Modeling road traffic flow with queueing theory
An exploration of performance for different scheduling disciplines in queueing networksSebastiaan Vermeulen
July 15, 2016
Bachelorproject wiskunde
Supervisor: prof. dr. Rudesindo N´
u˜
nez Queija
Korteweg-de Vries Instituut voor Wiskunde
Abstract
This paper considers traffic flow networks as queueing networks where bottlenecks in the traffic flow network are modeled as servers. Singular intersections and intersections chained by a green wave traffic flow are considered in more detail. The FIFO, priority and exponential polling serving disciplines are analytically analyzed under the assumption of exponential service times, Poisson arrivals and preemptive-resume polling. Expressions for the individual mean sojourn time are derived for all models. The individual queue length distribution is analytically analyzed for each serving discipline, and for all models except the green wave priority polling discipline with low priority for the green wave, expressions for the individual queue length distributions are derived. Simulations show that the influence on the queue length distribution and the mean sojourn time of using nonpreemptive polling and of different serving time distributions is marginal when considered separately, but can have dramatic effects when considered jointly.
Title: Modeling road traffic flow with queueing theory
Authors: Sebastiaan Vermeulen, mail@sebastiaanvermeulen.nl, 10834842 Supervisor: prof. dr. Rudesindo N´u˜nez Queija
Date: July 15, 2016
Korteweg-de Vries Instituut voor Wiskunde Universiteit van Amsterdam
Science Park 105-107, 1098 XH Amsterdam
Contents
Introduction
1
Performance measures 2
Preliminaries
3
Distributions 3
Transitions, Markov chains 4
Equilibrium distribution 5
PASTA 5
Little’s law 5
Kendall’s notation 6
Transforms and their notations 6
Notational conventions 6
Simple intersections
8
First-In-First-Out 8
Equilibrium probabilities 9
Expected sojourn time 10 Ranking the arrivals 10
Preemptive-resume scheduling 10
Expected remaining service time 11 Priority queueing 11
Expected sojourn time 11
Equilibrium distributions 12 Exponential polling 14
Equilibrium distributions 14
Expected sojourn time 15
The green wave
16
Priority queueing: green wave first 16
Expected sojourn time 16
Equilibrium distributions 17 Priority queueing: green wave last 17
Equilibrium distribution 17
Expected sojourn time 17 Exponential polling 18
Expected sojourn time 18
Equilibrium distribution 18
Simulations
19
influence of assumptions 19
Preemption 19
Exponential serving times 20
Discussion
22
1
Introduction
Between 1960 and present, the vehicle population has been growing steadily at a rate four times that of the human population (Dargay et al. 2007,United States Census Bureau 2016). To manage this increase of traffic load at minimal cost for governing authorities in such a manner that road users face the least delay and the impact for other parties is minimized, efficient traffic control systems are needed which optimize the sojourn time and other factors dynamically, adapting to changes in the traffic load.
When it comes to building new roads or updating existing infrastructure, authorities face design decisions which influence parameters such as construction and maintenance cost, but also expected sojourn time for commuters, sensitivity to congestion and impact on air quality. To be able to make optimal design decisions it is necessary to gain insight in how traffic network design influences these parameters.
Consequentially there is the desire to model traffic flow networks. Current literature employs different types of models. The first distinction I wish to make is between deterministic and stochastic models. Deterministic models often consider traffic flow either from the perspective of fluid dynamics (Jabari 2016,Ji & Geroliminis 2012) or as interacting particles, where the particles are made sentient beings by equipping them with a set of decision rules (Kerner 2004,Jin 2013). The former approach, known as macroscopic traffic flow modeling, describes the emergent behavior of traffic flow networks via differential equations which can be solved to find equilibrium states of relevant quantities such as the traffic density and speed on road segments. The latter approach, fittingly named microscopic traffic flow modeling, assumes that the behavior of individual road users can be modeled via functions of contextual parameters such as the distance to the preceding road user.
Microscopic traffic models offer the advantage of exposing emergent behavior over macroscopic models, which makes them more suitable for modeling for example pollutant emissions (Gan et al. 2011). However, such models pose computational difficulties when used to model larger traffic networks as shown byJha et al.(2004).
Micro- and macroscopic models require the use of either empirical or simulated data, from which results can be inferred with assumed certainty. This stringent dependency on the correct specification of input limits the predictive power of such models, thus limiting their use in decision processes (Jain & Smith 1997). To increase the robustness of microscopic traffic models the behavior of individual road users can be randomized (Kerner & Klenov 2006,Jabari & Liu 2012).
An altogether different modeling methodology is to model not the flow of traffic, but the delay traffic encounters. These queueing models can then be analytically assessed with results from the mathematical domain of queueing theory. An extensive review of queueing models used to describe traffic flow is given byVan Woensel & Vandaele (2007). This paper points out that queueing models do not curtail behavioral insights to the same level of detail as well-described microscopic models, in favor of a high predictive accuracy and robustness of the results. Moreover, the computations required for queueing models remain tractable as the complexity of the modeled network increases, and the analytic coupling between input parameters and results provides a direct explanation of the found results.
Queueing models of singular traffic bottlenecks can be combined to model traffic networks in the form of a queueing network. Queueing networks can be either closed, open or mixed. Open queueing networks assume that road users enter the network at some point, travel some route through the network and then leave the network again. In closed networks the road users never
enter or leave the system and their total number is fixed, while in mixed networks some road users behave as if they were in an open network, while other behave as if they were in a closed network. For the modeling purpose intended by this paper only open networks are applicable. Queueing networks are directed graphs of queue nodes representing bottlenecks, connected via arcs representing possible routes that road users may follow within the network. Bottlenecks can range from intersections to entrance- and exit ramps, or even straight sections of road. The latter can be viewed as a bottleneck in the sense that for any given maximum speed, the number of cars that can pass in any time window is limited. Consider Figure2.1for a graphical representation of a queueing network.
Two particular instances of such bottlenecks are studied in this paper. Firstly a single intersection with an arbitrary number of intersecting traffic flows is considered. Secondly a ‘green wave’ is modeled as a chain of intersections with on each intersection traffic flows that do not directly interact except via a single traffic flow passing through the chain of intersections, which is not allowed to advance except immediately to the end of the chain.
This paper consists of four parts. In chapter2, the preliminaries required for the analysis are introduced. In chapter3individual intersections are modeled and a number of control processes for these intersections are analyzed . Then the analysis is repeated for ‘green-wave’-systems in chapter4, where a number of intersections are chained such that one flow of traffic is guaranteed immediate passage at all but the first intersection in the chain. Lastly a number of simulations are run to quantify the effect of some of the simplifying assumptions that are made during the analysis. We end with a joint discussion of the results from theory and simulation.
1.1 Performance measures
To evaluate the models, two performance measures are considered in this paper. The first measure is the mean sojourn time per queue, which is the average time a customer spends in the system (both waiting in the queue and in service). The second measure is the equilibrium distribution of the number of customers in the system. This measure is highly relevant when studying traffic models because of the finite capacity of physical systems. While in our theoretical models we will allow for an infinite capacity of the queue, when a queue for one intersection extends to the preceding intersection, this intersection can no longer let traffic pass and this complication is not contained within the model.
