## Challenges in Frequency Allocation

This paper provides an explanation and analysis of the industrial application of graph theory. The paper relies mainly on secondary sources of the information and course knowledge gathered in discrete mathematics class. For, in Motorola company seeks to solve recoloring challenge in cell phone and communication system operation. Motorola poses a problem that is minimized by employing discrete mathematics knowledge. One channel undertaking is in operation and we want to switch to any other by changing one node at a time. In the course of this modification, while a few transmitters have long gone to their new channel and others still on their old channel, all the non-interference constraints have to be met. How do we determine an order on how to carry out the changes that achieves this? The result of the research shows that the problem can be solved through employment of dynamic channel assignment model; which make use of graph theory. Even so, it is unanimously agreed on that despite the solution applicable in the contemporary industrial processes, there is a need for future research as there are several difficulties being encountered in the course of implementing the dynamic channel assignment model.

For instance, do not forget the mistake-detecting and mistakes-correcting codes which can be utilized in all the virtual communications we make each day. The development of those codes is a basically discrete problem, but the purpose why such codes are wished arises from an underlying non-stop trouble of sign transmission over a radio hyperlink or optical fibre, and the probability distribution of errors in such transmissions (Ahuja, et al., 1991). to give another instance ,a timetabling trouble or a resource allocation problem will commonly take a in simple terms discrete form ,yet it has arisen from actual-world continuous constraints such as how lengthy it takes a records packet to get from A to B ,or a educate, or a class of schoolchildren (Damantis, et al., 2002).

Locating the most efficient frequency allocation of a mobile phone network base has been found to be very hard (Diestel, 2005). This papers objective is generally to deal with queries of similar nature. During certain periods of operation of a mobile phone network provider, it mostly become essential to adjust the frequency venture plan to align with the changes in traffic or any installation of a new hardware (Cerceda, et al., 2005). Most of the time switching to the new plan involves global rest of the network which leads to a crush of the network service being availed. It is therefore recommended that analysis of the opportunities of dynamic reassignment be done (Diestel, 2005). Such principle may be applied in switching to the new plan without interfering with the operations of the network. Such problems can be formulated in form of recoloring a graph. In this paper displays a simple but effective algorithm for dynamic reallocation based on a direct method of search and random coloring (Diestel, 2005).

## Graph Coloring and Frequency Allocation

So allow us to relate this instances to Motorola organization. The organization was acquired by Lenovo, a Chinese technological organization. Motorola was an American multinational telecommunications enterprise that was founded in September 25, 1928, based in Schaumburg, Illinois. Motorola depend on a network of radio transmitters and receivers communicating with cell devices within the surrounding place (Heater, 2012). There is probably the base stations of a cellphone community communicating with cell phones of their region or a wireless gate entry to points for the WLAN in an office building, communicating with the laptops, printers and so on inside the nearby rooms (van den Heuvel, et al., 1998). Each transmitter desires to operate on one among several “channels”, which may also, as an example, be extraordinary variations of a coding device. Neighboring transmitters want to apply exceptional channels so that a tool that is possibly midway among them is capable of locking unambiguously onto one of the incoming alerts despite that truth that they are incident with comparable depth. For that remember, a challenge is provided in coping with complexities involved (Preparata & Shanos, 1985).

There are numerous different constraints in practice, relying at the precise utility, which we might imagine of as “non-interference” constraints. Locating an awesome channel mission, that gives an amazing satisfactory of service to the users of the device, is an exciting discrete challenge in commercial mathematics that has been broadly studied in late years (Molloy, 2004). It is very intently associated with graph coloring, for we are able to think of the transmitters as nodes of a graph (discrete mathematics concept), and be a part of two nodes where their cells are geographical buddies. Then if we think about the present channels as colors ,of which we have to assign one to every node ,the simple constraint referred to earlier is that joined vertices ought to be extraordinary colors, this is, we should produce a vertex-coloring of the graph.

