Equipment selection heuristics for microwave fixed links

Authors


Abstract

Microwave fixed links use highly standardized radio equipment and, in the radio equipment standard referenced in this paper, there is a choice between exactly two radio systems when the assigner wishes to resolve a specific data rate exactly: one using a relatively lower-order modulation scheme and one a relatively higher-order scheme. Although the higher-order equipment requires less bandwidth for an isolated link, these systems radiate at higher powers and require larger protection ratios in the radio interference environment which lead to well-established trade-offs between modulation, bandwidth, equivalent isotropic radiated power, and frequency assignment criteria. Our earlier research showed that by extending the Frequency Assignment Problem to include equipment selection and using lower-order modulation equipment on selected links, we can actually reduce the overall span of frequencies required for a network frequency assignment. This work focused on the development of integer programming formulations and analyzed the exact solutions obtained. Exact solutions are impractical for real world problems; hence, here we focus on the development of heuristics for equipment selection. We can model the fixed link network as a complete graph where the vertices represent fixed link frequency assignment requests and the edges represent interference between pairs of vertices. Using a graph theoretic analysis of the interference problem, we propose heuristic techniques and discuss the relative success of our approach in this article.

1 Introduction

American engineers such as Metzger, Zoellner, Beall, and Hale recognized the significance of abstract mathematical graph-coloring problems in the 1970s and 1980s, pioneering an extended study of the Frequency Assignment Problem. Metzger's papers are difficult to obtain but Zoellner and Beall [1977], Hale [1980], and Hale [1981] give a good overview of this early work. Latter studies have been led by academia in the main, and an excellent body of work exists in the academic literature, see Aardal et al. [2007], Leese and Hurley [2002], and Hurley et al. [1997], for example, with some advanced algorithms now available and demonstrated on benchmark data sets. In general, professional practice and the wider engineering and scientific communities have not yet embraced these ideas fully and much of the expertise in this subdiscipline resides within academia, typically in the Computer Science or Mathematics fields.

A recent study by the authors [Flood and Allen, 2013] has investigated the nature of optimal frequency assignments and, in particular, the conflict between local and global spectral efficiency. This shows how the frequency assignment problem can be extended to include equipment selection and other practical planning considerations familiar to frequency assignment engineers.

The research first of all focused on the development of Integer Programming (IP) formulations with a minimum span objective function; that is, a set of frequency assignment and equipment selection problems were analyzed in order to find the smallest span of frequencies required to resolve a set of frequency assignment requests while satisfying the frequency separation constraints used to mitigate harmful interference. The problems were exposed to these formulations using a standard IP solver which is exhaustive and delivers exact, optimal solutions.

The study showed, contrary to intuition, that by using lower-order modulation equipment and so doubling the bandwidth requirement on selected links relative to a higher-order modulation equipment selection, we can, in some cases, actually reduce the overall span of frequencies required for a network frequency assignment. In other cases, it is possible to double the bandwidth requirement on a subset of links while maintaining the span obtained when higher-order modulation equipment is ubiquitous, thus improving the interference environment.

In this paper we investigate the development of heuristics. Using the exact solutions as a benchmark, these programs aim to optimize equipment selection and frequency assignment in a simulated off-line environment. The results obtained from using a range of equipment selection criteria are reported on here.

1.1 Inequalities in the Radio Interference Environment

Microwave fixed link operators require a high quality of service, and links are normally deployed in spectrum subject to detailed procedures such as the noise-limited frequency assignment methodology, for example, [Flood and Bacon, 2006]. The links are planned in order that a standard data rate is supported and, often, the planner has a choice of radio system types, each utilizing a particular modulation scheme. In Europe, for example, when seeking to resolve a standard data rate, the planner often has a choice between exactly two radio systems: one using a relatively lower order and one a relatively higher-order modulation scheme [ETSI, 2010].

There are well-established trade-offs between modulation, bandwidth, equivalent isotropic radiated power (EIRP), and protection ratios [Leuenberger, 1986], [Farrar and Hinkle, 1988]. While the higher-order modulation equipment requires less bandwidth to resolve a specific data rate on an isolated link, the higher S/N associated with these schemes means that the radios have higher receiver sensitivity levels (RSLs) and radiate at higher powers assuming that the higher RSL at the distant end of the link is the start point for calculation of EIRP. They require larger protection ratios in the radio interference environment when compared to equipment utilizing a relatively lower order of modulation.

