Queueing and Traffic
Graduation committee:
Chairman and secretary: prof. dr. P.M.G. Apers
Supervisor: prof. dr. R.J. Boucherie
Co-Supervisor: dr. J.C.W. van Ommeren
Members:
dr. techn. B.F. Nielsen Technical University of Denmark
prof. dr. R. N´u˜nez Queija University of Amsterdam
prof. dr. N.M. van Dijk University of Amsterdam
prof. dr. ir. E.C. van Berkum University of Twente
dr. ir. A. Al Hanbali University of Twente
dr. G.J. Still University of Twente
CTIT
CTIT Ph.D.-thesis Series No. 14-339
Centre for Telematics and Information Technology University of Twente
P.O. Box 217 – 7500 AE Enschede, The Netherlands
Beta Ph.D.-thesis Series No. D190
Beta Research School for Operations Management and Logistics
Eindhoven University of Technology P.O. Box 513 – 5600 MB
Eindhoven, The Netherlands
This work was financially supported by the Centre for Telematics and Information Technology (CTIT) of the University of Twente.
Typeset with LATEX. Printed by Ipskamp Drukkers.
Cover designed by Niek Ba¨er ISBN: 978-90-365-3812-1
ISSN: 1381-3617 (CTIT Ph.D.-thesis Series No. 14-339) DOI: 10.3990/1.9789036538121
Copyright c 2015, Niek Ba¨er, Deventer, the Netherlands
All rights reserved. No part of this publication may be reproduced without the prior written permission of the author.
QUEUEING AND TRAFFIC
PROEFSCHRIFT
ter verkrijging van
de graad van doctor aan de Universiteit Twente, op gezag van de rector magnificus,
prof. dr. H. Brinksma,
volgens besluit van het College voor Promoties, in het openbaar te verdedigen
op vrijdag 12 juni 2015 om 14.45 uur
door
Niek Ba¨er
geboren op 29 juli 1986 te Lelystad, Nederland
Dit proefschrift is goedgekeurd door: prof. dr. R.J. Boucherie (promotor)
Voorwoord
Ruim vier jaar geleden begon ik met het onderzoek voor dit proefschrift en er zijn momenten geweest dat ik niet had verwacht dat er ooit nog een proefschrift zou komen. Dat dit eindresultaat alsnog tot stand is gekomen komt door de steun van groot aantal mensen, te beginnen bij mijn promotor.
Richard, wij leerden elkaar kennen toen ik in mijn vierde jaar mijn Bachelor afs-loot bij SOR. Onder jouw en Judiths begeleiding hebben we een subsidi¨eringsmodel gemaakt voor het Nederlands Filmfonds. Ik raakte ge¨ınspireerd door jouw stijl van werken en toen ik na mijn studie besloot om te gaan promoveren, was de keuze voor de leerstoel SOR snel gemaakt. Door de jaren heen wist jij je altijd te interesseren voor mijn werk en wist je, door het stellen van kritische vragen, ons onderzoek vaak op een hoger niveau te tillen. Richard, wij konden niet altijd samen door ´e´en deur, maar ik ben je heel erg dankbaar dat jij mijn promotor hebt willen zijn.
Jan-Kees, vele malen ben jij voor mij een luisterend oor geweest. Als er weer een artikel afgewezen was, of wanneer de voorzitter van een sessie op een conferentie het op mij gemunt had, of gewoon zomaar; jouw deur stond altijd voor mij open, dank daarvoor. We hebben ook vele interessante discussies gehad over zowel jouw onderzoek als het mijne. Toch blijft er na al die jaren nog n vraagstuk onbeantwoord:
de eigenaardige overeenkomsten tussen q = k· v en de Wet van Little. Beide zijn
we er van overtuigd dat er een verband bestaat tussen die twee, en het lijkt mij leuk om hier samen eens een regenachtige zondagmiddag, of een zonnige (maar dan wel op het terras), aan te besteden.
Ahmad, every now and then, you joined us in our struggle to fully understand the threshold queues. Your constant enthusiasm is inspiring and it made me really enjoy the discussions we had. Thank you for the many fruitful discussions we had.
Bo, during the autumn of 2013 I have visited you for three months in Lyngby. I was reluctant to leave home for several weeks, but arriving in Denmark, I was warmly welcomed by you and David, the two of you made me feel right at home. David, I really enjoyed our coffee breaks and lunch times during my visit. I am still working on losing all the weight I gained by eating all those cookies. I am really grateful to the both of you: Tak!
Furthermore, I would like to thank the members of my graduation committee. I am honoured that you were all willing to participate in my graduation committee and I want to thank you for your careful reading of my thesis.
Mijn dank gaat uit naar al mijn collega’s aan de UT voor de leuke gesprekken bij de koffiemachine (ook al was de koffie niet te drinken) en tijdens de gezellige lunches. Jullie aanwezigheid zorgde voor veel plezier op de werkvloer.
viii Voorwoord
Maartje, ik kan er natuurlijk niet onderuit komen om jou bij naam te noemen in mijn voorwoord. Lange tijd hebben wij een kantoor gedeeld in de Zilverling en hebben we de grootste lol gehad om wie de slechtste muziek kon draaien. Door jou is mijn kennis van carnavalskrakers flink opgevijzeld, iets waarvoor ik nog steeds niet weet of ik daar nou dankbaar voor moet zijn of niet. Desalniettemin heb ik heel erg genoten van onze tijd als kamergenoten.
Een speciaal dankwoord voor mijn familie en vrienden, voor jullie steun en vertrouwen. In het bijzonder wil ik mijn paranimfen Hans en Chiel bedanken. Ik waardeer het enorm dat jullie mij willen bijstaan tijdens de afronding van mijn pro-motie.
Als laatste wil ik jou bedanken, Ellen. Jij weet als geen ander hoeveel moeite ik heb gehad om mijn promotie af te ronden en zonder jouw onvoorwaardelijke steun had ik het zeker nooit gehaald. Juist op die momenten dat ik jou het hardste nodig had, was jij er voor mij, no matter what. Daarvoor ben ik je voor altijd dankbaar.
Niek
Contents
1 Introduction 1
1.1 Motivation . . . 1
1.2 Outline of the thesis . . . 2
I
Traffic
5
2 Queueing models for highway traffic flows 7 2.1 Introduction . . . 72.2 Literature . . . 7
2.2.1 Microscopic, mesoscopic and macroscopic models . . . 8
2.2.2 Fundamental diagram . . . 10
Single-regime traffic models . . . 10
Multi-regime traffic models . . . 11
2.2.3 Queueing theory in uninterrupted traffic flows . . . 12
Heidemann’s model . . . 12
Jain and Smith’s Model . . . 15
2.3 Concluding Remarks . . . 15
3 Threshold queueing describes the fundamental diagram of unin-terrupted traffic 17 3.1 Introduction . . . 17
3.2 Two-stage M/M/1 threshold queue . . . 19
3.2.1 Model validation . . . 21
3.2.2 Sensitivity analysis of the fundamental diagram for the two-stage M/M/1 threshold queue . . . 22
3.3 Four-stage M/M/1 feedback threshold queue . . . 22
3.3.1 Model validation . . . 28
3.3.2 Sensitivity of the fundamental diagram for the four-stage M/M/1 feedback threshold queue. . . 29
3.4 Summary and Conclusion . . . 31
4 A tandem network of M/M/1 threshold queues 33 4.1 Introduction . . . 33
x Contents
4.2 A three queue tandem network of two-stage M/M/1 threshold queues 34
4.3 A three queue tandem network of four-stage M/M/1 feedback
thresh-old queues . . . 38
4.4 Summary and Conclusion . . . 42
II
Queueing
45
5 Matrix analytic methods 47 5.1 Introduction . . . 475.2 Phase-Type distribution . . . 47
5.3 Markovian Arrival and Service Processes . . . 50
5.4 Quasi-Birth-and-Death processes . . . 52
5.4.1 Stationary distribution for a QBD . . . 53
5.4.2 Stationary distribution for a finite QBD . . . 54
5.4.3 Fundamental matrix of a transient QBD . . . 54
5.5 Level Dependent QBD processes . . . 56
5.5.1 Stationary distribution for a LDQBD . . . 57
5.5.2 Stationary distribution for a finite LDQBD . . . 58
5.5.3 Fundamental matrix of a transient LDQBD . . . 58
6 The P H/P H/1 multi-threshold queue 61 6.1 Introduction . . . 61
6.2 Model description . . . 63
6.3 Steady-state analysis . . . 65
6.4 Examples . . . 69
6.4.1 Extended traffic model . . . 70
6.4.2 Le Ny and Tuffin [61] . . . 71
6.4.3 Choi et al. [21] . . . 73
6.5 Fundamental diagram of traffic . . . 74
6.6 Summary and Conclusion . . . 76
7 A successive censoring algorithm for a system of connected LDQBD-processes 77 7.1 Introduction . . . 77
7.2 Model description . . . 78
7.3 Successive censoring algorithm . . . 82
7.3.1 Reduction step k . . . 84
7.3.2 Intermediate step . . . 90
7.3.3 Expansion step k . . . 91
7.3.4 Inverse of−Qkk,k . . . 91
7.4 Simplified successive censoring algorithm . . . 92
7.5 Complexity analysis . . . 95
7.6 Demonstration of the successive censoring algorithm . . . 97
Contents xi
8 An iterative aggregation method for a queueing network with
finite buffers 101
8.1 Introduction . . . 101
8.2 Model description . . . 103
8.3 Iterative aggregation method . . . 104
8.3.1 Aggregation of a Markovian Arrival Process . . . 104
8.3.2 Iterative procedure . . . 109
8.3.3 Marginal distribution . . . 111
8.4 Numerical results . . . 112
8.4.1 A tandem of M/M/1/N queues . . . 113
8.4.2 A tandem of two-stage M/M/1/N threshold queues . . . 117
8.4.3 A tandem of four-stage M/M/1/N feedback threshold queues 120 8.5 Summary and Conclusion . . . 130
8.A Sets and Entrance states . . . 132
8.A.1 A tandem of M/M/1/N queues . . . 132
8.A.2 A tandem of two-stage M/M/1/N threshold queues . . . 133 8.A.3 A tandem of four-stage M/M/1/N feedback threshold queues 135
Concluding remarks 139
Bibliography 141
Summary 149
Samenvatting 151
CHAPTER 1
Introduction
1.1
Motivation
Few things unite cultures more than the frustration of sitting in a line of stationary traffic, with no discernible reason for the blockage and no end in sight.
