Evacuation Policies for Events

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Master Thesis

Evacuation Policies for Events

Jochem Braakman, 6035868

February 29th, 2016

Supervisors: Prof. dr. Rob van der Mei (VU) Dr. Erik Winands (UvA)

Daphne van Leeuwen (CWI) Frank Ottenhof (TrafficLink)

KdV Instituut voor Wiskunde

Faculteit der Natuurwetenschappen, Wiskunde en Informatica Universiteit van Amsterdam

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Abstract

Event traffic is the cause of many congestion problems. This is due to the fact that in a short time-period thousands of vehicles (belonging to event traffic) create extra pressure on the infrastructure around the event area. In this thesis we develop a model that guides event traffic to its destination, such that this congestion will be reduced. Two models will be presented. A simple assigning model and the extension of that model, that takes a more precise road structure around an event location into account. Finally the case study of the ArenA serves as an example to apply these models.

Data

Title: Evacuation Policies for Events

Author: Jochem Braakman, jochem.braakman@student.uva.nl, 6035868 Supervisor: Prof. dr. Rob van der Mei (VU)

Second Reader: Dr. Erik Winands (UvA)

Internship Supervisors: Daphne van Leeuwen (CWI), Frank Ottenhof (TrafficLink) Hand-in date: February 29th, 2016

Korteweg de Vries Instituut voor Wiskunde Universiteit van Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.science.uva.nl/math

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Acknowledgements

First of all I would like to thank Prof. Rob van der Mei for his guidance and support. Every time I enjoyed our constructive meetings, his good sense of humour and interest-ing discussions about football. There is no such a thinterest-ing, that I will ever become a PSV supporter, but even I as an Ajax supporter have to admit, that PSV will be likely the champion this season. I really hope, that I will be mistaken.

Furthermore I would like to thank Daphne van Leeuwen for her help, guidance and ideas during the process. Besides that, I got to know her research problems and appreciated our fruitful discussions.

My special thanks goes also to Frank Ottenhof for the guidance and support. Although there were difficult moments during the process, he never lost faith in me and showed patience. My gratitude also goes to Geert de Lepper and Coen van Haaster. We started together our internship, laughed a lot and had spectacular matches during breaks at the football table.

I would like to thank Dr. Erik Winands for being the second supervisor from the Univer-sity of Amsterdam.

At last I want to thank my parents for their support in difficult times. My love to them cannot be expressed in words.

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Introduction: Event traffic

Managing thousands of people going to the same place in short time is a challenging task for event organisers and local authorities. This is the reason why there is a growing interest in achieving an improved accessibility of the event area at all times. Redesigning the infrastructure of the road network is an expensive and time-consuming solution, which does not necessarily guarantee improvements. Dynamic Traffic Management offers a more cost effective solution, which optimises the usage of the existing infrastructure.

ArenA-Poort

The case of the ArenA-Poort, one of Amsterdam’s main poles of entertainment and sports events, provides an example for carrying out a DTM case study. The area, which is situ-ated in the South East of Amsterdam, incact includes three main event locations, namely the Amsterdam ArenA, the Ziggo Dome and the Heineken Music Hall. These structures have comprehensively a capacity of about 90,000 visitors. In addition to events locations, the area also contains shopping malls, offices and residential complexes. Understandably, the average traffic flow generated in the area is consistent and its optimal management constitutes a real challenge in occurrence of events such as football matches and music concerts.

Figure 1: A map of the ArenA-Poort, showing its main entertainment facilities and some parking facilities.

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Parking facilities ArenA-Poort

Besides shopping malls and event locations the area contains parking facilities. The total capacity of these facilities is 11,943 vehicles. A large part of them are owned by the municipality. Other facilities belong to companies which are located in this same area. Since the most events are outside office hours the companies put their parking spaces available during an event. The main parking facilities and their capacities are given in the table below.

Parking facility Capacity Parking facility Capacity Parking facility Capacity

P1-ArenA 2,400 P6 399 P22 228

P2 2,010 P10 814 P24 487

P3 366 P12 296 P-Amstelborgh 700 P4 1,248 P18 291 Dome-garage 550

P5 1,354 P21 450 Endemol 350

Table 1: Capacity of parking facilities in the ArenA Poort (total capacity 11,943 places)

Figure 2: Overview over the main parking facilities in the ArenA Poort

State of art

The figure below shows the main roads around the ArenA area. Drivers on the primary road network are able to take five exits towards this area as indicated in figure. People can use the road signs and/or a navigation system towards an event. According to the

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municipality of Amsterdam this leads to one main route; driving to highway A2 and taking Exit 1 towards the ArenA. [4] Consequently event traffic is the cause of traffic jams in that area.

Figure 3: Inner and outer network structure ArenA area, i.e. the yellow arrows describe the points at which event traffic enters, the green circles represent the exits that lead to ArenA-Poort.

Trinit´

e Automation and TrafficLink

Trinité Automation is the market leader in the field of dynamic traffic management. With the software of Trinité it is possible to manage and optimize the flow of traffic in a large area fully automatically. Trinité manages the rush-hour lanes, safety in tunnels, real-time video images of traffic, and route information on Dynamic Route Information Panels (DRIPs) and many more. All of this is done in an integral manner, from one interface. Trinité allows systems of different suppliers to work together, and makes automatic tuning between road managers from different areas possible. TrafficLink is a subsidiary of Trinité which sells a dynamic traffic management system that automatically controls all kinds of traffic managements tasks. With some basic functionalities an intersection can easily be controlled, but with advanced functionalities more complex networks like the area of Amsterdam are managed by TrafficLink.

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Approach

The amount of traffic all over the world is growing and the outcome has negative side effects, such as traffic jams, the increase of CO2, traffic victims, economic loss and so on.

Therefore it is feasible to look for possibilities to reduce these side effects. Therefore in the case of the ArenA-Poort we will have the following approach:

1. Develop a model, that takes the infrastructure of the ArenA into account. 2. Minimising queue length at parking destinations.

3. Minimising probability of traffic jams. 4. Determine optimal routing policies.

Organisation

A brief overview of this thesis is given below.

1. In Chapter 1 the basics of traffic flow, i.e. the idea of the fundamental diagram will be discussed. In addition we give an illustrative example for the ArenA, that uses Wardrop equilibria. In the end of this chapter Braess’s paradox will be pointed out. 2. In Chapter 2 basic queueing theory will be reviewed. Furthermore the idea to model a single road as a queue will be presented. This queue is called the M/G/c/c queue. 3. In Chapter 3 the idea of a fluid model will be carried out. Furthermore, two models will be presented. The first model will only show how to assign traffic, without precise knowledge of the road network. The second model is an extension of the first one and includes the whole network structure.

4. In Chapter 4 the performance and results of both models applied on the ArenA will be presented.

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Chapter 1 Preliminaries

1.1. Basic graph theory

Definition 1.1. A graph is an ordered pair G = (V, E), where 1. V is called the set of vertices or nodes,

2. E is called the set of edges, connecting the vertices in the graph G. In our thesis we will consider a traffic network as a graph, where

begin/end of a road respectively link will be represented by nodes roads respectively links will be represented by arcs

Example 1.2. An example of a graphical representation is the map of the ArenA.

s1 s2 s3 s4 e1 e2 e3 e4 e5 w1 w2 w3 w4 w5 w6

Figure 1.1: In this figure the network of the ArenA is represented as a graph. The yellow arrows in the left picture correspond to s1 − s4 From there on traffic will be passed on

to through to the set of exits e1− e5 indicated in green in the left picture. At the exits,

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1.2. Basics of traffic flow

Definition 1.3. Let us denote x for distance and t for time. The density of a traffic flow k(x0, t0) can be measured as the number of vehicles on a road section [x0− ∆x, x0+ ∆x]

at some time t = t0, i.e. taking the limit of ∆x to 0, we obtain the density in x0 by

k(x0, t0) = lim ∆x→0

number of vehicles from x0− ∆x to x0 + ∆x at t = t0

2∆x .

