# An Application Of Derivatives That I Might Use In My Professional Life Sample Assignment

A conceptual understanding of mathematical functions; their form and graphical representations; the slope, tangent, asymptote, and limit; as well as binomial expansion to compute the derivative of a function are essential foundations that can be applied in my future career. For instance, the knowledge that the derivative of a function is another function which measures the rate at which the original function is changing, becomes relevant in terms of business operations that have changing rates.

The ability to comprehend graphs of functions may also enable businesses to determine trends and calculate how their operations would behave, whether the slope is positive or negative from left to right.As an example, the profit rate via rental of condominium units can be computed. If a condominium corporation leases out a given number of its individual condominium units, we can denote a function for the amount of dollars they collect per month and a value for the total number of condominium units available.

The task is to find the total number of condominium units they should rent to maximize their profit. The solution to the problem, applying Calculus principles, is to capitalize on the profit function but with the limitation that the number of condominium units denoted earlier must be within the range.

First, the condominium corporation must calculate the derivative and the critical point or points falling within the range. To find the maximum value, the corporation should plug in the critical point calculated as well as the end points of the range in determining the number of units to lease.

REFERENCE

1. Hoffmann, L. and Bradley, Gerald. (1999). Calculus for Business, Economics, and the Social and Life Sciences. McGraw-Hill Science/Engineering/Math.

## Manipulative As Supplements In Teaching Mathematics

Manipulative as Supplements in Teaching MathematicsUsing manipulatives and technology as supplement to mathematics instruction is gaining momentum, popularity and much debate. Foundations of these concepts and value judgements on their positive or negative effects must consider the basic premise of mathematics, its learning and teaching strategies.1) The first computer was built during World War II, to help the British intelligence service crack military codes.

After the war there came a growing need to perform other complex calculations much more rapidly than could be done by paper and pencil or adding machine. To fulfill these needs technicians designed electronic calculators. Although slow and cumbersome by today’s standards, these devices represented a huge leap forward of information processing power. (Russel, 2006)2)Manipulative use in today’s mathematics classrooms is taken for granted.

Yet using manipulatives effectively to engage students in meaningful experiences that promote mathematical understanding is a function of the tasks for which a teacher conceives them being used (Moyer, 2001). Manipulatives are designed to represent explicitly and concretely abstract mathematical ideas. Research on their use has shown that students who use manipulatives during learning outperform those who do not (Driscoll, 1983; Greabell, 1978; Raphael ; Wahlstrom, 1989; Sowell, 1989; Suydam, 1985,1986). Some studies show that student achievement levels are related to teachers’ experience in using the manipulatives (Raphael & Wahlstrom, 1989; Sowell, 1989).

Although much of this research documents manipulative use in prescribed teaching and learning interactions (Parham, 1983), few reports tell us how teachers use the manipulatives (Moyer, 2001 ), nor do they describe how students use manipulatives when they are given a choice to do so. (Moyer, 2004)First Math is a language. Math is the universal language of science. If science is life, then math is the universal language of life.

Of the many languages in the world, mathematics is the most accurate.3) When I think of the development of Mathematics over the last 2500 years, I am less surprised that early mathematicians left lasting results than that, given the tools they possessed, they achieved anything at all that could have lived through centuries. Just think of it. Zero gained widespread use only in the last millennium.

Systematic introduction of modern algebraic notations began only in the sixteenth century and is most often associated with the French mathematician François Viète (1540-1603). René Descartes (1596-1650) was first to use letters at the end of the alphabet for unknowns. He also introduced the power notations: x2, x3. The sign of equality (two equal parallel strokes) has been invented by Robert Recorde (c.

1510-1558) in his The whetstone of witte (London, 1557 (:To help you appreciate the expressive power of the modern mathematical language, and as a tribute to those who achieved so much without it, I collected a few samples of (original but translated) formulation of theorems and their equivalents in modern math language.  (a + b)2 = a2 + b2 + 2ab. If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments. (Euclid, Elements, II.

4, 300B.C.)  (Bogomolny, 2006)Secondly, Math is both process and product of intelligent learning. Since learning is not confined within the four walls of the classroom, it is imperative to incorporate life and the immediate environments around the student in teaching math.

This is where manipulative are effective because they bring three dimensionalities to concepts taught and learned.4) Once a problem can be solved with manipulatives, students draw a picture and then write a number sentence to represent what was happening with the manipulatives as they solved the problem. This gradual 6 progression from concrete to pictorial to abstract representation provides a solid foundation of understanding upon which the students can build. Every method or algorithm can be understood, and even reinvented, with manipulatives, thus replacing rote learning of algorithms with understanding of concepts and ways to efficiently apply them.

Once the concepts have been firmly established and students understand how the algorithms work, they move away from using concrete manipulatives. However, manipulatives can be revisited at any time to remediate or extend a concept as needed. (Madden, 200.) Thirdly, the use of manipulates though treading in paths laden with positive and negative implications further develop the cognitive and affective responses in students.