The Economist, November 3rd 2014
Traffic jams are everywhere, some are caused by constructions or accidents but most occur naturally. These “natural” traffic jams may be a result of variable driving speeds combined with a high number of vehicles. To prevent these traffic jams, we have to understand traffic in general, and to understand traffic we have to understand the relations between the three key parameters of highway traffic, speed, the average speed of a vehicle, flow, the number of vehicles passing a reference point, and density, the number of vehicles on the road. These relations are often displayed in the so-called fundamental diagram of traffic, see Figure 2.1 for an example. A well-known relation between these parameters is that flow equals the product of speed and density. This thesis demonstrates that queueing theory can offer new insights in the remaining relations between these three parameters.
An important aspect of traffic is congestion. Congestion is a phenomenon that arises when roads become more and more crowded, speeds will decrease and travel times will increase. This phenomenon is well-known in queueing theory. In basic queueing models we observe that the average queue length and average time spent in the queue increases when the density increases. In the past, see Boon [14], queueing theory has shown its usefulness in the analysis of intersections where vehicles must wait before crossing, however, in the analysis of highway traffic, queueing theory has received little attention.
Another aspect of highway traffic is its hysteretic behaviour, explained by Helbing in [46]. He describes traffic by two different phases, non-congested and congested, and describes a hysteretic transition between these two phases. On a quiet highway, traffic is non-congested and average speeds are close to the speed limit. However, when density increases, vehicles will interact with one another until, at some critical point, a transition occurs and traffic becomes congested. In this phase, density is high and average speeds are well below the speed limit, i.e., a traffic jam emerges.
2 1. Introduction
This traffic jam is resolved when the number of vehicles on the road drops below a critical number and traffic becomes non-congested again. In general, it is assumed that the two critical numbers are unequal.
In this thesis we model the hysteretic behaviour of traffic on a single highway sec-tion by a so-called threshold queue. The service and arrival rates of these threshold queues are controlled by a threshold policy based on the queue length, i.e., they are changed when the number of customers in the queue reaches certain critical values. The analysis of these threshold queues results in a relation between the density of the queue and average time spent in the queueing system. With the help of Heide-mann’s model in [40] we are able to translate this to a relation between density and speed for highway traffic.
1.2
Outline of the thesis
This thesis is organised in two parts. Part I consists of the Chapters 2, 3, and 4 and discusses the application of queueing models in the analysis of highway traffic. Part II consists of the Chapters 5, 6, 7, and 8 and discusses the underlying queueing models in more detail.
Chapter 2 gives a broad overview of the literature on traffic models and provides the basis for Part I. In this chapter we give a historical overview of the traffic models used to create the fundamental diagram of traffic. Furthermore, we give an overview of the traffic models based on queueing theory. The traffic models in this thesis are based on Heidemann’s model which makes it possible to obtain the fundamental diagram from a queueing model.
In Chapter 3, we introduce two queueing models to model traffic on a single highway section, the two-stage threshold queue and the four-stage feedback thresh-old queue. The two-stage threshthresh-old queue models the hysteretic behaviour of traffic on a single section. However, the hysteretic behaviour is not restricted to a single section; once a certain highway section is jammed, it will eventually affect the pre-vious section. This feedback process is modelled by the four-stage feedback thresh-old queue. We obtain the fundamental diagram for each queue using Heidemann’s model. Next, both queueing models are fitted such that their fundamental diagrams resemble that of traffic on a Danish highway. Finally, we present a sensitivity analy-sis on the system parameters in which we investigate the effects of small changes in parameters on the shape of the fundamental diagram. Chapters 2 and 3 are based on [9] N. Baer, R.J. Boucherie, and J.C.W. van Ommeren. Threshold queueing describes the fundamental diagram of uninterrupted traffic. Memorandum 2000, Department of Applied Mathematics, University of Twente, Enschede, The Nether-lands, 2012.
In Chapter 4, we model a series of highway sections. We introduce a tandem network of two-stage threshold queues and a tandem network of four-stage feedback threshold queues. These networks are analysed numerically and the fundamental
1.2 Outline of the thesis 3
diagram of each separate queue is obtained using Heidemann’s model. We close this chapter with a sensitivity analysis on the system parameters of the network.
Chapter 5 is an introductory chapter to Part II. It presents known results from the field of Matrix analytic methods on Phase-Type distributions, Markovian Ar-rival Processes and Markovian Service Processes, and both Level Dependent and Level Independent Quasi-Birth-and-Death processes. The theory in this chapter is frequently used in Chapters 6, 7, and 8.
The introduction of the two-stage threshold queue and the four-stage feedback threshold queue in Chapter 3 leads to the introduction of the P H/P H/1 multi-threshold queue in Chapter 6. While the queueing models in Chapter 3 are limited to 2 or 4 stages and restricted to exponential distributions, the P H/P H/1 multi-threshold queue is a logical extension that relaxes both restrictions. In Chapter 6 we formally present the P H/P H/1 multi-threshold queue and we discuss why known results from Matrix analytic methods are not applicable. Next, we discuss how the stationary distribution of this queueing model can be obtained. Finally, we show that the fundamental diagram for the P H/P H/1 multi-threshold can be obtained and we perform a sensitivity analysis on the chosen Phase-Type distributions. Chap-ter 6 is based on
[10] N. Baer, R.J. Boucherie, and J.C.W. van Ommeren. The P H/P H/1 multi-threshold queue, volume 8499 of Lecture Notes in Computer Science, pages 95–109. Springer International Publishing, 2014.
The tandem networks discussed in Chapter 4 are special Markov chains that can-not be characterised as (Level Dependent) Quasi-Birth-and-Death processes. They turn out to be systems of connected Quasi-Birth-and-Death processes, which means that the Markov chain can be divided into subsets, each describing a Quasi-Birth-and-Death process. In Chapter 7 we extend this class of Markov chains and discuss a system of connected Level Dependent Quasi-Birth-and-Death processes, in which the Markov chain can be divided into subsets, each describing a Level Dependent Quasi-Birth-and-Death process. We provide a successive censoring algorithm to obtain the stationary distribution of such a system and investigate the possible connections between different subsets. Chapters 4 and 7 are based on
[7] N. Baer, A. Al Hanbali, R.J. Boucherie, and J.C.W. van Ommeren. A suc-cessive censoring algorithm for a system of connected qbd-processes. Memorandum 2030, Department of Applied Mathematics, University of Twente, Enschede, The Netherlands, 2013.
The analysis of the tandem networks in Chapter 4 is computationally very de-manding causing the networks to be limited to three queues. In Chapter 8 we present an iterative aggregation method which gives an approximation of a single queue in a larger tandem network. While focusing on a single queue in the network, the aggre-gation method aggregates all upstream network behaviour into a Markovian Arrival
4 1. Introduction
Process, and all downstream network behaviour into a Markovian Service Process. This is done in an iterative fashion, aggregating one queue in each iteration, until all upstream (or downstream) queues are aggregated. The resulting queueing model is then analysed using results from the field of Matrix analytic methods. The iterative aggregation method will be compared to simulation results of a tandem network of two-stage threshold queues, a tandem network of four-stage feedback threshold queues, and a tandem network of M/M/1/k queues. Chapter 8 is based on
[8] N. Baer, A. Al Hanbali, R.J. Boucherie, and J.C.W. van Ommeren. An iterative aggregation method for a tandem queueing network with finite buffers. Unpublished work, 2015.