These inequalities between radio systems can be given a more precise expression when we consider the radio system parameters used in the frequency assignment process. The assigner is primarily concerned with characterizing the radio as an interferer and as a victim in the radio interference environment. In both cases, we can actually use a receiver parameter to model the inequalities between radio system types.

The EIRP at a link end, expressed in dBW, is used to model interference at its source and a sophisticated frequency assignment method may include an EIRP assignment. Here RSL (dBW) is the receiver sensitivity level of the radio system at the distant end of the link and the start point for our calculation of EIRP at a link end. Working from RSL, we take account of all gains and losses on the wanted path arriving at an EIRP that, typically, is the minimum value required to support the link's availability requirement.

I (dBW) is the threshold for single-entry interference, and this parameter is used to characterize the victim receiver in pairwise interference scenarios. We have used a value for I associated with a noise limited frequency assignment criterion where the fully faded wanted signal is exposed to a model of the median interferer [Ofcom, 2012].

Using the syntax Mbit/s in MHz to describe the data rate and bandwidth of a radio system, Table 1 sets out values for RSL and I, illustrating the inequalities between the radio systems used in our study. Clearly, these inequalities are significant with ranges for RSL and I of 16.5 dB and 13.8 dB, respectively. Further, we can see that for a particular data rate, the higher-order modulation option uses half the bandwidth of the lower-order solution and, in general, has a higher RSL and a lower I.

Table 1. Radio Systems
Mbit/s in MHzRSL (dBW)I (dBW)
8 in 3.5−105.5−138.4
8 in 7−106.5−132.9
2 × 8 in 7−99.5−132.4
2 × 8 in 14−103.5−129.9
34 in 14−96.5−129.4
34 in 28−100.5−126.9
51 in 14−95.5−129.0
51 in 28−97.5−130.4
155 in 28−90−128.9
155 in 56−92.5−125.4

2 The Graph-Theoretic Model and Problem Sets

A precise mathematical description of the fixed links frequency assignment problem with equipment selection is set out in our earlier work [Flood and Allen, 2013]. We set out a very brief overview of our graph theoretic model and a description of the problem sets developed for the study here. For clarity, Table 2 sets out definitions for the mathematical notation used in this paper.

Table 2. Glossary of Terms and Parameters
Term/ParameterDefinitionUnit
S/NSignal to noise ratiodB
RSLReceiver sensitivity leveldBW
EIRPEquivalent isotropic radiated powerdBW
ISingle-entry interference thresholddBW
PSet of problems-
VSet of requests-
d(v)Data rate for request v-
GGraph-
EEdge set (representing interference between requests)-
riChannel raster-
fRadio frequencyGHz
bChannel bandwidthMHz
SSet of radio systems-
inline imageHigher-order modulation radio-
inline imageLower-order modulation radio-
GuAntenna gain at request udBi
λSignal wavelength.m
MFade margin for a fixed linkdB
inline imageInterfering signal powerdBW
D(θ)uAntenna discrimination at request udB
euvExcess interference at u sourced from vdB
N(u)Neighborhood of u, i.e., the requests constrained with u-
NFDNet filter discriminationdB
inline imageChannel separation constraint between u and v when u is assigned system s and v is assigned system tMHz
hHeuristic-
spSpan of a frequency assignmentMHz
GLFGeneralized largest first ordering technique-
RHORaster hierarchical ordering technique-
inline imageSmallest possible constraint between radio systems s and t due to raster organizationMHz
inline imageExcess constraint between u and v using radio systems s and t due to excess interferenceMHz
g(u)Equipment selection for request u-
gL(u)Lower-order modulation equipment selection for request u.-
gH(u)Higher-order modulation equipment selection for request u-
W(u)Weight of excess constraints incident to u in the higher-order modulation environment1.75 MHz bandwidth segments
W(u)Weight of excess constraints incident to u in the mixed modulation environment1.75 MHz bandwidth segments
ΔW(u)W(u) − W(u)1.75 MHz bandwidth segments
GLFEModified version of GLF using average excess weights-
TThreshold for equipment selection1.75 MHz bandwidth segments
CesSet of equipment selection criteria-
sp(hL,H)Span given by heuristic h in the mixed modulation environmentMHz
sp(kH)Span given by heuristic k in the higher-order modulation environmentMHz
Gij(hL,H,kH)Gain obtained from equipment selection for problem iP using criterion jMHz
IPHFrequency assignment method used to obtain exact solutions in the higher-order modulation environment-
IPFrequency assignment method used to obtain exact solutions in the mixed modulation environment-

