Optimising the Placement of Access Points for Smart City Services with Stochastic Demand

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Optimising the Placement of Access Points for Smart City

Services with Stochastic Demand

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Master’s Thesis Econometrics, Operations Research and Actuarial Studies Supervisor: prof. dr. R.H. Teunter

Co-assessor: dr. X. Zhu

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iii

Optimising the Placement of Access Points for Smart City Services

with Stochastic Demand

Maaike Verhoek s2421755

Abstract

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iv Contents

1 Introduction to Smart Cities

A relatively new development in the world is the use of big data. Data is collected wherever possible with the purpose of gaining insight into our environment and our lives. In a smart city, weather, traffic, air quality, and many other facets will be continuously monitored and the processed data is made available to its inhabitants to make their lives easier.

The process of monitoring the environment of cities has been done for a very long time. Organisations such as CBS and the OECD have been collecting and reporting data about socio-economic developments for many years now. Furthermore, data about the weather has been collected since halfway the 19th century by the Smithsonian Institute in Wash-ington, D.C. [2]. Over the course of a few years they accumulated a large network of volunteers across the United States and some parts of Latin America and the Caribbean. These volunteers provided daily observations of the local weather and sent their data by telegraph and mail. The raw data had to be processed manually and this created a large workload that was hard to keep up with, which was why the weather observation pro-gramme was eventually transferred to the government. Nowadays the collecting and the processing of such data is done by sensors and computers, which drastically reduces the manual labour involved. As a result it is possible to collect and process more diverse data, such as temperature, humidity, rainfall, speed and direction of the wind, and air pressure, but also more frequent data if it is collected continuously. This data is then either stored or transmitted over an airborne or cable network [13].

It would be simplest if all the monitoring is done from a single network such that all the observations are stored at the same location. Doing so will be most effective in a city, however building such a network in a city can be costly. It is cost efficient to use that network for more than just monitoring, for instance, it could also be used to provide citizens with internet access, or it can be used to accommodate air alarms or other such services. This type of technology is called Smart City Technology. A few cities have already implemented smart city technology. The ‘Array of Things’ is an example of smart city technology. According to their website, “The Array of Things (AoT) is an urban sensing project, a network of interactive modular sensor boxes ... to collect real-time data on the city’s environment, infrastructure, and activity for research and public use.” In the summer of 2016, 42 AoT nodes have been placed in Chicago with plans to increase this to 500 nodes by the end of 2018. These nodes, or fixtures, contain multiple sensors to measure temperature, air composition, decibel levels, count pedestrians and cyclists, and so on. The data can be used to look at the effect of rush hour on air quality, inform people with asthma about pollen counts in the area, and monitor traffic flows [16].

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2 1 Introduction to Smart Cities

in Portugal, and Songdo in South Korea. Because these cities are newly constructed, they have a pliable infrastructure which makes them better suited for experiments and this makes it easy to connect the IT environments from the start. Furthermore, because these cities view testing technologies as one of their main objectives, it is easier to adapt or alter policies or features than it is in a regular city.

Songdo was selected as a pilot by the government of Korea and is seen as a testing envi-ronment for new technologies with the goal of attracting technological companies to the area. However, most of the solutions that are tested there are pushed from the companies to the residents.

The IT company Living PlanIT relocated their company to PlanIT Valley where the em-ployees of the company and their families are the residents of the city. The intention for the city is to become a place where environmentally friendly IT solutions and urban services are developed, tested, fine-tuned, and showcased.

While there are upsides to building a city ‘from scratch’, there are also downsides. Much of the technology that is tested and used in these cities is developed elsewhere, taking away from their original goals. Secondly, the populations of these cities are very one-sided, leaving a socially homogeneous group of users for the products to be tested.

Furthermore, the sensors in the ‘boxed smart city solution’ such as the AoT are grouped together in a single unit. This unit is then placed at various locations in the city such that sufficient data can be collected. However, it might be that some sensors, e.g. temperature sensors, do not have to be placed as frequent as other sensors, such as ones that measure sound. Furthermore, it might be useful to include services such as Wi-Fi or communication with smart vehicles. Such services might have a non-uniformly distributed demand area and the demand might be of a stochastic nature. Creating units that contain every sensor results in a placement scheme over the city based on the requirements of the sensor with the smallest range. For clarity, solutions with a stochastic demand will be referred to as services, and solutions with deterministic demand will be referred to as sensors. While grouping the units in such a fashion is an easy solution, it is also costly and unnecessary. Optimally, each type of sensor or service should be placed in such a way that it is able to perform its task, whether that is gathering data or providing citizens with data. Many sensors or services have to have city-wide coverage to perform their task, while others have to be placed in accordance with the distribution of the population or usage.

While the placement of a sensor or service at a single location may not be very costly, a city-wide coverage or demand coverage can become quite expensive as cities tend to be large. An existing network that is already connected to a power source is the lighting system in cities. The smart city services and sensors can easily be placed on lampposts when the appropriate network is laid. Not only placement, but also combining several sensors and services at some locations can substantially reduce the costs of a smart city network.

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2 Literature 3

to this thesis. Very few research has been done on combining multiple facilities or services to decrease set-up costs. And nothing on that subject includes stochastic demand. Most research on stochastic demand has been done in the context of placing emergency services such as ambulances; however, the focus was always on a single emergency service. The combination of placing multiple services and stochastic demand is the topic of this thesis, which is especially interesting in the area of smart cities, because it can drastically decrease the costs associated with installing smart city services. And this reduction in costs will make the system more appealing to municipal councils.

In the next chapter, some relevant literature is reviewed. Chapter three describes the problem and its specifics. Chapter four introduces the models. Chapter five presents the results, and chapter six will discuss and conclude the thesis.

2 Literature

Optimal placement of facilities has been researched extensively in many variants. The general form of the facility location problem is that warehouses, plants or other facilities need to be placed on some out of a finite number of candidate locations. The locations are chosen such that transportation costs are minimised. A first distinction can be made in terms of uncapacitated facilities and capacitated facilities.