An intuitive explanation to this is that we are photographing a section of a road from above and that we count the amount of vehicles. Making the section around some fixed point arbitrarily small will result in the density in the point itself.

Definition 1.4. The intensity of a traffic flow q(x, t) is defined as the number of vehicles at some place x = x0 in the time interval [t0− ∆t, t0+ ∆t], i.e. taking the limit of ∆t to

0, we obtain the intensity in t0 by

q(x0, t0) = lim ∆t→0

number of vehicles crossing x0 from t = t0− ∆t to t = t0+ ∆t

2∆t .

Definition 1.5. As known from basic physics let v denote the velocity or speed of a vehicle. The unit of v is measured in distance per time. By the definitions of density and flow, we are able to relate v to q and k, namely

v = q/k ⇒ q = vk.

Further we are interested in the relations between v, k and q. Greenshield [1] was one of the first, who conducted research and developed a model of uninterrupted traffic flow that predicts and explains the trends that are observed in real traffic flows. Based on a very small dataset Greenshield concluded that the mean speed will decay as the density increases. The relation between v and k is described by a linear model, namely

v = v0− ck,

where c and v0 are constants that can be determined from field observations. As

q = vk,

we substitute insert the expression for v into the formula and obtain q = vk = (v0− ck)k = v0k − ck2.

As we are interested to obtain the the maximum flow, we find the maximum flow by solving

dq

dk = 0, ⇒ k = v0

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The maximum mean speed can now be expressed as vmax= v0 − kc =

v0

2.

Substituting this expression vmax into the quadratic equation for q, we find that

qmax= v0 2 · v0 2c = v2₀ 4c.

As Greenshield’s model is very simple, it does not describe the exact relation between v and k. There are better but more complex descriptions in the literature, that generalize the idea above. This leads to the so-called fundamental diagrams.

Fundamental Diagram 1.6. Road traffic is always in a specific state which is described by flow rate, density and mean speed . We combine all the possible homogeneous and stationary traffic states in an equilibrium function that can be described graphically by three diagrams. The equilibrium relations presented in this way are better known under the name of fundamental diagrams. The figure below sketches them and it shows the relationship between each of the diagrams, i.e. the relation between mean speed and flow, mean speed and density and flow and density respectively.

Figure 1.2: The three fundamental diagrams.

Conservation Law 1.7. The Conservation Law states that in a small region of the highway the difference of the number of verhicles in a certain time interval should equal the difference between vehicles entering the highway and vehicles leaving the highway in

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that time period. This means nothing else that that there exists no vehicle that during that time on the road can just appear or vanish.

{#vehicles in interval at t = t0+ ∆t} − {#vehicles in interval at t = t0− ∆t}

= {#vehicles crossing x0− ∆x from t = t0− ∆t to t = t0+ ∆t}

− {#vehicles crossing x0+ ∆x from t = t0− ∆t to t = t0+ ∆t}

By above definitions the expression can be rewritten to

2∆x(k(x0, t0+ ∆t) − k(x0, t0− ∆t)) = 2∆t(q(x0− ∆x, t0) − q(x0+ ∆x, t0)).

Using a Taylor expansion around (x0, t0) yields

2∆xk + ∆tkt+ 1 2(∆t) 2_k tt+ 1 6(∆t) 3_k ttt+ ... −k + ∆tkt− 1 2(∆t) 2_k tt+ 1 6(∆t) 3_k ttt+ ... = 2∆tq − ∆xqx+ 1 2(∆x) 2_q xx− 1 6(∆x) 3_q xxx + ... −q − ∆xqx− 1 2(∆x) 2_q xx− 1 6(∆x) 3_q xxx+ ... . This can be reduced to

kt+ O((∆t)2) = −qx+ O((∆x)2),

hence letting ∆t → 0 and ∆x → 0, we conclude that ∂k

∂t + ∂q ∂x = 0.

It is possible to express the balance law differently, by introducing velocity v(x, t) of the cars on the highway. However the velocity is an averaged quantity and should therefore be treated carefully. One way to measure the velocity of n cars is to take the average velocity. To this end

v(x0, t0) ≈ 1 n n X i=1 vi.

In the continuum model it is assumed that the limit of the average exists, by letting ∆x → 0, and the limit will be the velocity v(x0, t0). We are now able to relate the

velocity to the intensity, by the equation

q = vk.

Inserting our expression into the equation of the balance law yields, ∂k

∂t + ∂

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Assuming that the initial density is known, that is k(x, 0) = f (x), we could solve this equation.

Example 1.8 (Constant Velocity). To investigate the properties of the traffic flow prob-lem we assume that the velocity is constant, that is v(x, t) = a. The probprob-lem then can be stated as

∂k ∂t + a

∂k ∂x = 0, The solution to the partial differential equation is

k(x, t) = f (x − at).

1.3. Wardrop equilibria

There are several approaches known to tackle the theory of traffic flow. One of these possibilities is Graph theory. We introduce the following definitions and terms. Let us consider a routing network N = (V, E), where V represents nodes and E arcs that are directed.

Let r be a route. A route in N is a path that where all arcs are orientated in the same direction. Denote R as the set of roads;

The collection S is the set of source-destination pairs. An element s ∈ S is a route r that starts from a source point and ends in a destination point. Thus S is a collection of roads.

For any s, s0 ∈ S with s 6= s0 we assume that s ∩ s0 = ∅.

For every r ∈ R there is a flow xr ∈ R≥0, for each s ∈ S, there is a flow ys > 0

(otherwise we could leave s out of the network), and for each e ∈ E, there is an amount of flow ze ∈ R≥0.

For each arc e ∈ E we assign a costfunction ce(ze), that depends on the amount of

flow ze.

We assume that y = (ys : s ∈ S) is fixed and that x = (xr : r ∈ R) and z = (ze :

e ∈ E) are variables.

We can calculate y = (ys : s ∈ S) and z = (ze : e ∈ E) by x = (xr : r ∈ R) in the

following way: ys = X r:r∈s xr, ze= X e:e∈r xr. (1.1)

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Definition 1.9. For a source-destination flow vector y = (ys : s ∈ S) a Wardrop

Equilib-rium is a vector x = (xr : r ∈ R) and resulting rates z = (ze: e ∈ E), such that all x, y, z

satisfy the constraint (3.1) and for all s ∈ S and r ∈ s it holds, that xr > 0 =⇒ X a∈r ca(za) ≤ X a∈r0 ca(za).

Intuitively this means that if there is a flow xr > 0, then the Wardrop Equilibrium is

using the route with less costs than all the other routes over the same source-destination pair.

The Wardrop Equilibrium is based on the assumption that drivers will always make a selfish choice by choosing the route, which guarantees less cost. If for example there are two roads between one source-destination pair, it will be more attractive for a selfish driver to use the road which takes less time to reach the destination. However if every driver chooses route A, congestion will appear and drivers will automatically (to avoid congestion) chose route B. Thus after some time this will lead to a steady distribution of flow.

1.4. Wardrop’s equilibrium illustrated for the ArenA

case

Example 1.10. In this example we utilise the idea of Wardrop’s Equilibrium for the ArenA. It has been observed that [6] the main source of vehicles comes from the A2 highway with a flow of 3400 vehicles per hour. To keep it simple our source will be Knooppunt Holendrecht and our destination will be the Amsterdam ArenA. We assume that there are two roads connecting the source-destination pair. One road is the extension of the A2 highway that leads to the south entrance of the ArenA and the other road leads via the S112 to the north entrance. Thus we only have to determine the costs for both routes. The Bureau of Public Roads (BPR) developed in 1964 a link congestion function [5], that we will call ci(zi). It is of the form

ci(zi) = ti 1 + α z_i qi β! , where

ti denotes the free flow travel time

α > 0 and β > 0 are shape parameters for the model qi is the capacity of a link i per unit of time

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ci(zi) is the average travel time in minutes for a vehicle on link i and let f = za+ zb

be the total flow through the network

Knooppunt Holendrecht Amsterdam ArenA za+ zb = f za A2 zb S112

Thus we can determine Wardrop’s Equilibrium by solving

ta 1 + α za qa β! = tb 1 + α zb qb β!

towards the constraint

za+ zb = f.