As different learning theories are currently being studied and research, much of constructivist and Piagean educational advocates lobby use of manipulative in teaching math. Students of the modern world are grown amidst the background of technology. Manipulatives are part of their life and therefore incorporating it to the learning field is imperative. The way that teachers use these manipulatives in teaching mathematics will spell success of failure in the students’ progress.

5) Three types of choice are fundamental, those relative to the choice of content to be taught, the planning of interactions between learners and the knowledge to be learnt; the interventions and role of the teacher in a class situation. The choice of teaching content and its organization is based on epistemological-type hypotheses and learning hypotheses.  In the practice of teaching there is generally a process of contextualization of the knowledge taught – that is to say, the organization of a context which situates this knowledge, in which the activity of the pupils can operate. The interactions between knowledge and pupils operate through the context, the milieu (Brousseau, 1986).

(Wilder, Sue. 2004). Theories such as the Piagetian theory which is knowledge base finds the manipulative usage as effective for mathematics6) According to this emerging theory, students need to construct their own understanding of each mathematical concept. Hence, we believe that the primary role of teaching is not to lecture, explain, or otherwise attempt to “transfer” mathematical knowledge, but to create situations for students that will foster their making the necessary mental constructions.

A critical aspect of our approach is a decomposition of each mathematical concept into developmental steps following a Piagetian theory of knowledge based on observation of, and interviews with, students as they attempt to learn a concept. (Dubinsky, 2006)Lastly, research must be done to further realize insights in using manipulatives in teaching mathematics not only as supplement but as prime tool in at least showing how the concepts work. If science and life need to be understood, mathematics will be able to effectively explain the many concepts of science.7) In recent years, mathematics textbooks have changed to incorporate calculator-based lessons, and the College Board has begun to permit students to use graphing calculators while taking the SATs.

Teachers are gradually learning how to modify both the mathematical content of their curriculum and their approach to teaching to take advantage of these calculators. Some schools have equipment that links calculators to sensors for collecting data such as temperature, pH, movement, and light. The calculators can plot these data as a function of time and graph the results. They can also be connected to equipment that projects the work from any one calculator to a display so that everyone in the class can see it.

In these ways, graphing calculators can support inquiry, thinking, and dialogue about mathematics and scientific data, rather than the “plug and chug” routines that have characterized U.S. mathematics classrooms. (Wiske, 2006.

)Non advocates are afraid that the consequence of not memorizing the multiplication table is devastating. A person who relies on the calculators won’t rely anymore on his own mental multiplication skills. Whether a person who’s calculator conks out buys another set of batteries or mentally calculates the mathematical problem at hand is rooted out in values that have been taught in early life. Appreciation of mathematics that is the responsibility of the parents and creativity of the teachers is first step in helping a person to like math.

As it is, dogma and traditions have scared the wits of children towards math. Manipulatives help children realize that Math can be fun afterall.8) Most students benefit from visual representations of concepts.  While the research is scarce on mathematics achievement resulting from using virtual manipulatives in the classroom, Kelly Reimer’s and Patricia Moyer’s action research study (2005), Third-Graders Learn About Fractions Using Virtual Manipulatives: A Classroom Study, provides a look into the potential benefits of using these tools for learning.

Interviews with learners revealed that virtual manipulatives were helping them to learn about fractions, students liked the immediate feedback they received from the applets, the virtual manipulatives were easier and faster to use than paper-and-pencil, and they provided enjoyment for learning mathematics.  Their use enabled all students, from those with lesser ability to those of greatest ability, to remain engaged with the content, thus providing for differentiated instruction.  But did the manipulatives lead to achievement gains?  The authors do admit to a problem with generalizability of results because the study was conducted with only one classroom, took place only during a two-week unit, and there was bias going into the study.  However, results from their pretest/posttest design indicated a statistically significant improvement in students’ posttest scores on a test of conceptual knowledge, and a significant relationship between students’ scores on the posttests of conceptual knowledge and procedural knowledge.

(….., 2006)9) So why are children still taught mathematics as a pencil and paper exercise which is usually somewhat solitary? For most of us mathematics, like music, needs to be expressed in physical actions and human interactions before its symbols can evoke the silent patterns of mathematical ideas (like musical notes), simultaneous relationships (like harmonies) and expositions or proofs (like melodies). Regretfully I hold Mathematicians (with a capital M) largely to blame for this.

They are so good at making silent mathematics on paper for themselves and each other that they have put this about as what mathematics is supposed to be like for everyone. We are all the losers. Music is something which nearly everyone enjoys hearing at a pop, middle-brow or classical level. Those who feel they would like to learn to perform it are not frightened to have a go, and those who perform it well in any of these varieties are sure of appreciative audiences.

But Mathematicians have only minority audiences, consisting mostly or perhaps entirely of other Mathematicians. The majority have been turned off it in childhood. For these, the music of mathematics will always be altogether silent.(Skemp, 2006)  REFERENCES Bogomolny, Alexander.