In 2005 Motorola delivered to the United Kingdom mathematics-In-Industry Group to have a look at the organization’s recoloring challenge (Diestel, 2005): assume one channel undertaking is in operation and we want to switch to any other by changing one node at a time. In the course of this modification, while a few transmitters have long gone to their new channel and others still on their old channel, all the non-interference constraints have to be met. How do we determine an order on how to carry out the changes that achieves this? (Diestel, 2005)

## Dynamic Channel Assignment

In the mobile set, cells are defined as minute areas of coverage which when joined together deliver telecommunications service to a wider region. A network frequency plan involves a combination of frequencies into cells based on certain conditions.in the cause of network modelling the network cells are considered to be vertices of a graph and the adjoining are converged by use of a border.as the primary focus of this issue, we consider the cell network to be modelled via a basic graph F=(X, Y), where X represent the set and Y the edge in between (Preparata & Shanos, 1985) (Preparata & Shanos, 1985). The unique frequency q given to a vertex contains a couple: one channel plus one coloration code so that . One single vertex is related to a unique base station.in general most of the cells can be related to single base station. Finally, the base stations are arranged through a collection C of base station which controls the network (Molloy, 2004). Due to factors which are more elaborated below, it’s realistic to define a vertex v using the base station controller that controls the station base. Generally, we combine a tuple vertex v. The frequency plan is then defined as the set . The problem as stated here associates to the beacon frequencies which are used by cells to transport identity data. In normal situations, every cellular can even have one or more frequencies that carry traffic (Molloy, 2004).

When the traffic frequencies cross each other they can cause a problem, but that’s not discussed here. A frequency plan ought to abide by a spread of feasible constraints (Diestel, 2005):

2. a) The “*Neighbor-of-neighbor*” constraint: if v1 and v2 are each associates of v, then v1 and v2 must use exclusive frequencies from every other, i.e. fv1=fv2. Two frequencies are special in the event that they range both inside the channel or the color code or both.

3. b) The “*Neighbor*” constraint: if v1 and v2 are neighbors then they have to use special frequencies from every different, that is fv1 is not equal to fv2(Diestel, 2005)(Diestel, 2005).

c) “*Interference constraints*”: if v1 and v2 are geographically near each other, |fv1−fv2| is not supposed to be too small(Diestel, 2005).

d) Some neighboring cells, for example the ones corresponding to unique sectors on the same base station, should receive assignments which can be at the least two channels aside(Diestel, 2005).

## Limitations and Future Work

e) There may be different constraints depending on the type of hardware installed at a site(Diestel, 2005). The constraints essentially get up from the truth that a cell handset shifting through the coverage region of the network need to constantly be able to become aware of the neighboring cells uniquely; this allows for the easy handover of services from cell to cell. It is very important to realize that “*neighboring*” cells on this context are not necessarily geographically adjacent. “*Neighboring*” cells are ones between which a direct handover is feasible(Diestel, 2005).

The “*neighbor-of-neighbor*” constraint (a) is a difficult constraint inside the reallocation system. That means that it cannot be breached, and preserving this constraint is the key problem in this process. If cells have d neighbors, there are effectively constraints per cell at the frequency allocation. On the other hand, the “*interference constraints*” (c) are not important on the grounds that they are “*soft*” constraints. If they are violated, then they will now not be violated for extremely lengthy period of time, even though the period of awful interference conditions must ideally be minimized (ArbaughW, et al., 2005).

A common network may have two thousand to three thousand cells and every cell having fifteen to thirty neighbors with the frequencies being chosen from a set of 320, made from say 10 channels and 32 color codes as displayed below (Diestel, 2005).

Fig 1: A common graph with 500 vertices (Cerceda, et al., 2005).

Motorola Enterprise System and Success Assessment

To solve the above problem dynamic channel assignment was adopted. Apportionment of the dynamic channel can solve the interference problem inn networks in cases experiencing regular and dramatic variations in topology (Almeroth, et al., 2006). Dynamic channel venture entails constantly monitoring interference and reallocation channels as they should be. Arbaugh et. al. applied a type of dynamic challenge of their multi step greedy style of algorithms, it reassigns channels handiest to nodes affected by the highest interference (ArbaughW, et al., 2005). Almeroth et al. introduced dynamic allocation in their algorithm. Their allocation is called breadth first search channel assignment. This is abbreviated as BFS-CA. Breadth initial search-channel task which applies breadth initial search to determine nodes of fineness connectivity which mainly are concerned with the highest level of interference (Preparata & Shanos, 1985). Channels are reallocated on-the-fly, and connectivity is preserved through a certain “default channel” (Almeroth, et al., 2006).

