bl þ i E Bs : i ð3Þ i¼0 i¼0 . BP D be the allocated bandwidth on each link for As the call duration n is a random variable, the mean packet delivery of a call, bandwidth B is computed over all the call durations . BP E be the signaling bandwidth used per call for each path extension, n ¼ 1; . . . ; 1. With our assumptions, the probability that a . BP R be the signaling bandwidth used per call for call runs n periods is defined as P ðnÞ: each path reestablishment, P ðnÞ ¼ qf ð1 À qf ÞnÀ1 n ¼ 1; 2; . . . ð4Þ . DP D be the duration per call to allocate bandwidth BP D on a link of a new extended path and/or a such as reestablished path, . DP E be the signaling duration per call for a path X 1 X 1 n¼ nP ðnÞ ¼ nqf ð1 À qf ÞnÀ1 extension, and n¼1 n¼1 . DP R be the signaling duration per call for a path X 1 reestablishment. ¼ qf nð1 À qf ÞnÀ1 ¼ 1=qf Note that BP D and DP D are, respectively, the bandwidth n¼1 and the duration for packet delivery on each link of a path KARA: MOBILITY MANAGEMENT APPROACHES FOR MOBILE IP NETWORKS: PERFORMANCE COMPARISON AND USE... 1317 for an ongoing call. The BP D can take different fixed values reestablishment. The second term (q a q ð1ÀpÞð1Àq ÞH f BP D ) according to the traffic type carried through the path links: ½1Àð1Àpqa Þð1Àqf Þ f represents the additional bandwidth due to the path voice or data (e.g., 64 kbps). A link is the network qa connection between two network entities such as an FA extensions. The last term (qf ðBP E þ pBP R Þ) represents the and a router (e.g., in Fig. 3, there is three links between the signaling bandwidth due to the extensions and path F A1 and HA). The signaling duration is the time taken for reestablishments. the transmission and the execution of the different handoff In (9), the term qa =qf represents the mean number of signaling messages. The parameters BP E and DP E are, handoffs of a call. The second term ½DP D ½ð1 À pÞH þ pLp þ respectively, the bandwidth and the duration necessary to DP E þ pDP R represents the handoff delay which is the sum set up the path extension between two FAs involved in a of the resource reservation delay on the links of the extended handoff. The parameters BP R and DP R include the and the reestablished paths (DP D ½ð1 À pÞH þ pLp ), and the bandwidth and the duration, respectively, for 1) CoA signaling delay due to the path extensions and the path registration, 2) setting up the new portion of connection reestablishments (DP E þ pDP R ). between HA and FA in case of DHMIP and MIP, and The reader is referred to Appendix A.1 and B.1 for a between HA and GFA in case of MHMIP (e.g., path between detailed demonstration of these formulas. HA and F A4 in Fig. 2), and 3) terminating the old portion of a connection (e.g., path between HA and F A3 in Fig. 2). 3.3 MIP Analytic Model The MIP mobility approach is based only on the path 3.2 DHMIP Analytic Model reestablishment protocol. This latest allows maintaining The DHMIP mobility approach combines the path rees- the call connectivity when the MT moves between FAs. In tablishment and the connection extension protocols. The this case, events that may occur at each time i ¼ 1; 2; . . . path reestablishment protocol is invoked to set up a new are 1) path reestablishment and 2) call termination. Let FAs hierarchy. This protocol allows a path establishment between the HA and a new FA in the new hierarchy. In . qa be the probability that there is an inter-FAs this latest, the path extension is used to maintain the handoff and thus a partial reestablishment, mobile connection when mobile moves through the FAs . L be the number of links between the FA to which belonging to this hierarchy. The path reestablishment may the MT is attached and the remote end point with occur after each new FAs hierarchy setup. Events that may which the