Well-defined Uncapacitated Facility Location problems (UFL) include the p-center problem where p facilities are placed with the goal to minimise the maximum distance between con-sumer and its closest facility (summarised in Elloumi et al. (2004)), the p-median problem where the goal is to minimise the sum of demand-weighted distances (see Reese (2006)), and the Simple Plant Location Problem (SPLP) where a variable number of plants have to be placed to minimise the sum of construction costs and weighted travel distances (see the summary by Toregas et al. (1971)).

Capacitated Facility Location Problems (CFL) problems usually have similar objectives as the UFL problems, but are subjected to some constraints on the number of customers each facility can service.

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4 2 Literature

2.1 Stochastic Behaviour in Facility Location Models

When determining where to place a certain service, part of the placement problem involves some form of stochastic behaviour. According to Owen and Daskin (1998) two primary approaches are typically pursued when dealing with stochastic behaviour in capacitated location problems. One is a scenario planning approach which considers only a relatively small set of possible outcomes, and the other is a probabilistic approach which considers the underlying probability density. Scenario planning can be used when future demand has a limited amount of possible realisations. Each outcome has a different optimal solution and often the objective is to minimise some form of regret or the expected costs depending on the type of problem. The randomness in probabilistic approach can be dealt with in three ways: the randomness is placed in a constraint, in the objective function, or in a pre-processing step involving queueing theory. The following paragraphs elaborate on these three approaches.

2.1.1 Chance Constraints

Charnes and Cooper (1959) have been the first ones to suggest chance constrained pro-gramming in a conceptual framework for the area of inventory control. Since then it has been used for many fields of operations research where uncertainty and linear programming meet. To follow the notation of Miller and Wagner (1965):

max cx (2.1)

Ax ≤ b (2.2)

x ≥ 0 (2.3)

Pr{aix ≤ ¯bi} ≥ βi ∀i (2.4)

The chance constraint 2.4 can be replaced by

aix ≤ di ∀i (2.5)

where di is the largest number that satisfies

Pr(¯bi≥ di) ≥ βi (2.6)

For most known probability distributions, dican be found analytically. For others a solution

has to be found by means of numerical analysis.

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2 Literature 5

so this problem can be solved analytically. Furthermore, since the normal distribution is symmetric around its mean, the solution will be equal to that of the expected value solution.

2.1.2 Recourse Models

In the paragraph above, randomness in parameters was dealt with by modifying constraints as a constraint of meeting demand cannot always be satisfied. In this paragraph, some penalty for not meeting demand is added to the objective function. All random variables are in the objective function, and all constraints should be linear. In general formulations, the two-stage stochastic programming model is formulated as follows

min f (x) + Eξ[Q(x, ξ)] (2.7)

s.t. Ax ≤ b (2.8)

x ≥ 0 (2.9)

where Q(x, ξ) is the optimal value of the second stage problem

min q(ξ)y (2.10)

s.t. T (ξ)x + R(ξ)y ≤ h(ξ) (2.11)

y ≥ 0 (2.12)

Where T (·), R(·), and h(·) any function.

It is assumed that ξ has a known distribution, or even a relatively small set of outcomes. For the latter case there exists a deterministic equivalent of the model which can be solved using duality.

Louveaux (1986) reconstructs the p-median problem and the Simple Plant Location Prob-lem (SPLP) into a two-stage stochastic program with recourse. For the stochastic SPLP and the stochastic p-median problem, the first stage can be regarded as finding the locations for the facilities, and the second stage is to allocate demand to those facilities. Randomness is introduced in terms of demand, production and transport costs which yields an objective function of the form

max x,z EξU −f (x, y) + max y q(ξ)y (2.13) with U some utility function. If a pricing system is introduced and U is a linear function, then it turns out that both models will obtain the same solution.

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6 2 Literature

2.1.3 Emergency Service Modelling

Most of the research in the area of FL problems with stochastic demand is about locating emergency services. The aim of the articles, for instance ReVelle and Hogan (1989b) or Daskin (1982), is to find locations for emergency vehicle dispatch sites which ensure that a vehicle can arrive at the emergency location within an allocated standard time after an in-coming emergency call. Both ReVelle and Hogan (1989b) and Daskin (1982) use queueing theory to determine how many locations need to be within range of each demand point. Chapman and White (1974) were the first to introduce a so-called busy probability for emergency vehicles. Daskin (1983) used their concept in his Maximal Expected Covering Location Model (MEXCLP) where M locations are to be placed such that expected cov-erage is maximised. A facility is busy with probability p, and available with probability 1 − p. All facilities are assumed to be independent of each other, which is not realis-tic. ReVelle and Hogan (1989b) have replaced that system-wide busy probability with a location-specific busy probability. The aim is to minimise the number of vehicles such that all demand is covered with a certain reliability. Then the reliability constraint no longer has an analytic solution, but it can be used to calculate the minimum number of vehicles that should be within reach of each demand point. The reliability constraint can again be replaced by a linear equivalent. In the same article, they suggest two models derived from the model described above. In both of these models, the number of vehicles to be placed is fixed. These models are the Maximum Reliability Location Problem (MRLP) and the α-reliable p-center problem. The MRLP has the objective of maximising the reliability, whereas the α-reliable p-center problem attempts to minimise the time within which all demand nodes can be served with a reliability of α.

The Queueing Maximal Availability Location Problem (QMALP) by Marianov and ReV-elle (1996) extends the Maximal Availability Location Problem (MALP) from ReVReV-elle and Hogan (1989a) to include queueing theory. Both models maximise the population that can be served within the time standard with α reliability by locating a predetermined amount of servers. In the MALP the demand is random and demand i is fulfilled with reliability α if there are at least bi locations within the time standard, where bi is calculated in the

same way as in the article by ReVelle and Hogan (1989b). In the QMALP, the travel distances/times as well as demand are taken as random and the neighbourhood around i is modelled as a M/G/s-loss queueing system1. Note that in this system servers are not assumed to be independent. The number of servers within reach per demand point can be found analytically.