A set P of 50 problems were developed in the earlier stages of the research. For each problem iP, fifty 38 GHz fixed links were modeled on flat Earth using randomly generated values for link geometry and path length. A data rate requirement was randomly assigned from the set of standard data rates given in Table 1.

Here a fixed link vV is modeled as a request for equipment selection and frequency assignment. A link has two link ends, A and B. A is first of all assigned a random position in a simulation space with dimensions: 1.25 × 1.25 km. Then, we assign, randomly, a wanted signal angle incident to A in the range 0 to 360° and a wanted path length in the range 0.3 to 0.5 km. The position of B is then calculated. Each request vV is randomly assigned a data rate (from Table 1), denoted by d(v), which we have assumed may be resolved by exactly two radio system types (in agreement with the ETSI standard). We assign an antenna characteristic at each link end and calculate the EIRP for both radio system types.

The priority at this point in the research was to develop scenarios that delivered a set of requests and constraints for us to work with. Here we have generated links (and so a set of requests) using representative path lengths, density, and radio system parameters. While the deployments are a little abstract, the data and future planning for real-world networks is commercially sensitive. The set of requests and the associated constraints used in this study are entirely realistic from the perspective of the frequency assignment engineer.

We can construct a complete graph G = (V,E) to model the radio interference environment where a set of vertices V={v1,v2,…,vN} represents N fixed link frequency assignment requests (each vertex represents a duplex fixed link) and each request uV has a data rate requirement d(u) defined. The weighted edge set E represents interference between these requests; here an edge uv (connecting requests u and v) can be labeled with a constraint that is based on an analysis of the interference incident to u and sourced from v and vice versa. The weighted degree of u is the sum of the constraints on incident edges.

In general, frequency assignments for fixed links are made using highly structured channel plans and our study uses a Frequency Division Duplex channel plan for these 38 GHz links, specified by CEPT for use in Europe [CEPT, 2010]. On this basis, we consider interference sourced from a link-end at v and incident to a link-end at u when the transmitter at v and receiver at u operate in the same duplex sub-band.

A raster is a set of radio frequency channels with a common bandwidth b. Modern channel plans will often specify more than one raster and, when this is the case, the assigner may be said to operate in a multiraster environment. Let inline image denote a raster of radio frequency channels with bandwidth bi. Let a channel plan be an arrangement of n channel rasters {r1,r2,...rn} in the spectrum space such that bi=2·b(i − 1) for inline image.

Without loss of generality, in our model, the data rate d(u) is associated with exactly two radio systems from a set S={s1,s2,⋯,sm} where system si supports a data rate D(si). For a request u, inline image and inline image are the higher-order and lower-order modulation radio systems where D(si) = d(u). In the first instance, each uV is assigned a system inline image and EIRP is calculated for a typical availability requirement of 99.99% at each link end using the following:

display math(1)

where RSL(s)distantend is the receiver sensitivity level associated with radio system s at the distant receiver, Gu is the antenna gain available at the distant end of the link, the log term describes free space path loss, r is path length, λ is the wavelength of the radio signal, and M is the fade margin required to combat signal attenuation due to rain and atmospheric losses.

The network is tuned cochannel or near cochannel when u and v operate on alternative rasters. In order to generate a set of pairwise frequency separation constraints for the edge set E, our study uses a simple model of the radio interference path, assuming free space path loss between victim and interferer. On this basis, we calculate the interfering signal power inline image incident to a link end at request u and sourced from a link end at request v using the following:

display math(2)

where EIRPv is radiated power at the interference source, D(θ)v and D(θ)u are the antenna discrimination available at the two link ends under consideration, and Gu is boresight antenna gain at the victim link end. Then, excess interference at a link end of u is calculated using the following:

display math(3)

where I(s) is the single-entry interference threshold associated with s. The worst case value for euv incident to u and sourced from v is selected. Interference between every pair of requests is calculated using this approach, obtaining the most potent single-entry interference incident to u and sourced from v, for all vV that are constrained with u. If v is constrained with u then v is said to be in the neighborhood of u and we can denote the set of requests constrained with u using N(u) (the neighborhood of u).