Part I
CHAPTER 2
Queueing models for highway traffic flows
2.1
Introduction
In 1935, Greenshields [37] captured the empirical relation between speed, flow and density for uninterrupted traffic in the fundamental diagram, see Figure 2.1. Math-ematical models for uninterrupted traffic have been developed and the fundamental diagram in its basic form is now well-understood, see e.g. Newell [79, 80, 81] for a concise exposition.
Traffic jams are a major concern for highway operation and may occur in high density traffic due to variability in driving speed. A wide range of traffic models has been developed over the past decades. These models are mainly from statistical physics and non-linear dynamics, see [23, 46]. Congestion due to variable arrival and/or service processes is the main topic of queueing theory, that, however, has hardly been invoked to analyse the fundamental concepts of uninterrupted traffic flows. Notable exceptions are the models introduced by Heidemann [41] and Jain and Smith [50]. However, these models do not capture the empirical shape of the fundamental diagram for modern traffic as shown in Figure 2.2.
Section 2.2 gives a brief overview of the literature on the fundamental diagram for uninterrupted traffic flows and on queueing models for uninterrupted traffic flows.
2.2
Literature
Congestion is a key concept in queueing theory that models both the mesoscopic and macroscopic effects of randomness on delay and sojourn times. In interrupted traffic flows, where queues arise naturally at an intersection, queueing theory has been a popular tool since the early 1940s, see Boon [14] for a recent survey. In uninterrupted traffic, however, queueing models have received far less attention in literature.
In this section we first focus on the three levels of detail for traffic models. Next, we provide an overview of the results on the fundamental diagram in Section 2.2.2, starting from the literature surveys [83, 103, 104]. We close this section with an overview on queueing models for uninterrupted traffic flows.
Throughout the section, and in the remainder of this thesis, we use the following notation. Let k denote the traffic density, v the mean speed of a vehicle, q the flow
8 2. Queueing models for highway traffic flows 0 0.2 0.4 0.6 0.8 1 0 6 12 18 24 30 Density (k) Sp eed (v ) 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 Density (k) Flo w (q ) 0 2 4 6 8 0 6 12 18 24 30 Flow (q) Sp eed (v )
Figure 2.1: Fundamental diagram from the experimental data of Greenshields [37].
rate. By definition, these three parameters of traffic are related according to q = k· v.
Furthermore, let vf denote the desired mean speed or free flow speed, and kjam the jam or maximum density.
2.2.1
Microscopic, mesoscopic and macroscopic models
Uninterrupted or highway traffic flow models can be characterised by their level of detail: microscopic, mesoscopic and macroscopic.
In microscopic models a high level of detail is used in which each individual driver is characterised by its position and behaviour over time [47, 82]. In gen-eral, microscopic models lead to systems of (ordinary) differential equations [46].
2.2 Literature 9 0 50 100 150 0 50 100 150 200 250 Density (k) Flo w (q ) 0 50 100 150 0 2 4 6 8 10 Density (k) Sp eed (v ) 0 50 100 150 200 250 0 2 4 6 8 10 Flow (q) Sp eed (v )
Figure 2.2: Fundamental diagram from experimental data for modern traffic Sugiyama [93]. The flow-density diagram is fitted to the experimental data.
Well-known micropscopic models are the car-following model [15, 20], the cellular automata model [75] and the lane-changing model [1].
In mesoscopic models the individual drivers are not distinguished [47, 82]. The behaviour of drivers is characterised in terms of the probability density f (x, v, t) of vehicles at position x with speed v at time t. Examples of mesoscopic models are headway distribution models [17] and gas-kinetic continuum models [87, 88].
Macroscopic models have the lowest level of detail and consider only three vari-ables for each position x and time t: average speed v(x, t), traffic flow q(x, t) and spatial vehicle density k(x, t), that are related as q(x, t) = v(x, t)·k(x, t). These three variables are often presented in the fundamental diagram. Two classical examples of macroscopic models are the Lighthill-Whitham-Richards model [67, 68, 90] and the Payne model [84].
10 2. Queueing models for highway traffic flows
For more elaborate surveys on traffic models for uninterrupted traffic flows, see [12, 46, 47, 82, 100].
2.2.2
Fundamental diagram
The seminal work of Greenshields [37] has initiated the research to find a simple formula capable of capturing the fundamental diagram of traffic flows. As can be seen in Figure 2.1, the fundamental diagram proposed by Greenshields has a linear speed-density relationship given by a single formula, see Table 2.1. Greenshields model is an example of a single-regime traffic model. In multi-regime traffic models the speed-density relationship is a piecewise function, depending on the regime of the highway section, for instance free-flow or congested, see Table 2.2. These macroscopic traffic models all follow a key equation from
Below, we first consider single-regime models. Subsequently, we consider multi-regime models.
Single-regime traffic models
For an overview of single-regime traffic models that have been developed since Green-shields model, see Table 2.1 adapted from Wang et al. [103]. In this table we denote by vcand kcthe speed and density at capacity, i.e., when flow q equals the maximum
flow qmax. Furthermore, let ωv denote the jam wave speed. Wang et al. [103] fit
several single-regime models from Table 2.1 to empirical data. Greenshields model describes a linear relationship between speed and density, causing a parabolic rela-tionship between flow and density. Several models were introduced to better capture realistic traffic flows. The Greenberg model [36] proposes a logarithmic speed-density relationship based on fluid-flow analogies. This logarithmic approach performs well under congested conditions (v = 0 when k = kjam), but it does not satisfy boundary
conditions at low densities (v→ ∞ as k → 0). Underwood [94] proposes an
exponen-tial speed-density relationship which obeys the boundary conditions at low densities
but does not satisfy boundary conditions under congested conditions (v → 0 for
k → ∞). This limitation is also found in the Northwestern model [29] in which
a bell-shaped speed-density relationship was proposed. The Drew model [30] and Pipes-Munjal model [74] resemble the simple formulation of Greenshields with the introduction of an extra parameter n. This parameter n allows for extra degrees of freedom in fitting the models to empirical data. The general form of the Drew and Pipes-Munjal model is captured by the K¨uhne and R¨odiger model [56]. Recently, an extension was made to the Greenshields model by Mahmassani et al. [72], creating the modified Greenshields model for single-regime. In this model, which resembles
both the Drew and Pipes-Munjal model, a parameter v0denoting the minimal speed
at jam density was introduced, such that v = v0 when k = kjam. Based on generat-ing functions, Del Castillo and Benitez [27, 28] proposed a speed-density relationship based on the free flow speed vf, jam density kjam, and jam wave speed ωv. The au-thors note that their model can also be obtained from Newell’s car-following model in [78]. The Van Aerde model [95] was obtained from the Van Aerde car-following
2.2 Literature 11 Model Function Greenshields v = vf 1− k kjam , Greenberg v = vcln k jam k , Underwood v = vfe− k kc, Northwestern v = vfe− 1 2( k kc) 2 , Drew v = vf 1− k kjam n+12 , Pipes-Munjal v = vf 1− k kjam n ,
K¨uhne and R¨odiger v = vf
1− k kjam ab , Modified Greenshields v = v0+ (vf− v0) 1− k kjam n , Newell v = vf 1− e−vfλ 1 k− 1 kjam ,
Del Castillo and Benitez v = vf
1− e|ωv |vf 1−kjamk , Van Aerde k = c 1 1+vf −vc2 +c3v, MacNicholas v = vf kn jam−kn kn jam+ckn , Wang v = v0+ (vf− v0) 1 + ek−ktθ1 −θ2 .
Table 2.1: Single-regime speed-density relationships.
model and contains three extra parameters c1, c2 and c3. These parameters can be obtained by solving the boundary conditions, see [89]. It is shown by MacNicholas [71] that a model similar to the Van Aerde model can be obtained with fewer pa-rameters. A comparison between the MacNicholas and Van Aerde model is made in [71]. Recently, Wang et al. [104] proposed a logistic speed-density model based on five parameters. An important parameter in their model is kt, the density at which the transition from free-flow to congested traffic occurs. The parameters θ1 and θ2 are used to fit the traffic model. For a more concise overview and visual comparison of a selection of the models in Table 2.1, see [83, 103].
Multi-regime traffic models
In a multi-regime traffic model it is possible to use multiple single-regime models at the same time, one for each regime. This way, better fits compared to single-regime traffic models could be obtained. A drawback of multi-single-regime models lies in finding the point at which regimes change. Some multi-regime models occurring in literature are listed in Table 2.2. Since any combination of single-regime models can serve as a multi-regime model, this list is by no means extensive. The simplest
12 2. Queueing models for highway traffic flows Model Function Edie v = ( vfe− k kc, c lnkjam k , k≤ kc, k > kc. Triangular v = ( vf, vfkc kc−kjam 1−kjam k , k≤ kbp, k > kbp. Modified Greenshields v = vf, v0+ (vf− v0) k jam−k kjam−kbp α , k≤ kbp, k > kbp. Modified Greenberg v = ( vf, c lnkjam k , k≤ kbp, k > kbp.