2
Preliminaries
The analysis and simulations are based on results from queueing theory, an area of mathematical analysis that relies heavily on the theory of Markov processes. Specifically, a bottleneck is modeled as a node in a network of servers and road users arriving at intersections are modeled as customers joining the queue of the server. Consider Figure2.1for an exhibit. In this section some important results that will be used in this paper are treated. Because of the intimate connection with queueing theory, we will stick to using the generic queueing terminology as much as possible throughout this paper. As is customary in queueing theory, our model of an intersection can be described by two components. New customers arrive at a queue according to an independent stochastic process, which throughout this paper will be a Poisson process. Then the queued customers are served one-at-a time by the server, where the serving time is randomly distributed. When the queue is empty the server is idle, whilst if there are customers in the queue, the server does not pause until the queue is empty again.
server or bottleneck queue
direction of traffic
Figure 2.1 Left: Representation of a server and queue. Right: A queueing network to describe a traffic flow network consisting of 6 bottlenecks. Traffic enters the network at the left- and bottom side of the graph, then travels past a number of bottlenecks before leaving at the top- and right side of the figure.
2.1 Distributions
For the sake of completeness, we state here the functional specifications of the distribution families that wil be used thoughout this paper.
2.1: Poisson distribution
A random variable X is said to be Poisson distributed with rate λ if for k ∈ N
P[X = k] = e−λλ k k!
which is denoted as X ∼ Poisson (λ).
2.2: Exponential distribution
A random variable X is said to be ex-ponentially distributed with rate λ if for x ∈ [0, ∞)
P[X < x] = Z x
0
λe−tλdt which is denoted as X ∼ Exp (λ).
2.3: Geometric distribution
A random variable X is said to be geometrically distributed with probability p if for k ∈ N P[X = k] = (1 − p)k−1p
which is denoted as X ∼ Geom (p).
In addition, a Poisson process A(t) with rate λ is a function on [0, ∞) where A(t) ∼ Poisson (λt).
2.2 Transitions, Markov chains
A stochastic process describes the evolution of a random variable on a state space S in time. It can either be a sequence, in which case we speak of a discrete process, or a function [0, ∞) → S, in which case we speak of a continuous process. Markov chains are stochastic processes on a countable state space that respect the Markov property. The Markov property states that the evolution of the stochastic process at any time may depend on the current state of the process, but is independent from all the past states that the process has been in.
Formally we let for t ∈ [0, ∞) (resp. t ∈ N) M (t) be a stochastic process on the countable state space S ⊂ N. If the Markov property (equation2.4) holds, then M (t) is a continuous-time (resp. discrete-time) Markov chain.
2.4: Markov property (for Markov processes)
A process M (t) has the Markov property if
P[M (t + s) = j | M (s) = i] = P[M (t) = j | M (0) = i] = pij(t)∀s.
The transition probabilities pij(t) are the probabilities of the process M being in state j after t units of time, given that M started in state i at time 0. Let Jndenote the time at which the nth transition is made, then the sequence {Mn} = {M (Jn)}∞
n=0is an embedded Markov chain, which is a discrete Markov chain whose equilibrium distribution is directly related to the equilibrium distribution of M (t). Embeddings are not unique in the sense that any continuous chain has an arbitrary number of embeddings; consider that, {Mn} = {M (Jnk)} is an embedded Markov chain for each k ∈ N.
Let us return to the single server with single queue depicted in Figure2.1(Left). We assume Poisson arrivals with rate λ for the customers and we assume that the server finishes serving a customer in a period that is exponentially distributed with rate µ. The number of customers waiting in the queue now constitutes the continuous-time Markov chain Π(t) with state space N. Given the current queue length, with rate λ the queue length will increase by one and if the queue is not empty, with rate µ the queue length will decrease by one. Now we note that the exponential distribution has the following property: If A ∼ Exp (a) and B ∼ Exp (b), then P[A < B] = a+ba . If we define pij as the probability that state i is succeeded by state j, then the pij are given by pij = 1 if j − 1 = i = 0 λ λ+µ if j − 1 = i > 0 µ λ+µ if j + 1 = i > 0 0 else.
in Π(t) at the ‘jump moments’ Jn; Pn= Π(Jn). If a different embedding is chosen other transition probabilities arise. These probabilities are key to our analysis because via balance equations they lead to the steady-state distribution, whose definition and purpose are given in the next section.
2.3 Equilibrium distribution
If pij(t) is strictly positive for each i, j ∈ S, t ∈ R+ then the Markov chain admits an equilibrium distribution which equals the long-run proportion of time spent in each state. By definition, the in-and outflux of a state must match in the long-run equilibrium, where flux is given by the transition probabilities in the embedded chain and given by the transition rates in the continuous chain, in both cases weighted by the equilibrium distribution. This necessary condition on equilibrium distributions is known as the global balance condition (Chandy 1972,Ross 2010b):
2.5: Global balance condition
If π = {πi} is an equilibrium distribution of the Markov process Π(t) (resp. Pn), then the following holds:
X
k∈S\{i}
πiγik= X k∈S\{i}
πkγki for all i ∈ S
where the γij are transition rates of Π(t) (resp. transition probabilities of Pn). These equations can also be combined for different states to obtain balance equations between sets of states.
From these equations one can often infer a recurrence relation on the equilibrium distribution, which then sometimes can be solved to yield π. A practical tool we shall use to study these equations is the transition graph, which visualizes the possible one-step transitions from an arbitrary state. A necessary condition for balance at each node is balance at each cut of the transition graph, a result which will prove convenient in solving more complex balance equations. A more extensive treatise of Markov chains and equilibrium distributions can be found inNorris
(1998),Ross(2010a).
2.4 PASTA
Poisson Arrivals See Time Averages, or PASTA, is a specific case of the Arrival theorem which holds under the very basic assumptions that arrivals are Poisson and not anticipated by the observed process (Wolff 1982). The Arrival theorem states that open queueing networks (c.f. Figure2.1(Right)) with unrestricted queues and independent service times, observed at arrival moments exert the same equilibrium distribution as the continuous queueing network itself. By the virtue of Burke’s theorem (Burke 1956), the departure process in queueing networks with Poisson arrivals and unrestricted queues also constitutes a Poisson arrival process. Consequentially the expected number of customers and the expected sojourn time of a queueing network can be derived from the discrete Markov chain that is embedded at either the arrival- or departure moments in the continuous Markov chain describing the system. A more detailed exposition of the Arrival theorem with less strict sufficient conditions is given inBoucherie & van Dijk(1997).
2.5 Little’s law
Intuitively it makes sense that in the long run, the number of customers in a system L equals the number of arrivals per time unit λ multiplied by the time that one customer resides in the system W :
2.6: Little’s law
E[L] = λE[W ]
For a fixed finite time T the relationship is even trivial: T ET[L(t)] and λT ET[W (t)]1 are two ways of expressing the total time customers have spent in the system. InLittle(1961) it is proven that this rule holds for any system that deals with arrivals with average arrival rate λ that spend a stochastic length of time in the system. Little’s law is valid under remarkably loose assumptions: It holds as long as the arrival rate has an average and E[W ] is finite.