None of the above approaches perfectly represent the real world, because it is difficult to fully comprehend and model the dependencies between the demand points and the access points. Furthermore, all the problems described above guarantee that every demand is

1_{demands arrive according to a Memoryless distribution and ‘depart’ according to a General distribution,}

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2 Literature 7

satisfied with α100 percent reliability, while in most (telecom-) networks system-wide de-mand should be guaranteed with 100α percent. The difference is that the former considers only temporally aggregated demand per demand point, while the latter considers spatially as well as temporally aggregated demand.

2.2 Mobile Networks

Mobile networks offer a plethora of data for many research areas, because a large part of the interactions is logged, either for billing, resource management or other purposes. In their literature survey, Naboulsi et al. (2016) classify three major research areas where this type of data is used. These areas are Social Analysis, Mobility Analysis and Network Analysis. This thesis focuses on Network Analysis, and more specifically on Mobile Demand. Mobile demand can be viewed from two perspectives: the perspective of the subscriber and that of the access point, or base station.

Zhang and ˚Arvidsson (2012) research the difference between wireless and wireline data traffic. Data traffic is defined by the length and size of the session. Within such a session, multiple packets of data are sent in flows. The authors find that “a few long flows in fixed networks often translate to many short flows in cellular networks”. They also noted that “packet arrivals are burstier in wireless networks than in wireline ones.” They attribute this to radio access.

Oliveira et al. (2014) use clustering methods to divide the user base of a major Mexico City 3G network into four user profiles: light-occasional, heavy-occasional, light-frequent, and heavy-frequent. The light-occasional users generate traffic up to 20GB in a maximum of 278 sessions per day. This group contains almost 98% of the subscribers. A statistical distribution per user profile is fitted on the usage patterns of number of sessions per day, traffic volume per session, and interarrival times between sessions. Because of the large differences in usage for certain times, the authors choose to fit a distribution for peak and non-peak hours. Of the tested distributions (Log-Normal, Gamma, Weibull, Logis and Exponential), Gamma and Weibull were the distributions with the best fit depending on the profile and time.

Lee et al. (2014) fitted four distributions on spatial traffic density data from China. This data is in bytes per square kilometre They found that the gamma distribution provides the worst fit, and a log-normal mixture distribution performs best on the empirical data. The other two distributions, the Weibull and the log-normal distribution, are a good approx-imation to the spatial distribution of traffic density in cellular networks. The authors do note that the distribution changes with time and space.

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8 2 Literature

should generate synthetic smart phone traffic. The first model uses traffic that has been averaged over all users. The second model follows the patterns of the user whose character-istics follow the averages most closely. While the observed users are a heterogeneous group, 18 users is a rather small sample and the average user in this sample does not necessarily corresponds to the average user in the entire population. The authors conclude that the user patterns in off-work hours are of a longer duration, while the usage patterns during work hours are of a more bursty nature.

2.3 Multi Service

The PLANWAR model from Pirkul and Jayaraman (1998) solves a product, multi-stage location-allocation problem to place plants and warehouses. The plants produce multiple products which need to be shipped from the plant to the warehouse to the con-sumer. A subset of possible locations for the plants and warehouses are opened from which the products are shipped to the warehouses and the consumers, respectively. Both plant and warehouse are capacitated in terms of amount of products it can manufacture and hold, and the demand of the consumer is fixed. A paper relating to the research by Pirkul and Jayaraman (1998) is the master’s thesis of Vos (2016). In his research, the plants are lampposts that are connected to a network via a tangible connection (hubs), the ware-houses are lampposts that are connected to the hubs via a wireless connection, and the products are smart city sensors. This is where the similarities stop, however, because the Multi-Service Location Set Covering Problem (MSLSCP) of Vos is an uncapacitated prob-lem. Moreover, the hubs primarily act as service provider and the providing of a wireless connection to other lampposts is secondary. Another difference is that the MSLSCP is a location set covering model, whereas the PLANWAR model shares common features with the Capacitated Plant Location Problem (CPLP).

Another paper that incorporates multiple services is by Moore and ReVelle (1982) where a distinction is made between different levels of service. Each level of service incorporates all lower levels. These services are placed in facilities, which are also divided into several levels. The level of the facility determines what services can occupy them (only services of equal or lower level) and how far people will travel to get to the facility. The model places a given number of facilities that minimises the population which is not within reach of a facility.

2.4 Contribution

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3 Problem Outline 9

but is interesting to do. In the context of smart cities especially, since this is a fairly new concept. Determining an optimised placement scheme of smart city services and sensors will drastically reduce the costs associated with installing smart city services, which will make the system more appealing to city municipalities and coorporations or instances who are funding the smart cities.

Often, the reality is drastically simplified to apply some model, method or theory to it. While making simplifying assumptions is a necessary measure, simplifying too much yields results that are not applicable to reality. Frequently, demands are taken as deterministic, but using a stochastic distribution to describe the demand can make the model more realistic and does not complicate the problem too much.

This thesis extends the research of Vos (2016) to include services with stochastic demand. In most research, small datasets are used, whereas in this thesis, the proposed models are applied to actual cities. Results derived from the models provide insight into the behaviour of the model and solution method, as well as into the set-up of the cities.

3 Problem Outline

As mentioned in section 2.3, the problem of where to place the smart city sensors has been addressed in the thesis of Vos (2016). He emphasizes efficiency in his method which entails placing multiple sensors in a way that minimises costs and meets the coverage requirements. An example of such a requirement is that every spot in a city is within a certain distance of a sensor. The emphasis of this thesis will be on placing services that have to fulfil some demand which can be described by an underlying probability distribution. Combining the placement of sensors without capacity and services with capacity to decrease overall costs is another area of focus. Placing multiple sensors and services at the same location will be preferred. This will decrease the overall costs, since those grouped sensors and services will share the connection costs.

In the first section of this chapter some problem specifics are discussed and terms are introduced. In the second section the goal of this thesis is explained. The third section describes a solution method.