Research has shown [5] that the shape parameters α = 0.15 and β = 4 guarantee a good description for the average transportation time per vehicle. The traffic free flow time ta

via the A2 is 4 minutes and the length of the road is 5 km. The traffic free flow time tb via the S112 is 7 minutes and the length of that road is 7.5 km. Finally qa = 800,

and qb = 600. Let f = 3400, then this non-linear program can be solved numerically by

Mathematica for example. The solution yields

za≈ 1803 and zb ≈ 1597,

with an average travel time per user of

ca(1803) ≈ 34 minutes.

Thus if we would assign just about 56% of the vehicles via the A2 and 44% via the S112, we would reach an optimum in average travel time for every user.

Although this methods looks interesting it is just an illustration of what can be done with Wardrop’s equlibrium.

Assume now, that we would have two roads and between these roads there is a third road connecting these two roads. Wardrop’s equilibrium now represents the result of selfish choices by the players. Braess’s paradox will now show that a third route does not necessarily improve the average travel time for commuters.

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Braess’s Paradox

Let N = {1, 2, 3, 4}, A = {(1, 2), (1, 3), (2, 4), (3, 4)} and routes R = {r124, r134}. The

cost function for every arc is represented in the figure below. Our source destination pair is s = 14. Assume now that a flow of 6 units goes through the network constantly, i.e. y14 = 6.

4

2

1

3

6 10z 3 3 50+z 50+z 3 3 10z

4

2

1

3

6 10z 4 2 50+z 50+z 2 10+z 2 10z 4

Figure 1.3: The left network represents a network with two routes, whereas the right network describes inhabits the same routes as in the left network but with an additional route. The outer side of the arcs does show the cost function, whereas the inner side of the arcs describes the flow that has to be sent along the arcs to achieve the Wardrop equilibrium

If we compare the left network with the right network in Figure 1.2 we would say from an intuitive point of view that we would be better off with the extra route, because there are more possibilities to spread the flow along the routes with the result that the travel costs will decrease. This is not the case! This can be explained by determining the Wardrop equilibrium for both networks. Because of the symmetry of costs in the left network the Wardrop equilibrium will be achieved by sending 3 units of flow via route r124 and 3 units of flow via route r134 respectively. Thus the travel time via both routes

will be equal to 10 · 3 + 50 + 3 = 83. In the network to the right however we will achieve the Wardrop equilibrium if we send 4 units of flow from 1 to 2 and 2 units from 1 to 3 respectively. Thus as a result the travel time for route r1234, r124 and r134 will be equal to

10 · 4 + 50 + 2 = 10 · 4 + (10 + 2) + 10 · 4 = 92. Although a route has been added with the intention to decrease the travel time, we have achieved the opposite and every driver will have a longer average travel time. This phenomenon is called Braess’s paradox.

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Chapter 2 Queueing

Queuing theory is a branch of mathematics that studies and models the act of waiting in lines. Thus from a mathematical point of view Queueing theory enables us to study average queue lengths, average time of a customer spent in the system, equilibrium dis-tribution, etc. In our thesis we will consider queues as an analytical tool.

Definition 2.1. A queueing system can be fully described by Kendall’s notation, which is of the form

A/S/Queue discipline/Servers/Capacity, where

A denotes the type of the interarrival time distribution (general or exponential etc.) S denotes the service rate time distribution (general or exponential etc.)

The Queue discipline indicates the job priority of the queue. (FIFO - first in first out, LIFO - last in first out etc.)

The amount of servers describes the amount of customers that can be served at once, if the server is free

The capacity describes the amount of customers that can stay in the system.

λ µ

Waiting Area

Service Node

Figure 2.1: Visualization of a queue, where the rate of arriving jobs is λ. The capacity is equal to the Waiting Area in which jobs proceed to the service node, where they are served with rate µ.

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2.1. M/M/1

The simplest stochastic queue is the M/M/1, where we assume that inter-arrival times are distributed exponential(λ) and service times is exponential distributed with rate µ. For an M/M/1 queue we require that λ < µ, otherwise the queue would diverge.

0 1 2 3 · · · λ µ λ µ λ µ λ µ

Figure 2.2: A graphical representation of the M/M/1 queue with arrival rates λ and service rate µ.

The steady-state distribution

πn= P(n customers in system, while being in steady-state)

can be calculated by solving the so-called balance equations, i.e. λπ0 = µπ1

(λ + µ)πi = µπi−1+ λπi+1 i = 1, 2, 3, ...

Let us now define ρ = λ_µ < 1, then the well-known solution for πi can be expressed by

πi = (1 − ρ)ρi i = 0, 1, 2, 3, ...

Furthermore the average queue length EL of an M/M/1 queue is

EL = ρ 1 − ρ.

Theorem 2.2. Little’s law states that in any stable system the relation EL = λEW

holds, where EL represents the average number of customers in the queue and λ is the long-term average effective arrival rate and EW denotes the average waiting time of a customer spent in the system.

Proof. The famous proof found by John Little can be found in the literature. [7]

Thus for an M/M/1 queue the average waiting time EW is according to Little’s Law

EW = EL λ =

1 µ − λ.

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Transient Behaviour of M/M/1

As our queues will not start in steady-state, we are interested in the transient be-haviour of an M/M/1 queue at a certain time. Applied on a parking facility or traffic light, we would like to know what the distribution of the states in the queue is after some time t, starting in an empty queue at t = 0. In the literature there are many approaches known and a well known result is given by Bailey in 1954. Let ρ = λ/µ as before and describe the queue by Qt. Define then

π_ij(t) _{:= P(Q}t = j | Q0 = i)

Then Bailey’s formula states, that

π_ij(t) = 1 µtexp(−(λ + µ)t)ρ j ∞ X s=j+1 ρ−s/2sIs(2t p λµ) where Is(z) = X m≥0 (z/2)s+2m

m!(m + r)! is the modified Bessel function.

This expression is not very attractive as it inhabits Bessel functions whose solutions lead to complex numerical solutions and thus transient probabilities can be hard to calcu-late. Leguesdron, Pellaumail, Rubin and Sericola use a different approach, based on uniformization technique and generating functions.

Definition 2.3. Define ν(M ) = sup i X j |Mij|

and denote by M the collection of infinite matrices M , such that ν(M ) < ∞. The potential kernel of a matrix M ∈ M, denoted by ΦM is the complex function defined by

ΦM(z) = ∞ X k=0 Mkzk |z| < 1/ν(M ). Theorem 2.4. Let p = λ λ+µ and q = µ

λ+µ be the probabilities for jumping up respectively

down in the the M/M/1 queue, whereas the probabilities can be easily obtained by uni-formisation. Let C(z) the generating function of the sequence of the Catalan’s numbers, which satisfies the equation

C(z) = 1 + zC2(z). Define η(z) = C(pqz2_{). Let (P}n₎

ij represent the transient probabilities of the uniformized

Markov chain of the M/M/1 queue. Then the coefficients of the potential kernel of P are given by (ΦP(z))ij = (pzη(z))j−i i X k=0 pkqkz2kη(z)2k+1+ pjqi+1zi+j+1 η(z) 1 − qzη(z).

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Proof. The proof can be found in the article written by Leguesdron, Pellaumail, Rubin and Sericola. [9]

In our context we assume that the queue is initially empty. Thus by the coefficients above we are able to retrieve the transient probabilities starting from an empty queue. Before stating the analytical expression we need to prove the following Lemma.

Lemma 2.5. For every k ≥ 1 and for every z such that |z| ≤ 1/4, we have

Ck(z) = ∞ X k=0 s(k, n)zn, where s(k, n) = k(2n + k − 1)! n!(n + k)!

Proof. The proof can be found in the book of J.Riordan Combinatorial identities. [2] Theorem 2.6. For every j ≥ 0 we have that

(Pn)0j =    0 if n < j p q j Pbn−j2 c k=0 s(n + 1 − 2k, k)pkqn−k if n ≥ j.