2006. Mathematics as a Language. http://www.cut-the-knot.

org/language/index.shtml Dubinsky, Ed. 2006. Calculus, Concepts, Computer and Cooperative Learning.

http://www.pnc.edu/Faculty/kschwing/C4L.html Moyer, Patricia.

2004. Controlling Choice: Teachers, Students, and Manipulatives in Mathematics Classrooms. School Science and Mathematics, http://www.findarticles.

com/p/articles/mi_qa3667/is_200401/ai_n9348825 Madden, Nancy. 2001. MATHWINGS Effects on Student Mathematics Performance Robert E. Slavin, Johns Hopkins University Kathleen Simons, Success for All Foundation Report No.

39 (revised) July, 2001 Published by the Center for Research on the Education of Students Placed At Risk (CRESPAR), supported as a national research and development center by funds from the Office of Educational Research and Improvement (OERI), U.S. Department of Education (R-117-D40005). This work was also supported by the Charles A.

Dana Foundation, New American Schools, and the Carnegie Corporation of New York. However, any opinions expressed are those of the authors and do not necessarily reflect the policies or positions of our funders. Russel, Peter. 2006.

The Global Brain. http://www.peterussell.com/GB/Chap8.

htmlSkemp, Richard. 2006. The Silent Music of Mathematics. www.

tallfamily.co.uk/david/skemp/pdfs/silent-music.pdfSkemp, Richard R.

1987. The Psychology of Learning Mathematics. Lawrence Erlbaum Associates.Wilder, Sue.

2004. Fundamental Consructs in Mathematics Education. Routledge Falmer Wiske, Stone. 2006.

A New Culture of Teaching for the 21st Century http://www.edletter.org/dc/wiske.htm ….

, 2006. Math Manipulatives. http://www.ct4me.net/math_manipulatives.htm

## Mathematical Common Parameters

Common Parameters (Time for processing update): Time which is needed by each node in order to import a new entry in the binding cache. (Binding cache is a storage area kept by the home agent containing associations between the home addresses and the care-of addresses for the mobile nodes it serves)(Time for processing label packet): Time which is needed by each node in order to add a label to the packet which is used for the promotion/forwarding of packet.(Time for processing update table): The time which is needed by each node in order to renew its table, for importing a new entry that represents LSP that has been established between the two nodes.

(Time for processing check): The time that is required by the MAP or the HA or any similar node in order to look at its table and connect its binding cache RCoA with the LCoA of MN.(Time processing): This time is fixed as the biggest time that is required by a single node for the processing of some concrete activity.(Time for propagation): The time that is required for the distribution/propagation of message until a particular message is sent to its final destination. This time is influenced to a large extent by the number of nodes that is found in the path of transport.

The total time of distribution is calculated as follows: .  is fixed as largest time that is required by a single node for transmission of message.The time that is required for the establishment of LSP between the two nodes. This time is influenced to a large extent by the number of nodes that participates in the establishment of LSP.

The total time of establishment of LSP between two nodes is fixed as follows: .  is fixed as biggest time that is required by each node for the establishment of LSP.(Total time of propagation): The total time of distribution that is required for the complete operation of a concrete activity.(Total time for processing): The total time of processing that is required for the complete operation of a concrete activity.

MIPv6 1.         The Mobile IPv6 standard relies fundamentally on core IPv6 protocol functions and in particular the base IPv6 specification. Dunmore and Pagztis (2005) give the signal and analytical analysis of the protocol’s handover  a.                   MN establishes contact with the old AR and the new AR with messages RtSol and receives acknowledgements RtAdv from both of them.

The total time taken for this isb.                  MN sends out binding updates to CN/HA/AR in time . Therefore,c.                   MN stores the value of BU in the listd.

MN requests the receipt of the BU and HA/CN send BU acknowledgements in timee.                   BU is then removed from the receiver’s cache in time The total time for the signaling is     HMIPv6 2.         According to Howie, Sun, and Koivisto (n.d.

) Hierarchical Mobile IPv6 builds over the basic MIPv6 protocol. model. It adds an intermediary HA called a Mobility Anchor Point (MAP) to the basic model. The MAP is fewer router-hops away from the MN than the basic HA.

The exit point of the tunnel from the HA is now situated at the MAP. The MAP, in turn, has a bi-directional tunnel to the MN. Handover procedurea.                   Initially all ARs advertise the address of the MAP to attached MNs and MN establishes contact with the old AR and the new AR with messages RtSol and receives acknowledgements RtAdv from both of them.

The total time taken for this isb.                  The MN sends BU messages takingc.                    When the MN moves to a new subnet-prefix within the MAP domain, it just registers a new BU with the MAP taking a time ofd.                   The process for new CoA at the MN takese.