2.6 Kendall’s notation
The de facto notation to describe queueing nodes comes from Kendall(1953). The basic form reads A/S/n, where A denotes the type of arrival process, S denotes the serving process and n denotes the number of servers. In this paper, A and S will most often be Markovian processes, e.g. a Poisson arrival process and exponential service times, and a queue will face a single server. Such queues are denoted as M/M/1.
2.7 Transforms and their notations
We will employ the Laplace-Stieltjes Transform (LST) and the Probability Generating Function (PGF) in this paper, both of which are defined for non-negative random variables. Both transforms are injective mappings of the probability density resp. mass function such that we may switch arbitrarily2 back and forth to whatever form is most suitable for the calculations at hand.
2.7: Laplace-Stieltjes Transform (LST)
Let F be the cumulative density function of random variable X. The LST is a mapping X 7→ φ such that:
The LST of X is a function φ : [0, ∞) → R defined as φ(s) =R∞ 0 e
−stdF (t).
2.8: Probability Generating Function (PGF)
let the countable set {pi} be a probability mass of random variable Y . The PGF is a mapping Y 7→ ψ such that:
The PGF of Y is a function ψ : (−1, 1] → R defined as ψ(z) =P∞k=0z kp
k.
If a closed form is known for ψ(z), then pk can be found by determining the k-th derivative with respect to z and letting z ↓ 0.
2.8 Notational conventions
We conclude this chapter with an overview of the notation that is used throughout this paper. If a model with multiple queues is considered, each queue will be identified by a subscripted index and the queue obtained by joining all queues in a single queue is identified by the absence of a subscript. Moreover, the following processes are associated with queue i:
1
These averages are taken over time: ET[L] =
RT
0 L(t)/T dt and ET[W ] =
RT
0 W (t)/T dt.
2Limited only by our own algebraic abilities; retrieving a distribution function from the LST or PGF can be
• Ai(t), the arrival process with arrival rate λi; • Lqi(t), the queue length;
• Li(t), the number of customers in the system;
• Wi(t), the average sojourn time of customers in the system.
If an equilibrium distribution exists for the queue length, the latter 3 processes of the foregoing list have equilibrium distributions. Their random variables are denoted by Lqi, Liand Wirespectively.
3
Simple intersections
In this chapter we will review the properties of a model in which two classes of customers must compete for a single server in a network of servers, by which we represent an intersection in a traffic flow network. We assume that customers arrive at each node according to a Poisson process and that service times are exponentially distributed.
In fact, if customers enter the system according to a Poisson process and each node behaves as a single queue and server with exponential service times, Burke’s theorem (Burke 1956) states that indeed each node observes Poisson arrivals. These conditions however do not hold for all of the models we shall consider.
We start by reviewing a single node in this network, which by the preceding is a slight adaptation of an M/M/1 queue since a single server is shared by multiple queues. A graphical representation of the foregoing is given in figure 3.1. At queues 0 resp. 1, customers arrive according to a poisson process with arrival rates λ0 resp. λ1. The server X can only process one customer at a time, and does so at rate µ, independent of the class of the customer. In the coming sections we consider different methods to distribute the work to the server – that is, we consider different methods to direct traffic at intersections.
x
X(µ)
1(λ
1)
0(λ
0)
Figure 3.1 A single server X serving at rate µ in a queueing network and the two competing classes 0 and 1 with arrival rates λ0 resp. λ1 are considered as an independent segment of the network, which
we can then separately analyze.
3.1 First-In-First-Out
When we assume that the customers in queue 0 and 1 are processed by X on a First-In-First-Out (FIFO) basis, the model becomes essentially equivalent to a queueing process with arrival rate λ0+ λ1 and departure rate µ. This is due to the fact that after a customer joined the queue, it becomes irrelevant from which arrival process he spawned. Therefore in the FIFO model the two queues can be thought of as a single queue where customers of both classes join in their order of arrival. The number of customers in this queue thus is the sum of the number of customers in
queue 0 and the number of customers in queue 1. Furthermore, if we denote the Poisson arrival processes of queue i at time t with Ai(t), then
P[A0(t) + A1(t) = k] = k X n=0 P[A0(t) = n, A1(t) = k − n] = k X n=0 e−λ1t(λ1t) k−n (k − n)! e−λ0t(λ0t) n n! =e −(λ1+λ0)t k! k X n=0 k n tnλk−n1 λn0 = e−(λ1+λ0)t(λ0+ λ1) ktk k!
shows us that the sum of two Poisson distributions is again Poisson distributed. This statement can be generalized to countable sums of Poisson processes via induction. For now however, we focus on the case with two classes.
Equilibrium probabilities
Let us first find the proportion of time πn that there are n customers in the system. We start by describing the transition matrix, from there we find and solve the balance equations which give us the steady state equilibria. To this end, note that if there are n > 0 customers in the system, the possible transitions are to n − 1 or to n + 1. The transition rate to n − 1 equals the departure rate µ, and by the preceding result, the transition rate to n + 1 equals λ = λ0+ λ1. The state transitions and their rates are depicted in Figure 3.2.
0 1 . . . n − 1 n n + 1 . . . λ µ λ µ λ µ λ µ λ µ λ µ
Figure 3.2 State transitions in the FIFO model with total arrival rate λ and departure rate µ.
From Figure 3.2we can determine the balance equations: ∞
X
i=0
πi= 1, λπ0= µπ1, (λ + µ)πi= λπi−1+ µπi+1 for i > 0. These equations admit the solution
πi= ρiπ0 where ρ = λ
µ. (1)
The first equation now yields π0= 1 − ρ and the condition that ρ < 1, for otherwise the series will not converge. ρ is known as the occupation rate as it, when it is smaller than 1, denotes the fraction of time that the server is occupied. Let for future reference ρi denote λi/µ.
To find the equilibrium probabilities of each queue, consider the Markov chain Q describing the ordered queue: Define p = 1 − q = λ0
λ0+λ1, the probability that the next arrival is of type 0,
and denote the state space as
S = {(x1, x2, . . . , xn) | xi∈ {0, 1}, n ∈ N}
where xi denotes the type of the ithcustomer in the queue and n denotes the length of the queue. The following equality follows from independent arrivals, and because µ does not depend on the type of customer:
if xi= 0 for k distinct indices i. By deconditioning and counting the number of combinations with xi= 0 for k distinct indices we find
P[L0= k] = ∞ X n=k P[L0= k | L = n]πn= ∞ X n=k n k pkqn−kρn(1 − ρ) = (1 − ρ)(pρ) k k! ∞ X n=k (ρq)n−kn! (n − k)! =(1 − ρ)(pρ) k k! dk d(ρq)n ∞ X n=k (ρq)n ! = (1 − ρ)ρ k 0 (1 − ρ1)k+1 .
From this we can conclude that L0+ 1 ∼ Geom ρ0
1−ρ1
, and the distribution of L1now follows by symmetry.