3.1 Problem Specifics

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10 3 Problem Outline

service on a single access point are located within a certain radius of that access point. The size of that radius is dependent on the type of service. This radius will be refered to as the range of the service. Sensors also have a range. While a sensor can only measure or record at its location, the area within its range should give similar results. For example, due to the diffuseness of air the air quality sensor will not give very different results if it is moved 20 meters, but for sufficiently large distances it will. The range is chosen such that all points within range of the sensor do not give significantly different results from those at the location itself.

The access points that are equipped with sensors and services should be connected to a network. The sensors collect data which has to be communicated through this network. Services, such as a Wi-Fi connection to citizens, need this network to provide their service. The sensors would only need a small part of the available bandwidth, but the services consume a significantly larger part of the available bandwidth. The network does not have unlimited capacity as this would be quite expensive. Furthermore, the demand for the services is not known beforehand and likely exhibits peaks during the day. Instead of guaranteeing that the demand during those peaks is always satisfied, we strive to fulfil demand at least α100% of the time. α is a fraction which is yet to be determined.

A small note on how to fulfil demand: demand can be defined in two dimensions; space and time. Demand nodes can be considered separate or aggregate, and demand can be considered per point in time, or aggregated over the entire day or week. To be able to consider the aggregate demand, dependencies between demand points have to be known. Since dependencies between the demand points are not known in this situation and difficult to model, independence is assumed. Therefore, the choice is made to regard the demand nodes separately and guarantee that the demand at each node is fulfilled with α100% certainty at each point in time.

As services occupy a considerable part of the available bandwidth provided by the access point, a capacity on the bandwidth should be imposed. Furthermore, when a service unit is connected to too many devices, the performance of the network deteriorates due to radio interference2. Therefore, the model will impose a limit on the number of wireless connections a service can hold. Note that these limits do not apply to the sensors which do not occupy a considerable part of the bandwidth, nor make use of a wireless connection. The capacity parameters are assumed to be equal for every access point. It may be possible to include the option of having multiple types of network connection so that the capacity parameter is different for some access points. However, this will not be explored in this thesis.

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4 Model Formulations 11

3.2 Goal

The overall goal of this thesis is to create a model that assigns services and sensors to access points in a city in such a way that the services can fulfil demand with α100% reliability, that the sensors cover the city, and that services and sensors are grouped in a cost efficient manner. Since this is a rather large problem to solve at once, the modelling approach will start simple and extensions are consecutively included until we can solve the entire problem. The first step is to solve a model that can place a single service around the city such that it satisfies enough demand. The second step is to extend the model to accommodate multiple services, thus utilising economic dependencies of placing the services. As a final step sensors are added to the model. These models are used to gain insight into the placement of sensors and services in cities to make them ‘Smart’.

3.3 Solution Method

All the models that are proposed in the next chapter are Mixed Integer Linear Programs and can get large very quickly. Exact algorithms that attempt to find the optimal solution will run for a long time (hours at best). Therefore a heuristic solution approach might be more suitable than an exact solving method. The models will be solved by means of an algorithm that uses optimality cuts and the Branch-and-Bound method, which is readily implemented in software.

4 Model Formulations

We start this chapter by introducing some notation. Then the model and its extensions are consecutively introduced. As mentioned in the previous chapter, the goal is to optimally place smart city services in a city where demand points and possible access points are given. Any model that is discussed will adhere to the notation in Table 1.

The problem is simplified to only one service, then extended to consider more services. Finally, sensors that are not limited by some capacity are introduced to solve the entire problem.

4.1 Optimising for a Single Service

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12 4 Model Formulations

Table 1: Notation of parameters to be used in models. F : the set of all services

L: the set of all locations

Gu_: _{the set of demand points for service u ∈ F}

Cu

i: the set of locations within range of demand point i ∈ Gu for service u ∈ F .

co: costs of opening an access point

cu_f: costs of equipping an access point with service u ∈ F M : a large number

ηj: maximum number of connections for access point j ∈ L

au_ij: =

1 if demand point i ∈ Gu for service u ∈ F can be serviced by access point j ∈ L 0 if not

γj: capacity of data transmission for access point j ∈ L

α: reliability level

ω_iu: stochastic demand at demand point i for service u at time t

demand can be done per user, or aggregated over all users, and at every single point in time, or aggregated over a single day, and that here reliability is defined per user at a every point in time. Three types of variables are used, yj, xij, and sij. The model variable yj

denotes that a service is placed at location j ∈ L when it is equal to 1. xij denotes the

connection speed between access point j ∈ L and demand point i ∈ G. Lastly, sij denotes

an active connection between access point j ∈ L and demand point i ∈ G, thus sij = 1 if

xij > 0 and zero otherwise.

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The goal is to minimise the costs of building the network (4.1), we assume that there is no difference in opening costs between the different locations. In reality, this does not have to be the case, but we will leave it this way for simplicity. The opening cost parameter co and the equipping cost parameter cf are needed separately in the extended modelling

formulations, therefore they are not combined into a general cost parameter in this model. (4.2) constraints the number of connections for each access point to ηj and constraints

(4.3) limit the amount of data the access point can transmit. Do note that in this model, for every location either constraint (4.2) or (4.3) will be binding, the other is redundant. Constraints (4.4) assures that demand is satisfied at least α100 percent of the time. These chance constraints can be replaced by

X

j

aijxij ≥ bi (4.8)

where bi is the smallest number which satisfies Pr{ωi ≤ bi} ≥ α. This is done according

to what Miller and Wagner (1965) described. Lastly, (4.6) and (4.7) constrict the values these variables can attain.

This model guarantees that every demand point always has a certain amount connection speed at its disposal.

The chance constraints 4.4 are replaced by equivalent linear constraints to make the model linear, which simplifies solving it. For any problem, the model has (|L| + |G| +P

i|Ci|)

constraints and (2 ·P

i|Ci| + |L|) variables. SincePi|Ci| ≤ |L| × |G|, increasing the number

of locations by one will increase the number of constraints by at most (1 + |G|) and the number of variables by at most (1 + 2|G|). For non-trivial problems, the mixed integer linear problem (MILP) matrix will be large.