Proof. The proof can be found in the article as stated above. [9]

As the M/M/1-queue is the simplest of all queues, the expressions for the transient be-haviour are not. If we want to model a queue at a parking facility or on a road we need to consider queues with finite buffer size. However we also would like to know how this finite buffer queues will evolve over time. They can be used for an analysis of the stochastic behaviour.

2.2. Road modelling

This section will be focussed on how to model a single road link as a queue. The question arises which queue would be suitable in the context of traffic flow. In the literature Jain and MacGregor Smith [8] propose the M/G/c/c queue to model vehicular traffic flow on a road. In this article it is assumed that arrivals occur according to a Poisson process and that the service times are state-dependent. This is a realistic assumption, as people tend to arrive randomly and the service time depends on the amount of cars on a road according to the fundamental diagram. The capacity of a road is finite and will be described by the letter c measured in vehicles/km. As it is assumed that every car can receive service along a road, c servers will be needed. These assumptions justify the usage of an M/G/c/c queue. Performance measures, such as the probability of blocking can be measured and utilised to analyse road networks.

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2.3. Properties of M/G/c/c for a road

Before using M/G/c/c as a tool we need to determine certain properties. While it is assumed that arrivals occur according to a Poisson process, the Jain and MacGregor Smith propose the following service-rate, namely

µi = iµf (i) for i ∈ {1, ..., c},

where f is a function of speed, such that f will decrease if i (number of customers, vehicles) increases. The queue can be represented as follows:

0 1 2 · · · c − 1 c λ µ1 λ µ2 λ µ3 λ µc−1 λ µc

Figure 2.3: A graphical representation of the M/G/c/c queue with arrival rates λ and state dependent service rates µi.

2.3.1. Stationary distribution

The stationary distribution of the M/G/c/c queue as defined above can be found by solving the following set of equations:

λπ0 = µf (1)π1

(λ + iµf (i))πi = λπi−1+ (i + 1)µf (i + 1)πi+1 for i ∈ {1, · · · , c − 1}

λπc−1= cµf (c)πc.

The solution of this set of equations is

πi = i Y j=1 λ jµf (j)π0 for i ∈ {1, ..., c}. We are able to obtain π0 by normalisation, such that

π0 = c X i=0 i Y j=1 λ jµf (j) !−1 .

2.3.2. Linear model and exponential model

In the literature there are many models that describe the relation between the traffic density k and average speed v. The simplest version of this relation is introduced by Greenshield, where the average speed v as a function of k decreases linearly. However we

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will use also the exponential model, stated in the article written by Jain and MacGregor Smith [8]. The linear model is given by

flinear(i) =

c + 1 − i c

i ∈ {0, ..., c},

whereas the exponential model has the form

fexponential(i) = exp

− i − 1 β

γ

i ∈ {0, ..., c},

where β and γ are parameters obtained by certain assumptions about speed and length on a certain road, which will be stated in the next section.

2.3.3. Stationary distributions for the models

Before stating the stationary distribution for the linear and exponential model, we need to consider the mean service requirement 1_µ. Assume that a vehicle enters the road, then its mean service time will be

1 µ =

L v1

,

where L is the length of the road and v1 in the initial velocity. Thus µ = v_L1. Consequently

we are able to retrieve from the previous section the stationary distribution for the linear model, namely πlinear₀ = c X k=1 λ µ k k Y l=1 l c + 1 − l c −1!−1 , πlinear_i = λ µ i i Y j=1 j c + 1 − j c −1 π₀linear for i ∈ {1, ..., c},

and for the exponential model

π₀exponential= c X k=1 λ µ k k Y l=1 l exp − l − 1 β γ−1!−1 , π_iexponential= λ µ i i Y j=1 j exp − l − 1 β γ−1 πexponential₀ for i ∈ {1, ..., c}.

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2.3.4. Performance measures for M/G/c/c

Consider a road r with length l, initial velocity v1, jam-density kjam with number of lanes

N . Then the capacity c of a road r can be calculated by c = lN kjam.

An interesting performance measure in this queue is the stationary probability πc. It

indicates the probability of a road being occupied. The blocking probability can be calculated (dependent on the model choice) by

πc= c Y j=1 λ jµf (j)π0. Given arrival rate λ, we define the throughput Θ as

Θ = λ(1 − πc).

The throughput is a measure to determine the maximum arrival rate λ, that the M/G/c/c can handle, such that we can minimise the probability of blocking. Furthermore we can calculate the average queue length by

EL =

c

X

i=0

i πi.

The performance measures seem to be useful, however the computation of πcis very costly

in time for large c. For example if c = 500 it can take several minutes to determine λ, such that πc is small. On the other hand these calculations have to be made only once.

1000 2000 3000 4000 Λ 20 40 60 80 100 Average length Queue

1000 2000 3000 4000 Λ 500

1000 1500 Throughput - Q

Figure 2.4: Performance measures for a M/G/100/100 queue with speed v1 = 100 km/h

on a road with length l = 1 km for the linear model.

In the figures above we observe that there is around the maximum throughput a region that is very sensitive to small changes, such that the average queue length increases

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significantly. This can be explained more properly if we analyse the histograms of the probability distribution for these λ.

20 40 60 80 100 State 0.02 0.04 0.06 0.08 Probability 20 40 60 80 100 State 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Probability

Figure 2.5: Histograms for M/G/100/100 with different arrival rates λ = 1520 and 1720 respectively. We have obtained λ = 1520 as the maximum arrival rate, such that the blocking probability is bounded by πc(λ) ≤ 0.00001. While increasing λ it can be observed,

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Chapter 3 Deterministic Fluid Model

In the previous chapters we have treated the fundamental diagram, a small part of the ArenA problem and studied the M/G/c/c congestion model. However an integrated ap-proach is missing. The question arises how the problem could be translated into a model that includes queues, roads and travel time.

In the previous chapters it can be observed that the problem of the ArenA-Poort is a flow problem. This means that event traffic can be modelled as flow, that has to be directed via roads to its final destination. At the final destinations vehicles have to enter the parking facility. The inflow rate of vehicles that enter the parking facility will be at times larger than the service rate at a parking facility. Therefore queues appear. As in general stochastic queues are difficult to optimise, the problem will be approached by a deterministic fluid model in this chapter. The deterministic fluid model has the advantage that no stochasticity is included, which will ease model formulations and calculations.

A fluid queue can be viewed as a large tank, connected to a series of pipes that pour fluid in to the tank and a series of pumps which remove fluid from the tank. Let λi be the

inflow in state i and denote the service rate or outflow by µi. Then the net fluid arrival

rate will be ri = λi− µi. Let us denote by X(t) the fluid level at time t. Then

dX(t) dt =      ri if X(t) > 0 max(ri, 0) if X(t) = 0.

Let us now assume that we have discrete time steps ∆t. Given the fluid level of X at time t we are able to calculate X(t + ∆t) by

X(t + ∆t) = max(X(t) + ri∆t, 0). (3.1)

The above formula is easy to explain. It states that the fluid level at time t + ∆t is calculated by the fluid level at time t with in addition the net fluid arrival. As the outcome can also be negative, the maximum between the terms and 0 has to be taken.

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3.1. Modelling

We are going to model the network around the ArenA-area as a fluid model, where The water tank will be the amount of cars in a certain time entering the network Pipes will be represented by the roads connecting the inflow destination and parking destinations

The parking destination will be represented by a series of pumps that remove fluid from the tank. We expect on certain time steps that the demand will be higher than the total service rate of the parking lots, which means, that queues will certainly appear. The queue length will be calculated at every time step by the above formula.

3.2. Optimisation fluid model

In this section an optimisation, that can be applied onto the ArenA will be derived. The goal is to calculate a strategy to guide event traffic, such that

1. Queues at parking destinations will be minimised.

2. Queues at parking destinations will not exceed their buffer capacity. 3. Traffic will not be sent to occupied parking spots.