Expected sojourn time
In the FIFO model described above, the time average sojourn time E[W ] of customers can be determined using Little’s law. The time average total number of customers in the system E[L] can be determined from the steady states:
E[L] = ∞ X n=0 nπn = π0λ µ ∞ X n=0 n λ µ n−1 = π0λ µ d dλ µ ∞ X n=0 λ µ !n = π0λ µ(1 − λ/µ)2 = λ µ − λ. From Little’s law we now deduce:
E[W ] = E[L]
λ =
1 µ − λ.
Since in the FIFO model the origin of the customer is not related to the position in the queue of an customer, we find that the expected time spent in the system also does not depend on an customer’s origin, ergo E[W0] = E[W1] = E[W ]. Via Little’s law we find that E[L0] = λ0
λE[L] and E[L1] = λ1
λE[L], so the number of customers per queue in the system is proportional to the arrival rate of that queue.
3.2 Ranking the arrivals
One might wonder what happens to the mean sojourn time when the classes of arrivals are ranked, such that we distinguish between high-priority arrivals and low-priority arrivals. We distinguish between priority queues where one class is given absolute priority over the other, and polling systems in which the server distributes its attention between classes according to more arbitrary rules. But first there is a remark on preemption to be made.
Preemptive-resume scheduling
In priority- and polling systems the situation may arise that the server is triggered to switch to another queue while a customer is still in service. If this service is interrupted and later resumed, the scheduling discipline is considered preemptive-resume. If instead the current service is completed before the server switches to the next queue, the schedule is considered non-preemptive. For simplicity’s sake, we shall assume preemption during the remainder of the analytical treatise. The effect of this simplification will be analyzed in the chapter on simulations.
Expected remaining service time
In the preceding, the equilibrium distribution was used to determine the expected sojourn time. An alternative method is provided by the grace of the PASTA property, following the lecture notes ofAdan & Resing(2002): An arbitrary arrival will spend E[W ] time in the system including
his own (average) service time 1/µ, but also observes E[L] customers in the system upon arrival, each requiring on average 1/µ service time. Hence E[W ] = E[L]/µ + 1/µ, which via Little’s law returns the same result. Note that one of the E[L] is in service with probability ρ upon said arrival, however the expected remaining service time equals an entire service’s expected time by the memorylessness of the exponentially distributed service times. In the case of non-memoryless service times, the following general expression can be used for the expected remaining service time.
Let S denote the total service time of the service that has already started when our customer joins the queue, and let RB denote the remaining time on that service. Since we are randomly selecting a service time over time (consider Figure 3.3), the probability density of selecting a service time of length ` increases linearly with `. The probability density that a randomly selected service has length ` is of course still dB(`). It follows that dS(x) = xdB(x)C−1 where C−1=R∞
0 xdB(x) = E[B] is a scaling constant. Given that a Poisson arrival happens during a fixed time interval, the time of arrival is uniformly distributed within this period, such that
3.1: Distribution of remaining service time
dRB(s) = Z ∞ s 1 tdS(t)ds = R∞ s dB(t) E[B] ds =P[B > s] E[B] ds, and we have
3.2: Expectation of remaining service time
E[RB] = Z ∞ 0 tdRB(t) = R∞ 0 t − tB(t)dt E[B] = t2 2(1 − B(t)) ∞ 0 + R∞ 0 t2 2dB(t) E[B] = E[B 2] 2E[B]. Arrival moment R
Bi−1 Bi Bi+1= S Bi+2
Figure 3.3 The random remaining service time R upon the arrival of an agent into the queue. The probability of selecting a service interval depends on the probability B(t) of a service interval of that length occurring, as well as the length t of that interval.
3.3 Priority queueing
Expected sojourn time
In priority queues arrivals of class 0 are given total priority; this means queue 1 is only served if queue 0 is empty, and is left to grow without bounds as long as the server is not done processing the entire queue 0. Customers within the same class are served FIFO. This amendment with
respect to the regular FIFO 2-class model does not influence the total arrival rate or departure rate, nor does it impose restrictions on when the server may serve customers (just on the order in which the customers are served). Therefore the quantities E[L] and E[W ] haven’t changed with regard to the FIFO model. Moreover, the customers in queue 0 observe a regular M/M/1 queue with arrival rate λ0 and departure rate µ.1 This fact becomes apparent if we realize that
customers of class 1 are queued separately from customers of class 0, and they are processed precisely when there are no customers of class 0 in the system. We can therefore conclude that the number of type-0 customers in the system and their average time spent in the system are given by the equivalent quantities from an M/M/1 queue. To summarize:
E[W ] = 1
µ − λ, E[W0] = 1 µ − λ0
.
The customers in queue 1 however do not follow a regular M/M/1 queue. The class-1 customers observe a server with exponential serving times, but also suffer from random service intermissions. Queueing models equipped with this property are denoted vacation models for the obvious reason. We can nevertheless find the time average number of customers and time spent in the system with relative ease. First of all, we have L0+ L1= L, such that
E[L1] = E[L] − E[L0] = λ µ − λ−
λ0 µ − λ0 =
µλ1 (µ − λ)(µ − λ0). Now, we may apply Little’s law again to find
E[W1] = E[L1] λ1 =
µ
(µ − λ)(µ − λ0).
This expression quantifies the effect of an increase in the type-0 rate of arrival in priority queues. in FIFO queues, the average time spent in the system was equal for customers of all classes and an increase of the arrival rate of one class had the same effect on all classes. The fractional time-penalty for second tier customers is Eprio[W1]/Ef if o[W1] = λ(1−ρ1
0), implying that the
low-priority penalty on the waiting time for type-1 customers grows hyperbolically with the proportion of the server’s capacity that is occupied by type-0 customers. Furthermore, since λ ≥ λ0 we have µ − λ ≤ µ − λ0, thus Eprio[W1] is finite if and only if the average time a customer would spend in the equivalent FIFO system is finite.
Equilibrium distributions
The customers in class 0 observe a regular M/M/1 queue, and therefore their equilibrium probabilities are as was earlier derived in (1): πi,0 = ρi0(1 − ρ0). Since the total arrival- and departure rates haven’t changed, the equilibrium probabilities for the total number of customers remains πi= ρi(1 − ρ).
The type-1 customers on the other hand observe random interruptions of their service whenever a type-0 customer arrives. During these vacations the type-1 queue grows with A1(BP ) customers, where BP is the length of a type-0 busy period and A1(t) is the type-1 Poisson arrival process. Using that conditional on BP , A1(BP ) is Poisson distributed, the conditional PGF of A1is given by e−λBP (1−z). Thus, denoting the LST of the busy period by φ(t), the PGF of the number of type-1 arrivals during a vacation is given by
∞ X k=0 zkγk := Z ∞ 0 e−λ1t(1−z)dBP (t) = φ(λ 1(1 − z)).