Table 2.2: Multi-regime speed-density relationships.
multi-regime traffic model is the Triangular model, named after the triangular shape of the flow-density plot, as used by Newell [80]. The corresponding speed-density relationship is given by the Triangular traffic model in Table 2.2. Here, kbp denotes the break-point density, the density at which the regime changes. Similar speed-density relationships are obtained with the modified Greenberg model [29], and the modified Greenshields model [72]. All three models describe a constant speed when k ≤ kbp after which the speed decreases to zero (or v0). The Edie model [31] is a combination of the Underwood model and the Greenberg model of Table 2.1. The Underwood model is used to describe traffic flows at low density (k ≤ kc) and the Greenberg model is used to describe traffic flows during congestion (k > kc). In [29] two more multi-regime models were introduced: one consisting of two regimes while the other has three regimes. Both models assume a linear speed-density relation during the regimes and both models were fitted to empirical data in [29, 73].
2.2.3
Queueing theory in uninterrupted traffic flows
Two main queueing theoretic approaches can be identified to model uninterrupted traffic: the queue with waiting room of Heidemann [41] and the queue with blocking of Jain and Smith [50].
Heidemann’s model
Heidemann [41] introduces an M/G/1 queueing system to model highway traffic. The
server in the queueing system corresponds to a highway segment of length 1/kjam,
which is the minimal part of the highway each vehicle requires. The mean service time in the queue is the average time it takes a vehicle in free flow traffic to cross the segment: E[B] = 1/(kjam· vf). The traffic density outside the chosen segment is k, so that the mean time between two arrivals is E[A] = 1/(k· vf). The probability
2.2 Literature 13 0 50 100 150 200 250 0 2500 5000 7500 10000 Density (k) Flo w (q ) cs= 0.25 cs= 0.50 cs= 1.00 cs= 2.00 cs= 4.00 0 50 100 150 200 250 0 20 40 60 80 100 120 140 Density (k) Sp eed (v ) cs= 0.25 cs= 0.50 cs= 1.00 cs= 2.00 cs= 4.00 0 2500 5000 7500 10000 0 20 40 60 80 100 120 140 Flow (q) Sp eed (v ) cs= 0.25 cs= 0.50 cs= 1.00 cs= 2.00 cs= 4.00
Figure 2.3: Fundamental diagram obtained with Heidemann’s M/G/1 queue, kjam= 200, vf = 120, and varying coefficient of variation cs.
of the server being busy is defined by 1− π0= E[B]
E[A] =
k kjam
, (2.1)
where π0denote the probability of the queue being empty.
In the M/G/1 queue, an arriving vehicle may find the server busy upon arrival and must wait for service. The total time required to cross the segment is the sojourn time, E[S], which is the sum of the waiting time and the service time. For the M/G/1 queue the Pollaczek-Khintchine formula [105] gives
E[S] = E[B] 1 + ρ 1− ρ· (1 + c2 s) 2 ,
14 2. Queueing models for highway traffic flows 0 250 500 750 1000 0 750 1500 2250 3000 Density (k) Flo w (q ) Exponential Linear 0 250 500 750 1000 0 15 30 45 60 Density (k) Sp eed (v ) Exponential Linear 0 750 1500 2250 3000 0 15 30 45 60 Flow (q) Sp eed (v ) Exponential Linear
Figure 2.4: Fundamental diagram obtained with Jain and Smith’s M/G/c/c queue for a linear and exponential decreasing speed, kjam= 200, vf = 55 mph.
where cs is the coefficient of variation of the service time. The speed, v, of a vehicle passing the segment then is
v =1/kjam
E[S] . (2.2)
Figure 2.3 gives the fundamental diagram for the M/G/1 queue for various choices of cs.
Generalisations of Heidemann’s model include the transient analysis of the M/G/1 queue [42, 43, 44]. Vandaele et al. [102] and Van Woensel [98] consider the G/G/s queue. Validation of their queueing model [99, 101] shows that the M/G/1 queue models non-congested traffic and that the G/G/s is a more suitable model for
con-gested traffic. Accidents were incorporated by Baykal-G¨ursoy et al. [11] in an
2.3 Concluding Remarks 15
Jain and Smith’s Model
An alternative approach to a queueing model for highway traffic is the M/G/c/c model of Jain and Smith [50], where an arrival that finds all servers occupied is
blocked and cleared (lost). Their model is based on pedestrian flows in
emer-gency evacuation planning [107]. The servers correspond to a road segment. As the M/G/c/c model does not incorporate waiting, the speed of a vehicle is obtained by the service time that equals the sojourn time in the queue. The capacity C of a road segment equals the number of vehicles that fit in this segment, i.e., the product of the jam density, kjam, the length of the road segment, L, and the number of lanes,
N , C = kjam· L · N. The mean speed of a vehicle, Vn, depends on the number of
vehicles n on the road segment and is now a function that is input for the model. Two functions for Vn are considered [50, 107]
Vn= vf
C (C + 1− n) ,
that linearly decreases in the number of vehicles on the segment, and for suitable constants γ and β, see [107],
Vn = vf· exp
− n− 1β γ
,
that exponentially decreases with the number of vehicles. In Figure 2.4 we present the fundamental diagram obtained with the M/G/c/c queue for both speed functions Vn.
A network of M/G/c/c queues was considered by Cruz et al. [25] and Cruz and Smith [24]. In this network a blocked customer will occupy its server until it is no longer blocked. Simulation techniques and approximations were used to derive blocking probabilities, throughput, mean queue length and mean waiting times.
2.3
Concluding Remarks
The queueing models in literature result in a fundamental diagram similar to the fundamental diagram by Greenshields. However, these models do not capture the hysteresis effect as seen in modern-day traffic flows. In Chapter 3 we introduce the two-stage M/M/1 threshold queue and a four-stage M/m/1 feedback threshold queue which mimic the hysteresis effect and we show that they capture the resulting capacity drop.
CHAPTER 3
Threshold queueing describes the
fundamental diagram of uninterrupted
traffic
3.1
Introduction
Chapter 2 gives an overview of queueing models in literature to analyse highway traffic flows. These queueing models do not capture the capacity drop, an important aspect of modern traffic flows characterised by the sharp descent in the fundamental diagram, see Figure 3.1. Helbing [46] explains the capacity drop as the transition from non-congested traffic to congested traffic. When the density of vehicles reaches a certain critical value, ρ2, traffic will become congested and the average speed is significantly lower than in non-congested traffic. When density decreases again and reaches another critical value, ρ1 ≤ ρ2, a transition from congested traffic to non-congested traffic occurs and traffic flows recover. In the density interval [ρ1, ρ2] both congested and non-congested traffic flows exist, which indicates the existence of hysteresis. As is shown in our numerical results, it is precisely this hysteresis effect captured by the threshold queue introduced in this chapter that results in the capacity drop in the fundamental diagram of Figure 3.1 observed in empirical data for speed, flow and density.
In this chapter we discuss two special cases of the P H/P H/1 multi-threshold queue which is introduced in Chapter 6, namely the two-stage M/M/1 threshold queue and the four-stage M/M/1 feedback threshold queue. We model the hysteretic behaviour of traffic, which follows from Helbing [46], by adjusting the service rates in the two-stage M/M/1 threshold queue. Since this hysteretic behaviour is not restricted to a single section alone, we also introduce the four-stage M/M/1 feedback threshold queue. In this queueing model we adjust both the service rates and the arrival rates and model both the hysteretic behaviour of traffic on a single highway section, as well as the hysteretic interaction between a highway section and the preceding section.
Both queueing systems have exponential service times and Poisson arrivals, and the arrival and service rates are controlled by a threshold policy. Once the queue length reaches certain upper thresholds, the arrival and/or service rates are adjusted.
18 3. Threshold queueing model for uninterrupted traffic 0 50 100 150 0 50 100 150 200 250 Density (k) Flo w (q ) 0 50 100 150 0 2 4 6 8 10 Density (k) Sp eed (v ) 0 50 100 150 200 250 0 2 4 6 8 10 Flow (q) Sp eed (v )
Figure 3.1: Fundamental diagram from experimental data for modern traffic Sugiyama [93]. The flow-density diagram is fitted to the experimental data.
They are changed to their original values once the queue length drops below the corresponding lower threshold. We will modify Heidemann’s model [41], explained in detail in Section 2.2.3, to obtain the fundamental diagram for both queueing models and give the best-fit, using the method of least squares on both the flow and on the speed, to empirical data for Danish highways [86].