The MN then receives BU Acknoledgement takingThe time taken for handover is as below Packet Delivery Procedurea.                   The original HA at the highest hierarchical level can now tunnel a packet to a new HA (MAP) one level lower in the hierarchy taking a time ofb.                  This in turn in turn, tunnels the packet to the MN and takes a time of  The time taken for packet delivery is as belowThe total time for signaling would be FMIPv6 3.         According to Dunmore and Pagztis (2005), the aim of FMIPv6 protocol is to allow MN to configure a new CoA before it moves and connects to a new network.

It also allows the MN to use the new CoA immediately upon connecting to the new network. Furthermore, the FMIPv6 protocol seeks to eliminate the latency involved during the MN’s BU procedure by providing a bi-directional tunnel between the old and new networks while the BU procedures are being performed. Thus, compared to normal MIPv6 operation, the FMIPv6 protocol claims to be more efficient as it removes IPv6 configuration delay introduced by DAD, router discovery and address configuration. It also removes the delay introduced by the MN performing BU procedures with its HA and CNs   a.

MN first sends a RtSol message to old AR and receives RtAdv message from it which takes a total timeb.                  When a HO-warning occurs, old AR acts as proxy to MN by calculating the new CoA taking a time ofc.                   Old AR uses this mapping to determine the address of the targeted new AR in the neighboring domain taking a time ofd.                  The newCoA is sent simultaneously  to both the MN and newAR (PrRtAdv and HI messages, respectively) takinge.

Their responses (F-BU and HAck) verify the validity of the newCoA .f.                    Old AR sends simultaneous binding acknowledgements (F-BAcks) to both MN and the new AR taking a time ofg.                  Before the packets addressed to the Old CoA are routed by old AR to MN taking a timeh.

MN then sends a RtSol message to new AR and receives RtAdv message from it which takes a total time Total time taken for the signaling is Mobile MPLS (With HMIP architecture) (Integration of MPLS with MIP) 4.         Um and Choi (2001) in their paper give the MPLS based Hierarchical Mobile IP architecture. Their paper proposes a way to reduce the LSP set up latency by integrating the MPLS with mobile IP. In this way the IP-IP tunnels are replaced by one or more LSP setups on the MPLS network.

They have given the following figure for the registration and LSP set up. In the figures below GFA stands for Gateway Foreign Network and RFA stands for Regional Foreign Agent.  The above figure holds true when after the FA has advertised the addresses of hierarchical FA and MN determines that it is in a foreign network. The figure is divided into two parts for analysis: registration request and LSP set up.

The packets are assumed to arrive, after which the LSP set up procedure takes place. Registration Requesta.                   MN sends a registration request to the FA, the time it takes to do so is given byb.                  FA needs some time for analyzing the registration request since it is an LER, this time is given byc.

FA then sends the registration request to the next RFA, the time it takes to do so is given byd.                  The RFA which is an LSR receives the registration request and updates it in the node table. The time taken for this is given bye.                   The RFA then sends the registration request to the GFA, time taken for this is given byf.

The GFA after receiving the request caches this information, the time taken for this is given byg.                  The GFA then sends the registration request to the HA, and time for this is given byh.                  The HA sends the registration reply to the GFA, the time for which can be given byi.                    The GFA in turn sends the registration reply to the RFA, the time taken for this is given byj.

The RFA then sends the registration reply to the FA, and time for this is given byk.                  The FA being the lowest in the hierarchy checks its cached information, the time for this is given byl.                    The FA then send the registration reply to the HA, the time taken for this is given byHence the final equations can be given as below:Hence, LSP Set up Between HA and GFAa.                   The HA receives the packet and sends the label acknowledgement with care-of-address as FEC.

The time for this is given byb.                  The time taken for LSP to establish between the GFA and HA is given by . Since GFA assigns label during this time frame there is no need to give additional time for the table update here.c.

After receiving the label mapping message, the HA updates the table, the time taken for this is given byd.                  The HA then sends the user packets to GFA along the LSP from HA to GFA, the time for this is given by Between GFA and RFAa.                   When GFA receives the label, it writes the label value attached to the user packets on an incoming value of corresponding MN. The time for this is given byb.

The GFA sends the label request to the next RFA with the CoA as the FEC. The time for this is given byc.                   The RFA updates the information in the table taking time ofd.                  The LSP is established when RFA responds back with the message i.

e. the time ise.                   The GFA then updates its table which takes a time off.                   Finally the GFA sends the user packets to the RFA through the LSP taking a time of Between RFA and FAThe authors say that this above process is repeated in every FA till the lowest FA next to the MN, which takes an additional time of .

In the figure there is only one such FA and hence the time for the LSP to set up would be Total time for LSP set upThe total time taken for LSP set up is given as below: Handover Delaya.                   MN sends a request for handover to the root RFA, the time for this is given byb.                  The root RFA then sends message to MN to inform about the next BS that MN should be attached, the time for this is given byc.                   When data arrives for the MN, the root RFA multicasts the data to all the members in the multicast group in time , where the #nodes gives the number of base stations that form a multicast LSP group.