4. Extra walking time from a parking destination to the event location will be penalised. In this chapter we will see, that this optimisation can be written into a linear program. A linear program (also abbreviated as LP) is a mathematical model whose requirements are represented by linear relationships. In canonical form it can be represented as

maximise (minimise) cTx subject to Ax ≤ b

x ≥ 0.

Here x represents the vector of variables, that have to be determined. The vectors c and b are known. The expression to be maximised (minimised) is called the objective function. In the fluid setting we need to do some additional work to carry out the optimisation. Also the main idea and notational work need to be explained. This will be done in the next section.

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3.2.1. Assumptions

In the context of the fluid model we assume that we know the inflow at all times. In the case of the ArenA however this is not a realistic assumption. On the other hand information about event traffic based on empirical data, could lead to predictions, that could work out well. Until now there is still little known about the exact distribution of event traffic. Before stating how the fluid model works, we introduce some necessary notation.

Notation

Variable Description

I Index set inflow sources. An element of I will be denoted by i.

J Index set parking destinations. An element of J will be denoted by j. T Set of time points , i.e. T = {t0, t1, t2, ...}.

t Time step. T Last time point.

Pj Capacity at destination j.

Ltk

j Queue length at time tk at destination j ∈ J .

Bj Buffer at destination j ∈ J .

µj Service rate at destination j ∈ J .

Λi Inflow vector at inflow point i.

λtk

i Inflow at source i ∈ I at time tk, i.e. the k’th coordinate of Λi.

xijtk Decision variables assigning flow from source i to destination j at time tk.

yij 1 if there is a connection from i to j, 0 otherwise.

lj Walking time from destination to ArenA

As the notation seems quite abstract we will illustrate the idea of the fluid model as a graph as shown on the next page.

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s1 s2 w1 w2 w3 Λ1 Λ2

Figure 3.1: A graphical representation of the fluid model. At the source nodes s1-s2 the

fluid will be poured into the network. The parking destinations w1-w3 inhabit certain

properties, such as queue length, buffer capacity, service rate and capacity. A source-destination pair is connected by lines. The variable ysiwj represents the connection of a

source-destination pair. For example as there is a line between s1 and w1, the variable

ys1w1 = 1. From s2 to w3, there is no line connecting the s-d pair. Therefore ys2w3 = 0.

Example 3.1. To illustrate how the fluid model works, we will state an example that is related to Figure (3.1). Assume that Λ1 = (2, 3) and Λ2 = (4, 1). Let t = 1, Pw1 = Pw2 = 3

and Pw3 = 4. Furthermore µw1 = µw2 = 2 and µw3 = 1. The buffer for the queues is

given by Bw1 = Bw2 = 4 and Bw3 = 3 respectively. Let xs1w1tk = xs1w3tk = 0.5 and

xs2w1tk = xs2w2tk = 0.5 for all tk. Assume that the queues are empty in the beginning. To

this end the queue length can be calculated by the formula Ltk wj = max(L tk−1 wj + X i∈I xsiwjtkysiwjλ tk si − µwjt, 0) Thus Lt1 w1 = max(0 + 0.5 · 1 · 2 + 0.5 · 1 · 4 − 2 · 1, 0) = 1. Lt1 w2 = max(0 + 0.5 · 1 · 4 − 2 · 1, 0) = 0 Lt1 w2 = max(0 + 0.5 · 2 − 1 · 1, 0) = 0,

and at time t2, the queue length will be

Lt1 w1 = 1 Lt1 w2 = 0 Lt1 w2 = 1.5.

Thus at the end of the inflow period Pw1 = −1, Pw2 = 0.5 and Pw3 = 1.5. As the queues

are at all times below their buffer, the choice of xsiwjtk however leads to negative capacity.

Our model will inhabit the restriction, that the capacity should not be negative. Could another choice of xsiwjtk lead to a better solutions, that minimises the total queue length,

such that the capacity will not be exceeded? This optimisation problem will be tackled in the next sections.

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Under the parameters as stated above, the fluid model is nothing more than an assign-ment model. This means that we are assigning a certain amount of flow from the inflow source to its destination. To this end, restrictions, such as a more precise structure of the pipes, that connect the source-destination pair are not taken into account yet.

3.2.2. Rewriting Lindley’s recursion

Before stating the optimisation it is necessary to study the recursion in (3.1). This recursion can be rewritten in a different way, which will be shown in the following lemma. Lemma 3.2. Let Lindley’s recursion be given, that is

Ltk _{= max L}tk−1 _{+ η}tk, 0 ,

where ηtk ∈ R and the initial value of the recursion is Lt0 _{= L}

0. Then it follows that

Ltk ₌ _max u∈{k,...,0} u X l=0 ηtk−l _{+ L}t0₁ {u=k} ! .

Proof. The proof can be obtained by straightforward calculation, namely Ltk _{= max L}tk−1 _{+ η}tk_{, 0} = max max(Ltk−2 _{+ η}tk−1_{, 0) + η}tk_{, 0} = max max Ltk−2 _{+ η}tk−1 _{+ η}tk_{, η}tk , 0 = max Ltk−2 _{+ η}tk−1_{+ η}tk_{, η}tk_{, 0} · · · = max ηtk_{, η}tk _{+ η}tk−1_{, ..., η}tk _{+ η}tk−1 _{+ ... + η}t0 _{+ L}t0_{, 0} = max u∈{k,...,0} u X l=0 ηtk−l _{+ L}t0₁ {u=k} ! .

In our setting a queue at parking destination j at time tk can be calculated by the

recursive formula for the fluid queue (3.1), namely

Ltk j = max L tk−1 j + X i∈I λtk i yijxijtk− µj, 0 ! , where Lt0 j = L j

0, ∀j ∈ J . Consequently by Lemma 3.1, with

ηtk j = X i∈I λtk i yijxijtk− µj,

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the queue Ltk j can be expressed as Ltk j = max u∈{k,...,0} u X l=0 ηtk−l j + L t0 j 1{u=k} ! .

3.3. Linear Program

In the setting of our fluid model we want to minimise queue lengths at parking destinations. Thus we would like to find

min x_ijtk X tk∈T X j∈J Ltk j ,

according to the previous section this is equivalent in finding

min x_ijtk X tk∈T X j∈J max u∈{k,...,0} u X l=0 ηtk−l j + L t0 j 1{u=k} ! .

In order to omit the maximum term in the optimisation, the problem can be written into a constrained optimisation, that is

min x_ijtk X tk∈T X j∈J zjtk subject to zjtk ≥ u X l=0 ηtk−l j + L t0 j 1{u=k} ! ∀j ∈ J , ∀tk ∈ T , ∀u ∈ {k, ..., 0}.

Consequently as zjtk depends linearly on xijtk, the objective function is linear as well. As

a result optimising the queue length in a fluid model is an optimisation problem, that can be formulated as a linear program.

3.4. Mathematical formulation of the assignment model

Summarizing the previous sections we are able to formulate the LP for our fluid model. To this end we want to find

min x_ijtk X tk∈T X j∈J zjtk+ xijtklj

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such that X j∈J xijtk = 1 ∀tk ∈ T , ∀i ∈ I (3.2) Pj − X tk∈T X i∈I xijtkyijλ tk i ≥ 0, ∀j ∈ J (3.3) zjtk ≥ u X l=0 ηtk−l j + L t0 j 1{u=k} ∀j ∈ J , ∀tk∈ T , ∀u ∈ {k, ..., 0} (3.4) 0 ≤ zjtk ≤ Bj ∀j ∈ J , ∀tk ∈ T (3.5) xijtk ∈ [0, 1], ∀i ∈ I, ∀j ∈ J , ∀tk ∈ T (3.6)

The objective function is a linear combination of zjtk, that represents the queue-length

at destination j at time tk and the walking time from the destination to the ArenA-area.

Note that zjtk depends on xijtk. The walking time will be influenced by the factor xijtk,

that penalises at time tk the decision to choose from i destination j. Constraint (3.2)

describes that at every time step and from every source the inflow will be distributed over the destinations. Constraint (3.3) tells, that at the very end of the inflow-period the initial capacity of a parking facility should not be exceeded. Constraint (3.4) is necessary to rewrite the minimax problem into an LP. Constraint (3.5) ensures, that over all destinations and all time periods the queue will not exceed buffer level Bj. Constraint

(3.6) states, that xijtk are decision variables.