The remainder of this chapter is divided into two sections, Section 3.2 dis-cusses the two-stage M/M/1 threshold queue, and the four-stage M/M/1 feedback threshold queue is discussed in Section 3.3.
3.2 Two-stage M/M/1 threshold queue 19 0 1 · · · L− 1 L · · · U L · · · U U + 1 · · · N− 1 N λ λ λ λ λ λ λ λ λ λ λ λ λ µH µH µH µH µH µH µL µL µL µL µL µL µL
Figure 3.2: Transition Diagram for the two-stage M/M/1 threshold queue.
3.2
Two-stage
M/M/1 threshold queue
Consider a single server queue with finite buffer N in which service rates are con-trolled by a threshold policy. Customers arrive according to a Poisson process with rate λ and require, upon service, an exponentially distributed service time, depend-ing on the stage of the queue. In stage 1, the non-congested stage, the service rate is µH, and in stage 2, the congested stage, the service rate is µL, with µL≤ µH. The transition from the non-congested stage to the congested stage, and back, are con-trolled by a threshold. Once an arrival occurs while the queue length is U , the stage changes from non-congested to congested. The stage changes back from congested to non-congested when the queue length is L and a departure occurs. The state space and transition rates of this threshold queue are depicted in Figure 3.2. The stationary queue length probabilities π for the two-stage M/M/1 threshold queue can readily be obtained from standard Markov chain analysis, see also Le Ny and Tuffin [61]. Let ρ = λ
µH and δ =
λ
µL and π(i, j) the probability of having i customers in the queue in stage j. Then
π(i, 1) = π(0, 1) ρi, i = 1, . . . , L − 1, π(i, 1) = π(0, 1) ρ i − ρU +1 1− ρU−L+2, i = L, . . . , U, (3.1) π(i, 2) = π(0, 1)δ− δ i−L+2 1− δ ρU − ρU +1 1− ρU−L+2, i = L, . . . , U, π(i, 2) = π(0, 1)δ i−U − δi−L+2 1− δ ρU − ρU +1 1− ρU−L+2, i = U + 1, . . . , N,
with π(0, 1) such that
" U X i=0 π(i, 1) + N X i=L π(i, 2) # = 1. (3.2)
The mean sojourn time is then given by
E[S] = 1 Λ " U X i=0 iπ(i, 1) + N X i=L iπ(i, 2) # , (3.3)
20 3. Threshold queueing model for uninterrupted traffic
where Λ is the effective arrival rate to the queue Λ = λ " U X i=0 π(i, 1) + N−1 X i=L π(i, 2) # .
Remark 3.1 (Infinite buffer). Note that if we would consider a two-stage M/M/1
threshold queue with an infinite buffer, i.e., N =∞, the stationary distribution in (3.1) still holds but with π(0, 1) such that
" U X i=0 π(i, 1) + ∞ X i=L π(i, 2) # = 1. The mean sojourn time is
E[S] = 1 Λ " U X i=0 iπ(i, 1) + ∞ X i=L iπ(i, 2) # , and effective arrival rate is
Λ = λ.
In order to use the threshold queue with a finite buffer as well as with an infinite buffer we slighty alter Heidemann’s model. Recall from (2.1) and (2.2)
k = (1− π(0, 1)) kjam, v =1/kjam
E[S] , q = k· v =
1− π(0, 1)
E[S] . (3.4)
Note that these three parameters are completely determined by the system variables λ, µL, µH, L, U , and N . They can therefore be interpreted as functions of these system parameters. In the two-stage M/M/1 threshold queue with an infinite buffer, k→ kjam when the queue approaches instability, i.e., λ→ µL. To obtain the same result for the two-stage M/M/1 threshold queue with a finite buffer, we adjust Heidemann’s model and define
k = (1− π(0, 1)) C, v = 1/C
E[S], q = k· v =
1− π(0, 1)
E[S] . (3.5)
Here, we use 1/C instead of 1/kjamas in Heidemann’s model. The parameter C is
obtained by fitting the queueing model to empirical data. In this adjusted model we define
kjam= lim
λ→µL k.
We may now obtain the capacity, qmax, the critical density, k∗, and the jam wave speed, ωv, as follows. Let us denote by λ∗the value of λ, with 0≤ λ ≤ µL, for which the maximum flow, qmax, is obtained, i.e., the value λ = λ∗ for which
∂ ∂λq = 1 E[S] ∂ ∂λ(1− π(0, 1)) + (1 − π(0, 1)) ∂ ∂λ 1 E[S] = 0.
3.2 Two-stage M/M/1 threshold queue 21
From (3.5), we now obtain
qmax= 1− π(0, 1) E[S] λ=λ∗ , k∗= C (1− π(0, 1))|λ=λ∗, (3.6)
where |λ=λ∗indicates that we insert λ = λ∗ in these expressions. Finally, we determine the jam wave speed, ωv,
ωv = lim k→kjam ∂ ∂kq = limλ→µL ∂ ∂λq ∂ ∂λk = lim λ→µL n 1 E[S] ∂ ∂λ(1− π(0, 1)) + (1 − π(0, 1)) ∂ ∂λ 1 E[S] o C ∂ ∂λ(1− π(0, 1)) . (3.7) The above expressions can readily be evaluated numerically for the M/M/1 threshold queue. In Section 3.2.1 we fit the two-stage M/M/1 threshold queue to experimen-tal data obtained for Danish highways [86] and we determine the capacity, critical density and jam wave speed.
Remark 3.2 (Multi-lane traffic). Our model also captures multi-lane traffic. On a
multi-lane highway, vehicles switch from high density lanes to low density lanes to improve their driving speed. To maintain safe driving distance other vehicles decel-erate and the average speed decreases. Using Helbing’s description of the hysteretic transition from non-congested to congested traffic we distinguish three density re-gions: light, medium and heavy traffic. In light traffic (k≤ ρ1) vehicles will switch lanes but they will not greatly affect other vehicles. In heavy traffic (k≥ ρ2) vehi-cles will not be able to switch lanes due to high densities on all lanes. In medium traffic (ρ1 < k < ρ2) vehicles will switch lanes causing average speed to decrease
and density to increase. Once the density reaches ρ2 the system becomes congested
and the average speed is considerably lower than in non-congested traffic. When we aggregate all lanes the hysteresis in this multi-lane system resembles that in our
single server threshold queue.
3.2.1
Model validation
In Figure 3.3, the two-stage M/M/1 threshold queue is fitted to empirical data points obtained from measurements made in September and October 2013 on three different locations on the Helsingørmotorvejen (two-lane) in Denmark made available by DTU Transport, Denmark [86]. DTU Transport provided three data sets with measurements on different time intervals: hourly measurements near Hørsholm, 15-minute measurements near Sandbjerg, and 5-15-minute measurements near Kokkedal. These data sets were uniformised to represent hourly measurements on all three locations. The data points in Figure 3.3(a) refer to the measurements near Hørsholm, the data points in Figure 3.3(b) refer to the measurements near Sandbjerg, and the data points Figure 3.3(c) refer to the measurements near Kokkedal. The curves in each subfigure show the best-fit using the two-stage M/M/1 threshold queue for
22 3. Threshold queueing model for uninterrupted traffic 0 40 80 120 160 200 0 1,000 2,000 3,000 4,000 Density (k) Flo w (q ) (a) Hørsholm N = 10 N = 20 N =∞ 0 50 100 150 200 250 300 0 1,000 2,000 3,000 4,000 Density (k) Flo w (q ) (b) Sandbjerg N = 10 N = 20 N =∞ 0 50 100 150 200 250 300 0 1,000 2,000 3,000 4,000 Density (k) Flo w (q ) (c) Kokkedal N = 10 N = 20 N =∞
Figure 3.3: Flow-Density diagram obtained with a fitted two-stage M/M/1 threshold queue. The flow-density curves were fitted to empirical flow-density points of a Danish motorway.
three different buffer sizes, N = 10, N = 20, and N =∞. The parameters for these
curves, as well as the jam density, kjam, the capacity, qmax, critical density, k∗ and jam wave speed, ωv, are given in Table 3.1. We show that the qualitative behaviour of the two-stage M/M/1 threshold queue captures the behaviour of the fundamental diagram of highway traffic.
Figure 3.3 and Table 3.1 show that both the jam density, kjam, and jam wave
speed, ωv, can be regulated by the buffer size N . Decreasing the buffer size results in a decreasing jam density and decreasing jam wave speed.
3.2.2
Sensitivity analysis of the fundamental diagram for the
two-stage
M/M/1 threshold queue
Figure 3.4 characterises the fundamental diagram of the two-stage M/M/1 threshold queue for four different scenarios. Each scenario is based on the basic scenario with
C = 1, µH = 25, µL = 15, L = 15 and U = 10 but one of the four parameters is
modified: (a) the lower threshold, L, (b) the upper threshold, U , (c) the high service rate, µH and (d) the low service rate, µL. C is kept unchanged since it is merely a scaling constant.