Fig 2: Network topology with varying channel allocation (Almeroth, et al., 2006).

Figure 2 (a) depicts a sample topology of the network where all nodes present broadcast over one frequency. . Suppose you take a node C for channel reassignment by adding two new radios that broadcast and receive through channel two and three, through this method of casting off the risk of interference, there is a break in connection between nodes A, B, and D. Hence communication is only through the use of one hope-over node C. Despite this, problems still arise due to change in topology. The result of that is the inability of nodes A, B, and D to communicate despite Bering within each other’s communication range. For all the alterations in topology, failure of the node is an actual hazard in breaking conversation (Preparata & Shanos, 1985). With a pre-set channel in vicinity, an intensive connection of the node failure can be solved by minimum downtime. As an addition to the way BFS-CA will operate in very congested topologies, indicating just how lengthy the downtime caused by node failure will go on in an open issue, and in addition coloring algorithm are works still under development due to their NP-HARD and NP-Complete algorithmic problems (ArbaughW, et al., 2005).

If we have got a frequency plan A and want to alternative to another new plan B, we assume that each plan satisfies the applicable constraints (Diestel, 2005). One method of alternating is to develop and downfall all the new databases using the brand-new frequency plan within the system which is followed by a reset of all the cells in one instant. But, this results into an outage of 20 minutes to the service (Diestel, 2005). To prevent this outage, we will choose to alternate the channel or color code at a time in skilful predetermined way, waiting to alternate every mobile up to the time when no calls are not required. The change overs are then unrecognized by the subscribers. Within each step, we are permitted to insert either CH or CC for k vertices where k is |c|, types of base station controllers within a different phrase, each base station controller can make one change independently provided we don’t breach the constrains within the process. In a perfect state of affairs, it's going to take two steps to alternate from “*f= (CH, CC) to f= (CH, CC); either via (CH, CC) or (CH, CC)” (Cerceda, et al., 2005).*

Furthermore, the network should function successfully during the entire changeover operation. In other words, the preset sequence of steps have to be such that the constraints are satisfied in all the intermediate states, where a few cells are using the authentic frequency, others have changed both channel or color code, and others have already changed both channel and color code (Almeroth, et al., 2006). For this reason, the direct transition may not be feasible, and a few cells might also need to exchange from *fi *to* f’i* in more than a single step (Molloy, 2004). Consequently, the main difficulty is coming up with the smallest steps in the changeover process without negatively affecting the entire process (Ahuja, et al., 1991).

References

Ahuja, R. K., Magnanti, T. L. & Orlin, J. B., 1991. Some Recent Advances in Network flows. *SIAM, *33(2), pp. 175-219.

Almeroth, K. C., Belding, E. M., Budhikot, M. M. & Ramachandra, K. N., 2006. *ucsb. *[Online]

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[Accessed 12 4 2017].

ArbaughW, Banerjee, S. & Mistra, A., 2005. Weighted coloring based channel assignment for WLANs. *ACM Mobile, *9(3), pp. 19-31.

Cerceda, L., Van den Heuvel, J. & Johnsom, m., 2005. *Connectedness of the graph of vertex colouring, *LSE: LSE-CDAM.

Damantis, Z. G., Tsahalis, D. T. & Borchers, I., 2002. Optimization of an active noise control system inside an aircraft based on the simultaneous optimal positioning of mcrophones and speakers with the use of a genetic algorithm. *Comput. Optim., *23(1), pp. 65-76.

Diestel, R., 2005. *Graph Theory. *3rd ed. Verlag: Springer.

Heater, B., 2012. *engadget. *[Online]

Available at: __https://www.engadget.com/2012/05/30/motorola-a-brief-history/__

[Accessed 12 4 2017].

Molloy, M., 2004. The Glauber dynamics on colouring of a graph with high girth and maximum degree. *SIAM jJournal of Computing, *33(1), pp. 721-737.

Preparata, F. P. & Shanos, M. I., 1985. *Computational Geometry: An Introduction. *1st ed. Verlag: Springer.

van den Heuvel, J., Leese, R. A. & Shepherd, M., 1998. Graph labelling and radio channel assigment. *Journal of Grpah Theory, *29(1), pp. 263-283.

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