Net Filter Discrimination (NFD) is defined as the discrimination available (dB) when an interferer is offset in frequency from a victim receiver [Ofcom, 2012]. In order to map euv to a channel separation constraint, we first of all construct a look-up table of the NFD available between each pair of radio system types covering all of the possible frequency offsets between this pair using a methodology specified by ETSI 2005. We define a constraint inline image which gives the minimum frequency separation required between u and v when u is assigned equipment s and v is assigned equipment t. We have calculated euv in a tuned environment where u and v are cochannel or near cochannel and the look-up table allows us to map this excess interference to a value for inline image such that NFD ≥euv. Interference between u and v is likely to be asymmetric, and on this basis we label edge uv with the worst case value for inline image.

This initial model of the all-higher-order modulation environment can be modified by introducing lower-order modulation equipment selections and so a mixed modulation environment. Without any loss of generality, we have exactly two equipment options per data rate and we can formulate equations that will support alternative equipment selections and adjust the excess interference at all victim receivers accordingly.

If we consider the pair u and v, revised excess values can be calculated if u is switched from higher-order to lower-order modulation. We can first of all develop our notation so that, in the higher-order modulation environment, the excess value at u and sourced from v is denoted by inline image (here the arrow points away from the source of interference and toward the victim) and the excess at v, sourced from u, by inline image. If we select inline image at u, then using the values for I and RSL given in Table 1, revised values for excess interference may be calculated using the following:

display math(4)

and

display math(5)

and these equations can easily be reformulated to accommodate lower-order equipment selections at v or at both u and v. We can select the most potent values incident to each request and rerun the mapping inline image, again selecting the largest constraint when labeling edge uv. These procedures can be used to support the assignment of inline image for a specific request or can be applied to the entire network in which case we will have established an all lower-order modulation environment.

The frequency assignment methods used in our study are required to resolve all requests while satisfying all of the frequency separation constraints with zero violations. That is, frequency assignments for u and v denoted by f(u) and f(v) must satisfy the condition:

display math(6)

then the interference between the two links is acceptable in both directions and at all receivers.

3 Heuristics

The study has investigated the development of equipment selection heuristics through experimentation including the use of ordering techniques and the evaluation of interference across the entire simulated radio interference environment. On this basis, problems are exposed to an objective analysis where the complete graph G is analyzed by the equipment selection heuristic.

We report on some results for heuristics operating in a simulated off-line environment. The aim was to obtain results for span that were equal to or close to the exact solutions, obtained in the earlier stage of this research, for a significant number of problems. We evaluate these heuristics using a set covering analysis and the exact solutions as a benchmark to measure performance.

3.1 Ordering Techniques

The study uses three approaches to ordering the request queue:

Generalized Largest First (GLF). This classical technique orders requests according to assignment difficulty and uses the weight of channel separation constraints incident to a vertex to do this.

Raster Hierarchical Ordering (RHO). While GLF orders requests using the weight of constraints, the new RHO technique does this but first of all takes account of raster organization and so the impact that channel bandwidth can have on the minimum span problem.

Random Ordering. The request queue is exposed to random orderings. This provides a useful comparison to more analytical techniques.

GLF sums the weights on edges incident to a request and orders the queue accordingly, breaking ties using maximal weight and then the initial order, but there is no account taken of radio system bandwidth. This can be important when operating in a multi-raster environment and addressing the minimum span problem. The RHO technique, developed during this study, is a listing of V by raster order such that requests mapped to the raster with largest bandwidth are first in order with ties broken by the (now subordinate) GLF technique.