MT is communicating, and . Lr be the number of links between the HA and the occur at each time i ¼ 1; 2; . . . are 1) path reestablishment, new FA to which the MT moved (e.g., the number of 2) path extension, and 3) call termination. Let links between the HA and the F A3 following the . p be the probability that a new FA hierarchy is set and handoff from F A1 to F A3 in Fig. 1). consequently a path reestablishment is performed, L and Lr are random variables with general distributions . L be the number of links between the FA to which and with mean L and Lr , respectively. the MT is attached and the remote end point with The mean bandwidth per call is which the MT communicates, . Lp be the number of links between the HA and the 1 qa Br ¼ LBP D þ BP R : ð10Þ initial FA through which a new hierarchy is set (e.g., qf qf F A1 and F A4 in Fig. 2), and . H be the number of links of the path extension In (10), the first term q1f LBP D is the bandwidth of the (e.g., in Fig. 2, this number is equal to 1 when MT original connection and the reestablished paths. The second qa moves from F A1 to F A2 and becomes connected term qf BP R is the signaling bandwidth due to the path to F A2 ). reestablishments. p L, L , and H are random variables with general distribu- The mean handoff duration per call is tions and with mean L, Lp , and H, respectively. qa r P D The mean bandwidth per call is Dr ¼ ðL D þ DP R Þ: ð11Þ qf L PD qa ð1 À pÞð1 À qf ÞH qa In (11), the term qf represents the mean number of handoffs Bp ¼ B þ BP D qf qf ½1 À ð1 À pqa Þð1 À qf Þ for a call. The term Lr DP D þ DP R represents the handoff ð8Þ qa delay which is the sum of the delay for resource allocation on þ ðBP E þ pBP R Þ; qf the reestablished path (Lr DP D ) and the signaling delay (DP R ). while the mean handoff delay per call is Details on these computations are given in Appendix A.2 and B.2. qa P D Dp ¼ D ½ð1 À pÞH þ pLp 3.4 MHMIP Analytic Model qf qa ð9Þ The MHMIP mobility approach is based on the path þ ½DP E þ pDP R : qf reestablishment and the multicast protocols. When the MT moves within a GFA group, the mobile connection is L In (8), the first term (qf BP D ) represents the bandwidth maintained using the multicast protocol. When the MT used on the original path and the paths resulting from the moves outside this hierarchy, a combination of the path 1318 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 10, OCTOBER 2009 reestablishment and the multicast protocols allows main- taining the call’s connection. Events that may occur at each time i ¼ 1; 2; . . . are 1) path reestablishment and 2) call termination. 0 We define qa as the probability that there is an inter- GFAs handoffs and thus path reestablishments such as 0 qa ¼ qa with 0 1. is the fraction of inter-GFAs MHMIP handoffs on the whole possible handoffs qa (intra- and inter-GFAs). The inter-GFAs handoff arrivals are modeled using a Bernoulli process. For each mobile connection, we define . Lh as the number of links between the GFA to which the mobile is currently attached and the remote end point with which the MT is communicating, . Lhp as the number of links between the HA and the GFA to which the mobile is currently belonging, and . Lhr as the total number of links in the GFA hierarchies. Lh , Lhp , and Lhs are random variables with general Fig. 5. Symmetric hierarchical IP network architecture. distributions and with means Lh , Lhp , and Lhs , respectively. The mean bandwidth per call is Fig. 5 shows an architecture with Lp ¼ Lr ¼ 7, Lhp ¼ 3, and 1 h PD q 0 Lhr ¼ 240. Bh ¼ L B þ Lhr BP D þ a BP R : ð12Þ For comparison purpose, we take the number of links qf qf between the HA and the end point the same for the three In (12), the first term q1f Lh BP D is the bandwidth used on mobility management approaches. For a fixed remote end