The effects of modifying L are minimal, as can be seen in Figure 3.4(a). Fig-ure 3.4(b) shows that the steepness of the capacity drop increases by increasing U . Figures 3.4(c) shows that qmax increases and k∗ decreases when µH increases. Increasing µL results in increasing both qmaxand k∗ as is shown in Figure 3.4(d).
3.3
Four-stage
M/M/1 feedback threshold queue
Consider a single server queue with finite buffer N , exponential service times and Poisson arrivals. The arrival rates and service rates are stage-dependent and con-trolled by a threshold policy. This threshold policy determines the stage of the queue
3.3 Four-stage M/M/1 feedback threshold queue 23 (a) Hørsholm (b) Sandb jerg (c) Kokk edal N 10 20 ∞ 10 20 ∞ 10 20 ∞ L 1 1 1 2 2 2 1 1 1 U 3 3 3 2 2 2 2 2 2 µH 26190.13 21301.42 20984.62 103728.36 55584.00 51689.51 86078.55 50447.65 47969.29 µL 5896.21 6899.43 6970.39 6160.00 7605.49 7775.93 5719.23 6701.10 6782.52 C 234.80 187.81 184.75 1146.85 598.78 554.46 938.01 538.23 510.46 kjam 86.34 128.25 184.75 122.24 259.07 554.46 144.16 271.82 510.46 qmax 2635.07 2652.65 2653.70 2882.80 2922.13 2923.28 2503.89 2504.77 2500.60 k ∗ 35.83 37.20 37.36 48.08 51.61 52.19 42.83 45.59 45.89 ωv -14.18 -9.17 -3.37 -8.38 -2.30 -0.53 -4.49 -1.95 -0.65 T able 3.1: P arameters for the tw o-stage M / M / 1 thresh old queues in Figure 3.3.
24 3. Threshold queueing model for uninterrupted traffic 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 Density (k) Flo w (q ) (a) L = 0 L = 3 L = 6 L = 9 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 Density (k) Flo w (q ) (b) U = 6 U = 10 U = 15 U = 20 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 Density (k) Flo w (q ) (d) µL= 12 µL= 16 µL= 20 µL= 24 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 Density (k) Flo w (q ) (c) µH= 16 µH= 20 µH= 24 µH= 28
Figure 3.4: Flow-Density diagram for the two-stage M/M/1 threshold queue with varying (a) L, (b) U , (c) µH and (d) µL.
based on its queue length. Let (n, s) denote the state of this Markov chain in which n, with n = 0, . . . , N , and s, with s = 1, . . . , 4, denote the queue length and stage, respectively, of the queue. If the queue is in stage s and a departure or an arrival causes the queue length to drop below Ls or to exceed Us, the stage of the queue changes. These changes are depicted in the state diagram in Figure 3.5. In this section we assume
L1= 0, U1= U3= Uµ,
L2= L4= Lµ, U2= Uλ,
3.3 Four-stage M/M/1 feedback threshold queue 25
and
0 < Lλ< Lµ≤ Uµ< Uλ< N.
The stage-dependent service and arrival rates are given in Table 3.2. Finally, we define κ to be a constant such that
λL µL
= κλH µH
. (3.8)
The stationary queue length probabilities π for the four-stage M/M/1 feedback
Stage Arrival Rate Service Rate
s = 1 λH µH
s = 2 λH µL
s = 3 λL µH
s = 4 λL µL
Table 3.2: The stage dependent service and arrival rates for the four-stage M/M/1 feedback threshold queue.
threshold queue can readily be obtained from standard Markov chain analysis. Let
α = λH µH, β = λH µL, γ = λL µH, δ = λL µL,
and let π(i, j) be the probability of having i customers in the queue in stage j, then:
π(i, 1) = π(0, 1) αi, π(Lλ− 1, 1)α i+1(1+Z) −αLµZ αLλ(1+Z)−αLµZ, π(Lµ− 1, 1)α i+1 −αUµ+2 αLµ−αUµ+2, π(i, 2) = π(Uµ+ 1, 2) β Lµ−βi+1 βLµ−βUµ+2, π(Uµ, 1)( βLµ−Uµ−β2)( βi−βUλ+1) (1−β)(βLµ−βUλ+2) , (3.9) π(i, 3) = π(Uλ, 2) α(1−γi+1−Lλ) 1−γ , π(Lµ− 1, 3)γ i+1 −γUµ+2 γLµ−γUµ+2,
26 3. Threshold queueing model for uninterrupted traffic 0 1 · · · L λ − 1 L λ · · · L µ − 1 L µ · · · U µ L µ · · · U µ U µ + 1 · · · U λ L λ · · · L µ − 1 L µ · · · U λ L µ · · · U µ U µ + 1 · · · U λ U λ + 1 · · · N − 1 N λ H λ H λ H λ H λ H λ H λ H λ H λ H λ H λ H λ H λ H λ H λ H λ H λ L λ L λ L λ L λ L λ L λ L λ L λ L λ L λ L λ L λ L λ L λ L µ H µ H µ H µ H µ H µ H µ H µ H µ H µ L µ L µ L µ L µ L µ L µ H µ H µ H µ H µ H µ H µ L µ L µ L µ L µ L µ L µ L µ L µ L µ L Figure 3.5 : T ransition diagram for the four-stage M / M / 1 feedbac k thresho ld queue.
3.3 Four-stage M/M/1 feedback threshold queue 27 π(i, 4) = π(Uµ+ 1, 4) δ Lµ−δi+1 δLµ−δUµ+2, π(Uλ, 2)β (
γUµ+2−Lµ−γUµ+2−Lλ)(δi−Uµ−1−δi+1−Lµ) (1−γUµ+2−Lµ)(1−δ) + (1−γUµ+2−Lµ)(1−δ i+1−Lµ) (1−γUµ+2−Lµ)(1−δ) , π(Uλ, 2)βδi−1−Uλ (
γUµ+2−Lµ−γUµ+2−Lλ)(δUλ−Uµ−δUλ+2−Lµ) (1−γUµ+2−Lµ)(1−δ) + (1−γ(Uµ+2−Lµ1 )(1−δUλ+2−Lµ) −γUµ+2−Lµ)(1−δ) , with Z = α Uµ+2−LµβUλ−Uµ 1− βUµ+2−Lµ (1− αUµ+2−Lµ) (1− βUλ+2−Lµ) , and π(0, 1) such that
Uµ X i=0 π(i, 1) + Uλ X i=Lµ π(i, 2) + Uµ X i=Lλ π(i, 3) + N X i=Lµ π(i, 4) = 1.
The mean sojourn time is given by
E[S] = 1 Λ Uµ X i=0 iπ(i, 1) + Uλ X i=Lµ iπ(i, 2) + Uµ X i=Lλ iπ(i, 3) + N X i=Lµ iπ(i, 4) ,
where Λ is the effective arrival rate to the queue
Λ = λH Uµ X i=0 π(i, 1) + Uλ X i=Lµ π(i, 2) + λL Uµ X i=Lλ π(i, 3) + N−1 X i=Lµ π(i, 4) .
Remark 3.3 (Infinite buffer). Note that if we would consider a four-stage M/M/1
feedback threshold queue with an infinite buffer, i.e., N =∞, the stationary distri-bution in (3.9) would still hold but with π(0, 1) such that
Uµ X i=0 π(i, 1) + Uλ X i=Lµ π(i, 2) + Uµ X i=Lλ π(i, 3) + ∞ X i=Lµ π(i, 4) = 1.
The mean sojourn time is
E[S] = 1 Λ Uµ X i=0 iπ(i, 1) + Uλ X i=Lµ iπ(i, 2) + Uµ X i=Lλ iπ(i, 3) + ∞ X i=Lµ iπ(i, 4) ,
and effective arrival rate is
Λ = λH Uµ X i=0 π(i, 1) + Uλ X i=Lµ π(i, 2) + λL Uµ X i=Lλ π(i, 3) + ∞ X i=Lµ π(i, 4) .
28 3. Threshold queueing model for uninterrupted traffic 0 150 300 450 600 750 0 1,000 2,000 3,000 4,000 Density (k) Flo w (q ) (a) Hørsholm N =∞ 0 150 300 450 600 750 900 0 1,000 2,000 3,000 4,000 Density (k) Flo w (q ) (b) Sandbjerg N =∞ 0 50 100 150 200 250 300 0 1,000 2,000 3,000 4,000 Density (k) Flo w (q ) (c) Kokkedal N =∞
Figure 3.6: Flow-Density diagram obtained with a fitted four-stage M/M/1 feedback threshold queue. The flow-density curve was fitted to empirical flow-density points of a Danish motorway.