Following the work ofN´u˜nez Queija(2000), we will first consider L1 only during non-vacation periods by considering arrivals during a vacation as a batch arriving at the end of that vacation. Notice that the time until a vacation period is exponentially distributed with rate λ0 such that the preceding constitutes a Markov chain that sees departures at rate µ, single arrivals at rate λ1 and batch arrivals at rate λ0. The corresponding balance equation is given by
πn,1(λ0+ λ1+ µ) = πn−1,1λ1+ πn+1,1µ + λ0 n X
k=0
πk,1γn−k.
Multiplying by zn+1, summing over n and splitting the Cauchy product of the rightmost summand yields a functional equation for the PGF of L1during non-vacation periods which solves into
E[zL1 | L0= 0] =
(1 − ρ1)(z(µ + λ0− ν) − λ0φ(λ1)) + λ1− µ z(λ0+ λ1+ µ) − µ − z2λ1− zλ0φ(λ1(1 − z)).
During vacation periods the following can be said about L1: the PASTA property implies that the number of customers at the start of a vacation has the same distribution as L1 during non-vacations.
Next we wish to know the PGF ψ of the number of arrivals during the backwards recurrence time of a vacation – that is, during the time since the start of the vacation. By symmetry, the backward recurrence time of a vacation at a random arrival has the same distribution as the residual vacation length at a random arrival, which we derived in (3.1). This gives the distribution of the number of arrivals during the backwards recurrence time as
P[A(RBP) = k] = Z ∞ 0 P[A(t) = k]P[RBP = t]dt = Z ∞ 0 P[BP > t] E[BP ] e−λ1t(λ1t) k k! dt.
Multiplied by E[BP ]λ1(1 − z)zk, summed over k ∈ N and after bringing the summand inside the integral the foregoing reads
E[BP ]λ1(1 − z)ψ(z) = λ1(1 − z) Z ∞
0
P[BP > t]eλ1t(z−1)dt = 1 − φ(λ1(1 − z)), where the last equality is obtained by partial integration. Hence we find that
ψ(z) = 1 − φ(λ1(1 − z)) E[BP ]λ1(1 − z)
.
Thus, by independence of arrivals and because L1at any point during a vacation is the sum of L1 at the start of the vacation plus the arrivals in the backward recurrence time up to that point, we obtain the PGF of L1 during vacations:
E[zL1 | L0> 0] = 1 − φ(λ1(1 − z)) E[BP ]λ1(1 − z)E[z L1 | L0= 0]. Hence, because P[L0> 0] = ρ0, E[zL1] = ρ0(1 +1 − φ(λ1(1 − z)) E[BP ]λ1(1 − z)) (1 − ρ1)(z(µ + λ0− ν) − λ0φ(λ1)) + λ1− µ z(λ0+ λ1+ µ) − µ − z2λ1− zλ0φ(λ1(1 − z)). (2) A type-0 idle period lasts on average 1/λ0. Since type-0 busy periods and idle periods alter-nate, 1/λ0
1λ0+E[BP ] and 1 − ρ0 are equivalent expressions for the fraction of idle time, such that
E[BP ] = ρ1−ρ0/λ00. Lastly (see for example Adan & Resing (2002) for an elegant derivation)
φ(s) = (2λ0)−1(λ0+ µ + s −p(λ0+ µ + s)2− 4λ0µ), by which the equilibrium distribution of L1 is defined. Alternatively a computationally more efficient (when approximations suffice) recursive relation for the γk can be derived, of whichSleptchenko et al.(2015) contains a detailed exhibit.
3.4 Exponential polling
We have now considered two extremes, the FIFO model where we showed that a customer’s class effectively means nothing and a priority model where classes are used as a strict total ordering. In many practices these models are inapplicable due to various limitations. Often there is a penalty on switching the server between different queues, making the FIFO approach infeasible as it requires unconstrained switching between queues. There may also be a maximum vacation time for each non-empty queue, which cannot be modeled in a priority queueing system. The general polling system does allow to model these demands: in a polling system the server distributes its time between the queues according to an arbitrary scheme, possibly depending on the time, the arrival time of customers or the queue that is being served. Note that both FIFO and priority models are specific cases of polling systems. In this section we introduce exponential polling, which allows both control over the average switching frequency and the fraction of time spent on each queue.
Equilibrium distributions
The exponential polling scheme devotes some exponentially (with polling rate τ0) distributed period to queue 0 after which a τ1-exponentially distributed period is devoted to queue 1. Because the fraction of time available to serve type-i customers reduces from 1 to 1/τi
1/τ0+1/τ1, the queue
length of class i is finite if ρi< 1/τi
1/τ0+1/τ1. This system is suboptimal in the sense that it might
devote its capacity to type-1 customers at times when the type-1 queue is empty whilst the type-0 queue is not. This polling scheme is independent from the queue length, time, and the arrival-and serving process, by the grace of which it is relatively easy to write down the possible state transitions and their transition rates. We only have to consider whether or not a queue is being polled when we wish to write down the transition matrix for that queue. Each queue i therefore has state process (Li, δi) where Li are the number of customers in the system of type i and the indicator δi∈ {0, 1} denotes whether queue i is being polled. Let us derive the equilibrium probabilities for type-0 customers. The equilibrium probabilities for type-1 customers follow by a symmetry argument. For future reference, define
P[(L0, δ0) = (n, 1)] = πn, and P[(L0, δ0) = (n, 0)] = π¯n.
Of course, if a queue is not being polled, it can not reduce in size. Figure3.4contains a graphic overview of the state transitions. Because of the exponential polling times, the exponential service times and the Poisson arrivals the evolution of this process is Markovian, such that we may solve the balance equations to find the equilibrium probabilities.
The red resp. green cut in Figure 3.4yield for n > 0 the balance equations λ0(πn+ π¯n) = µπn+1 and (λ0+ τ0)πn+ λ0πn−1= µπn+1+ τ1π¯n.
We point out that the balance equation for π0 reads (λ + τ1)π0= τ0π0 and that queue 0 is being polled a fraction equal to τ1
τ0+τ1 of the time
2. First we rewrite the red cut balance equation
as πn+1= ρ(πn+ π¯n), and substitute this twice in the green cut balance equation: (λ0+ τ0)πn+ λ0πn−1= µπn+1+ τ1π¯n= λ0(πn+ πn¯) + τ1πn¯
τ0πn+ λ0πn−1= (λ0+ τ1)π¯n τ0ρ(πn−1+ πn−1) + λ0πn−1= (λ0+ τ1)πn¯
2One polling cycle spends on average 1/τ
Hence the balance equations describe the bivariate homogeneous recurrence relation πn πn = Anπ0 π0 = An 1 τ0 λ0+τ1 π0, with A = ρ ρ τ0ρ λ0+τ1 λ0+τ0ρ λ0+τ1
where we must have lim n→∞A n= 0 , ∞ X n=0 πn πn = τ1 τ0+τ1 τ0 τ0+τ1 and π0+ π¯0= π0 τ0+ τ1+ λ0 τ1+ λ0 = 1 − ρ,
which completely defines the individual equilibrium distribution of class 0 via the identity P[L0 = n] = πn+ π¯n = ρ−10 πn+1 given by the balance equation at the red cut. The balance equations of class 1 follow by symmetry.
