A roundabout model with on-ramp queues: exact results and scaling approximations
AA roundabout model with on-ramp queues: exact results and scaling approximations
P.J. Storm, S. Bhulai, W. Kager ∗ Vrije Universiteit, Amsterdam
M. Mandjes
University of Amsterdam † (Dated: January 31, 2020)This paper introduces a general model of a single-lane roundabout, represented as a circular latticethat consists of L cells, with Markovian traffic dynamics. Vehicles enter the roundabout via on-rampqueues that have stochastic arrival processes, remain on the roundabout a random number of cells,and depart via off-ramps. Importantly, the model does not oversimplify the dynamics of traffic onroundabouts, while various performance-related quantities (such as delay and queue length) allowan analytical characterization. In particular, we present an explicit expression for the marginalstationary distribution of each cell on the lattice. Moreover, we derive results that give insighton the dependencies between parts of the roundabout, and on the queue distribution. Finally, wefind scaling limits that allow, for every partition of the roundabout in segments, to approximate1) the joint distribution of the occupation of these segments by a multivariate Gaussian distribution;and 2) the joint distribution of their total queue lengths by a collection of independent Poissonrandom variables. To verify the scaling limit statements, we develop a novel way to empiricallyassess convergence in distribution of random variables. I. INTRODUCTION
Over the past decades, a broad class of models hasbeen proposed to better understand and control trafficstreams in road traffic networks. This has led to math-ematical models that help shed light on the propertiesof the underlying traffic dynamics. In particular, thesemodels allow for studying the influence of the model’sparameters, which in turn allows for developing effectivedesign and control rules. For reviews on traffic flow theory,see, e.g., [1, 2], and for cellular automata models used inthis area, see [3]. In the literature on traffic flows, mostmathematical analyses are done for road segments andseveral forms of intersection traffic control, i.e., signal-ized intersections and unsignalized intersections with orwithout priorities.Roundabouts are a type of intersection that is noto-riously hard to analyze mathematically. Fouladvand etal. [4] studies the delay experienced by traffic on round-abouts in relation to their geometry by simulating astochastic cellular automata model. Wang and Ruskin [5],Wang and Liu [6], and Belz et al. [7] study the capacity ofcellular automata roundabout models incorporating thetraffic behavior of individual cars in a more sophisticatedmanner. In these models, the analysis focuses on therelationship between the circulating flow, and the capac-ity of an entry road at the roundabout. The conclusionsare primarily based on simulation results, and hence donot provide explicit insight into, e.g., the way the systemparameters affect the capacity or delay.In addition, there are a number of analytical papersstudying the relationship between circulating flow and ∗ { p.j.storm, s.bhulai, w.kager } @vu.nl † [email protected] capacity at an entry road. For example, Flannery etal. [8, 9] have obtained an analytical approximation ofthis relationship based on earlier work for unsignalizedintersections by Tanner [10] and Heidemann and Weg-mann [11]. For these results, vehicles are assumed to beseparated by i.i.d. distributed gaps, so that on-ramps canbe modeled as M/G/ a r X i v : . [ n li n . C G ] J a n tion: we develop a novel procedure to statistically assessconvergence in distribution.The outline of the paper is as follows. In Section II,we introduce the model. In Section III, we identify theexact marginal stationary distribution for the occupationof the roundabout, and discuss why it is difficult to derivefurther analytic results. Our methods, which are usedin later sections, are explained in Section IV. Section Vcontains results on intrinsic model properties, whereas inSections VI and VII we study scaling results. II. MODEL DESCRIPTION
The model we consider is a road traffic model for aroundabout with (on-ramp) queues at the points of en-trance onto the roundabout. The exit point of a car fromthe roundabout is random and depends on its point ofentry. The roundabout is modeled as a stretch of roadconsisting of L cells numbered 1 , . . . , L , which we assumeto be arranged in a circle, so that cell 1 is adjacent tocell L . Making use of this circularity, we will also use theindex L + i to refer to cell i , for 1 ≤ i ≤ L , to simplify no-tation. Each cell can contain at most one car, and to keeptrack of the cars on the roundabout, we attach the statespace { , , . . . , L } to each cell: state 0 indicates that acell is vacant, and a state j ∈ { , . . . , L } indicates thatthe cell is occupied by a car that entered the roundaboutat cell j . For ease of reference, we will also say that a cellis occupied by a car of type j if the state of the cell is j .The main characteristics of the evolution of our stochas-tic system are the following. To model how cars get ontothe roundabout, we assume that there is an on-rampqueue in front of each cell i . At every time step, a new cararrives at the queue of cell i with probability p i ∈ [0 , i is empty. Ifcell i is occupied by a car of type j at a specific momentin time, then with probability q ij ∈ [0 ,