the original path and the reestablished paths. The second point, the number of links between the HA and this end term Lhr BP D is associated to the multicast resources used by 0 point do not change for an ongoing call of an MT. Then, we qa the call in the GFA hierarchies. The last term qf BP R is the consider that the end point is directly connected to HA signaling bandwidth due to the path reestablishment (e.g., Lh ¼ Lhp ¼ 3 and L ¼ Lp ¼ Lr ¼ 7 for the example following the GFA handoffs. given in Fig. 5). The mean call duration per call is Two types of configurations are considered for the network given in Fig. 5: 0 qa hp P D Dh ¼ ½L D þ DP R : ð13Þ . Configuration 1: the average number of links are qf 0 Lh ¼ Lhp ¼ 3 and Lr ¼ Lp ¼ L ¼ 7. These values q In (13), the term qf is the mean number of handoffs of a a result in the number of link where the resources call. The second term ½Lhp DP D þ DP R is the handoff delay were allocated Lhr ¼ 240. which is the sum of the delay of resource allocated on the . Configuration 2: the average number of links are reestablished path (Lhp DP D ) and the signaling delay (DP R ). Lh ¼ Lhp ¼ 1 and Lp ¼ Lr ¼ L ¼ 7. From these The details on these computations are given in Appendix A.3 and B.3. values, we obtain Lhr ¼ 252. For each configuration, two cases are analyzed: realistic and critical. In the realistic case, the inter-GFAs handoffs may 4 RESULTS ANALYSIS occur less frequently than the intra-GFAs handoffs In this section, we compare the performance in terms of 0 (qa ¼ 0:1 Â qa ). In the critical case, the intra- and the inter- mean bandwidth and mean handoff delay per call of the GFAs handoffs may occur with the same probability three mobility management approaches MHMIP, DHMIP, 0 0 (qa ¼ qa , where qa and qa are variables). and MIP. For both cases, the path extension for the DHMIP 4.1 Numerical Data mobility management approach should occur after each handoff and the path reestablishment should occur after The mean call holding time is a random value chosen each two consecutive handoffs (p ¼ qa =2). For p > qa =2, the between 60 and 120 seconds for voice traffic and between mean bandwidth and mean delay is higher than that get 900 and 1,200 seconds for data traffic. For simplification with p ¼ qa =2 (see Section 4.2). purpose of the mean number of links computation (Lr , Lp , We suppose that the MT handoff to a new FA involves a L, H, Lh , Lhp , and Lhr ), a symmetric hierarchical IP network path extension of mean length H ¼ 1. For length greater architecture is considered (Fig. 5). Symmetric architecture than this value, the mean bandwidth and the mean handoff means that the number of links between the HA and each delay are high. FA is the same (e.g., there is five links between the HA and We rewrite (8), (12), (10), (9), (13), and (11) to obtain the each F Ai ; fi ¼ 1; . . . ; 32g in Fig. 5). The example given in ratios Bj R ¼ Bj =BP R and Dj R ¼ Dj =DP R , where j ¼ p; r; h. P P KARA: MOBILITY MANAGEMENT APPROACHES FOR MOBILE IP NETWORKS: PERFORMANCE COMPARISON AND USE... 1319 Fig. 6. Mean bandwidth per call Bp R and Bh R for voice traffic with 1=qf ¼ P P Fig. 7. Mean bandwidth per call variation Bp R for voice traffic with 1=qf ¼ P 60 seconds, BP D =BP R ¼ 0:5. 