Expected sojourn time
The expected sojourn time is found as follows. An arriving type-0 customer observes E[L0] customers already queued who, including the arrival, require 1
µ(1 + E[L0]) service time. During this period the server switches τ0
µ(1 + E[L0]) times to serve type-1 customers, switching back after on average τ1
1. With probability
τ0
τ0+τ1 the server is at queue 1 upon the customers arrival,
from which it returns after 1
τ1. Setting τ = τ0+ τ1, altogether this yields
E[W0] = 1 µ(1 + E[L0]) + τ0 µτ1(1 + E[L0]) + τ0/τ1 τ0+ τ1 so that E[W0] = τ /µ + τ0/τ τ1− ρ0τ
with a symmetrical expression for E[W1], and by using L = L0+ L1 and two applications of Little’s law E[W ] = λ1 λ τ /µ + τ1/τ τ0− ρ1τ + λ0 λ τ /µ + τ0/τ τ1− ρ0τ . 0, 1 . . . n − 1, 1 n, 1 n + 1, 1 . . . 0, 0 . . . n − 1, 0 n, 0 n + 1, 0 . . . τ0 τ1 τ1 τ0 τ1 τ0 τ1 τ0 τ1 τ0 λ0 µ λ0 µ λ0 µ λ0 µ λ0 µ λ0 λ0 λ0 λ0 λ0
Figure 3.4 State transitions in the exponential polling model for queue 0. Each node corresponds to a state (L0, δ0) as defined above.
4
The green wave
Consider a network of intersections as depicted in Figure 4.1. We shall study a traffic flow 0 passing through n intersections without interruption, where at each intersection another flow of traffic competes for service time. Throughout this chapter we will distinguish between ‘green wave’ type-0 customers and ‘cross traffic’ type-k customers, where, unless otherwise stated, it may be assumed that k > 0. Let us assume that the time required by type-0 customers from exiting intersection k − 1 to arriving at intersection k is fixed for each k, and that the service time at each subsequent server is determined by and equal to the first service time. Since the arrival- and departure times are independent of time, without any more loss of generality we can now assume that a type-0 customer passes all servers at the same time and that all type-k customers experience exactly the same interruption in their service whenever a type-0 customer is served. Furthermore, let the interarrival and service times be exponentially distributed. We also once more assume that the service time is the same for each queue. We will develop different methods to divide service time amongst type-0 customers and other-type customers and study the implications on the mean time spent in the system for each type and for the system as a whole.
0(λ0) X1 X2 X3 Xn
1(λ1) 2(λ2) 3(λ3) n(λn)
Figure 4.1 The elementary network model. Type-0 customers must pass intersections X1, . . . , Xn
where they must compete with type-k customers over server Xk.
4.1 Priority queueing: green wave first
Expected sojourn time
If we prioritize type-0 customers and we assume preemption then no time is needed to clear the servers before a type-0 customer arriving at an empty type-0 queue can be served. Prioritization of type-0 customers thus works the same as in single queue M/M/1 models, where for each k > 0 the queue formed by adding type-0 and type-k customers is again a regular M/M/1-queue. This latter queue is indexed by the subscript {0, k}. Hence we find
E[W0] = 1 µ − λ0, E[Wk] = E[L{0,k}] − E[L0] λk = µ (µ − λ0− λk)(µ − λ0). Since E[L] =Pnk=1E[Lk], two applications of Little’s law and setting λ =
Pn k=0λk gives us E[W ] = λ−1 n X k=0 λkE[Wk].
Equilibrium distributions
Since the type-0 customers observe a regular M/M/1 queue, the equilibrium distribution of L0 is given by P[L0 = n] = ρn0(1 − ρ0). Each type-k customer observes the same system as the type-1 customer in the single-server priority system presented in3.3, such that the equilibrium distribution of Lk may be derived from (2):
E[zLk] = (ρ0+
(1 − ρ0)(1 − φ(λk(1 − z))) λ0λk(1 − z) )
(1 − ρ1)(z(µ + λ0− ν) − λ0φ(λ1)) + λ1− µ z(λ0+ λk+ µ) − µ − z2λk− zλ0φ(λk(1 − z)) where φ(s) is, as before, given by φ(s) = (2λ0)−1(λ0+ µ + s −p(λ0+ µ + s)2− 4λ0µ).
4.2 Priority queueing: green wave last
If the type-0 customers are given the lower priority, the type-k customers observe a regular queue with the by now common expressions for the equilibrium distribution and expected sojourn time. For type-0 customers, consider the following.
The system cycles through two periods: either one or more type-k customers are being served or no type-k customers are being served. During the former period (the Joint Busy Period with length J BT ) the type-0 queue sees only arrivals whilst during the latter (the Joint Idle Period with length J IP ), type-0 customers can also be served.
Equilibrium distribution
The type-0 queue length is itself not a Markov chain because it depends on the state of the other queues. The smallest continuous-time Markov chain describing L0therefore has state space Nn+1, rendering a derivation of the equilibrium distribution via solving the balance equations virtually impossible. Alternatively we may embed a Markov chain at the transition moments of L during the joint idle periods much like our work in section3.3, but this method requires the distribution of J BP . Note that at the start of a joint busy period only a single queue is in use and there is no limit to the number of individual busy periods during a a J BP , such that J BP is not related to the maximum of all busy periods, or any comparable quantity. Moreover when one busy period ends, the probability that other busy periods have ended depend on the length of the busy period that just finished. This dependence makes the distribution of J BP far from trivial1. This problems is also closely related to determining the return time to the origin of
n-dimensional random walks in the positive orthant. However, using an indirect argument it is possible to derive the expectation of J BP .
Expected sojourn time
We find that the fraction of time that the system is in a joint idle period equals the fraction of time that all type-k queues are idle:
γ := E[J IP ] E[J IP ] + E[J BP ] = n Y k=1 (1 − ρk).
Furthermore, the joint idle period finishes upon the first non-type-0 arrival. By memorylessness the time until the first type-k arrival after the start of a joint idle period is exponentially distributed with rate λk, and since the minimum of exponentially distributed stochastics is exponentially distributed with the sum of the individual rates as its rate, defining λ¯0=Pnk=1λk we find
E[J IP ] = 1 λ¯0 and E[J BP ] =1 − γ γλ¯0 .