The total time for handover is given by The total time will be given by Time for registration Time for handover Mobile MPLS (with MIP architecture) (Integration of MIP with MPLS) 5.         Ren, Tham, Foo and Ko (n.d.) propose an architecture which uses MPLS in Mobile IP so that the Mobile IP has some benefits by MPLS in a heterogeneous network environment comprising IP, ATM and wireless networks.

Registration requesta.                   MN receives the advertisement messages broadcast by the mobility agents and determines whether it is at home or the foreign network, the time taken for this isb.                  If the MN is in a foreign network it acquires a temporary CoA from FA taking time , and sends the registration request to the FA taking timec.                   The FA updates its routing table to add a host specific row with the value of MN home address taking a time ofd.

The FA forwards the registration request to HA taking a time ofe.                   When HA gets the registration request, and learns the CoA, it searches its routing table to find the row with the MN home address as FEC, the time for thisf.                   HA then sends a label request to FA with CoA as FEC taking a time ofg.                  FA replies with a label mapping message and LSP is established between the FA and HA, this takes a time ofh.

HA adds a new table which takes a time of , and maps the MN home address row in routing table to the label table row taking a time ofi.                    HA then sends a registration reply to FA which takes a time of , FA then forwards the registration reply to MN taking a time of  and the registration endsTotal time for registration Datagram delivery procedurea.                   CN sends the packets to MN addressed to the MN home address taking a time ofb.                  HA looks up its table value using the label index value as the index, taking a time ofc.

HA strips the label and sends the packet to the IP layer taking a time of +d.                  HA searches the routing table to find the mapping in a time of  and sends the packet back to the MPLS layer taking a time ofe.                   HA inserts the label value in the new row of the label value into the packet taking a time of  and forwards it to the MN taking a time off.                   The packet is delivered from HA to FA along the LSP by label swapping with a time ofg.

FA looks up its label table in a time  and sends the packet to the IP layer with a time ofh.                  FA forwards the packet to MN in a time ofThe total time taken for this process isTotal time for the signaling is  Time for registration Time for handover  Hierarchical Mobile MPLS6.         According to Yang and Makrakis (2001) conventional mobile MPLS does not support mobility, and hence researchers look into ways to integrate Mobile IP with MPLS to support mobility in MPLS networks. However, the methods of the integration prove to be obsolete in the latest wireless networks which are implemented on micro-sized cells.

For overcoming the limitations due to micro-cells, Yang and Makrakis have proposed a new scheme to support local mobility MPLS which they have named as hierarchical mobile MPLS. Hierarchical MPLS, as will be seen in the analysis below, handles the movement of the mobile host locally, that is to say in the same MPLS domain there is no need to send the location update messages to the remote home agent. In other words, in the Hierarchical Mobile MPLS, the movements of a mobile host moving within a domain are transparent to the home agent of the mobile host. Mobile host registration in Hierarchical Mobile MPLSa.

The mobile host receives agent advertisement message and determines whether it is at home or in a foreign domain, the time for this is given byb.                  If the mobile host is in a foreign domain, it acquires a temporary CoA from the FAWhy do we use tprocc again?c.                   The MN then sends a registration request to FA taking a time ofd.                  The FA forwards the registration request to FDA using normal IP forwarding taking a timee.

In case the mobile host first moves in this domain, the FDA will forward this registration message to the HA taking a time ofTime for registration is given by LSP set upa.                   When HA gets the registration message, it sends a label request message using the LDP protocol to the FDA with the IP address of FDA as FEC. The time for this is given byb.                  The FDA replies with the label mapping message to the HA and sends a label request message to the FA where the mobile is currently located.

The total time for this is given byc.                   When the label mapping message arrives at HA the LSP is established in time and is send from FA to FDA in timed.                  Also FA replied with the LDP label mapping message back to the FDA in time  and the LSP is established in timee.                   The HA searches the label table to find the rows with the mobile host’s home address as FEC and changes the out going port and out label with the same values as the LSP from HA to FDA taking a timef.

When MH handoffs from one sub-network to another sub-network within the foreign domain, it sends a registration request to the new FA taking timeb.                  MH also sends a binding update to the old FA which takes timec.                   The FA forwards the registration request to FDA taking timed.                  The FDA sends back the label request to the FA taking timee.

The FA responds back with a label swapping message taking timef.                   A new LSP is then established between FDA to FA in timeg.                  FDA sends registration reply to new FA. New FA in turn sends registration reply to old FA and the old FA sends the registration reply to MH.

The entire process takes timeTime for handover is given byTotal time for signaling is given by  Time for registration Time for handover  Micro-Cell Mobile MPLS (incorporation of MPLS forwarding)Yang, Dong, Zhang and Makrakis (n.d.), have described this protocol in their paper. It is an incorporation of MPLS forwarding, localized signaling, and soft-state location management.

Registrationa.                   When a mobile first moves into a foreign domain it sends a registration request message to the nearest FA in timeb.                  The FA relays this registration request message to the FDA of this MM-MPLS domain in timec.                   FDA sets up an LSP between itself and the HA following the process in H-MPLS which takes a time ofd.