60 seconds, BP D =BP R ¼ 0:5. Thus, these equations contain the following ratios: BP D =BP R , BP E =BP R , DP D =DP R , and DP E =DP R . We expect that the signaling bandwidth for the path extension in DHMIP is smaller than that of the path reestablishment (BP R > BP E and DP R > DP E ). Then, BP R =BP E < 1 and DP R =DP E < 1. As an example, we take BP R =BP E ¼ DP R = DP E ¼ 0:2. Moreover, we expect that BP R > BP D and DP R > DP D . Indeed, the BP R represents the sum of the allocated bandwidth on each link of the path over which the handoff signaling is transferred (the path between HA and FA for DHMIP and MIP, and between HA and GFA for MHMIP). This bandwidth could be greater than BP D which is the Fig. 8. Mean bandwidth ratio Bp R =Bh R and Br R =Bh R for voice traffic bandwidth allocated on a path link for packet transfer. P P P P with p ¼ qa =2; BP D =BP R ¼ 0:8. According to the BP D and the signaling bandwidth values, we could have 0 < BP D =BP R < 1. In the same way, DP R bandwidth represents a performance measurement that an represents the sum of the delays for bandwidth allocation on IP network operator can use to determine the needed each link of the path carrying the signaling traffic and the resources to be deployed in the network to service a certain delay for signaling messages processing. It is greater than number of MTs. The MHMIP mobility management DP D , which represents the delay for BP D allocation. Thus, we approach is the method that allows cost reduction in terms take 0 < DP D =DP R < 1. In this analysis, we show an example of resources usage compared to the DHMIP approach. of the results for BP D =BP R ¼ DP D =DP R ¼ 0:5 and 0.8. Fig. 7 illustrates the Bp =BP R ratio variation for different values of the probability p. We note that lower is p higher is 4.2 Numerical Results the mean bandwidth per call. Moreover, we note a different We propose to compare the performance of the MHMIP behavior of this bandwidth between the intervals qa 0:3 handoff approach with those obtained with DHMIP and and 0:3 qa 1. For 0:3 qa 1, the mean bandwidth MIP approaches in terms of mean bandwidth and mean value decreases while it increases in the interval qa 0:2 for handoff delay per call. For summarization purpose, we different values of p (p ¼ qa =6; qa =4; qa =2) and still increasing compute the ratios Bp R =Bh R , Br R =Bh R , Dp R =Dh R , and P P P P P P in the interval 0:2 qa 0:3 for p ¼ qa =6. This is in fact due Dr R =Dh R . These ratios allow a simple and direct reading of P P to the low probability of path reestablishment p and the the different performance between the tree mobility frequent use of path extension in the interval qa 0:3. management approaches. Hence, less frequent path reestablishment usage for DHMIP Figs. 6 and 7 give an example of mean bandwidth mobility management approach involves a high mean variation per call Bp R and Bh R for the DHMIP and MHMIP P P bandwidth per call consumption. handoff approaches. 4.2.1 Mean Bandwidth Fig. 6 illustrates the mean bandwidths per call for MHMIP and DHMIP mobility management approaches. It Figs. 8 and 9 show examples of the mean bandwidth ratio shows that the MHMIP mean bandwidth per call is smaller variation Bp R =Bh R and Br R =Bh R for the realistic and the P P P P than that obtained with the DHMIP approach. This mean critical cases, respectively. 1320 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 10, OCTOBER 2009 TABLE 2 Mean Delay the partial reestablishment is more frequently used due to 0 the high probability qa of inter-GFAs handoffs. Fig. 9. Mean bandwidth ratio Bp R =Bh R and Br R =Bh R for data traffic with P P P P p ¼ qa =2; BP D =BP R ¼ 0:8. 4.2.2 Delay Comparison The ratio values of the mean delay per call Dp R =Dh R and P P Note that the ratio Bp R =Bh R is much higher than the P P Dr R =Dh R are summarized in Table 2 for the Realistic Case P P ratio Br R =Bh R for different call holding time duration P P (RC) and the Critical Case (CC). These values represent the (1=qf ¼ 60; 90; 120 seconds) specifically for small probabil- arithmetic average of the ratios Dp R =Dh R and Dr R =Dh R P P P P ity qa . This means that the combination of the path over the set of qa values. In the realistic case, the MHMIP extension and the path reestablishment for handoff mean delay is smaller than those of the DHMIP and MIP management