3.5. Application of the mathematical model onto the

ArenA

How can we relate the above optimisation to the case of the ArenA? Firstly it should be noted, that our model so far can only tackle source destination problems. In the example of the ArenA however, vehicles should first be assigned from inflow points to exits. From there on the flow should be distributed over all parking facilities. To this end, we obtain an assignment problem in two parts. This is illustrated in the figure below.

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s1 s2 s3 s4 e1 e2 e3 e4 e5 w1 w2 w3 w4 w5 w6 Λ1 Λ2 Λ3 Λ4

Figure 3.2: In this figure the idea of a fluid model for the ArenA is illustrated. The fluid Λi is poured into the set of sources s1− s4. From there on it passes on via the routes, that

are connecting the source points with the exit points e1− e5. From the exit points again

the flow will be distributed over the parking facilities w1− w6. Note that this figure does

not correspond one-to-one with the case of the ArenA, as we have left out some parking destinations.

Our assignment problem can be described as follows. First we will assign from source points traffic to exit points. This procedure will be called optimisation in the exterior network. As a result we can calculate the flow at the exits. From there one traffic has to be redistributed over the interior network. Details about the exterior and interior network in the case of the ArenA can be found in the appendix.

3.5.1. Exterior assignment

The question arises in the context of the ArenA, under which constraints we are able to assign the traffic flow to the exits. In some sense there will not change that much in terms of optimisation, however some subtle remarks should be made. As J will represent parking facilities, the collection E will represent the collection of exits. Let us define the matrix Y = (yej)e∈E,j∈J, where

yej =     

1 if there is a connection between e and j 0 otherwise

The idea is to view the collection of exits as deterministic queues. The only thing that rests is to define the capacity, service-rate and buffer-size of the exits. To this end, we introduce the relative capacity of an exit given by

P_e∗ =X j∈J 1 P e∈Eyej yejPj.

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The relative capacity can be explained as follows. As it depends on yej, if a parking

destination j can be reached from an exit e, the capacity of one parking facility j has to be divided over the number of connections from exits e to j. In the same way we introduce the relative service rate of an exit by

µ∗_e =X j∈J 1 P e∈Eyej yejµj. Define

Die = travel distance from source i to exit e

The buffer capacity of an exit represents the number of cars in the queue that may at most appear. As all parameters are determined, we are able to assign traffic from inflow points to exits by optimising

min x_ietk X tk∈T X e∈E zetk+ xietkDie such that X e∈E xietk = 1 ∀tk∈ T , ∀i ∈ I (3.7) P_e∗− X tk∈T X i∈I xietkyieλ tk i ≥ 0, ∀e ∈ E (3.8) zetk ≥ u X l=0 ηtk−l e + L t0

e1{u=k} ∀e ∈ E, ∀tk ∈ T , ∀u ∈ {k, ..., 0} (3.9)

0 ≤ zetk ≤ Be ∀e ∈ E, ∀tk∈ T (3.10)

xietk ∈ [0, 1], ∀i ∈ I, ∀e ∈ E, ∀tk ∈ T (3.11)

Given that

Λi = (λtik : tk ∈ T ) ∀i ∈ I

we can retrieve the new inflow from the exits towards the parking spots by the policy generated by the optimisation above. Thus

˜ Λe= λ˜tek = X i∈I xietkyieλ tk i : tk ∈ T !

3.5.2. Interior assignment

As the solution The new inflow vector from the exits will be used as new input to assign from the exits to the parking facilities. This is done by the optimisation

min x_ejtk X tk∈T X j∈J zjtk + xejtklj

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such that X j∈J xejtk = 1 ∀tk ∈ T , ∀i ∈ I (3.12) Pj − X tk∈T X e∈E xejtkyej˜λ tk e ≥ 0, ∀j ∈ J (3.13) zjtk ≥ u X l=0 ηtk−l j + L t0 j 1{u=k} ∀j ∈ J , ∀tk∈ T , ∀u ∈ {k, ..., 0} (3.14) 0 ≤ zjtk ≤ Bj ∀j ∈ J , ∀tk ∈ T (3.15) xejtk ∈ [0, 1], ∀e ∈ E, ∀j ∈ J , ∀tk ∈ T (3.16)

In the end we have presented a methodology for designing optimal routing policies. As the model uses general input parameter input, it can be applied onto the ArenA as well. The procedure to calculate optimal routing policies for the the ArenA case can be respresented in the flowchart below.

Figure 3.3: A flowchart, that indicates the procedure to calculate the optimal route in the case of the ArenA.

Routes and links

A network can be divided into links. Therefore a route from some point i to a point j can be described by a route-link matrix. The rows of the matrix are represented by the

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collection of links and the columns are represented by the several routes. An illustrative example to this idea can be found below.

1

2

3

4

5

l1 l2 l3 l4 l5 l6 l7

Figure 3.4: An oriented graph with links li.

In our figure the collection of links is given by L = {l1, , ..., l7}.

As an example, we will state all the routes from point 1 to 5. After that we will be able to represent the routes in a route link matrix.

r151= {l1− l3− l5− l7}

r152= {l1− l3− l6}

r153= {l1− l4− l7}

r154= {l2− l5− l7}

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The routes can be represented in the route-link matrix R by R =                1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 1 1 0                .

3.6. Extension of the assignment model

In the previous sections so far we have shown, how to use the fluid model for the case of the ArenA. However in the model, we just assigned traffic flow from A to B without taking routes into account. Routes consist of links. Links have a capacity and this capacity may not be exceeded. The question arises, how much flow may be poured into one link, such that the probability of blocking (we refer to Chapter 2) will be small. The congestion model that has been discussed in Chapter 2 (M/G/c/c queue) gives us an answer to that question. To this end, we can extend our model including more precise information about the infrastructure of a road network. Before stating the model it is necessary to introduce some additional notation.

Extended notation

Variable Description

rijkt Percentage that chooses k’th route from i to j at time t

Tijkt Travel time choosing route rijtk.

αwa

ijkt Entry of route-link matrix. 1 if link wa lies on route rijkt, zero otherwise.

λmax_w_a Maximum number of cars that is allowed on link wa to avoid blocking.

Using information about congestion, it makes sense to introduce the term travel time. The total travel time Tijk from point i to j via route k can be defined as the sum of the

travel times of the links, that a vehicle has to pass on his route. The travel time on a single link can be estimated by the linear function

Twa _{= t}free flow

wa +

λwa

λmax wa

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where tfree flow_w_a is the minimal travel time on a link. The congested travel time tcongested_w_a on a link can be retrieved by the M/G/c/c congestion model. Furthermore it can be observed, that all parameters except λwa are fixed numbers. Consequently T

wa _{is a linear}

function that depends on λwa. The inflow λwa can be retrieved by

λwa = X i∈I λt_iX j∈J X k rijktαwijkta ,

that is a linear function of the decision variables rijkt. To this end, does the travel time

Twa _{depend linearly on r}

ijkt. Finally the travel time at time t from i to j via the k’th

route rijkt can be calculated by

T_ijkt = X

wa∈L

αwa ijktT

wa_.

It can be observed, that the travel time from point i to j via route k is a linear function as well. As this linear function depends linearly on rijkt, the travel time is a linear function

as well. Therefore our objective, i.e. the total travel time is a linear function that depends on rijkt. Thus again we obtain a linear problem.