An arriving type-0 customer will find E[L0] type-0 customers already queued, such that he leaves again after E[L0] + 1 services of length 1/µ. During this time, joint busy periods start at a rate λ¯0each of which lasts E[JBP ]. Moreover, the arriving customer may find the system to be in a joint busy period upon arrival with probability 1 − γ such that he must wait the remaining duration of the joint busy period E[RJ BP]. Unfortunately, an expression for E[RJ BP] eludes the author. Yet, putting everything together using (3.2) and Little’s law:
E[W0] = 1 µ(E[L0] + 1)(1 + λ¯0E[J BP ]) + (1 − γ) E[J BP2] 2E[JBP ] = (µγ)−1+ γλ¯0E[J BP2]/2 1 − ρ0/γ . For stability we thus require that the fraction of time available for serving is greater than the fraction of time that the system isn’t idle, that is :γ > ρ0. Via the inequalities 0 ≤ E[RJ BP] ≤ E[J BP ] a somewhat loose bound is obtained on E[W0]:
1 µ γ − ρ≤ E[W0] ≤ 1 µ+ (1−γ)2 λ0 γ − ρ
4.3 Exponential polling
Expected sojourn time
Suppose we let the server switch from serving type-0 customers to serving type-k customers and back after exponential times τ0 resp. τ1. The theory from section 3.4can be reapplied since the switching happens independently from server occupation, yielding
E[W0] =τ /µ + 1 − τ1/τ τ1− ρ0τ , E[Wk] = τ /µ + 1 − τ0/τ τ0− ρkτ , E[W ] = λ −1 n X k=0 τ ρk+ λk− λkτδ0k/τ τδ0k− ρkτ where δjk is used as the kronecker delta. Moreover, for stability we require τ1/τ > ρ0 and τ0/τ > ρk. If we let τ0= ατ for some fixed α ∈ (maxk>0ρk, 1 − ρ0), then the previous expression reads as E[Wk] =αµ−λ1
k+ τ
−1 1−α
α−ρk, which shows that increasing the total cycle rate τ decreases
the waiting time for each queue. If we let τ approach infinity, the queues even become unaware of each others presence in terms of expected sojourn time, but instead observe a server that operates at rate αµ. Rewriting the expression as E[Wk] = α−ρ1
k
τ +µ−λk
µτ − τ
−1on the other hand shows us that for fixed τ , E[Wk] grows hyperbolically as the server shifts its attention more often to queue 0. The mirrored results follow for queue 0 by symmetry.
Equilibrium distribution
The equilibrium distributions for type 0 are given by πn,0= ρn0(1 − ρ0)τ1+ λ0 τ + λ0 1 1 1 1 τ0 λ0+τ1 µ+τ0 λ0+τ1 n 1 τ0 λ0+τ1
and for type k by
πn,k= ρnk(1 − ρk) τ0+ λk τ + λk 1 1 1 1 τ1 λk+τ0 µ+τ1 λk+τ0 n 1 τ1 λk+τ0 .
Again we write τ0= ατ and let τ approach infinity. This time the matrix expression for πn,k collapses into the familiar expression πn,k = αnρnk(1 − αρk), such that the queue length even converges in distribution to a regular M/M/1 queue with serving rate αµ.
5
Simulations
In this section we will review via simulations the effect on the performance measures of a number of assumptions that were made. In particular, we will study the network model with high and low priority for the green wave and with exponential polling. Since the indices are of inferior imporance in this chapter, we will refer to the type-0 customers as green wave customers and to the type-k customers as cross queue customers. Models with low and high saturation are simulated, where saturation is measured as the green-wave arrival rate plus the greatest cross-queue arrival arrival rate as a ratio of the departure rate. Saturation thus coincides with the average occupation ρ of the M/M/1 queue comprised of the green wave and the busiest cross queue. For a given saturation, the simulated models are either ‘heavy green wave’, ‘balanced’ or ‘heavy cross queue’. In heavy green wave models the bulk of the arrivals are green wave customers, and so forth.
5.1 influence of assumptions
The simulations were run for a period of 20,000 time units with arrival rates between .5 and 6 such that the number of arrivals ranged from approximately 10,000 to 120,000 per queue.
Figure 5.1 Simulations of different queueing disciplines in the saturated, heavy cross queue system setup. The exponential polling model switches service to the green wave at rate 3 and switches service to the crossing queues at rate 7.
Preemption
To study the effect of preemption on the queue length distribution and expected sojourn time, a queueing network is simulated with Poisson arrivals and exponential departures according to the proposed network queueing schedules both with and without preemptive-resume. Instances were considered with low (20%) or high (80%) saturation. Both instances were considered with a relatively high load for either cross-queues or the green wave, as well as a balanced load.
Instances with a low saturation showed insignificant differences between models with and without preemption. In the balanced model and the model with a high load green-wave arrivals, notable differences in expected sojourn time and queue length distribution are present for the green wave between models with and without preemption, but the differences are again insignificant for the cross-queues.
In the saturated model with high-load cross queues large differences are found in expected sojourn time and queue length distribution, where the effect is proportionally larger for cross queues with a lower arrival rate. The queue lengths are reported in Figure5.1 and the mean sojourn time and total queue size per queue and queueing discipline are reported in Table5.1.
CQ 1 (1.5) CQ 2 (3) CQ 3 (6) GW (2)
exponential polling with preemption 0.234 0.328 1.28 1.73
exponential polling without preemption 0.183 0.253 0.913 1.07
prioritized cross queues with preemption 0.118 0.142 0.252 6.28 prioritized cross queues without preemption 0.191 0.212 0.289 1.48
prioritized green wave with preemption 0.19 0.249 0.616 0.125
prioritized green wave without preemption 0.211 0.267 0.692 0.222
Table 5.1 Mean sojourn time and number of arrivals per queue (CQ for crossing queue and GW for green wave) and per queueing discipline in the saturated, heavy cross queue system setup.
The difference in sojourn time is in each instance the largest for the green wave. In the exponential polling model, the green wave is confronted with a higher frequency of residual service time for a cross queue upon a switch than vice versa due to the higher greatest arrival rate of the cross queue.
If the green wave is prioritized, non-preemption influences the green wave by adding residual service of the cross queues to the waiting time. Meanwhile the effect of preemption on cross queues is more indirect: By the PASTA property the average number of queued cross queue customers at the start of a green wave busy period is the same with or without preemption. However, at the start of the green wave busy period the green wave queue has length 1 with preemption but is longer without preemption. Consequently, without preemption the green wave sojourn time is higher and the green wave busy period is longer, also causing a higher sojourn time for the cross queues.
When cross queues are prioritized, the increase on waiting time without preemption is only marginal for cross queues because the low arrival rate for green wave customers implies that the probability that the first cross queue arrival after a cross queue idle period finds a green wave customer in service is low. The green wave customers meanwhile find their sojourn time quartered because with preemption they were interrupted with high probability, given the high arrival rate of the crossing queues.
Exponential serving times
To test the effect of the choice of serving process, the analysis for preemption is repeated with a different feasible serving process: a deterministic departure time. Again the low-saturation system is insensible to modeling choices, having more or less the same distribution and sojourn time regardless of departure process, load distribution and preemptiveness. The simulation results for the saturated system with balanced green wave arrival rate and cross queue arrival rate are depicted in Figure5.2and Table 5.2.
When exponential polling and preemption is used, it is apparent that the stability condition alone does not to tell the whole story. The number of customers that get serviced during a polling period decreases hyperbolically with the polling rate. Unlike with exponential services, with
Figure 5.2 Simulations of different queueing disciplines in the saturated, balanced system setup. The exponential polling model switches service between the green wave and cross queues at rate 5.