involves higher mean bandwidth per call approaches. The mean delay differences are 8 and 21 than that used by the approaches based only on the path compared to DHMIP and MIP, respectively. They become reestablishment (such as MHMIP and MIP). This behavior high, if we consider the configuration 2 where this difference is noticed in all the analyzed cases. The main obtained reaches 40. This result was expected because in the MHMIP results are summarized in Table 1 that give the ratio values approach, the path reestablishment is performed through a of the mean bandwidth per call for the Realistic Case (RC) shorter path than that of the DHMIP and MIP approaches. and the Critical Case (CC) for both Types of Configuration The mean delay per call of the MHMIP approach in the (ToC). These values represent the arithmetic average of the critical case is smaller than that of the MIP and DHMIP ratios Bp R =Bh R and Br R =Bh R over the set of qa values P P P P approaches unless for configuration 1 where this delay is (0 qa 1). These results show that the MHMIP mean greater than that of the DHMIP approach, because the 0 bandwidth is smaller than those of the DHMIP and MIP path reestablishment is not only more frequent (qa ¼ qa ) approaches. This bandwidth difference is higher in the but also the number of links involved in the path configuration 2 than in the configuration 1 because the reestablishment is greater than that of configuration 2 MHMIP reestablishment is performed over small number (Lh ¼ 3 compared to Lh ¼ 1). of links in the configuration 2, yielding to a smaller mean 4.3 Recommendation bandwidth per call than that computed with the config- Hence, we can derive the following recommendations that uration 1. However, this mean bandwidth difference is indicate when one would use the MHMIP, the DHMIP, or small in the critical case than in the realistic case because the MIP mobility management approaches: . The MHMIP versus DHMIP usage: TABLE 1 Mean Bandwidth 1. If the inter-GFAs handoffs are not frequent then we suggest to use the MHMIP approach that provides a best mean handoff delay and mean bandwidth per call for voice and data traffic. 2. If the inter-GFAs handoffs occur frequently 0 (such as qa ¼ qa ), then we have two cases. If the mean handoff delay per call is more important than the mean bandwidth per call and if the number of links involved in MHMIP path reestablishment is high, then we suggest using the DHMIP handoff approach; otherwise, we suggest using the MHMIP handoff approach. KARA: MOBILITY MANAGEMENT APPROACHES FOR MOBILE IP NETWORKS: PERFORMANCE COMPARISON AND USE... 1321 . The MHMIP versus the MIP usage: We suggest The mean bandwidth used by a call in a time interval using the MHMIP handoff approach that gives a best i; i þ 1 is given by mean handoff delay per call and a mean bandwidth per call for voice and data traffic, and for both Bl ¼ Bl ½ð1 À Ki Þ þ Ki ð1 À Ji Þ i iÀ1 frequent and nonfrequent GFAs handoffs. þ BKi ½Ji Li ðiÞ þ ð1 À Ji ÞHi ðiÞ ð16Þ ¼ Bl ½1 À Ki Ji iÀ1 5 CONCLUSION þ BKi ½Ji Li ðiÞ þ ð1 À Ji ÞHi ðiÞ: In this paper, we have proposed an analytical model We define new random variables sj ¼ ðKj ; Jj Þ, where sj which evaluates the mean handoff delay per call and the represents the pair of variables K and J at the period j and mean bandwidth per call of three mobility management the vector sj ¼ ½s1 ; s2 . . . sj . approaches: MIP, DHMIP, and MHMIP. Numerical results At each interval, Li and Hi represent, respectively, the show that the MHMIP mobility approach compares very number of links of the original path or a path resulting favorably with the previously considered mobility ap- from a path reestablishment and the number of links of proaches. More specifically, our analysis gives in almost all the extended path. In general, these