1] the car will leavethe roundabout in the next time step (and otherwise itmoves to the next cell). The fact that the probability q ij depends on j reflects that, in general, the positionwhere a car leaves the roundabout can depend on whereit entered. Note that by setting p i = 0 or q ij = 0 wecan remove on-ramps and off-ramps from the system, andthus flexibly model their position.Now that we have sketched the main principles behindour model, we proceed by providing a more precise accountof the dynamics. A key feature of the model is that theupdate rules (given in detail below) are local , meaning thatat each time step, we can consider what happens at eachof the cells of the model independently, and then updateall the local states in parallel (in accordance with thecellular automata paradigm). Thus it suffices to describewhat happens at a single cell and the corresponding queue.We distinguish between the following cases: Case 1: cell i and queue i are both empty. In thiscase, if no new car arrives at cell i (which happens with probability 1 − p i ), then cell i + 1 and queue i will bothbe empty at the next time step. Otherwise, the newlyarrived car immediately enters the roundabout and moveson to cell i + 1, meaning that cell i + 1 will be in state i at the next time step, and queue i will still be empty. Case 2: cell i is empty and queue i is not empty.In this case, the first car waiting in queue i enters theroundabout and moves on to cell i + 1. Thus, cell i + 1will be in state i at the next time step, and the length ofqueue i will either decrease by one (if no new car arrivesat cell i ), or otherwise stay the same. Case 3: cell i is occupied by a car of type j . In thiscase, queue i is blocked, and hence its length will stay thesame if no new car arrives at cell i , or otherwise grow byone. Meanwhile, the car of type j can decide to leave theroundabout (which it does with probability q ij ), in whichcase cell i + 1 will be empty at the next time step, or thecar decides to drive on, in which case cell i + 1 will be instate j at the next time step. III. PRELIMINARIES
The model under consideration is a discrete-timeMarkov chain, the state of which is a vector describing thestate of each cell and the length of each queue. We willdenote the Markov chain by X = { X t : t ∈ Z + } . It is notdifficult to see that X is irreducible and aperiodic, sincewith positive probability, by choosing the right events, wecan empty the system in a finite number of steps, keepit in the empty state for an arbitrary number of steps,and then send it to any state we like in a finite numberof steps.We say that the model is stable if the Markov chain X is positive recurrent, and hence has a unique stationarydistribution. As our first result, we will now show that,under the assumption of stability, the marginal stationaryprobability π ij that a given cell i is in state j is given by π ij = p j (cid:81) i + L − (cid:96) = j +1 ¯ q (cid:96)j − (cid:81) L(cid:96) =1 ¯ q (cid:96)j , if 1 ≤ i ≤ j ≤ L,p j (cid:81) i − (cid:96) = j +1 ¯ q (cid:96)j − (cid:81) L(cid:96) =1 ¯ q (cid:96)j , if 1 ≤ j < i ≤ L, (1)where ¯ q (cid:96)j := 1 − q (cid:96)j , and π i = 1 − L (cid:88) j =1 π ij , ≤ i ≤ L. (2) Proposition III.1 (Marginal stationary distribution) . If the model is stable, then the marginal stationary prob-ability that cell i is in state j is given by (1)–(2). Proof.
Assume that the model is stable, and first considerthe case that i and j satisfy 1 ≤ j < i ≤ L . Then theprobability that a car that enters the roundabout at cell j will leave at cell i (potentially after first completing n ≥ q ij i − (cid:89) (cid:96) = j +1 ¯ q (cid:96)j ∞ (cid:88) n =0 (cid:18) L (cid:89) (cid:96) =1 ¯ q (cid:96)j (cid:19) n = q ij (cid:81) i − (cid:96) = j +1 ¯ q (cid:96)j − (cid:81) L(cid:96) =1 ¯ q (cid:96)j . We conclude that this expression multiplied by p j is therate at which cars arrive that are of type j , and thatintend to leave the roundabout at cell i . But if themodel is stable, then the rate at which such cars leavethe system must be equal to π ij q ij , where π ij denotes themarginal stationary probability that cell i contains a carof type j . This proves (1) when j < i . The proof in thecase 1 ≤ i ≤ j ≤ L is similar.Even though we now have an exact expression for themarginal stationary distribution of the cells, the full jointstationary distribution of the Markov chain X cannotbe found. In particular, the stationary distribution willnot be the product distribution of the marginals of thecells and queues. Indeed, consider the event that queue i and cell i + 1 are both empty for some i ∈ { , . . . , L } .Then, one time unit earlier queue i must have been empty,because otherwise, either queue i would now still be non-empty, or a car from queue i would now be in cell i + 1.This shows that there is a dependency in the model be-tween adjacent cells and queues, ruling out a product-formstationary distribution.To conclude this section, we discuss the model’s stabilitycondition. We have shown above that when the model isstable, π i is the stationary probability at which cell i isempty. Since cars arrive at cell i with probability p i , andcan only enter the roundabout when the cell is empty, itis conceivable that the model cannot be stable if p i ≥ π i for some cell i . Conversely, one suspects that if p i < π i for all cells i , then the cells will be vacant often enough toprevent the queue lengths from growing arbitrary large,and hence the model will be stable. We have testedthis conjecture using extensive simulation experimentsin which we replace p i by αp i and increase α (startingfrom α = 0). The experiments confirm that a systembecomes unstable when α exceeds the smallest value forwhich αp i ≥ π i ( α ) for some i ∈ { , . . . , L } . ThroughoutSections IV–VII, we therefore restrict ourselves to caseswhere p i < π i for all i ∈ { , . . . , L } . IV. METHODS