3.6.1. Linear Programme with full network topology

As a corollary the mathematical model in Section 3.4 can be extended by optimising over travel time, queueing length and walking time, taking congestion constraints into account. This multi-objective is proposed as

min rijkt X j∈J X t∈T zjt+ rijktlj + X t∈T X i∈I X j∈J X k T_ijkt such that 0 ≤ zjt ≤ Bj (3.17) X i∈I λt_iX j∈J X k rijktαijktwa ≤ λ max wa ∀t ∈ T (3.18) X j∈J X k rijkt = 1 (3.19) Pj − X t∈T X i∈I X k rijktyijλti ≥ 0, ∀j ∈ J (3.20) zjt ≥ u X l=0 ηk−l_j + Lt0 j 1{u=k}∀j ∈ J , ∀t ∈ T (3.21) rijkt∈ [0, 1] (3.22)

Our objective function is a linear combination of the total queueing length over time and destinations, total travel time from source i to destination j and the walking time penalty.

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Constraint (3.17) ensures, that queues at parking facilities will be bounded by their buffer. Constraint (3.18) tells, that all road links can take a certain maximum inflow. Constraint (3.19) and (3.22) state that rijktare decision variables. Constraint (3.20) guarantees, that

vehicles will not be sent to occupied parking destinations. Constrained (3.21) is related to Lindley’s recursion as stated before.

3.7. Application extended LP for ArenA

As the extension of the assignment model has been presented, we are able to integrate the network structure of the ArenA into our optimisation. The idea, that has been developed in Section 3.5 for the ArenA will be reworked, in fact the procedure remains the same and differs only in the optimisation parts.

3.7.1. Exterior assignment

To assign traffic in the exterior network, taking the full network topology into account, can be done by optimising

min riekts X t∈T X e∈E zets + X ts∈T X i∈I X e∈E X k Tts iek such that X e∈E yie X k riekts = 1 ∀ts∈ T , ∀i ∈ I (3.23) X i∈I λts i X e∈E X k riektsα wa iekts ≤ λ max wa ∀ts ∈ T (3.24) P_e∗−X ts∈T X i∈I yie X k riektsλ ts i ≥ 0, ∀e ∈ E (3.25) zets ≥ u X l=0 ηts−l e + L t0

e 1{u=s} ∀e ∈ E, ∀ts ∈ T , ∀u ∈ {s, ..., 0} (3.26)

0 ≤ zets ≤ Be ∀e ∈ E, ∀ts∈ T (3.27)

riekts ∈ [0, 1], ∀i ∈ I, ∀e ∈ E, ∀ts∈ T (3.28)

Given that

Λts _{= (λ}ts

i : i ∈ I) ∀ts∈ T

we can retrieve the new inflow from the exits towards the parking spots by the policy generated by the above algorithm. Thus

˜ Λts ₌ _λ˜ts e = X i∈I rietsyieλ ts i : e ∈ E ! ∀ts ∈ T

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3.7.2. Interior assignment

The new inflow vector from the exits will be used as new input to assign from the exits to the parking facilities. This is done by the optimization

min rejkts X ts∈T X j∈J zjts + rejktslj + X ts∈T X e∈E X j∈J X k Tts ejk such that X j∈J X k rejktsyej = 1 ∀ts∈ T , ∀i ∈ I (3.29) X i∈I ˜ λts e X e∈E X k riektsα wa iekts ≤ λ max wa ∀ts ∈ T (3.30) Pj− X ts∈T X e∈E X k rejktsyejλ˜ ts e ≥ 0, ∀j ∈ J (3.31) zjts ≥ u X l=0 ηts−l j + L t0 j 1{u=s} ∀j ∈ J , ∀ts ∈ T , ∀u ∈ {s, ..., 0} (3.32) 0 ≤ zjts ≤ Bj ∀j ∈ J , ∀ts∈ T (3.33) rejkts ∈ [0, 1], ∀e ∈ E, ∀j ∈ J , ∀ts∈ T (3.34)

This procedure is the same as in section 3.5 and differs only in the optimisation part.

Choice of software

In order to solve huge LP-problems, we need a suitable choice of LP-software, that can be easily combined with a programming language like MATLAB or Python. One could use CPLEX, LINDO for example, but GUROBI has been the most suitable choice in our situation. This is due to the fact, that it can handle a huge amount of variables and constraints, whereas it is easy to combine with Python, C++, MATLAB and other programming languages. As a programming language we utilised Python, as it is very easy to learn and in addition very handy in combination with GUROBI.

3.8. Online algorithm

Event traffic creates extra pressure to the road network. Although a solution to relax the pressure has been treated in this thesis, there is still the possibility that incidents occur, that cannot be predicted beforehand. Examples are accidents on the roads, unexpected change of weather and so on. The methods that have been developed cannot take care of these coincidences. This justifies the fact to develop an online algorithm, that could handle these occurrences. The online algorithm is suggested below.

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Initialise: t, λexpected_i (t), L, P Network structure, intitial weather and road conditions

while T > 0 do

Input: λi(t)expected,Estimate/Prediction of Λts, µ∗, P∗,t,L,T

Exterior Optimization and Assigning with Full Network Topology Calculate policy xiets

Output: Policy (xiets : i ∈ I, e ∈ E , ts ∈ T )

Calculate ˜Λts _{by previous policy}

Input: ˜Λts_{, µ, P ,t,L,T}

Interior Optimisation and Assigning Full Network Topology Calculate policy xejts

Output: Policy (xejts : e ∈ E , j ∈ J , ts ∈ T )

T := T − t P := Pmeasured

L := Lmeasured

λi(t)expected := λi(t)expected_measured

Update Network structure, update weather and road conditions end

Algorithm 1: Pseudocode online algorithm with full network topology. Note that updating the network structure corresponds to the change of weather, availability of roads/routes and so on. Parameter P , µ and queue lengths L at parking facilities can be obtained online. The most uncertain parameter is the prediction of the inflow Λ.

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Chapter 4 Results ArenA

Since the pilot has not been started yet, while this thesis was finished, there are no prac-tical results to discuss. However we will present results obtained by simulation. As we have pointed out the idea of an online algorithm, this means that it can not be carried out in this chapter. This is due to the fact, that knowledge is needed to predict event traffic. Furthermore live measures are not available.

In this chapter we are going to present the results of the models, that have been stated in chapter 3. As our models have been developed in a more or less general setting, we are able to apply these onto the ArenA area. Therefore several parameters have been measured for parking facilities, i.e. service rate µ, that represents the amount of cars a parking facility can take in a certain time, the capacity and the buffer for the queue. Furthermore geographical parameters, such as walking distance, distance of inflow points to exits, exits to parking facility respectively need to be obtained as well. To execute the model that takes all the routes and links into account, a congestion model such as the M/G/c/c is required. By M/G/c/c the maximal inflow rate to avoid blocking has to be calculated. Consequently the route-link matrix as stated in the previous chapter needs to be determined. The results of the calculations for M/G/c/c can be found in the appendix. Furthermore we will consider two special scenarios. One will be related only to the parking facility P1, located below the football stadium. This is due to the fact, that this parking location seems to be the hot spot at events. In addition, the problem is very illustrative, as it depicts only two queues and the effect under optimisation will be easy to perceive. In further parts of the chapter the comparison between the assignment model and routing model will be treated and results will be discussed.

4.1. Noord-Zuid case

Noom [6] has shown that one of the main hot spots is the parking facility P1 under the stadium. As the majority of visitors tends to drive via Exit 1 (we refer to Figure 3

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in the introduction), it would be interesting to illustrate the effect under optimisation. Therefore we introduce a scenario in which we will illustrate the effect of a unfavourable policy and that of an optimal policy. This will be done by comparing waiting times and observing the queue length, i.e. does the queue length exceed its buffer. Furthermore the relative difference in terms of optimisation will be presented as well. In addition, it would be interesting to see if more possibilities to change the strategy will lead to better optimisation results.

Example 4.1. The assignment model, as it has been formulated in section 3.4, will be applied on this case. Assume that the inflow pattern for users, that want to park their car in P1, is known.

Total number of vehicles that has reservated for P1: 2280. Inflowtime: 200 minutes.

Service rate P1-Zuid, P1-Noord will be tested for

µ = (µNoord, µZuid) = (4, 4), (6, 6), (8, 8) and (12, 12) respectively.

The buffer capacity is BZuid = 126 and BNoord = 120. Violation of the buffer is

equivalent to congestion on the A2, S112 respectively.