values are depen- cases a lower mean handoff delay per call and a mean dent in complex way of events that occurred before the instant i. They are generally random variables which bandwidth per call than those offered by the DHMIP and should be designated by Li ðsi Þ and Hi ðsi Þ. Then, we can MIP approaches. It also shows the robustness of the rewrite (16) more explicitly such as MHMIP approach in the sense that for critical scenario corresponding to the extreme situation where all handoff Bl ðsi Þ ¼ Bl ðsiÀ1 Þ½1 À Ki Ji i iÀ1 ð17Þ events are localized at the multicast group borders, this þ BKi ½Ji Li ðsi Þ þ ð1 À Ji ÞHi ðsi Þ: approach essentially yields to 1) a lower mean bandwidth per call than the DHMIP and MIP approaches; 2) a lower As the events are independent in each interval, we can write for si mean handoff delay per call than that offered by the MIP approach; 3) a lower mean handoff delay than that offered pðsi Þ ¼ pðsiÀ1 ÞpðKi ; Ji Þ; by the DHMIP except in case of frequent inter-GFAs where pðKi ; Ji Þ is the probability to have a given config- handoffs with a network configuration having a high uration of the random variables Ki and Ji . Note that number of links involved in MHMIP path reestablishment pð0; 1Þ ¼ 0 because we could have a path reestablishment such as the configuration 2. Since we expect a diversity of only after a handoff. Let define the two auxiliary entities: multimedia applications for future IP mobile networks, we recommend using the MHMIP approach in networks parts gðKi ; Ji Þ ¼ ½1 À Ki Ji ; ð18Þ carrying delay sensitive and/or low mean bandwidth Â Ã consumption type of applications and this according to hðsi Þ ¼ BKi Ji Li ðsi Þ þ ð1 À Ji ÞHi ðsi Þ : ð19Þ the mobility type. The mean bandwidth over the handoff events is given by APPENDIX A Â Ã X i X E Bl ¼ i pðs ÞZi ¼ pðsiÀ1 ÞpðKi ; Ji ÞZi EVALUATION OF THE MEAN BANDWIDTH PER CALL s i i X Xs ð20Þ iÀ1 By using the discrete time model given in Fig. 4, we ¼ pðs Þ pðKi ; Ji ÞZi ; estimate the mean bandwidth per call for the DHMIP, MIP, siÀ1 Ki ;Ji and MHMIP mobility management approaches. where Zi ¼ ½BiÀ1 ðsiÀ1 ÞgðKi ; Ji Þ þ hðsi Þ. As Li and Hi are A.1 DHMIP Approach dependent on si in a complex way and for computation simplification purpose, a set of assumptions are made. We In this case, we take into account the following events: the suppose these variables constant from the fact that in inter-FAs handoffs (path extension) that occur with prob- general we can expect that the connection paths and the ability qa and the path reestablishment which is executed path extensions do not vary a lot from one handoff to with probability p. another. From this assumption, we can continue our For each time interval i; i þ 1, we define two random computations such as (20) becomes variables corresponding to the occurrence or not of a Â Ã X iÀ1 X handoff event and the path reestablishment. E Bl ¼ i pðs ÞBiÀ1 ðsiÀ1 Þ pðKi ; Ji ÞgðKi ; Ji Þ siÀ1 Ki ;Ji & X X Ki ¼ 1; a handoff occurs; ð14Þ þ pðsiÀ1 Þ pðKi ; Ji ÞhðKi ; Ji Þ 0; otherwise: siÀ1 Ki ;Ji Â ÃX ð21Þ & ¼E Bl iÀ1 pðKi ; Ji ÞgðKi ; Ji Þ 1; a path reestablishment is executed; X Ki ;Ji Ji ¼ ð15Þ 0; otherwise: þ pðKi ; Ji ÞhðKi ; Ji Þ; Ki ;Ji We also suppose that E½Ki ¼ qa and E½Ji ¼ p. 1322 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 10, OCTOBER 2009 P X 1 X 1 with siÀ1 pðsiÀ1 Þ ¼ 1. We can compute the two terms: n P ðnÞ ¼ qf n ð1 À qf ÞnÀ1 X n¼1 n¼1 pðKi ; Ji ÞgðKi ; Ji Þ ¼ ð1 À qa Þgð0; 0Þ þ qa pgð1; 1Þ qf X 1 Ki ;Ji ð22Þ ¼ ½ ð1 À qf Þn ð31Þ 1 À qf n¼1 þ qa ð1 À pÞgð1; 0Þ; ! ¼ qf : X 1 À ð1 À qf Þ pðKi ; Ji ÞhðKi ; Ji Þ ¼ ð1 À qa Þhð0; 0Þ þ qa phð1; 1Þ Ki ;Ji ð23Þ By replacing in (30), we obtain þ qa ð1 À pÞhð1; 0Þ; ! l M 1 qf B ¼ þK 1À with gð0; 0Þ ¼ 1; gð1; 1Þ ¼ 0, gð1; 0Þ ¼ 1, hð1; 1Þ ¼ L, hð1; 0Þ ¼ qa pqf qa p 1 À ð1 À qf