Since we do not have a closed-form expression for thejoint stationary distribution, we resort to finding ap-proximations for the stationary distributions of cells andqueues. More specifically, in Section II we introduced the p i and q ij , which can be seen as discrete profiles of arrivaland departure probabilities (as a function of the posi-tion i between 1 and L ). In Section IV A, we introducetheir continuous counterparts, so that for finite L , the p i and q ij are obtained as discretizations of these continuum profiles. The continuous setting allows explicit analysis,with which we can approximate our discrete model.Later in the paper (in Sections VI and VII) we stateclaims on, respectively, the number of empty cells andtotal queue length for each section of the roundabout inthe regime L → ∞ . To verify these claims from simulationexperiments, we develop a novel methodology, which isdescribed in Section IV B. A. Continuum Profiles and Parameters
We proceed by introducing the continuum profiles ofarrivals and departures. We start with the arrivals. Let (cid:37) : (0 , → R + be an integrable function that satisfies (cid:82) (cid:37) ( x ) d x = 1. For given L ∈ Z + , θ >
0, and i ∈{ , . . . , L } , we set p i ≡ p i ( L ) = θ (cid:90) ( i +1) /Li/L (cid:37) ( x ) d x. This construction can be interpreted as follows. Whentaking the limit L → ∞ , the circular stretch of roadis mapped onto the unit interval (0 , θ > u, v ] ⊂ (0 , (cid:82) vu (cid:37) ( x ) d x represents the rate at which cars arrive in thatinterval. Informally, for L large, p i is roughly proportionalto L − . Note that in this setup, the arrival rate over everysegment of the roundabout is invariant in L .To describe the continuum profile for the departures, weintroduce a family ( F x ( · )) x ∈ (0 , of cumulative distributionfunctions on [0 ,
1] (which are non-decreasing with F x (0) =0), and denote by F cx ( · ) ≡ − F x ( · ) their complementarydistribution functions. The idea is that in the limit L →∞ , F cx ( u ) represents the probability that a car that entersthe roundabout at point x , travels at least a distance u along the roundabout before leaving. For each finite L ∈ Z + and i, j ∈ { , . . . , L } , we now set q ij ≡ q ij ( L ) = − F cj/L (( i − j + 1) /L ) F cj/L (( i − j ) /L ) , i ≥ j ;1 − F cj/L (( L + i − j + 1) /L ) F cj/L (( L + i − j ) /L ) , i < j. Since we interpret F j/L as the distribution function ofthe driving distance for cars arriving at j/L , 1 − q ij ( L ) for i ≥ j can be seen as the conditional probability that such acar drives at least distance ( i − j +1) /L on the roundabout,given that the car has driven distance ( i − j ) /L , andsimilarly for i < j . Hence the above definition of the q ij guarantees that the distribution of driving distance ofcars remains invariant in L . We further assume that F cx ispiecewise continuous as a function of x , meaning that carsthat arrive at roughly the same place on the roundabout,also have roughly the same distribution of driving distance.This condition is natural, and guarantees the existence oflim L →∞ π (cid:100) uL (cid:101) , .To summarize: for given (cid:37) and a family of F x , weobtain a sequence of models in L , which can be viewed asdiscrete representations of the same roundabout. In theremainder of the paper, we consider two specific cases ofcontinuum profiles and the discrete models they producefor different values of L , in order to support the claimswe make in Sections V, VI, and VII.Most of the arguments by which we arrive at our claimsare based on the symmetric case where p i = p ∈ (0 ,
1) and q ij = q ∈ (0 ,
1) for each i, j ∈ { , . . . , L } . We thereforechoose a parameter setting in this symmetric case suchthat π i > p i . More specifically, we choose (cid:37) ( x ) = 1 with θ = 1, and F cx ( u ) = exp( − u ) for each x ∈ (0 , L we have p ( L ) = 1 /L and q ( L ) = 1 − exp( − /L ).We refer to this choice as the homogeneous setting or homogeneous case .To illustrate that our claims are also supported in a re-alistic non-homogeneous case, we use an example from [13,Ch. 21], namely Example Problem 1. This example de-scribes a roundabout with four on/off-ramps, and givesfor each on-ramp (1) the number of arrivals per hour,and (2) the fraction of arriving cars that depart via eachof the four off-ramps. To choose a (cid:37) and family of F x that correspond to this example, we start by calibratinga finite L model that has a realistic size. Using the cali-bration in [7, Section 3.1], we take the length of each cellto be about 7 meters and our time steps to be 1 second,and find that L = 20 is a suitable choice. The result-ing model has geometric features and car velocities thatmatch the realistic ones described in [13] and [14]. Welet the on/off-ramps be located at cells i = 1 , , , p i and q ij to zero. We calculate the arrival probabilities p i at the four on-ramps from the given number of arrivalsper hour in the example problem. The departure probabil-ities q ij are analogously obtained from the given fractionsof arriving cars that depart via the off-ramps. The latterrequires that we first fix the probability (cid:81) L(cid:96) =1 ¯ q (cid:96)j that acar completes a full circle on the roundabout; we set thisprobability equal to 1% for every type of car, and thendetermine the q ij to reproduce the departure behavior ofthe example.Now that we have the p i and q ij for L = 20, we canchoose our continuum profiles (cid:37) and F x accordingly. Re-call that we map the full roundabout to the interval (0 , L = 20, each cell corresponds to an intervalof length 0 .