We obtain the speed-flow-density relations, and thus the fundamental diagram, using the result from the altered Heidemann’s model in (3.5). To this end, we vary 0 ≤ λH < µκH such that λµLL < 1, see (3.8), and such that the four-stage M/M/1 feedback threshold queue with infinite buffer is stable. We obtain the capacity, qmax, critical density, k∗, and jam wave speed, ωv, following equations (3.6) and (3.7). These expressions can readily be evaluated numerically for the four-stage M/M/1 feedback threshold queue. In Section 3.3.1 we fit the four-stage M/M/1 feedback threshold queue to experimental data obtained for Danish highways and we determine the capacity, critical density and jam wave speed.
3.3.1
Model validation
In Figure 3.6, the four-stage M/M/1 threshold queue is fitted to empirical data points obtained from measurements made in September and October 2013 on three different locations on the Helsingørmotorvej (two-lane) in Denmark made available by DTU Transport, Denmark [86]. DTU Transport provided three data sets with measurements on different time intervals: hourly measurements near Hørsholm, 15-minute measurements near Sandbjerg, and 5-15-minute measurements near Kokkedal. These data sets were uniformised to represent hourly measurements on all three loca-tions. The data points in Figure 3.6(a) refer to hourly measurements near Hørsholm, the data points in Figure 3.6(b) refer to 15-minute measurements near Sandbjerg, and the data points Figure 3.6(c) refer to 5-minute measurements near Kokkedal. The curves in each subfigure show the best-fit using the four-stage M/M/1 feedback
threshold queue. The parameters for these curves, as well as the capacity, qmax,
critical density, k∗and jam wave speed, ω
v, are given in Table 3.3. In the sensitivity analysis in Section 3.3.2 we show that the effects of the buffer size N and the con-stant κ, which links λL to λH, see (3.8), are minimal. Therefore, we assume here
3.3 Four-stage M/M/1 feedback threshold queue 29
(a) Hørsholm (b) Sandbjerg (c) Kokkedal
Lλ 1 1 1 Lµ 3 5 2 Uµ 5 5 5 Uλ 6 6 6 µH 74951.98 78651.14 25834.87 µL 1.00 1.00 86.88 C 702.97 891.57 278.83 kjam 702.97 891.57 278.83 qmax 2598.17 2849.88 2500.57 k∗ 34.39 44.86 41.18 ωv -88.29 -43.96 -92.03
Table 3.3: Parameters for the three four-stage M/M/1 feedback threshold queues in Figure 3.6.
3.3.2
Sensitivity of the fundamental diagram for the
four-stage
M/M/1 feedback threshold queue.
Figures 3.7, 3.8, and 3.9, characterise the fundamental diagram of the four-stage M/M/1 feedback threshold queue for nine different scenarios. Each scenario is based on the basic scenario with
Lλ= 3, Uµ= 9, µH= 25000,
Lµ= 6, Uλ= 12, µL= 10,
C = 400, N =∞, κ = 1,
which is based on the results in Section 3.3.1.
In Figure 3.7 we modify: (a) the lower threshold Lλ, (b) the lower threshold
Lµ, (c) the upper threshold Uµ, and (d) the upper threshold Uλwhile keeping other
parameters fixed. The effects of modifying Lλ or Lµ are minimal, as can be seen
in Figure 3.7(a) and Figure 3.7(b). Figure 3.7(c) shows that modifying Uµ greatly
affects the fundamental diagram. Increasing Uµ results in a greater qmax and k∗. The same effect is achieved by increasing Uλ but to a lesser extent as is shown in Figure 3.7(d).
30 3. Threshold queueing model for uninterrupted traffic 0 100 200 300 400 0 1,000 2,000 3,000 4,000 5,000 Density (k) Flo w (q ) (a) Lλ= 1 Lλ= 3 Lλ= 5 0 100 200 300 400 0 1,000 2,000 3,000 4,000 5,000 Density (k) Flo w (q ) (b) Lµ= 4 Lµ= 6 Lµ= 8 0 100 200 300 400 0 1,000 2,000 3,000 4,000 5,000 Density (k) Flo w (q ) (c) Uµ= 7 Uµ= 9 Uµ= 11 0 100 200 300 400 0 1,000 2,000 3,000 4,000 5,000 Density (k) Flo w (q ) (d) Uλ= 10 Uλ= 12 Uλ= 14
Figure 3.7: Flow-Density diagram for the four-stage M/M/1 feedback threshold queue with varying (a) Lλ, (b) Lµ, (c) Uµand (d) Uλ.
µL, (c) the constant C, and (d) the buffer size N while keeping other parameters
fixed. The latter has little effect as is shown in Figure 3.8(d) in which N is mod-ified. Figure 3.8(c) shows that C serves as a scaling parameter for the density. In Figure 3.8(a) we modify µH and it shows that by increasing µH, we increase qmax while keeping k∗unchanged. Finally, Figure 3.8(b) shows that both q
maxand k∗are increased by increasing µL.
In Figure 3.9 we modify the parameter κ. Figure 3.9 shows that both qmaxand
3.4 Summary and Conclusion 31 0 100 200 300 400 0 1,500 3,000 4,500 6,000 Density (k) Flo w (q ) (a) µH= 15000 µH= 25000 µH= 35000 0 100 200 300 400 0 1,500 3,000 4,500 6,000 Density (k) Flo w (q ) (b) µL= 1 µL= 10 µL= 100 µL= 10000 0 200 400 600 0 1,000 2,000 3,000 4,000 Density (k) Flo w (q ) (c) C = 200 C = 400 C = 600 0 100 200 300 400 0 1,000 2,000 3,000 4,000 Density (k) Flo w (q ) (d) N = 15 N = 20 N =∞
Figure 3.8: Flow-Density diagram for the four-stage M/M/1 feedback threshold queue with varying (a) µH, (b) µL, (c) C and (d) N .
3.4
Summary and Conclusion
In this chapter we have introduced the two-stage M/M/1 threshold queue and the four-stage M/M/1 feedback threshold queue to study the parameters of traffic that influence the shape of the fundamental diagram including the capacity drop in this diagram observed in empirical data for modern traffic flows.
The two stage M/M/1 threshold queue has two service regimes: high and low service rates, and switches from high rates to low rates when the queue length exceeds the upper threshold, and returns to high rates when the queue length falls below the lower threshold, where the lower threshold is smaller than, or equal to,
32 3. Threshold queueing model for uninterrupted traffic 0 100 200 300 400 0 1,000 2,000 3,000 4,000 Density (k) Flo w (q ) κ = 0.5 κ = 1.0 κ = 1.5
Figure 3.9: Flow-Density diagram for the four-stage M/M/1 feedback threshold queue with varying κ.
the upper threshold. The two-stage M/M/1 threshold queue was successfully fitted to experimental data and the capacity, critical density and jam wave speed were computed for three different data sets.
The four-stage M/M/1 feedback threshold queue has four regimes with either a high or low arrival rate, and either a high or low service rate. The service rate switches from high rates to low rates when the queue length exceeds the upper threshold Uµ, and returns to high rates when the queue length falls below the lower threshold Lµ. Similarly, the arrival rate switches from high rates to low rates when the queue length exceeds the upper threshold Uλ, and returns to high rates when the queue length falls below the lower threshold Lλ, where Lλ < Lµ ≤ Uµ < Uλ. The four-stage M/M/1 feedback threshold queue was successfully fitted to experimental data and the capacity, critical density and jam wave speed were computed for three different data sets.
Comparing the three different fits of each queueing model we can conclude that the four-stage M/M/1 feedback threshold queue gives a better fit to experimental data. Sensitivity analysis of the two-stage M/M/1 threshold queue reveals that steepness of the capacity drop, the capacity and the critical density are determined by the value of the higher threshold, U , and the mean service times, µL and µH. Sensitivity analysis of the four-stage feedback threshold queue reveals that the shape
of the fundamental diagram is determined by the upper threshold Uµ and the high
CHAPTER 4
A tandem network of
M/M/1 threshold
queues
4.1
Introduction
In Chapter 3 we have introduced and analysed the two-stage M/M/1 threshold queue and the four-stage M/M/1 feedback threshold queue. These queueing systems were controlled by a threshold policy based on the queue length of the queue. In the two-stage M/M/1 threshold queue only the service rates changed when the queue length reached certain thresholds while in the four-stage M/M/1 feedback threshold queue, both the arrival rates and service rates changed. Both queueing models modelled a single highway section. The two-stage M/M/1 threshold queue modelled the hysteric behaviour of traffic on the highway section while the four-stage M/M/1 feedback threshold queue, in addition, also models the hysteretic behaviour of arriving traffic, i.e., traffic on the preceding highway section. Finally, an adjusted version of Heidemann’s method was used to create the fundamental diagram of uninterrupted traffic for both queueing systems.