FDA then sets up an LSP between the FDA and current FA in time  using RSVP-TE, a soft-state signaling protocol.e.                   FDA sends a label request object to FA with a time off.                   After receiving this message the FA responds back with a received message which takes timeg.

Finally FDA relays the registration replay message to the LSP between FDA and current FA this takes time Total time taken for registration is as below: Handoff ProcedureWhen the mobile host handoffs from one access network to another within the same foreign domain, it will send a registration request to the new FA. This takes timeThe new FA relays the registration message to FDA of the domain which takes timeLSR checks its label table to see whether there is an entry for the mobile host which takes timeThe LSR sets up a new LSP to the new FA using the mobile host’s home address as FEC which takes timeThe LSR then updates its table which takes timeThe LSR redirects the LSP back to the new FA which takes timeThe LSR generates an RESV message which takes time  and then sends this message to the FDA which takes timeThe MH deletes the entries from their label table which takes timeTotal time taken for handoff procedure is as below: The total time taken for signaling is as below  Time for registrationTime for handover  Micro-Mobile MPLS (Integration of MIP & MPLS)7.         Langar, Bouabdallah and Tohme (2006) give the analysis of Micro-Mobile MPLS, which integrates Mobile IP and MPLS protocols using a two level hierarchy. Their proposal has two variants.

In the first variant, called FH-Micro Mobile MPLS, we consider the fast handoff mechanism, which anticipates the LSP procedure setup with the neighboring subnets to reduce service disruption. In the second variant, called FC-Micro Mobile MPLS, the forwarding chain mechanism, which is a set of forwarding paths, is provided to track efficiently the host mobility within a domain. The forwarding chain can reduce registration update cost and provide low handoff latency. Mobile Node Registration in Micro Mobile MPLS a.

MN sends solicitation message to LER/FA and receives Advertisement message from it which takes timeb.                  When an MN moves for the first time into a Micro-mobile MPLS foreign domain, it sends a IP registration request message to the nearest LER/FA taking a timec.                   The LER/FA records this MN home address in its routing table taking a time ofd.                  It relays the registration message to the LERG of this domain taking a time ofe.

When LERG gets the registration message it gets the RCoA taking a time off.                   The LERG sends a registration message to the HA of the MN taking a time ofg.                  The LERG uses the IP address as the CoA to perform the global registration for the inter-domain mobility taking a timeh.                  LERG establishes an LSP between it and the current LER/FA with the RCoA as the FEC taking a timei.

LERG then relays the registration reply message sent from the HA to the MN along the established LSPs taking a time Total time for registration is given as below Handoff Support procedure in Micro Mobile MPLSThere are two types of handoff in the scheme: Intra-LER and Inter-LER handoffs. An Intra-LER handoff occurs when the MN moves between two BSs managed by the same LER/FA. An Inter-LER handoff occurs when a new BS and the old BS are under different LER/FAs. Intra-LER handoffa.

Once the association to the current BS is lost, the MN sends a Movement signaling message to the current LER/FA with a time of , which initiates the buffering mechanism and stores in-flight packets. The time taken for this is given by +b.                  The MN issues a local interface-update message in the subnet to which the MN belongs, so that all stations in the same subnet, especially the current LER/FA, update their ARP (Address Resolution Protocol) cache, taking time .c.

The current LER/FA stops the buffering mechanism and forwards in-flight packets destined for the MN toward the new BS. This takes timed.                  The LERG continues to use the old LSP between it and the current LER/FA to send packets to the MN taking a time of Total handoff time in this case is given by  Inter-LER Hand off a.                   When MN finds that it has entered a new IP subnet , it sends a registration request to the new LER/FA in time  and performs steps which is same as in the registration procedureb.

MN sends a handoff notification to the old LER/FA with a time of and there is a handoff acknowledgement which takes a time ofc.                   When the old LER/FA receives his notification message it stops the buffering mechanism and forwards the in-flight packets to the new subnet. This takes a time +d.                  LSP is then establish which takes a time ofe.

After this the corresponding table is updated by the LER/FA which takes a time The total time except the registration period time is  If registration is also added to this then the total time becomes Handoff in FH (Fast Handoff) Micro Mobile MPLSThis is explained by Langar, Bouabdalaah, Tohme (2006) give the mechanism as below:a.                   Once an MN enters an overlapped area of the boundary cells of two subnets, it receives a new beacon from the possible new AP in timeb.                  MN notifies the current LER/FA for the possible handoff by sending a handoff initiate signaling message in timec.                   LER/FA looks into its neighbor mapping table to get the new LER/FA’s IP address in timed.

LER/FA then informs the LERG for the possible handoff operation in timee.                   FH-Micro mobile MPLS will pre-establish an extra passive LSP between the LERG and the new subnet that the MN is likely to visit in timef.                   The current LER/FA informs the MN for the new CoA by sending a neighbor advertisement message which takes timeg.                  FA forwards the mobile IP registration request message to the LERG in timeh.