Þ ! H, and hð0; 0Þ ¼ 0. By replacing each of these terms in (21), M 1 qa p ¼ þK ð32Þ we obtain qa pqf qa p 1 À ð1 À qf Þ ! Â Ã Â Ã M 1 E Bl ¼ ð1 À qa pÞE Bl PD ¼ þK : i iÀ1 þ qa ½ð1 À pÞH þ pLB : ð24Þ qa pqf 1 À ð1 À qa pÞð1 À qf Þ If we suppose that E½Bl ¼ LBP D , we can solve the 0 By replacing M with its value and by reorganizing the recurrence (24) and we obtain terms we finally find Â Ã ! E Bl ¼ ð1 À qa pÞi LBP D l L 1 qa ð1 À qf Þ i B ¼ BP D þ BP D ð1 À pÞH : qf qf ð1 À ð1 À qa pÞð1 À qf ÞÞ þ ½1 þ ð1 À qa pÞ þ Á Á Á þ ð1 À qa pÞiÀ1 M ð25Þ ð33Þ i 1 À ð1 À qa pÞ ¼ ð1 À qa pÞi LBP D þ M; Using (28) and (33), we compute Bp ¼ Bl þ Bs qa p L PD qa ð1 À pÞð1 À qf ÞH where M ¼ ½qa ð1 À pÞH þ qa pLBP D . Bp ¼ B þ BP D qf qf ð1 À ð1 À qa pÞð1 À qf ÞÞ ð34Þ The mean signaling bandwidth due to handoff in the qa þ ðBP E þ pBP R Þ: interval i; i þ 1 is given by qf Bs ¼ Ki ðBP E þ Ji BP R Þ; i i ! 1; ð26Þ A.2 MIP Approach An inter-FAs handoff occurs with probability qa . For each with the mean of Bs over the handoff events is i time interval i; i þ 1, we define the random variable E½Bs ¼ qa ðBP E þ pBP R Þ; i ! 1: ð27Þ & i 1; a handoff occurs; Ii ¼ ð35Þ 0; otherwise; As E½Bs is not a variable, then we use (7) and (27) to i s compute B . which corresponds to the occurrence of a handoff followed Â Ã by a path reestablishment of the MT’s ongoing connection. s E Bsi 1 Note that E½Ii ¼ qa . B ¼ ¼ qa ðBP E þ pBP R Þ : ð28Þ qf qf The mean signaling bandwidth due to mobility in the interval i; i þ 1 is given by We use (25) such as Bs ¼ Ii BP R ; i i ! 1: ð36Þ X nÀ1 !X nÀ1 M M The mean bandwidth corresponding to the handoff events Bl ðnÞ ¼ þ LBP D À i i¼0 qa p qa p i¼0 is given by ! ð29Þ Mn M 1 À n Â Ã ¼ þ LBP D À ; E Bs ¼ qa BP R ; i i ! 1: ð37Þ qa p qa p qa p The bandwidth used in the time interval i; i þ 1 is where ¼ ð1 À qa pÞ. given by Using (6) and (29), we find Bl ¼ ð1 À Ii ÞBl þ Li ðiÞIi BP D ; i iÀ1 i ! 1: ð38Þ X 1 X Â Ã nÀ1 l B ¼ qf ð1 À qf ÞðnÀ1Þ E Bl Note that Bl depends on all the random variables Ij ; j ¼ i i n¼1 n¼1 1; . . . ; i À 1 and on the variable Ii of the current interval. ð30Þ MX 1 1 X 1 The length Li ðiÞ depends in complex manner on the events ¼ nP ðnÞ þ K ð1 À n ÞP ðnÞ; that occurred before i. It is represented by Li ðsi Þ with qa p n¼1 qa p n¼1 si ¼ ½I1 ; I2 ; . . . ; Ii . We can then rewrite (38) as where K ¼ ½LBP D À qMp. We evaluate the following a Bl ðsi Þ ¼ ð1 À Ii ÞBl ðsiÀ1 Þ þ Li ðsi ÞIi BP D ; i iÀ1 i ! 1: ð39Þ equation: KARA: MOBILITY MANAGEMENT APPROACHES FOR MOBILE IP NETWORKS: PERFORMANCE COMPARISON AND USE... 1323 Let pðsi Þ ¼ pðsiÀ1 ÞpðIi Þ, where pðIi Þ is the probability of the A.3 MHMIP Approach occurrence or not of a handoff such as pð1Þ ¼ qa and In this case, two types of handoff can occur in each time interval: inter and intra-GFAs handoffs. From the fact that pð0Þ ¼ 1 À qa , and let the auxiliary entities gðIi Þ ¼ ð1 À Ii Þ intra-GFAs handoff doesn’t generate signaling traffic. This and hðsi Þ ¼ BP D Ii Li ðsi Þ. The mean value on the handoff type of events is not considered in our computations. events is given by In each time interval i; i þ 1, we define a random Â Ã X i X variable corresponding to the inter-GFAs handoff occur- E Bl ¼ i pðs ÞF ¼ pðsiÀ1 ÞpðIi ÞF i i rence and consequently to a path reestablishment. s X Xs ð40Þ & iÀ1 ¼ pðs Þ pðIi ÞF ; 1; an inter-GFAs handoff occurs; Ii ¼ ð49Þ siÀ1 Ii 0; otherwise: where F ¼ ½BiÀ1 ðsiÀ1 ÞgðIi Þ þ hðsi Þ. As Li depends in com- with a mean value E½Ii ¼ qa . 