Vehicles will enter from the inflow entrances as shown in the map below Take unfavourable policy in order to simulate worst case scenario, i.e.

• 90% Amstel – P1-Zuid (10% to P1-Noord)

• 60% Watergraafsmeer – P1-Zuid (40% to P1-Noord) • 60% Diemen – P1-Zuid (40% to P1-Noord)

• 90% Ouderkerk a/d Amstel – P1-Zuid (10% to P1-Noord)

Furthermore the the amount of cars, that enters the network at the inflow point is for Amstel: 710 cars.

Watergraafsmeer: 504 cars. Diemen: 510 cars.

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Figure 4.1: unfavourable policy that distributes traffic at the inflow points to Zuid, Noord respectively.

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4.1.1. Results Noord-Zuid case

The effects on the queue size at the parking facilities P1-Zuid, P1-Noord obtained by a unfavourable policy and an optimal policy are presented in the figures below. Tables show the difference of the two policies in terms of waiting time and violation of the buffer size. Furthermore the effect under optimisation in terms of the optimisation function will be demonstrated below.

Figure 4.3: The queue length, obtained under the constant policy with µ = (4, 4). Note that the one-sided inflow at P1-Zuid leads to violation of the buffer. The optimisation gave no solution.

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Figure 4.4: The result of the constant policy above and the result of the optimized policy below for µ = (6, 6). Note that the one-sided inflow at P1-Zuid, leads to congestion, as the queue is violating the buffer constraint. The optimization led to a solution under the constraints.

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Figure 4.5: The result of the constant policy above and the result of the optimized policy below for µ = (8, 8). Note that the one-sided inflow at P1-Zuid, leads to congestion, as the queue is violating the buffer constraint. The optimization led to a solution under the constraints.

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Figure 4.6: The result of the constant policy above and the result of the optimized policy below for µ = (12, 12). Note that although there is one-sided inflow at P1-Zuid, it does not lead to congestion as the service speed can keep up with the queue. The optimization led to a solution under the constraints.

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Waiting Times in minutes (Average, Maximum) and buffer violation under unfavour. policy µ W¯Zuid W¯Noord WZuidmax WNoordmax Violation BZuid Violation BNoord

(4, 4) 131.48 0.07 237.90 1.08 yes no (6, 6) 55.15 0 103.60 0 yes no

(8, 8) 18.54 0 39.44 0 yes no

(12, 12) 0.97 0 6.85 0 no no

Waiting Times in minutes (Average, Maximum) and buffer violation under optimal policy µ W¯Zuid W¯Noord WZuidmax W

max

Noord Violation BZuid Violation BNoord

(4, 4) - - -

-(6, 6) 6.45 6.25 20 18.17 no no (8, 8) 0.93 0.36 6.50 3.75 no no

(12, 12) 0 0 0 0 no no

Table 4.1: Average waiting time (in min) ( ¯W∗), maximum waiting time (W∗max) (in min)

and Buffer violation are indicated for different values of µ under the unfavourable and optimal policy.

Value of the objective function for Unfavourable vs Optimal Policy and Relative Change µ Unfavourable Policy Optimal Policy Relative Change

(4, 4) 21049.40 -

-(6, 6) 13237.50 3127 -76,4 %

(8, 8) 5931.50 421 -93 %

(12, 12) 469.80 0 -100%

Table 4.2: Value of the objective function under the unfavourable Policy compared to the Optimal Policy and their relative change. The effect of the optimal policy in this example is significant.

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Optimal policy Noord-Zuid scenario Source/Destination P1-Noord P1-Zuid

Amstel 0.08 0.92 Watergraafsmeer 1 0

Diemen 1 0

Ouderkerk a/d Amstel 0.005 0.995

Table 4.3: Average policy that has to be applied to obtain the optimal solution for µ = (6, 6).

Waiting Times in minutes (Average, Maximum) ∆t W¯Zuid W¯Noord WZuidmax WNoordmax Calculation time

2 6.85 6.18 18.88 18.45 58.23 4 6.83 6.43 18.73 19.07 6.55 5 6.45 6.25 20 18.17 3.42 8 6.67 6.53 18.59 19.07 1.03 10 6.27 6.76 18.4 18.93 0.66 20 7.88 6.06 20.5 18.33 0.28 25 8.0 6.52 20.75 19.38 0.3 40 7.54 5.82 21.05 16.55 0.2 50 8.92 5.31 24.36 14.33 0.17 100 11.14 3.85 28.32 10.26 0.17

Table 4.4: Average waiting time ( ¯W∗) (in min), maximum waiting time (W∗max) (in min)

and calculation time (in sec) are indicated for µ = (6, 6) and different values of ∆t under Algorithm 1. It can be observed, that the less decisions are taken, the more uneven the algorithm will assign traffic.

The results in the previous example show, that as the service rate will be increased, the queue lengths trivially will decrease. However in real life, there is a limit for the service-rate at a parking facility. A service service-rate of 4 vehicles per minute per barrier would be already an achievement. To this end, a distribution of traffic is needed to control the lengths of the queues. Furthermore an optimal policy will always decrease the queues and to this end waiting times significantly, which can be seen in Table 4.2.

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4.2. Scenarios

As the Noord-Zuid case was an illustrative example, the model developed in section 3.5 will be applied onto the parameters of the ArenA in this section. Figures, showing every queue, could puzzle the reader and will not be stated. Therefore information about the waiting time and the policy will be shown.

4.2.1. Scenario I: event with 6300 cars (not uniform inflow)

scenario with properties

Total number of vehicles that needs to be assigned: 6300 Inflowtime: 200 minutes

Parking Locations will be denoted by P , i.e.

Pstring =(P 1 − Noord, P 1 − Zuid, P 2, P 3, P 4, P 5, P 6, P 10, P 12, P 18, P 21, P 22, P 24,

Amstelborgh, Dome, Endemol)

The capacity is for all parking facilities is given by

P = (1200, 1200, 2010, 366, 1248, 1354, 399, 814, 296, 291, 450, 228, 487, 700, 550, 350)

The model will be tested under the service rate µP_i , i.e.

µP₁ = (4, 4, 6, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 4, 4, 4), µP₂ = (6, 6, 8, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 4, 4, 4), µP₃ = (8, 8, 11, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 4, 4, 4).

The buffer size for P is given by

BP = (126, 120, 20, 20, 20, 20, 20, 20, 20, 20, 10, 10, 10, 10, 10, 10). The amount of cars, that enters the network at hthe inflow point is for Amstel: 1670 cars

Watergraafsmeer: 1300 cars Diemen: 1420 cars

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4.2.2. Scenario II: event with 8600 cars (uniform inflow)

scenario with properties

Total number of vehicles that needs to be assigned: 8600 Inflowtime: 240 minutes

Parking Locations will be denoted by P , i.e.

Pstring =(P 1 − Noord, P 1 − Zuid, P 2, P 3, P 4, P 5, P 6, P 10, P 12, P 18, P 21, P 22, P 24,

Amstelborgh, Dome, Endemol)

The capacity is for all parking facilities is given by

P = (1200, 1200, 2010, 366, 1248, 1354, 399, 814, 296, 291, 450, 228, 487, 700, 550, 350)

The model will be tested under the service rate µP i , i.e.

µP₁ = (4, 4, 6, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 4, 4, 4), µP₂ = (6, 6, 8, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 4, 4, 4), µP₃ = (8, 8, 11, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 4, 4, 4).

The buffer size for P is given by

BP = (126, 120, 20, 20, 20, 20, 20, 20, 20, 20, 10, 10, 10, 10, 10, 10). The amount of cars, that enters the network at the inflow point is for Amstel: 2400 cars

Watergraafsmeer: 1000 cars Diemen: 2000 cars

Ouderkerk a/d Amstel: 3200 cars

4.3. Results assignment model

In this section we are going to apply the assignment model to scenario I and II as described above. The results for scenario I are stated for the unfavourable policy and the optimal policy.

Evacuation Policies for Events

Master Thesis