In this chapter we extend the results from Chapter 3 and model a sequence of highway sections by a tandem queueing network. In a tandem queueing network, a customer is served at multiple queueing systems in a line, see Figure 4.1 for a tandem queueing network of 5 queues. Throughout this chapter we will assume Poisson arrivals to the first queue and exponentially distributed service rates. Furthermore, the buffer at each queue is finite and we assume a Blocking Before Service (BBS) protocol, i.e., a server becomes blocked, and is unable to serve customers, if its departing customer fills the downstream queue. Due to the exponential growth of the state space, we are limited to tandem networks of three queues. In Chapter 8 we introduce an iterative aggregation method enabling the approximative analysis of larger tandem networks
In Section 4.2 we introduce a three queue tandem network of the two-stage M/M/1 threshold queues in which the service rates are controlled by a threshold policy based on the queue length at each station. Each queue in the network models the hysteretic behaviour of traffic on a highway section. We extend this tandem network of two-stage M/M/1 threshold queues to a 3-tandem network of four-stage M/M/1 feedback threshold queues in Section 4.3. This network models the hysteric
34 4. A tandem network of M/M/1 threshold queues
N1 µ1 N2 µ2 N3 µ3 N4 µ4 N5 µ5
λ
Queue 1 Queue 2 Queue 3 Queue 4 Queue 5
Figure 4.1: A tandem network with 5 queues.
0 1 · · · L− 1 L · · · U
L · · · U U + 1 · · · N− 1 N
µH µH µH µH µH µH
µL
µL µL µL µL µL µL
Figure 4.2: Transition Diagram for a single two-stage M/M/1 threshold queue.
behaviour of each highway section, as well as the hysteric interaction between two consecutive sections, i.e., once a highway section becomes incredibly crowded, the vehicles in the preceding section will drive slower. Both tandem networks are anal-ysed numerically and the fundamental diagram is obtained for each seperate queue using the adjusted Heidemann’s method in equation (3.5). Finally, we perform a sensitivity analysis on the system parameters for both tandem networks. Section 4.4 gives concluding remarks.
4.2
A three queue tandem network of two-stage
M/M/1 threshold queues
We consider a tandem queueing network of three identical two-stage M/M/1 thresh-old queues with finite buffer N , with exponential service times at each queue, and with Poisson arrivals with rate λ to the first queue. Each queue is controlled by a threshold policy which determines the stage of the queue based on the queue length. Let (n1, s1, n2, s2, n3, s3) denote the states of this Markov chain in which ni, with ni= 0, . . . , N , and si, with si= 1, 2, denote the queue length and stage, respectively,
of queue i. The stage of queue i becomes congested, denoted by si = 2, once the
queue length grows beyond the upper threshold, U , and it becomes non-congested again, denoted by si= 1, if the queue length drops below the lower threshold, L, as depicted in Figure 4.2. In this section we assume
0 < L≤ U < N.
The arrival rate to the first queue is λ but the service rate in each queue depend on the stage and is given in Table 4.1. Let Q be the generator of this Markov chain and let π be its stationary distribution such that πQ = 0 for πe = 1. By modelling this
4.2 Two-stage M/M/1 tandem threshold queue 35
Stage Service Rate
si = 1 µH
si = 2 µL
Table 4.1: The stage dependent service rates for queue i, for i = 1, 2, 3.
Markov chain as a finite Level Dependent Quasi-Birth-and-Death process (LDQBD) we can obtain the stationary distribution using the Successive Censoring Algorithm from Chapter 7.
Let the element π(n1, s1, n2, s2, n3, s3) of π be the (stationary) probability that the Markov chain is in state (n1, s1, n2, s2, n3, s3). Furthermore, let πi be the marginal queue length distribution of queue i, i.e.,
π1(n1) = 2 X s1=1 2 X s2=1 2 X s3=1 N X n2=0 N X n3=0 π(n1, s1, n2, s2, n3, s3), π2(n2) = 2 X s1=1 2 X s2=1 2 X s3=1 N X n1=0 N X n3=0 π(n1, s1, n2, s2, n3, s3), (4.1) π3(n3) = 2 X s1=1 2 X s2=1 2 X s3=1 N X n1=0 N X n2=0 π(n1, s1, n2, s2, n3, s3). The mean queue length of queue i, E[Li], is now given by
E[Li] = N X ni=0
niπi(ni), (4.2)
and the mean sojourn time in queue i, E[Si], is obtained using Little’s Law [69] E[Si] = E[L
i]
Λ , (4.3)
where Λ is the effective arrival rate to the tandem network
Λ = (1− π1(0)) λ. (4.4)
Finally, we apply the adjusted Heidemann’s method from equation (3.5) to obtain the fundamental diagram for each highway section. Recall from (3.5) the definition of the traffic parameters for density, k, speed, v, and flow, q, in terms of the marginal queue length distribution and mean sojourn time for each queue i, i = 1, 2, 3,
ki= (1− πi(0)) Ci, vi= 1/Ci E[Si]
, qi= ki· vi =1− πi(0) E[Si]
36 4. A tandem network of M/M/1 threshold queues (a) (b) (c) 0 0.25 0.5 0.75 1 0 2 4 6 8 10 Density (k) Flo w (q ) Queue 1 L = 2 L = 3 L = 4 0 0.25 0.5 0.75 1 0 2 4 6 8 10 Density (k) Flo w (q ) Queue 2 L = 2 L = 3 L = 4 0 0.25 0.5 0.75 1 0 2 4 6 8 10 Density (k) Flo w (q ) Queue 3 L = 2 L = 3 L = 4 0 0.25 0.5 0.75 1 0 2 4 6 8 10 Density (k) Flo w (q ) Queue 1 U = 5 U = 6 U = 7 0 0.25 0.5 0.75 1 0 2 4 6 8 10 Density (k) Flo w (q ) Queue 2 U = 5 U = 6 U = 7 0 0.25 0.5 0.75 1 0 2 4 6 8 10 Density (k) Flo w (q ) Queue 3 U = 5 U = 6 U = 7 0 0.25 0.5 0.75 1 0 2 4 6 8 10 Density (k) Flo w (q ) Queue 1 N = 8 N = 10 N = 15 0 0.25 0.5 0.75 1 0 2 4 6 8 10 Density (k) Flo w (q ) Queue 2 N = 8 N = 10 N = 15 0 0.25 0.5 0.75 1 0 2 4 6 8 10 Density (k) Flo w (q ) Queue 3 N = 8 N = 10 N = 15
Figure 4.3: Flow-Density diagram for a tandem network of three two-stage M/M/1 threshold queue with varying (a) L, (b) U , and (c) N .
In addition, we denote the capacity of the highway section, i.e., the maximum flow
that can be achieved, by qmax. The density at which this capacity is achieved is
called the critical density and is denoted by k∗. Finally, we denote by kjamthe jam density.
Figures 4.3 and 4.4 characterise the fundamental diagram for each queue in a tandem network of three two-stage M/M/1 threshold queues for different scenarios. The results for each of the three scenarios are displayed in three horizontal plots, characterising the fundamental diagram for the first queue (left), the second queue
4.2 Two-stage M/M/1 tandem threshold queue 37 (a) (b) (c) 0 0.25 0.5 0.75 1 0 2 4 6 8 10 Density (k) Flo w (q ) Queue 1 µL= 15 µL= 20 µL= 25 0 0.25 0.5 0.75 1 0 2 4 6 8 10 Density (k) Flo w (q ) Queue 2 µL= 15 µL= 20 µL= 25 0 0.25 0.5 0.75 1 0 2 4 6 8 10 Density (k) Flo w (q ) Queue 3 µL= 15 µL= 20 µL= 25 0 0.25 0.5 0.75 1 0 2 4 6 8 10 Density (k) Flo w (q ) Queue 1 µH= 25 µH= 30 µH= 35 0 0.25 0.5 0.75 1 0 2 4 6 8 10 Density (k) Flo w (q ) Queue 2 µH= 25 µH= 30 µH= 35 0 0.25 0.5 0.75 1 0 2 4 6 8 10 Density (k) Flo w (q ) Queue 3 µH= 25 µH= 30 µH= 35 0 1 2 3 0 2 4 6 8 10 Density (k) Flo w (q ) Queue 1 C = 1 C = 2 C = 3 0 1 2 3 0 2 4 6 8 10 Density (k) Flo w (q ) Queue 2 C = 1 C = 2 C = 3 0 1 2 3 0 2 4 6 8 10 Density (k) Flo w (q ) Queue 3 C = 1 C = 2 C = 3
Figure 4.4: Flow-Density diagram for a tandem network of three two-stage M/M/1 threshold queue with varying (a) µL, (b) µH, and (c) C.
(middle) and the third queue (right) in the tandem network. Each scenario is based on the basic scenario with
L = 3, U = 5, N = 8, µL= 20, µH= 30, C = 1.
In each of the six scenarios in Figures 4.3 and 4.4 we change one of the above parameters, while keeping the others unchanged.
In Figure 4.3(a) it is seen that the effects of L are minimal. Figure 4.3(b) shows that increasing U , also increases qmax and k∗ for all three queues. Finally,