05. We further split each of the four cells i = 1 , , ,
16 into two halves, where the half that isadjacent to the previous cell corresponds to the off-rampof the cell, and the other half to the on-ramp. We nowchoose (cid:37) proportional to p i at the on-ramps and zeroelsewhere, and we choose θ = (cid:80) i =1 p i , so that for L = 20,integration of (cid:37) gives us the correct p i . As for the de-parture profiles, we choose the F cx to be exponentiallydecreasing at the off-ramps in analogy with the homoge- . . . .
81 0510152025 % %F cx F cx u FIG. 1. Graphs of F cx for x ∈ (0 . , . (cid:37) , in theheterogeneous setting. neous case, and constant in between. Here, the rate ofthe exponential decrease is chosen such that we obtainthe correct q ij for L = 20. In Fig. 1 we have plotted theresulting profiles (cid:37) and a representative from the family( F cx ) x ∈ (0 . , . for illustration. We refer to the profiles (cid:37) and F x thus obtained as the heterogeneous setting or heterogeneous case . We stress that, although these profileswere calibrated for L = 20, we use the same (cid:37) and F x inour simulations of the heterogeneous case for other valuesof L .In the remainder of the paper, we present various claimsabout the model. We cannot prove these claims, as we lackan analytic expression for the joint stationary distribution.Instead, we will support our claims using simulation incombination with statistical evidence. We throughoutuse the following structure: first we state the claim andgive intuition behind it based on properties of the model,then we describe an experiment by which we aim tosupport the claim, provide our support, and finally, wedraw our conclusions. In each simulation experiment, weinitialize (1) the cells according to the marginal stationaryprobabilities π ij , and (2) the queues empty. We then letthe system run for 4 L units of time, as we have observedthat this is a sufficiently long time interval to safely assumethe system has entered the stationary regime. B. Supporting Convergence in DistributionStatistically
In Section VI, we consider the number of empty cells ona segment, for a sequence of models in L that we obtainfrom the continuous arrival and departure profiles, asexplained in the previous section. Among other things,we claim that this quantity converges in distribution to anormal random variable as L → ∞ . To empirically verifythis claim, we use two methods. The first, which is classi-cal, is to show that the (empirical) distribution functionsconverge pointwise. The second uses statistical tests andis, to the best knowledge of the authors, a novel methodto numerically support convergence in distribution. Weexplain the second method in this section.For our explanation, we consider the situation where { ξ L } L is a sequence of random variables that convergein distribution to a N ( µ, σ ) random variable. In ourmethod, we use the chi-squared goodness-of-fit test with aconfidence level equal to 0 .