4.2 Optimising for Multiple Services

We broaden the model to jointly place multiple services. A subscript letter u is added to the variables to denote the type of service, thus yj becomes yuj and denotes that service

u ∈ F is placed at location j ∈ L when it is equal to 1. The variable zj is included to denote

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14 4 Model Formulations Minimise X j,u cu_fy_ju+X j cozj (4.9) subject to: X i,u su_ij ≤ ηjzj ∀j (4.10) X i xu_ij ≤ γ_jy_ju ∀j, u (4.11) Pr    X j aijxuij ≥ ωui    ≥ α ∀i, u (4.12) xu_ij ≤ M su_ij ∀i, j, u (4.13) X u y_ju≤ |F |z_j ∀j (4.14) xu_ij ≥ 0 ∀i, j (4.15) su_ij, yu_j, zj ∈ {0, 1} ∀i, j (4.16)

All constraints (4.9-4.16), except for constraint (4.14), are adapted from constraints (4.1-4.7) to accommodate the multi-service nature. The cost function (4.9) includes costs of connecting access points, as well as the costs of placing a service unit at an access point. Each access point is restricted in number of connections in order to avoid interference between too many wireless connections (4.10). Every service unit is restricted in bandwidth (4.11). Similarly as in the single service model, the chance constraint (4.12) can be replaced by

X

j

aijxuij ≥ bui (4.17)

where bu_i is the smallest number which satisfies Pr{ω_iu ≤ bu

i} ≥ α. Constraint 4.13 forces

su_ij = 1 if xu_ij > 0, constraint (4.14) forces zj = 1 if any service is placed at access point j,

and constraints (4.15) and (4.16) restrict the outcomes of the variables.

This model has ((2+|F |)·|L|+P

u|Gu|+

P

i,u|Ciu|) constraints and (2·

P

i,u|Ciu|+(1+|F |)·

|L|) variables. Compared to the single service model, if one service of equal size is added, the number of constraints will be increased by (3 · |L| + |G| +P

i|Ci|), and the number of

parameters will be increased by (2 ·P

i|Ci| + 2 · |L|). This will increase the MILP matrix

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4.3 Including Sensors in the Model

As a last step, sensors are included to be placed together with the services. Sensors should be placed such that they provide city-wide coverage, therefore the demand points will make up an appropriately spaced grid over the city. To differentiate, services will be denoted by superscript u and sensors will be denoted by superscript q.

The model is as follows

Minimise X j,u cu_fy_ju+X j,q cq_fyq_j +X j cozj (4.18) subject to: X i,u su_ij ≤ ηjzj ∀j (4.19) X i xu_ij ≤ γ_jy_ju ∀j, u (4.20) X j aijxuij ≥ bui ∀i, u (4.21) X j aijsqij ≥ 1 ∀i, q (4.22) xu_ij ≤ M su ij ∀i, j, u (4.23) X i sq_ij ≤ M y_jq ∀j, q (4.24) X u y_ju+X q y_jq≤ |F |z_j ∀j (4.25) xu_ij ≥ 0 ∀i, j (4.26) su_ij, yu_j, yq_j, zj ∈ {0, 1} ∀i, j, u, q (4.27)

As mentioned in section 3.1, the sensor unit will be placed on the access point and will not provide a wireless connection to some demand point, furthermore, the sensor will send data continuously when possible and can save some data if the entire bandwidth is occupied by the services. As a result, sensors are not included in constraints (4.19) and (4.20). Constraint (4.21) is the same as constraints (4.4) and (4.12) in the previous models, where bu_i is the smallest number which satisfies Pr{ω_iu≤ bu

i} ≥ α. Constraint (4.22) ensures

that the city is covered. Constraint (4.23) forces su_ij = 1 if xu_ij > 0, constraint (4.24) forces y_jq = 1 if a sensor is placed at access point j, and constraint (4.25) forces zj = 1 if any

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16 5 Results

This model has ((2+|F |)·|L|+P

u,q|Gu|+

P

i,u|Ciu|) constraints and (2·

P i,u|Ciu|+ P i,q|C q i|+

(1 + |F |) · |L|) variables. Compared to the multi service model, there are |Gq| + |L| more constraints andP

i,q|C q

i| + |L| more variables when one sensor is added. Dependent on the

range of the sensor, this is not a large increase relative to the increase of adding a service.

5 Results

First, the experimental design is presented and parameter values are explained, after which the software and algorithms are discussed. Then two small test instances are introduced to validate the models. Last, the models are run on data from actual cities.

5.1 Experimental Design

The services that we consider are Wi-Fi connection in the first model, Wi-Fi and Smart Vehicle Communication (SVC) in the second model, and the third model includes an alarm as sensor.

The variables that will be compared are total costs, the fraction opening costs to total costs, the fraction equipping costs to total costs, and number of opened access points, which will be compared to the minimum number of access points that should be opened. The number of connections ηj, the capacity per access point γj, the equipping costs cu_f,

the range, and the demand distribution are set fixed. The values of these parameters are designed to be as realistic as possible. The equipping costs cu_f and range are taken from the thesis by Vos (2016). The distribution for the demand for Wi-Fi was based on the study by Oliveira et al. (2014). The fitted distribution for the session size of the light-occasional cluster in peak hours is the Gamma distribution with shape parameter αsh3= 0.10 and

scale parameter β = 0.0000002. The session volume was recorded in kilobytes (kB). To convert this to a speed measure in megabits (Mb), the scale parameter has to be multiplied by a measure of session length. The only measure of session length that had been reported was the median, which was 63 seconds. The converted scale parameter is 0.0016128 and the mean connection speed is 62 megabit per second (Mbps).

The distribution for smart vehicle communication demand is not based on literature, but chosen with regard to the distribution of Wi-Fi. A right tailed distribution is most accurate in demand, as there can be outliers to the right but no values smaller than zero. Taking a mean speed around half of the mean speed of Wi-Fi, the Weibull distribution with shape parameter k = 1 and scale parameter λ = 30 is chosen.