3.2 Frequency Assignment Heuristics

The study first of all investigated some frequency assignment heuristics operating in the All-higher-order modulation environment where all requests are mapped to inline image. This approach was taken in order to test the new RHO technique against GLF, to measure the performance of both RHO and GLF against random orderings, and to establish some useful benchmarks for the performance of more complex heuristics with equipment selection procedures. We describe these heuristics here:

hglf maps all requests to higher-order modulation equipment, applies the GLF ordering technique to V and exposes V to a sequential frequency assignment.

hrho maps all requests to higher-order modulation equipment, applies the RHO ordering technique to V and exposes V to a sequential frequency assignment.

hrand maps all requests to higher-order modulation equipment, applies a random order to V and runs a sequential frequency assignment.

A sequential greedy assignment procedure is used with the experimental heuristics. The objective is to minimize the span of the frequency assignment via the low end packing approach described by Frazier [1975]. On this basis, the sequential assignment works up-band from the smallest frequency on the appropriate channel raster, assigning the first frequency that satisfies all of the constraints associated with the request under consideration.

The problem set was exposed to the simple heuristics, and the results are set out here. The heuristic hrand was configured to deliver 100 random orderings of V and so 100 network frequency assignments; this configuration of the heuristic is denoted by inline image. Figure 1 shows, for each of the 50 problems (each with 50 requests) in our problem set, a plot of the spans delivered by inline image using a box plot representation and with line graphs showing the exact spans delivered by hglf and hrho. The spans given are for the entire request queue, and the data is ordered according to the median spans delivered by inline image.

Figure 1.

Box plot representation of the spans delivered by inline image, hglf, and hrho.

The vertical lines associated with each box plot indicate the range of spans delivered by inline image while the box indicates the range of spans between the 25th and 75th percentiles denoted here by sp(25%) and sp(75%). The horizontal lines that divide the boxes indicate the median value. In the one case where the box representation is collapsed, the median value is exactly equal to sp(25%) and sp(75%). In cases where the box is fully developed and the median value is not apparent, the median is exactly equal to sp(25%) or sp(75%).

These results show that random orderings can outperform highly organized ordering techniques. The minimum of 100 spans delivered by inline image is smaller than the spans given by hglf and hrho for 16% and 8% of problems, respectively. However, in general, hglf and hrho deliver spans that are less than or equal to the minimum given by inline image.

Clearly, the new RHO technique is able to outperform GLF with hrho delivering a smaller span than hglf for 30% of the problem set. Conversely, hglf improves on the solutions given by hrho for just 4% of problems.

We may conclude that good ordering techniques are required if we are to develop heuristics for use in an off-line environment that are able to approach the optimal solutions. However, in agreement with Hurley et al. [1997], the results also indicate that the best heuristic for a particular problem is not easy to assess without experimentation and we note that this heuristic is not always the best performing over all problems.

3.3 Equipment Selection

Using the heuristics operating in the all higher-order environment as the basis for further development, hglf,hrho, and hrand can be refined by adding a subordinate equipment selection heuristic allowing for a request to be mapped to either inline image or inline image. Here the heuristics operate in the mixed modulation environment.

The frequency separation constraint inline image gives the minimum separation required between u and v in order that the interference calculated in a tuned environment is mitigated when the constraint is satisfied. However, the constraint also captures the minimum frequency separation required when u and v operate on alternative rasters and there is no excess interference incident to the receivers. When evaluating these constraints, it is useful to separate the contribution given by excess interference and that attributable to the subdivided channel plan. We can determine the minimum constraint between radio systems using the following:

display math

Values for inline image are due to excess interference incident to u, v, or both u and v and an excess channel separation constraint can be calculated:

display math(7)

Summing these excess constraints over all incident edges, we can determine the weight of u. Developing the notation for equipment selection, let g:VS denote an equipment selection for the request queue and let g(u) denote an equipment selection for request u. Then, let gL(u) and gH(u) denote lower-order and higher-order modulation equipment selections for u, respectively. Then, the weight of u in the all higher-order modulation environment is given by

display math(8)

Here gH(u,v) denotes a higher-order equipment selection at u and v; that is, at request u and all requests v in the neighborhood of u. We can consider the impact on weight when lower-order selections are made. Let W(u) denote the sum of excess constraints under the assumption that u is assigned lower-order modulation equipment and that all previous equipment selections at vV are respected. Then the revised weight is given by

display math(9)

Here gL(u) denotes a lower-order selection at u and g(v) denotes an equipment selection at v in agreement with the decisions made by the equipment selection algorithms. ΔW(u) denotes the difference between these two weights:

display math(10)

The exact solutions obtained at the earlier stages of this research were analyzed using these equations [Flood and Allen, 2013]. The study determined that values for ΔW(u) were an important indicator of lower-order modulation equipment selection with the selection gL(u) being made when the weight of constraints is reduced relative to a selection gH(u). This important result has informed the design of our heuristics.