99. We take 10 bins, the bound-aries of which are chosen such that every bin contains10% of the probability mass of the N ( µ, σ ) distribution.The naive idea for testing convergence to a normaldistribution would be to take L large, and apply the χ -test with the ( L -dependent) hypotheses H ( L ) : ξ L d = N ( µ, σ ); H ( L ) : ξ L d (cid:54) = N ( µ, σ ) . (3)However, a χ -test with these hypotheses does not giveuseful information on convergence because, in practice,one expects that ξ L does not have a N ( µ, σ ) distributionfor finite L , and therefore, one will always reject H ( L )if the sample size is large enough. The underlying issueis of course that to support convergence in distribution,it is not sufficient to consider a single ξ L , but one hasto consider the full sequence. Our method exploits thefact that we can always reject H ( L ) by increasing thesample size. The basic idea is that we compare the samplesizes M ( L ) for which we first reject H ( L ). If ξ L convergesin distribution, M ( L ) should diverge to ∞ with L .To put this idea into practice, we start our procedureby drawing a sample of 50 independent copies of ξ L . Weperform the chi-squared test for goodness-of-fit, with thehypotheses as in (3), which is significant for 50 samples(taking into account the expected counts in each bin).If we reject H ( L ), we set M ( L ) = 50; otherwise, weadd another independent copy of ξ L to our sample, andperform the chi-squared test again. We keep addingindependent copies of ξ L until we reject H ( L ), at whichpoint we record the size of our sample M ( L ). Note that M ( L ) is itself a random variable, so we run this proceduremultiple times to estimate the mean E M ( L ). Finally, weuse linear regression to test whether E M ( L ) increaseslike a power law with L , which implies that as L → ∞ , adiverging number of samples is required to reject H ( L ),thus supporting convergence in distribution.Our method can, in theory, be applied to every limitingdistribution with a set of hypotheses as in (3), using anygoodness-of-fit test. For practical applications, however,one has to be able to compute an estimate of E M ( L ).For instance, in Section VII, we claim convergence indistribution of the total queue length on a segment to aPoisson random variable. There is, however, no statisticaltest that is powerful enough to distinguish the specific al-ternative distribution that we are considering. Therefore,one has to use a huge sample size M ( L ) to reject H ( L ),even for small L , which makes estimating the E M ( L )computationally infeasible in this particular case. V. MODEL PROPERTIES
In this section, we study the spatial correlations andmarginal queue distributions of our model in the finite L ce ll s qu e u e s cells queues Corr. ce ll s qu e u e s cells queues p ce ll s qu e u e s cells queues Corr. ce ll s qu e u e s cells queues p ce ll s qu e u e s cells queues Corr. ce ll s qu e u e s cells queues p FIG. 2. Heatmaps of correlations (Corr.) on the left andcorresponding p -values ( p ) on the right, for the homogeneouscase. From top to bottom we have the heatmaps for L = 32, L = 64 and L = 128. regime in equilibrium. Our results also provide informa-tion about the behavior in the regime L → ∞ , which westudy in more detail in Sections VI and VII. A. Spatial Correlations
Our roundabout model can be seen as a system ofparticles moving over a one-dimensional circular lattice.Moreover, the update rules are local, so that correlationsin the model arise via nearest-neighbor interactions. It is,therefore, conceivable that the correlations decay geomet-rically in the distance between cells. To investigate thisidea, denote by C i the state of cell i and by Q i the stateof queue i in equilibrium. In order for states of the cellsto contribute symmetrically to the correlations, we let (cid:101) C i := (( C i − i ) mod L ) + 1when C i (cid:54) = 0, and (cid:101) C i := C i if C i = 0. Thus, (cid:101) C i measuresthe forward distance to the cell where the car that occupiescell i entered the roundabout. Now, our claim is as follows: Claim 1.
The correlation between the random variables (cid:101) C i and (cid:101) C i + k decreases in k for each i ∈ { , . . . , L } , andis bounded above, uniformly in both i and L , by a functionthat decreases geometrically with k . The same statementis true for the correlations between (cid:101) C i and Q i + k , and forthe correlations between Q i and Q i + k . l l l l l l l −10−8−6−4−20 1 2 3 4 5 6 7 8 9 10Distance k away l og ( C o rr e l a ti on ) l cells vs. cellscells vs. queuesqueues vs. queues l l l l l l l l −10−8−6−4−20 1 2 3 4 5 6 7 8 9 10 11Distance k away l og ( C o rr e l a ti on ) l cells vs. cellscells vs. queuesqueues vs. queues l l l l l l −10−8−6−4−20 1 2 3 4 5 6 7 8 9 10 11Distance k away l og ( C o rr e l a ti on ) l cells vs. cellscells vs. queuesqueues vs. queues FIG. 3. Decay of correlations, on a log-scale, for the homo-geneous case. From top to bottom we have L = 32, L = 64and L = 128. For clarity of the figure, the graphs have beenshifted horizontally by a small value. To support Claim 1 it is sufficient for the sample corre-lation to be geometrically decreasing, starting from somedistance k ≥