The capacity on the number of connections and on the connection speed is chosen high

3_{To distinguish between reliability level and distribution parameter, the subscript sh is added to denote}

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5 Results 17

enough such that the algorithm can always find a feasible solution, but low enough to be capacitative. The values of all the fixed parameters are in Table 2. The location of the demand points is determined by addresses. This is realistic for the Wi-Fi service, but not for the service SVC and the sensor alarm. The demand points for both are grid points determined by the range of the service or sensor. For a large area, however, the number of grid points are disproportional to the number of addresses resulting in not being able to fulfil demand. Hence, for the SVC service, some of the grid points are deleted leaving a number of grid points that is somewhere between 0.6 and 1.2 times the amount of addresses. Such a measure is not necessary for the alarm function, because there are no capacity constraints for sensors.

Table 2: Fixed parameters in the Experiments

Parameter General Wi-Fi SVC Alarm

ηj 20

γj (Mbps) 3000

cu_f (e) 350 200 150

Range (m) 100 50 300

Distr. Gam(0.1, 0.00161) Wbl(30,1)

-The parameters that will be varied are the reliability level α and the opening costs co.

Both reliability level and opening costs will be set to a high value and a low value to infer how they influence the final solution. See Table 3 for the experimental parameter settings. Experiments 1 and 2 are performed on the Single Service model for each city, because there is no need to vary the opening costs in this model. All four experiments are performed on the Multi Service model, and only experiments 2 and 4 are performed on the model that includes sensors, because varying the reliability level has no added value there.

The results that are collected from the Single service model are the number of opened locations, and the total costs. The results collected from the other two models also include the number of service units placed, the costs for opening the locations and the costs for placing the units. For all models, the relative gap between the lower and upper bound is also presented. The relative gap is calculated as follows

Gap = U − L

1 + U (5.1)

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18 5 Results

Table 3: Experimental setup

Experiment α co 1 0.70 1000 2 0.95 1000 3 0.70 25 4 0.95 25 5.2 Software

Matlab is used to solve all the models. Existing tools within matlab which makes use of several heuristics is applied to find solutions. First, the algorithm tries to reduce the prob-lem size in a preprocessing step. In this step, some variables or constraints are identified as redundant and can be removed to make subsequent steps run faster and more efficient. Also, sparsity of the MILP matrix can be improved upon, infeasibility of the primal or dual model can be detected, and bounds of variables can be relaxed. For in depth explanation of these problem reduction techniques see M´esz´aros and Suhl (2003) or Andersen and An-dersen (1995).

Second, the algorithm solves a relaxed problem to get a lower bound on the objective value. This problem is relaxed in the sense that the integer constraints are removed, but the other constraints remain.

Third, the algorithm further reduces the problem with an integer preprocessing step. In addition to finding infeasibility, improving bounds, and deleting redundant variables and constraints, the algorithm performs a probing technique in which it fixes a certain variable and explores the consequences. The goal of this probing is to simplify branch-and-bound calculations that will ensue in a coming step.

Fourth, some cuts are generated to constrict the feasible region so that solutions will be closer to integers. The different types of cuts are described in the paper by Cornu´ejols (2008).

Fifth, an upper bound must be found. Several heuristics are employed to find a feasible solution. Depending on the choice of the user, the algorithm uses a combination of a local search heuristic, a heuristic that rounds the LP relaxation, and a diving algorithm. The latter is somewhat similar to a branch-and-bound method, but only creates one branch at each node. When a feasible solution is found, the algorithm moves on to the sixth and last step.

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5 Results 19

5.3 Model Validation

To validate the models, they are tested on two small test instances. The test instances respectively contain 24 and 9 lampposts, and 67 and 41 addresses. The 41 addresses in the second test instance are located in two flats, which will give insight into if and how the model handles situations where many demand points are reachable by only a few access points. A visual representation of the two test instances can be seen in Figures 1 and 2, respectively. Do note that these points are located in close proximity to one another, which will have influence on the results.

The Single Service model will only place one type of service. A single service can reach almost all demand points because the test areas are relatively small compared to the range of 100 m. Therefore, in this situation it is more a case of placing enough service units such that capacity is sufficient. The minimum number of required service units is

max bi· |G| γj ,|G| ηj (5.2)

With 67 demand points there have to be at least 67₂₀ = 3.35, thus 4, opened facilities. With a reliability level of 0.95 there have to be at least 359.89·67₃₀₀₀ = 8.03, thus 9, opened facilities. With a reliability level of 0.70 there have to be 10.81·67₃₀₀₀ = 0.24, thus 4 opened facilities. Similarly for the second test area, there have to be at least 3 opened facilities to fulfil the connection constraint and at least 5 opened facilities to fulfil the speed capacity constraint at 95% reliability. These numbers correspond to the results found in Table 4.

Table 4: Results Single Service Model for test areas

Test Area 1 Test Area 2

size 1 246 × 2 286 357 × 602

α 0.70 0.95 0.70 0.95

# Opened Loc. 4 9 3 5

Total Costs (e) 1 400 3 150 1 050 1 750

Rel. Gap 93.90% 10.69% 94.97% 1.63%

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20 5 Results

Figure 1: Mapped Locations and Demand Points of Test Area 1

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5 Results 21

as determined by capacity constraint (4.3), which is why the gap does not converge to zero. The second and third model include an additional service with stochastic demand. In these models, the placement costs start to play a factor. Equation (4.10) dictates that the services have to share the available connections. The minimum number of required opened locations is Omin= max b1 i · |G1| γj ,b 2 i · |G2| γj , P u|Gu| ηj (5.3) The number of demand points for the SVC service in the test areas are 13 and 9 respectively. Based on equation 5.3, the SVC will not be the limiting factor. Furthermore, according to that equation, the number of locations should be the same as in the Single Service model for each realisation of α. For this model which is slightly larger than the Single Service model, the cut-off time is set at two hours.