FA notifies the old LER/FA for the handoff event in timei.                    As soon as LERG receives the mobile IP registration request it activates the pre-established passive LSP in time  and the traffic is delivered through thisj.                     Once the old LER/FA is notified the in-flight packets are forwarded to the MN through the new FA in timeTotal time for handover is given by  Handoff in FC (Forward Chain) Micro Mobile MPLSThis is explained by Langar, Bouabdalaah, Tohme (2006) give the mechanism as below:a.                   When MN moves to a new subnet a new RCoA is registered at the old LER/FAb.

The existing LSP between the LERG and the old subnet will be extended from the old FA to the new one in timec.                   MN has a buffer for storing IP address of the LER/FAs which is updated in timed.                  Packets traveling towards the MN are intercepted by the first FA in the chain and are then forwarded along the chain of FAs. It takes timee.

To avoid delays a threshold on length is set. When this threshold is reached, the MN is reached and deletes all addresses in the buffer. This takes timeThe total time for handover will be as given below:  Time for registration Time for handoverCase- I Intra-LER HandoffCase- II Inter-LER HandoffCase III FH MM MPLSCase IV FC MM MPLS  Hierarchical MPLS8.         Nguyen, Li and Xie (2003) give a new approach for IP/MPLS for providing micro-mobility in an efficient way with QoS.

They give a common framework of hierarchical IP/MPLS based for 3G-Radio Access Networks. An enhanced label edge router LER called local mobility agent LMA is introduced to setup two-stage label switched paths LSPs between radio network controller RNC and base station BS for reducing handover latency caused by local mobility. Registration and LSP setup  a.                   When MT (mobile terminal) receives an advertisement message from a BS, it sends registration message to that BS with its home IP address, home agent IP address.

This takes a timeb.                  The BS registers with its LMA via the pre-established LSP, taking a time ofc.                   When the corresponding LMA receives the request message from the lower level BS, it adds its record about that MT and attaching BS taking a time ofd.                  LMA then sends the registration message to all the RNCs in the network via the pre-established LSP taking a time ofe.

When the RNC receives the request message from the lower-level LMA it adds its record about that MT and attaching BS taking a time off.                   RNC sends mobility binding message to MT home agent taking a time ofg.                  When the acknowledgement is received the RNC will send registration reply message back to the requesting LMA taking time ofh.                  The registration message is then sent back to BS taking a time of  and finally back to the mobile terminal MT taking time ofTotal time for handover LSP setupLSP is set up when any connection is initiated from or to the MT.

Hence a set up time of  is added every time a connection is initiated. HandoffsThere are three types of handoff in the considered hierarchical radio access network: Intra-BS which is a link-layer handoff, Intra-LMA, Inter-LMA Intra-LMA handoffIn the intra LMA handoff both the old and new LMA are under the same LMA.a.                   When the LMA receives the registration from BS, it checks its records taking a time of .

It finds that the MT is already recorded with a different BSb.                  The LMA then updates its record of MT taking a time ofc.                   It then sends a registration reply to the new BS taking a time ofd.                  LSP is established in the meantime with resource reservation with a time ofe.

LMA then sends the registration reply to the old BS taking a time of  Total time taking for handoff is Inter-LMA HandoffIn inter-LMA handover old and new BS are under different LMAsa.                   When the new LMA receives the registration message from the new BS, it checks its records taking a time of . It finds that there is no record for that MT.b.

The LMA then adds its record for that MT taking a time ofc.                   The LMA sends a registration request message to all RNCS, taking a time ofd.                  When the RNC receives this request message it checks its record taking a time of , it finds that a different LMA is attached.e.

The RNS updates its record taking a time off.                   The RNC sends back a reply to the new LMA taking a time ofg.                  When the new LMA receives the reply message from RNS it sends reply message to the new BS with resource reservation taking time of Total time for handover in this case is  Total time for the whole process in case of Intra-LER handoff is Total time for the whole process in case of Inter-LER handoff is Time for registration Time for handoverCase- I Intra-LMA HandoffCase- II Inter-LMA Handoff  Overlay MMPLS (HMIP over MPLS in overlay environment) 9.         Vassiliou and Pitsillides (2007) analyze the mobility events, and particularly handovers in an overlay environment, where Hierarchical Mobile IP is operating over MPLS.

a.         CoA and HA Discovery This is represented in the figure belowMN acquires a new address to represent the new region which takes a time ofAfter this the MN sends a Discovery Request to the HA, which takes a time ofThe HA would then respond with a reply message which takes time b.         Registration processThe MN sends a MA Registration BU in the MAP in time .The MAP imports the new entry in binding cache in order to connect the MN with the new address which it has acquired, this takes timeMAP sends BU acknowledgement to the MN which takes timeMN sends Registration BU to the HAin time  .