1. To verify this, we estimate the mean sam-ple correlation coefficient between pairs of cells and/orqueues, from a sample of 100 correlation coefficients, eachestimated from a simulated data set of size 64 · . To an-alyze the decay of the, potentially negative, mean samplecorrelation coefficient on a log scale, we take the abso-lute value of the 100 samples and consider their mean.We then verify that on a log scale, these mean absolutesample correlations are bounded by a decreasing linearfunction. However, from known results on the asymp-totic distribution of the sample correlation [15, Example10.6], we expect that the variance becomes constant asthe correlation tends to zero. As a consequence, in ourexperiment, the mean absolute sample correlation willnot be a good estimator of the absolute correlation whenthe correlation is small. To ensure that our estimates areaccurate, we therefore only consider points for which themean sample correlation is at least two standard errors(as determined from the 100 samples) away from zero.If the correlations do decay geometrically, cells and/orqueues that are ‘sufficiently far apart’ are approximatelyindependent. To verify this, we also perform a statis-tical test of independence. We use the statistic t = r (cid:112) ( n − / (1 − r ), where r is the correlation coefficient,and n is the sample size. For generally distributed in- ce ll s qu e u e s cells queues Corr. ce ll s qu e u e s cells queues p ce ll s qu e u e s cells queues Corr. ce ll s qu e u e s cells queues p ce ll s qu e u e s cells queues Corr. ce ll s qu e u e s cells queues p FIG. 4. Heatmaps of correlations (Corr.) on the left andcorresponding p -values ( p ) on the right, for the heterogeneouscase. From top to bottom we have the heatmaps for L = 32, L = 64 and L = 128. dependent random variables and large n , the t statisticcan be shown to have a Student’s t distribution with n − n = 10 and use t to test whether thesample correlations are significant. We aim to show thatthe correlations are significant over a constant distanceindependent of L , thus further supporting the claim ofgeometric decay, uniformly in L . Support (of Claim 1) . We consider the homogeneouscase first. In Fig. 2 we show heatmaps of the correlationsand their corresponding p-values between cells and queuesfor L = 32 , , L , and then thequeues, indexed from 1 through L . First of all, notice thatnon-trivial correlations do exist, and that for each L theyare significant for certain pairs of cells and queues. Thisconfirms that a product-form stationary distribution doesnot apply, as pointed out earlier. However, we also seethat the dependence is not very strong, since (althoughthey are significant according to the p-values) the correla-tions between neighboring cells and/or queues are small.Furthermore, we observe that p-values are only significantfor correlations between cells and queues that are at most(about) distance 10 away from each other. This distanceis more or less constant in L , which supports our claimthat the rate of the decay is uniform in L .In Fig. 3 we have plotted the mean absolute value of the l l l l l −10−8−6−4−20 1 2 3 4 5Distance k away l og ( C o rr e l a ti on ) l cell 1cell 5cell 9cell 13 l l l l l l l −10−8−6−4−20 1 2 3 4 5 6 7Distance k away l og ( C o rr e l a ti on ) l cell 1cell 9cell 17cell 25 l l l l l l l l −10−8−6−4−20 1 2 3 4 5 6 7 8Distance k away l og ( C o rr e l a ti on ) l cell 1cell 17cell 33cell 49 FIG. 5. Decay of correlations, on a log-scale, for the hetero-geneous case. From top to bottom we have L = 32, L = 64and L = 128. For clarity of the figure, the graphs have beenshifted horizontally by a small value. −15−13−11−9−7−5−3−1 0 2 4 6 8 k l og ( λ k ) −19−17−15−13−11−9−7−5−3−1 0 20 40 60 80 k l og ( λ i k ) i = 13i = 77i = 141i = 205 FIG. 6. Marginal queue distribution on log scale for thehomogeneous case (left) and heterogeneous case (right), alongwith the best-fit line for the homogeneous case. sample correlations between a cell/queue and neighboringdownstream cells and/or queues, for L = 32 , ,
128 start-ing from distance k = 1, with the corresponding standarderror represented by error bars. We have tested for alinear relationship, by applying linear regression, yielding R ≈ .
99 for every line. We therefore deduce that thedecrease is linear, and we can conclude that the meanabsolute correlations decay geometrically. Furthermore,we see that the behavior is homogeneous in L . Basedon the above, we conclude that our experiments supportClaim 1 numerically in the homogeneous case.For the heterogeneous case, we likewise present a set of heatmaps of the correlations and their correspondingp-values in Fig. 4. As some queues are by constructionempty in the heterogeneous case, their correlations aredepicted in gray in the heatmaps. As in the homogeneouscase, the results of our simulations numerically supportClaim 1. To analyze the decay of the correlations, we havealso plotted on a log scale, for the first four cells that area distance L/ R ≈ . L = 64,which has R ≈ .
94. The results confirm a linear decayon a log scale. Therefore, as in the homogeneous case, wefind numerical support for Claim 1.
B. Queue distribution
A natural quantity to study is the (marginal) queuelength distribution of the system. Because of the depen-dencies in the system, one cannot derive the marginalqueue length distribution analytically; likewise, no mean-value analysis is possible to capture the mean queue length.However, because of the weak dependence, one wouldexpect the queue distribution to approximately have ageometric tail. We, therefore, claim the following:
Claim 2.
All queues have marginal stationary distribu-tions with a tail that is close to geometric.