Table 5: Results Multi Service Model for test areas Test Area 1

size 1 381×2 482

α 0.70 0.95 0.70 0.95

co 1000 1000 25 25

No. Opened Loc. 4 9 7 9

No. Wi-Fi Units 4 9 4 9

No. SVC Units 4 4 4 4 Costs Loc. 4 000 9 000 175 225 Costs Units 2 200 3 950 2 200 3 950 Total Costs 6 200 12 950 2 375 4 175 Rel. Gap 20.78% 10.03% 58.46% 9.82% Test Area 2 size 413×683 α 0.70 0.95 0.70 0.95 co 1000 1000 25 25

No. SVC Units 3 3 3 3

Costs Loc. 3 000 5 000 75 125

Costs Units 1 650 2 350 1 650 2 350

Total Costs 4 650 7 350 1 725 2 475

Rel. Gap 21.46% 0%* 47.82% 0%*

*: Solution found in 3182s, **: Solution found in 487s minutes.

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22 5 Results

much higher. When this result is compared to the result of experiment 1 for Test Area 1 the conclusion may be drawn that some improvement might still be possible, this conclusion can also be drawn from the number of units per service. The optimal solution may be found when the cut-off time for the experiment is increased. For the other experiments there is no difference in solution which vary only in opening costs. Of course, the solutions in Figures 3-10 in Appendix A do give different solutions, but those are all optimal solutions because the spatial element plays no role in the test areas. Furthermore, when the reliability level is low (α = 0.70), every location is equipped with both services. Therefore optimally sharing the opening costs. The relative gap of experiments 2 and 4 for Test Area 2 is zero because Omin= 4.918 is very close to the integer solution of 5.

The third model includes an alarm which needs to cover the area. To do so, grid points based on the range of the sensor are placed on the areas. All grid points need to be within reach of at least one sensor. Test Area 1 and 2 both contain 16 grid points for the sensor alarm. It may be the case that small parts of the areas are not covered because of this design. If the grid points are placed closer to each other, the program becomes larger and requires more computational effort to solve. The experiments now vary only in opening costs co because the reliability level α has no effect on the placement of sensors. α is fixed

at 95%.

Considering that there were no differences between experiments with equal reliability levels in the Multi Service model for the test areas, similar results are expected for the model that includes sensors.

Table 6: Results for the Multi Service Model Including Sensors for test areas

Test Area 1 Test Area 2

size 1 421 × 2 889 438 × 836

co= 25 co= 1000 co = 25 co = 1000

No. SVC Units 4 4 3 3

No. Alarm Units 1 1 1 1

Costs Loc. 225 9 ˙000 125 5 000

Costs Units 4 100 4 100 2 500 2 500

Total Costs 4 325 13 100 2 625 7 500

Rel. Gap 9.99% 4.10% 0%* 1.47%

*: Solution found in 189s.

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5 Results 23

Appendix A for the graphical representation of these solutions. The difference in relative gap can be explained by the height of the opening costs. The absolute gaps are likely to be equal.

When the spatial element plays a role, different results are expected. Equations 5.2 and 5.3 will still hold, but they are no longer binding for the optimal solution. In the next section, the implications of adding a spatial element are examined.

5.4 Problem Specifics

The models are applied to three cities in the Netherlands with varying sizes Addresses in the city will serve as demand points with stochastic demand, and the locations of lamp posts will be used as possible access locations. The number of such points per city are given in Table 7.

Table 7: Number of possible access points and demand points per city. No. access points No. demand points per function

Cities Wi-Fi SVC Alarm

Schiermonnikoog 233 809 147 253

Rozendaal 523 627 188 141

Noordwijk 1 162 9 061 1 626 532

See Figures 15 and 16 in Appendix B for the access points and demand points in Schier-monnikoog, respectively. Similar plots are made for Rozendaal (see Figures 17 and 18 in Appendix B), and Noordwijk (see Figures 19 and 20in Appendix B). Since these problems are larger in size and computational difficulty than the validation problems, the program is cut off after four hours for the single service model and eight hours for the other two models.

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24 5 Results

Table 8: Results for the Single Service Model

Schiermonnikoog Rozendaal Noordwijk

size 8 050 × 13 793 13 750 × 24 677 105 662 × 189 716

α = 0.70 α = 0.95 α = 0.70 α = 0.95 α = 0.70 α = 0.95

No. Opened Loc. 73 107 74 87 No Feasible

Total Costs 25 550 37 450 25 900 30 450 Solution

Rel. Gap 54.15% 4.29% 12.53% 93.03%

As expected, the program found that the Noordwijk problem was infeasible for both of the reliability levels. This implies that the problem is also infeasible in the next two models. The relative gap at a reliability level of 95% is for both other cities smaller than that for a lower reliability level, similar to the test areas. Furthermore, the difference in number of opened locations for different reliability levels is larger in Schiermonnikoog. This could be due to the dense dispersion of access points and demand points in one region and a few access and demand points widely dispersed in another region. The points in Rozendaal are more evenly dispersed over the entire area. The graphical representation of the solutions is not clear because of the large amount of points on a sheet of paper, therefore it is not included.

In the previous section there was no difference in optimal solution in terms of costs for problems with different reliability levels. Therefore we expect no such difference in this section either. The results of the Multi Service model are in Table 9.

For both cities, there is a difference in optimal solution for experiments with the same reliability level. For Rozendaal, no feasible solution was found within the 8 hours cut-off time for experiment 4. Hence, there is no comparison to be made in the case of α = 0.95. When the opening costs are relatively low in Schiermonnikoog (experiments 3 and 4), more locations are opened and less units are placed compared to the previous model. This implies that the height of the opening costs does have an influence on the solution in the sense that the model favours opening more locations to decrease the number of placed units when opening costs are relatively low. However, for Rozendaal the opposite is the case. The reason for the differences between these cities may be their respective shapes and sizes. Schiermonnikoog has a stretched out shape, which would limit the number of feasible solutions, whereas Rozendaal has a more oval shape and contains more access points. These differences combined would result in the algorithm needing more time for Rozendaal to find a better solution. For the same reason as the previous model, the graphical representation of the results are not included.