CQ 1 (1) CQ 2 (2) CQ 3 (4) GW (4)
exponential polling with preemption 0.474 0.686 307 413
exponential polling without preemption 0.231 0.29 0.645 1.05 prioritized cross queues with preemption 0.105 0.113 0.133 2497.76 prioritized cross queues without preemption 0.134 0.139 0.152 0.769 prioritized green wave with preemption 0.318 0.413 1.2 0.134 prioritized green wave without preemption 0.217 0.259 0.475 0.176
Table 5.2 Mean sojourn time and number of arrivals per queue (CQ for crossing queue and GW for green wave) and per queueing discipline in the saturated, balanced system setup.
deterministic services no customers are served if the server switches after a period that is shorter than the expected service time. The fraction of polling cycles of type k with rate τk that last shorter than the service time 1/µ is given by 1 − eτk/µ, hence as the polling rates increase this
fraction approaches 1 and each service is preempted before it is finished.
Also in the high-saturation system with preemption, the green wave has an unstable queue length distribution regardless of load balance when the cross queues are prioritized. The reasoning behind this is similar to that of the exponential polling: The fraction of cross queue idle periods which last shorter than 1/µ is given by 1 − ePnk=1λk/µ, such that again the fraction of preempted
green wave services quickly increases as the number of cross queues and/or their service rates increase.
6
Discussion
In this paper we have modeled traffic flow networks under the assumption that each bottleneck sees Poisson arrivals and that queues have unlimited capacity. Burke’s theorem indeed states that in networks of intersections with unlimited queues, exponential service and serving disciplines which act as if there is only a single queue at each intersection, all intersections actually see Poisson arrivals. A more general version of Burke’s theorem however has not been proven. Moreover, the queues between subsequent bottlenecks have a finite capacity such that the arrival process of subsequent bottlenecks deviates more from Poisson arrivals as the occupation rate of the previous bottleneck approaches 1, since this in general leads to queues hitting their capacity limit more frequently. An investigation on the validity of Poisson arrivals in a more general setting as well as a review of the effect of the Poisson arrivals-assumption provides an interesting topic for future research.
One distinct advantage of the microscopic traffic flow model over the queueing network model is that the former can elicit emergent behavior - that is, typical patterns that are clearly observable on an aggregate scale, but which are very hard to relate to decisions of individual traffic participants. Incorporating emergent behavior in the queueing network model might lead to interesting new insights. One method we would like to suggest for future research is by incorporating a parametrized estimation of empirical arrival processes in queueing networks with general arrival processes.
The concepts from this paper also lend themselves for an interesting marriage with game theory: If the traffic participants are aware of the design of a queueing network, they can alter their route to minimize their sojourn time. This sentience should be taken into account when the optimal network design is sought. This requires determining the Nash equilibria, a game-theoretical concept that depicts systems in which each participant cannot further improve his or her situation by altering their route.
7
Populaire samenvatting
Beleidsmakers hebben tal van ontwerpkeuzes te maken bij de aanleg van nieuwe wegen en kruisingen, en bij deze keuzes moeten ze rekening houden met de wensen van de weggebruikers, milieugroeperingen, de bewoners, enzovoorts. Om zo goed mogelijk aan ieders wens te voldoen is het belangrijk om te weten wat de gevolgen zijn van alle ontwerpkeuzes. Er zijn verschillende soorten modellen beschikbaar om deze gevolgen te onderzoeken. Een belangrijke klasse modellen gebruikt wachtrijtheorie. Deze modellen delen een wegennetwerk op in bottlenecks en vervolgens modelleren ze zo’n bottleneck door iedereen die er voorbij wil aan te laten sluiten in een wachtrij en ze er ´e´en voor´e´en langs te laten. Door een aantal aannames te maken over hoe weggebruikers aankomen bij de bottleneck en hoe ze vervolgens langs de bottleneck kunnen komen lukt het om de verdeling te vinden van de lengte van de wachtrij en de verwachtte tijd die een weggebruiker nodig heeft om langs de wachtrij en de bottleneck te komen. We bekijken twee soorten bottlenecks in detail:
KRUISINGEN •De eerste is een kruising van een aantal verkeersstromen. Voor drie verschillende voorrangsgregels kunnen we per verkeersstroom de verwachte wachttijd en de verdeling van de wachtrijlengte bepalen. Deze regels zijn:
First In First Out Hier maakt het niet uit vanaf welke kant je komt aanrijden, er is maar ´e´en wachtrij waar iedereen achteraan aansluit.
Priority polling Hier moet elke verkeersstroom wachten tot er niemand van een hogere prioriteit de kruising wil oversteken. Vervolgens mogen ze vrij oversteken tot er weer iemand van een hogere prioriteit aankomt. Voor elke verkeersstroom geldt dat de wachtrij verkregen door zijn wachtrij met de hogere-prioriteits wachtrijen samen te nemen zich gedraagt als een enkele wachtrij.
Exponential polling Bij exponential polling mogen de verschillende verkeersstromen afwisselend oversteken. De tijd die elke verkeersstroom achter elkaar krijgt is onafhankelijk van de lengte van de wachtrijen. Elke wachtrij heeft dus te maken met onafhankelijke onderbrekingen waarin die rij niet mag oversteken.
Bij de laatste twee regels moet er onderscheid gemaakt worden tussen nonpreemption en preemptive-resume. In het eerste geval mag iedereen die al begonnen is om de kruising over te steken gegarandeerd de hele kruising over steken, ook als het halverwege het oversteken eigenlijk tijd is om een andere verkeersstroom over te laten steken. In het tweede geval wordt de beurt van de persoon die aan het oversteken is onderbroken, en mag hij verder gaan waar hij gebleven was zodra zijn verkeersstroom weer aan de beurt is.
GROENE GOLVEN •De tweede bottleneck is een keten van kruisingen, waarbij er verkeersstromen zijn die de keten dwars over steken en er een enkele verkeersstroom is die langs alle kruisingen in de keten gaat. Bovendien hoeft deze ene ‘groene golf’ verkeersstroom alleen bij de eerste kruising te wachten, daarna mogen ze vrij doorrijden. Voor de groene golf-bottleneck bekijken we twee voorrangsregels:
Priority polling Ditmaal moeten we een verschil maken tussen voorrang voor de groene golf of voorrang voor de kruisstromen. In het eerste geval gedraagt elke kruisstroom zich samen met de groene golf alsof ze samen een enkele kruising zijn met priority polling. Als de kruisstromen voorrang hebben dan gedragen de kruisstromen zich als afzonderlijke enkele kruisingen met maar 1 verkeersstroom, maar het gedrag van de groene golf wordt nu erg complex.
Exponential polling Elke kruisstroom gedraagt zich nu samen met de groene golf net zo als exponential polling in de enkele kruising.
Bij nonpreemption moet er nu ook rekening mee gehouden worden dat er meerdere personen nog aan het oversteken kunnen zijn als de beurt wisselt van de kruisstromen naar de groene golf.
Tijdens het modelleren worden een aantal aannames gemaakt. Simulaties laten zien dat het wel of niet gebruiken van preemptive-resume en de verdeling van de tijd die nodig is om de kruising over te steken los van elkaar geen grote invloed hebben op de verwachte wachttijd en de verdeling van de wachtrijlengte. Als ze samen worden bekeken dan kunnen ze wel een erg grote invloed hebben.
Bibliography
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