To verify Claim 2, we have simulated the roundabout for L = 256. To estimate the tail of the queue distributionsin a sample of size n = 10 sufficiently accurately, we haveto scale the p i by a factor α , in both the homogeneousand the heterogeneous case. We choose α such that π i ( α ) − αp i ≈ . ≤ i ≤ L . From our data,we estimate the marginal distributions of a set of queueswith equal distance between them, and analyze the tail. Support (of Claim 2) . The results in the homogeneouscase are shown in Fig. 6 (left), where λ ik on the verticalaxis denotes the stationary probability of the event thatqueue i has length k . The figure shows the distributionon a log scale along with its regression line. The slightdeviation from the linear relation for small k shows thatthe distribution is not exactly geometric. However, weobserve that the tail is indeed geometric, as the plot isvery close to the regression line and linear in the tail, untilthe estimation errors kick in, thus confirming Claim 2.For the heterogeneous case, Fig. 6 (right) shows theresults on a log scale. That is, we have plotted thedistribution of one queue in each of the four arrival zones(i.e., the four on-ramps on the roundabout) for L = 256.The legend indicates which queues are considered. Wesee that each distribution is close to a linear decay on alog scale for k above, say, 4. For i = 205, there seems tobe a small deviation from a linear line in the tail of thedistribution, though performing linear regression yields l l l l l l l l l l l l l l lllllllllllllllll −5−4−3−2−1 6 7 8 9 10log ( L ) l og ( d M ( L )) l M = 128M = 256M = 512M = 1024
FIG. 7. Graphs of d M ( L ) in the homogeneous case, for M ∈{ , , , } , along with their best-fit line. an R equal to 0 . VI. SCALING LIMIT FOR CELLS
In this section, we formulate claims about the stationarystate of the cells in the regime L → ∞ . More specifically,we claim that for each division of the roundabout intosegments, the occupation of these segments follows a jointGaussian distribution in the limit. This Gaussian limitprovides an approximation to the stationary distributionof the number of occupied cells, on every segment ofthe roundabout. This knowledge is particularly usefulwhen designing the roundabout; for instance, a perfor-mance target could concern the maximum utilization ofthe roundabout.We first introduce some notation. Let T Lk be the ran-dom variable that counts the number of vacant cells up tocell k : with C i the state of cell i , and δ jk the Kroneckerdelta (i.e., δ jk = 1 if j = k , and δ jk = 0 if j (cid:54) = k ), T Lk := k (cid:88) i =1 δ C i , . Observe that this is a sum of 0-1 random variables withexpectations π i . For x ∈ (0 , π L ( x ) := π (cid:100) xL (cid:101) , and σ L ( x ) := π L ( x )(1 − π L ( x )). Put s L := 1 L L (cid:88) i =1 σ L ( i/L ) = (cid:90) σ L ( x ) d x, s L ≥ , and t Lk := 1 Ls L k (cid:88) i =1 σ L ( i/L ) = 1 s L (cid:90) k/L σ L ( x ) d x. Now let T L : [0 , → R be the random continuous func-tion that is linear on each interval [ t Lk − , t Lk ], k = 1 , . . . , L , l l l l l l l l l l l l l l lllllllllllllllll −5−4−3−2−1 6 7 8 9 10log ( L ) l og ( m a x k j d k j M ( L )) l M = 128M = 256M = 512M = 1024
FIG. 8. Graphs of max kj d Mkj ( L ) in the heterogeneous case, for M ∈ { , , , } , along with their best-fit line. and has values T L ( t Lk ) = T Lk − E ( T Lk ) (cid:112) Ls L at the points of division. Then our claim is as follows: Claim 3. As L → ∞ , T L converges in distribution toa time-inhomogeneous Brownian motion (cid:98) T on [0 , withthe representation (cid:98) T ( t ) = (cid:90) t η ( u ) d B u , interpreted as an Itˆo integral with respect to a standardBrownian motion B , where η is a deterministic continuousfunction on [0 , . We write ‘time-inhomogeneous’, where obviously in thiscontext ‘time’ refers to the position on the roundabout.
Remark VI.1.
Instead of counting vacant cells, onecould also count cells containing a car of a type between (cid:100) aL (cid:101) and (cid:100) bL (cid:101) , for fixed a and b satisfying 0 < a < b ≤ T Lk were independent, T L would converge to standard Brownian motion by anextension of Donsker’s theorem [16, Exercise 8.4]. Unfor-tunately, as stressed before, the cells are not independent.However, we have seen in Section V that the correlationsbetween cells are geometrically decaying in the distancebetween them, and that cells that are ‘sufficiently farapart’ are nearly independent. Hence, one still expectsconvergence to a (time-inhomogeneous) Brownian motion.In particular, we expect that non-overlapping incre-ments of the random function T L become asymptoticallyindependent (as L grows). Moreover, since the centrallimit theorem still holds for sequences of random variablesthat are nearly independent when they are far away fromanother (e.g., see [17, Thm. 27.4] for the stationary case),we expect that the increments converge in distribution tozero-mean normal random variables. −7−6−5−4−3−2 5 6 7 8 9 10log (L) l og ( d s up ( L )) N = 1N = 2N = 4N = 8