HA imports the new entry in binding cache in order to connect the MN with the new address which it has acquired in timeHA sends a BU acknowledgement to MN which takes time The figure below shows the remaining sections which are analyzed below: c.         Establishment of LSP between the CN and the ??a.       The CN checks if an entry exists in binding cache taking time . If there is no such entry, then it sends the packet to the address of MN after an LSP is established between the CN and HA.

b.      The CN sends a label request to the HA and HA responds with label mapping. The time for establishing an LSP isc.       HA and CN’s LERs renew their tables adding a new entry which represents the particular LSP established between them.

The time for each renewal of table takes . Since there are two renewals the total time would bed.      LER adds a packet to the label taking a time of . The packet is then sent from CN to the HA taking a time of;d.

Packet reception from the ??a.       When the HA receives a packet it checks its binding cache in order to find an entry with the CoA that represents the MN in the new region that has been moved. This takes a time of;e.         LSP establishment between the ?? and the MAPa.

An LSP is first established between HM and MAP in timeb.       After this, the HA and the MAP renew their tables adding a new entry representing the particular LSP established between them. Time for processing each table is . Hence, time for renewing both the tables will be .

c.       HA adds a packet to the label and takes a time of .d.      The packet is then sent form HA to the MAP which takes time of;f.

Packet reception from the MAPa.       The MAP receives the packet from the HA and connects the RCoA with the LCoA of the MN which takes time .b.      MAP also needs to renew its table to import an entry representing the LSP for sending the packets to MN .

;g.         LSP establishment between the MAP and the MN’s LERa.       The MAP sends a label request to MN’s LER a label request which takes a time of .b.

After this, the MN’s LER and the MAP renew their tables adding a new entry representing the particular LSP established between them. Time for processing each table is . Hence, time for renewing both the tables will be .c.

MAP adds a packet to the label and takes a time of .d.      The packet is then sent form MAP to the MN’s LER which takes time of h.         Packet reception from MN’s LERa.

MN’s LER removes the label from the packet and sends in the IP Layer which takes a time of ..b.      Message IP Packet is delivered from MN’s LER to the MN.

This time is fixed as i.               Briefing of CN from the MN on the new address that it has acquireda.       An LSP is established between CN and MN and the time for this is .b.

After this the MN sends a Binding Update in the CN which takes time .c.       Some time is required for the processing by the CN in order to renew its binding cache and connecting MN with the new address which it has acquired. This takes a time .

d.      CN responds back with BU Ack which takes time . j.                                            LSP establishment between the CN’s LER and the MAPa.

CN’s LER sends in the MAP a label request and MAP corresponds with label mapping which takes time .b.      After completing the process of exchange of messages of establishment of LSP, MAP and CN’s LER should renew their tables by adding a new entry which will represent the particular LSP that has been established between them. Time of renewing each table is , and time taken for renewal by both would bec.

After this CN’s LER takes some time for processing i.e  which will be used for adding a to the label and to promote via the recent LSP that has been established. The time that is required for the promotion of packet from CN’s LER to the MAP is fixed as . k.

LSP establishment between the MAP and the MN’s LERa.       The MAP sends in MN’s LER a label request and MN’s LER corresponds with label mapping. Time for this is .b.

After completing the process of exchange of messages of establishment of LSP, MAP and MN’s LER should renew their tables by adding a new entry which will represent the particular LSP that has been established between them. Time of renewing each table is , and time taken for renewal by both would be .c.       After this MN’s LER takes some time for processing i.

e  which will be used for adding a to the label and to promote via the recent LSP that has been established. The time that is required for the promotion of packet from MAP to MN is fixed as . The total time required for the completion of above process is given below:  Time for registration           ComparisonThe following are the total time for all the mobility events1.                  MIPv62.

HMIPv63.                  FMIPv64.                  Mobile MPLS (with HMIP architecture)5.                  Mobile MPLS (With MIP architecture)6.

Hierarchical Mobile MPLS7.                  Hierarchical; MPLSa.       With Intra-LER Handoffb.      With Inter-LER Handoff                   References Howie D, Sun J, Koivisto A, (n.d.), A Hybrid Model for Wireless Mobility Management using IPv6, University of Oulu, FinlandDunmore M, Pagztis T, (2005), Mobile IPv6 Handovers: Performance Analysis and EvaluationLangar R, Bouabdallah N, Tohme S, (2006), On the Analysis of Mobility Mechanisms in Micro Mobile MPLS Access Networks, Journal of CommunicationLangar R, Tohme S, Bouabdallah N, (2006), Mobility management support and performance analysis for wireless MPLS networksRen Z, Tham CK, Foo CC, Ko CC, Supporting Multi-protocol Label Switching In Mobile IP,Um TW, Choi JK, (2001), A Study on path Re-routing Algorithms at the MPLS based Hierarchical Mobile IP networkYang T, Dong Y, Zhang Y, Makrakis D, Practical Approaches for Supporting Micro Mobility with MPLSYang T, Makrakis D, (2001), Hierarchical Mobile MPLS: Supporting Delay Sensitive Applications over Wireless Internet……..