0 plex manner on si , we assume that Li is constant. This The computation of the mean bandwidth per call follows assumption is valid since the connection length after a path the same steps than those explain for the MIP handoff 0 approach by replacing qa by qa and Li by Lh . i reestablishment doesn’t change a lot from one handoff to The mean bandwidth per call for the MHMIP handoff another. Consequently, we can write approach is given by Â Ã X iÀ1 X E Bl ¼ i pðs ÞBiÀ1 ðsiÀ1 Þ pðIi ÞgðIi Þ 1 h PD siÀ1 Ii Bh ¼ ðL B þ qa BP R Þ: ð50Þ X X qf iÀ1 þ pðs Þ pðIi ÞhðIi Þ ð41Þ siÀ1 Ii To this bandwidth, we add a fixed bandwidth due to Â ÃX X resources allocation in the GFAs hierarchies Bhs given by ¼ E Bl iÀ1 pðIi ÞgðIi Þ þ pðIi ÞhðIi Þ; Ii Ii Bhs ¼ Lhs BP D : ð51Þ P iÀ1 with siÀ1 pðs Þ ¼ 1. We can compute the two terms Lhs is the number of links in the GFAs hierarchies. Then, X pðIi ÞgðIi Þ ¼ ð1 À qa Þgð0Þ þ qa gð1Þ; ð42Þ the mean bandwidth per call is given by Ii 1 h PD 1 Bh ¼ L B þ Lhs BP D þ qa BP R : ð52Þ X qf qf pðIi ÞhðIi Þ ¼ ð1 À qa Þhð0Þ þ qa hð1Þ; ð43Þ Ii APPENDIX B with gð0Þ ¼ 1; gð1Þ ¼ 0; hð0Þ ¼ 0, and hð1Þ ¼ L. By replacing EVALUATION OF MEAN HANDOFF DELAY PER CALL these terms in (41), we find Â Ã Â Ã As in Appendix A, we consider the temporal diagram given E Bl ¼ ð1 À qa ÞE Bl i iÀ1 þ qa LB PD : ð44Þ in Fig. 4. The handoff delay is defined as the sum of the Let E½Bl ¼ LBP D , we can solve the recurrence and we delay due to bandwidth allocation BP D on the new path 0 obtain links and the signaling delay. In each interval i; i þ 1, let . Dl be the delay to allocate the bandwidth BP D on a Â Ã 1 À ð1 À qa Þi i E Bl ¼ ð1 À qa Þi LBP D þ i X ¼ LBP D ; ð45Þ link of the new path resulting from a handoff event qa during a call, with X ¼ qa LBP D . . Ds be the signaling delay due to a handoff that i As E½Bs given in (37) is not variable, then we compute i occurred in the time interval i; i þ 1, and s B using (7): . Di be the total delay for a call in the time interval i; i þ 1. s 1 Dl and Ds are the random variables which depend on the B ¼ qa BP R : ð46Þ i i qf occurrence of handoff events in the interval i; i þ 1. Di Since E½Bs given in (45) is not variable, we use (7) and (45) represents the sum of the delay due to the bandwidth i l allocation on the new path following a handoff event and to compute B . the signaling delay. l 1 B ¼ LB ð47Þ qf Di ¼ Dl þ Ds : i i ð53Þ We use (46) and (47) to compute the mean bandwidth per B.1 DHMIP call such as We reuse the same procedure than that used for the 1 computation of the DHMIP mean bandwidth per call Br ¼ Bl þ Bs ¼ ðLBP D þ qa BP R Þ: ð48Þ qf (Appendix A.1) by replacing B by D. 1324 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 10, OCTOBER 2009 The signaling delay due to handoff event in the time The mean handoff delay per call is given by interval i; i þ 1 is given by X 1 qa p P D D¼ ðDs ðnÞ þ Dl ðnÞÞP ðnÞ ¼ ½L D þ DP R : ð65Þ Ds i ¼ Ki ðDP E þ Ji DP R Þ; i ! 1: ð54Þ qf n¼1 The mean Ds , on the handoff events is given by i B.3 MHMIP Â Ã The handoff delay computation procedure is similar to that E Ds ¼ qa ðDP E þ pDP R Þ; i i ! 1: ð55Þ of MIP approach. By replacing Lp by Lhp , the mean handoff The delay for bandwidth Bl allocation in the time i delay per call becomes interval i; i þ 1 is given by X 1 Dl ¼ Hi DP D Ki ð1 À Ji Þ þ Lp DP D Ki Ji : i i ð56Þ Dp ¼ ðDs ðnÞ þ Dl ðnÞÞP ðnÞ n¼1 ð66Þ Note that Dl depends on the random variables Ki and Ji in qa i ¼ ½Lhp DP D þ DP R : the current interval. If we assume that Lp and Hi do not i qf change a lot according to i and in general they remain constant, we obtain REFERENCES Â Ã E Dl ¼ qa ðð1 À pÞH þ pLp ÞDP D : ð57Þ [1] C.E. Perkins, “IP Mobility Support for IPv4,” IETF RFC 3344, Aug. i 2002. [2] D. Johnson, C. Perkins, and J. Arkko, “Mobility Support in IPv6,” The handoff delay DðnÞ during the n periods of a call is IETF RFC 3775, June 2004. [3] R. Caceres and V.N. Padmanabhan, “Fast and Scalable Handoffs X Â Ã X Â Ã nÀ1 nÀ1 for Wireless Internetworks,” Proc. ACM MobiCom, pp. 56-66, 1996. l s Dp ðnÞ ¼ D ðnÞ þ D ðnÞ ¼ E Dl þ i E Ds i [4] C. 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