The pre-equipment selection ordering technique, GLFE, is a modification of GLF that calculates the order of V by the average excess weight of u across lower-order and higher-order modulation environments. Again, ties are broken using maximal weight (taking account of both lower-order and higher-order environments) and then the initial order.

The hes(T).hglf orders V using the GLFE technique and the set of excess constraints. The heuristic is configured to run the subordinate equipment selection heuristic with a criterion for selection T which denotes the threshold for ΔW(u). The GLF ordering technique is applied to the request queue delivered by hes(T) using the full set of constraints, and the GLF ordered request queue is exposed to a sequential frequency assignment procedure.

The hes(T).hrho heuristic orders V using the GLFE technique and the set of excess constraints. The heuristic is configured to run the subordinate equipment selection heuristic with a criterion for selection T. The RHO ordering technique is applied to the request queue delivered by hes(T) using the full set of constraints, and the RHO ordered request queue is exposed to a sequential frequency assignment procedure.

The hres(p).hrand selects lower-order modulation equipment randomly according to a probability p. The heuristic then applies a random ordering to the request queue and runs a sequential frequency assignment.

With hes(T).hglf, and hes(T).hrho the subordinate equipment selection heuristic works through the GLFE ordered request queue calculating the excess weight of u in the all higher-order environment, then the revised excess weight when lower-order equipment is selected at u or at uand v (in cases where request v has already been processed and lower-order equipment has been selected). The difference between these two weights is calculated. If ΔW(u) is less than or equal to the threshold T, the heuristic selects lower-order equipment at u. The subordinate equipment selection heuristic can be described using pseudo-code:

image

Since these algorithms are presented without any loss of generality, it should be noted that Algorithm 1 could be adapted to make a selection from more that two radio systems.

The heuristic hres(p).hrand is configured such that, on average, lower-order equipment selections are obtained for a proportion of the request queue given by p·|V|. For the initial run, p was set to a value of 0.5.

image

Once the equipment selection procedure has run, the request queue is ordered according to GLF or RHO procedures for hes(T).hglf, and hes(T).hrho and a random ordering is generated for hres(p).hrand. The request queue is then exposed to a sequential frequency assignment.

Again, with hrand, 100 orderings are applied to V ahead of frequency assignment and this configuration of the heuristic with equipment selection is denoted by inline image.

4 The Equipment Selection Criteria and Results

Here we discuss our results and the specification of the equipment selection criteria used with hes(T).hglf and hes(T).hrho.

If the excess weights are normalized to a count of 1.75 MHz bandwidth segments then the smallest possible constraint greater than zero is 2, corresponding to inline image = 3.5 MHz. Therefore, all ΔW values are divisible by 2 and the equipment selection criteria has been developed on this basis. Ten criteria are specified where the threshold T =− 2·k for k={1,2,...,10}. Therefore, each run of the heuristic delivers 10 results per problem and we denote these particular configurations using inline image and inline image.

The numbers of lower-order selections across the range of equipment selection criteria used in these runs is evaluated, and the gain associated with each criterion calculated. Figure 2 shows a CDF of the lower-order selections made for both inline image and inline image. The x axis is a count of requests where the equipment selection heuristic selects lower-order modulation equipment and the y axis is the probability that this count will occur, e.g., when T =− 2, an analysis of the selections made over 50 problems gives the one hundredth percentile p1.0 a value of 18, the ninetieth percentile p0.9 a value of 15 and so on.

Figure 2.

CDF of lower-order selections delivered by the equipment selection criteria.

From a start point of T =− 2 the criteria becomes progressively more conservative, and adjustments have a radical effect on the number of lower-order selections made. With T =− 2, then p0.5=11 but when T =− 20, then p0.5=0.