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7 Discussion 25

also have to be placed in the less densely populated areas. Hence, an increase in opened locations relative to the second model is expected. The sensor that is added in this third model is subject to deterministic demand, therefore changing the reliability level provides no additional value. The reliability level is fixed at 95% and the experiments can only be compared to experiments 2 and 4 of the Multi Service model. The results are in Table 10. The algorithm was not able to find a feasible solution within 8 hours for Rozendaal when opening costs were set to 1000. When the opening costs were set to 25, the number of opened locations is increased by 9 compared to experiment 2 of the previous model. For Schiermonnikoog there is no increase in the number of opened locations compared to experiment 4 of the previous model, there is however a small increase in the number of units that are placed for both of the services. For experiment 2 on Schiermonnikoog, we find an increase in the number of opened location, while there is a small decrease in the number of placed units for the services. These changes are minor for all experiments and this could imply that they are caused by the solution method in stead of the model. Similarly to the two prior models, the graphical representation of the solutions is not included.

6 Conclusion

The test areas gave insight in how the models behaved when there were no spatial limita-tions. Without spatial limitations, the number of opened locations would depend on the capacity of either the number of connections or the bandwidth availability and could be calculated beforehand. Introducing a sensor would not influence this number.

In larger areas the placement is more important, but the algorithm will not find large improvements when the costs of opening a location are drastically decreased. Furthermore, the shape of the area and the dispersion of the demand points also plays a role in the solutions that are found. We found that for an area that is somewhat oval in shape with no large differences in dispersion of access and demand points benefits less from economic dependencies than a stretched out area with large differences in dispersion of access and demand points between regions when a sensor is added.

Overall, the results are unsatisfactory. Specifically, the gaps between the lower bounds and the current solutions is large. This might be due to the value of the parameter γj. This is

also the reason that the gap is lower for the reliability level of α = 0.95.

7 Discussion

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26 7 Discussion

First, the actual connection costs per access point are based on their proximity to the existing network, as no such data was available the connection costs were taken as equal for all locations. Changing the connection costs to differ between access points will naturally affect the optimal solution. Furthermore, some lamp posts might be more suitable for a larger connection than others because of a difference in height or location. In this research, the implicit assumption was made that all lamp posts will have the same type of connection. In the experimental setting, it was mentioned that the demand distribution was based on the paper by Oliveira et al. (2014). The data that was used in that paper was Mexican data for a 3G network. Using Dutch data would be more representative, however, such data was not available. The model and program can easily be altered to use such data if it were available. There was no approximation for the distribution for Smart Vehicle Communication available.

GSMA (2015) addresses the increase of demand for bandwidth as well as usage of mobile networks. This growing demand can make current networks insufficient. No exact prog-noses of such growth are available at the moment. It will be interesting to measure the impact of such growth on the smart city network.

Furthermore, the ranges of the services and sensor were assumed to be fixed. In fact, range for wireless connection can be influenced by many different factors, such as surroundings, number of connections and nearby posts. Range for an alarm can be decreased when there is noise pollution in an area.

Furthermore, the models in this thesis are rather restrictive in the sense that they allocate bandwidth to a demand point, rather than let the demand point be serviced by any nearby access point. This restriction does not only have implications for the value of the optimal solution, but also on the ability of the model to find feasible solutions. A model that tries to maximise the probability of an available connection such as the MALP (ReVelle and Hogan (1989a)) or QMALP (Marianov and ReVelle (1996)) may be more appropriate to ensure feasibility in all problems. Such a model will not allocate resources to particular demand points, which will be beneficial for the cutting heuristic as well, resulting in lower optimality gaps.

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7 Discussion 27

Table 9: Results for the Multi Service Model Schiermonnikoog

size 8 920 × 14 763

α 0.70 0.95 0.70 0.95

co 1000 1000 25 25

No. Opened Loc. 114 129 122 134

No. Wi-Fi Units 78 111 78 109

No. SVC Units 89 88 87 89 Costs Loc. 114 000 129 000 3 050 3 350 Costs Units 45 100 56 450 44 700 55 950 Total Costs 159 100 185 450 47 750 59 300 Rel. Gap 23.56% 5.75% 35.06% 7.75% Rozendaal size 15 824 × 27 403 α 0.70 0.95 0.70 0.95 co 1000 1000 25 25

No. Opened Loc. 119 138 117

No. Wi-Fi Units 63 87 64 No

No. SVC Units 95 104 102 Feasible

Costs Loc. 119 000 138 000 2 925 Solution

Costs Units 41 050 51 250 42 800 Found

Total Costs 160 050 189 250 45 725 Rel. Gap 54.23% 30.80% 69.61% Noordwijk size 115 602 × 204 026 α 0.70 0.95 0.70 0.95 co 1000 1000 25 25

No. Opened Loc.

No. Wi-Fi Units No Feasible

No. SVC Units Solution

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28 7 Discussion

Table 10: Results for the Multi Service Model Including Sensors

Schiermonnikoog Rozendaal Noordwijk

size 9 406 × 18 928 16 488 × 36 079 117 301 × 221 620

co = 25 co= 1000 co= 25 co= 1000 co = 25 co = 1000

No. Opened Loc. 134 134 147

No. Wi-Fi Units 111 110 91 No No

No. SVC Units 88 87 103 Feasible Feasible

No. Alarm Units 24 24 14 Solution Solution

Costs Loc. 3 350 134 000 3 675 Found

Costs Units 60 050 59 500 54 550

Total Costs 63 400 193 500 58 225

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References 29

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A

Graphical Representation of Solutions to Test Areas

Figure 3: Solution Multi Service model Test Area 1 Experiment 1

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32 A Graphical Representation of Solutions to Test Areas

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Figure 11: Solution Multi Service model including Sensors Test Area 1 Experiment 2

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Figure 13: Solution Multi Service model including Sensors Test Area 2 Experiment 2

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B Graphical Representation of City layout 37

B

Graphical Representation of City layout

Figure 15: Access points in Schiermonnikoog

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38 B Graphical Representation of City layout

Figure 17: Access points in Rozendaal

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B Graphical Representation of City layout 39

Figure 19: Access points in Noordwijk