The number of problems where a specific criterion delivers a reduction in span can be counted and expressed as a gain in MHz. That is, the reduction in span when results are compared with those delivered by the complimentary heuristic operating in the all higher-order modulation environment.

Definition 1. Let inline image denote the set of equipment selection criteria available. Let sp(hL,H) and sp(kH) denote the spans obtained from heuristics operating in the mixed modulation and all higher-order modulation environments respectively. Then, for a pair of heuristics h, k addressing a problemiP and using a criterion jCes, let the gain obtained by the equipment selection criterion be calculated using

display math(11)

and, neglecting any reductions in span, let the mean gain obtained when using a criterion j in combination with a heuristic hL,H, evaluated over all of P be given by

display math(12)

Therefore, inline image denotes the mean gain over P for a criterion j where gain is considered in cases where sp(hL,H) < sp(kH) only.

The pairs hglf, inline image and hrho, inline image are defined and Figure 3 shows the mean gain delivered by the equipment selection criteria over the problem set.

Figure 3.

Mean gain delivered by inline image and inline image.

The graph shows that inline image delivers more gain than inline image over the complimentary heuristic when the equipment selection threshold is from {T1,T2,...,T6} but gives a lower gain when the more conservative criteria {T7,T8,...,T10} is applied.

Earlier, it was established that the RHO technique outperforms GLF and these results for gain suggest that there is less potential for the equipment selection heuristic to reduce span when it is paired with a superior ordering technique. Figure 3 shows that the gains only begin to correlate when the equipment selection criteria is conservative. Further, this suggests that an on-the-fly equipment selection heuristic has the potential to obtain radical reductions in span because in an online environment any optimization of the frequency assignment available from an ordering of the request queue must be neglected by the assigner.

A more precise measure of performance can be obtained using a set covering analysis [Cameron, 1977; Allen et al., 2008] where the spans obtained by each pair of heuristics over the problem set P are compared and between each heuristic and the frequency assignment methods delivering exact solutions.

Using the approach taken by Hurley et al. [1997] we make full use of the off-line environment and select the best span delivered by each heuristic. That is, in cases where the heuristic is configured to deliver more than one solution per problem, we take the smallest span from this set of solutions: for inline image and inline image, we take the smallest span from 10 solutions while for inline image and inline image, we take the smallest span from 100 solutions.

Two configurations of the frequency assignment method associated with the IP solver and exact solutions are included in the set covering analysis.

Definition 2. Let IPH and IP denote configurations of the frequency assignment method delivering exact optimal solutions constrained to the all-higher-order modulation and mixed modulation (where equipment selection is possible) environments, respectively. Let h and k denote a pair of frequency assignment methods from {hglf, hrho, inline image, inline image, inline image, IPH, and IP}. Let sp(h) denote the span delivered by h and let the set covering c(h, k) denote the percentage of problems where sp(h)≤sp(k). Let inline image denote the average of c(h, k) for h over all other frequency assignment methods.

Table 3 sets out the results of the set covering analysis including the average cover afforded by a frequency assignment method over all other methods inline image.

Table 3. Set Covering Analysis for the Frequency Assignment Methods
Methodhglfhrhoinline imageinline imageinline imageinline imageIPHIPinline image
hglf-70847262100362063.4
hrho96-928686100482876.6
inline image3832-323098201237.4
inline image907890-8498502874.0
inline image90929094-98543278.6
inline image441046-224.6
IPH1001001009494100-5491.7
IP100100100100100100100-100

The frequency assignment methods can be ranked by the average cover.

Table 4 ranks IP first. We have established that the IP frequency assignment method delivers a set of exact solutions and, on this basis, there is an expectation that IP will attract a 100% cover for all other frequency assignment methods. Our results confirm this.

Table 4. Ranking of the Frequency Assignment Methods
RankingMethodinline image
1IP100.0
2IPH91.7
3inline image78.6
4hrho76.6
5inline image74.0
6hglf63.4
7inline image37.4
8inline image4.6

IPH is ranked second. We expect IPH to deliver a 100% cover for all other frequency assignment methods operating in the all-higher-order modulation environment, and the results confirm that this is the case. However, IP, inline image and inline image, operating in the mixed modulation environment, are able to improve on the spans delivered by IPH for a subset of P. Therefore, the cover afforded to these frequency assignment methods by IPH is less than 100%. This is considered to be a very important result and strong evidence in support of an equipment selection paradox: Exact, optimal solutions have been obtained by IPH while constrained to the all-higher-order modulation environment, but these can be improved upon by a configuration of the method delivering exact solutions and heuristics that double the bandwidth requirement for selected requests in the mixed modulation environment.

We have described how the equipment selection heuristic works through a GLFE ordered request queue and, for some criterion, makes precisely the same selections whatever ordering technique is applied to V ahead of frequency assignment. It can be concluded that the ranking of the heuristics inline image and inline image is determined by the ordering technique applied to the request queue ahead of frequency assignment. Further, both of the heuristics using RHO dominate the pair using GLF. The study has established that RHO can outperform GLF, and this set covering analysis shows that a heuristic constrained to the all-higher-order modulation environment and using a superior ordering technique can outperform a heuristic using equipment selection combined with a less effective ordering technique.

The inline image and inline image heuristics are ranked last. These results suggest that a random ordering of V in the all-higher-order modulation environment or a random equipment selection, with the number of lower-order selections set arbitrarily and combined with a random ordering of V in the mixed modulation environment are not profitable. This underlines the importance of well-designed equipment selection criteria.

4.1 Benchmark Solutions

Some further analysis allows for a comparison to be made between the spans obtained by IP and all other frequency assignment methods. A scatter or line graph of these spans is difficult to interpret (or many graphs are required). Therefore, a simple approach is presented here where we report on the mean difference in the spans given by IP and all other methods.

Definition 3. Let Min(sp(h)i) denote the smallest span given by frequency assignment method h for problem iPand let sp(IP)i denote the exact solution for problem iP in the mixed modulation environment. Let the difference between these spans be calculated using the following:

display math(13)

Figure 4 gives the mean of Δsp(h,IP) over all of P and so an understanding of how close the frequency assignment methods are to the exact solutions on average. We can see that the data have been ordered and that this order follows the rankings given in Table 4 exactly.

Figure 4.

Mean difference in span.

5 Conclusion

We have shown that it is possible for a greedy heuristic to reduce the span of a frequency assignment by taking account of raster in the ordering technique applied to the request queue. The new RHO technique has outperformed GLF even when the heuristic using GLF runs a subordinate equipment selection heuristic and the heuristic using RHO is constrained to an all-higher-order modulation environment.

It may be possible to apply RHO or other ordering techniques to particular real-world problems including, perhaps, retuning campaigns where there could be opportunities for the assigner to reorder the request queue. However, these ordering techniques are used here for experimental purposes, primarily to investigate the scope for using equipment selection heuristics.

Equipment selection heuristics can improve on the gains offered by heuristics operating in the all-higher-order modulation environment. This confirms our earlier analysis of the exact solutions delivered by the IP solver. It has been demonstrated, again, that the use of lower-order modulation equipment on specific links, while doubling the bandwidth requirement for these requests, can actually reduce the span of the (network) frequency assignment.

An analysis of the gain offered by these equipment selection heuristics shows that there is less potential for the heuristic when combined with a good ordering technique. While RHO outperforms GLF, the equipment selection heuristic combined with GLF achieves greater gain over its complimentary heuristic than the equipment selection heuristic combined with RHO. The findings, including the use of random orderings, suggest that there is a great deal of scope for an online equipment selection heuristic; an environment where requests are handled sequentially in the order that they arrive and where there is usually little or no opportunity for the assigner to make use of ordering techniques.

The spans achieved by the heuristics are often suboptimal, and there was no expectation that these methods would cover all of the exact solutions. However, the results are very encouraging and, when using a range of equipment selection criteria in a simulated off-line environment, these heuristics are often able to obtain the spans given by the exact solutions.

Professional practice (frequency assignment) utilizes complex radio propagation models, where considerable efforts are made at optimization. However, despite extensive research, the frequency assignment methods generally remain simple and questions such as spectrally efficient equipment selection are neglected entirely. Our research aims to provoke some discussion on interdisciplinary working and closer collaboration between experts in radio propagation, frequency assignment, and related subjects.

Acknowledgments

The data generated for the simulations reported on in this article are available from the authors on request.

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