FIG. 9. Graph of d sup1 ( L ) for the homogeneous case. As for the covariance matrix between increments, weexpect first of all thatVar (cid:0) T L ( t ) (cid:1) = 1 Ls L (cid:98) tL (cid:99) (cid:88) i =1 Var ( δ C i , )+ 2 Ls L (cid:98) tL (cid:99) (cid:88) i =1 (cid:98) tL (cid:99) (cid:88) j = i +1 Cov (cid:0) δ C i , , δ C j , (cid:1) ∼ (cid:98) tL (cid:99) L + 2 (cid:98) tL (cid:99) a ( t ) L → t + 2 ta ( t ) , where a ( t ) is a constant representing the row averageof all correlations in the upper triangular part of thecorrelation matrix. This sum should be finite because ofthe geometric decay of correlations. Finally, we expectthat the covariances between increments converge to zero,sinceCov (cid:0) T L ( t ) − T L ( s ) , T L ( s ) (cid:1) ∼ Ls L Cov (cid:18) (cid:98) tL (cid:99) (cid:88) i = (cid:100) sL (cid:101) δ C i , , (cid:98) sL (cid:99) (cid:88) j =0 δ C j , (cid:19) ∼ L (cid:98) tL (cid:99) (cid:88) i = (cid:100) sL (cid:101) (cid:98) sL (cid:99) (cid:88) j =0 ρ C i ,C j → . Here, ∼ means that both sides have the same limit as L → ∞ , ρ C (cid:96) ,C k denotes the correlation coefficient, andthe limit is zero since the double sum in the third line is ofconstant order in L by the geometric decay of correlations.In view of the above, to support Claim 3, we aim totest (1) that increments of T L become asymptoticallyindependent as L → ∞ , and (2) that they converge indistribution to zero-mean normal random variables. A. Independence of Increments
To verify that the increments of T L become independentas L → ∞ , we compare the joint distribution of twoincrements to the product distribution of the marginals. −7−6−5−4−3−2 5 6 7 8 9 10log (L) l og ( m a x k d k s up ( L )) N = 1N = 2N = 4N = 8
FIG. 10. Graph of max k d sup k ( L ) for the heterogeneous case. We divide the roundabout of size L into four segmentsof equal length, and denote the four increments of T L on these segments by I Lk , where k ∈ { , , , } and thesuperscript L indicates the dependence on L .As a measure of the distance between the joint distribu-tion of increments k and j and the distribution one wouldhave if these increments were independent, we define d Mkj ( L ) = sup | A | = M (cid:88) a ∈ A (cid:12)(cid:12) f kj ( a ) − f k ( a ) f j ( a ) (cid:12)(cid:12) , (4)Here, the supremum is taken over sets A consisting of M distinct outcomes of the random vector ( I Lk , I Lj ), f kj is thejoint density of I Lk and I Lj , and f k and f j are the respectivemarginal densities. We note that for a = ( a k , a j ) ∈ A ,we interpret f k ( a ) as f k ( a k ) and f j ( a ) as f j ( a j ). Toensure that the product sample space of I Lk and I Lj hasat least M elements, L must be large enough (to beprecise, ( L/ ≥ M ). To support Claim 3, we wishto empirically show that d Mkj ( L ) → k (cid:54) = j as L → ∞ .Note that if we replace the supremum in (4) by asupremum over sets A of arbitrary size, then (4) becomesthe total variation distance. That distance is not suitedfor our purposes, because we have to estimate the densitiesin (4), and the total estimation error grows faster thanthe total variation distance decreases. This is why werestrict the sum to the M largest contributions in (4).In our experiment, we take M ∈ { , , , } and evaluate d Mkj ( L ) by estimating the densities f kj ( a )and f k ( a ) using a simulated sample of size 10 . Support (of Claim 3) . We first consider the homogeneouscase. Since neighboring increments have a stronger de-pendence, as shown above, and because of symmetry, theresults of d M ( L ) are representative for all d Mkj ( L ). Fig. 7shows the estimated d M ( L ) as a function of L on a log-logscale, together with the best-fit line. For every M , weobtain an R between 0 .
78 and 0 .
92, and a negative slope.Thus we conclude that for each M ∈ { , , , } ,the estimated d M ( L ) decreases in L according to a powerlaw. This is sufficient to also conclude that d M ( L ) → L → ∞ , which supports our claim that the incrementsof T L become independent as L becomes large.0 TABLE I. Results linear regression of L versus estimated E M ( L ) (homogeneous case). N k
Rsq Rsq adj
F p intercept slope1 1 0.8134 0.8072 130.7484 1.8457e-12 1.0406 1.00762 1 0.7128 0.7033 74.4703 1.2586e-09 0.7283 1.02612 2 0.7197 0.7104 77.0325 8.7121e-10 0.7331 1.02744 1 0.5816 0.5677 41.7071 3.9042e-07 1.4803 0.78424 2 0.5690 0.5546 39.6035 6.1631e-07 1.3822 0.80334 3 0.5728 0.5586 40.2324 5.3689e-07 1.4932 0.78144 4 0.5797 0.5657 41.3778 4.1895e-07 1.4924 0.78118 1 0.5509 0.5359 36.8022 1.1580e-06 0.1515 0.92818 2 0.5479 0.5328 36.3542 1.2841e-06 0.1686 0.92528 3 0.5435 0.5282 35.7116 1.4914e-06 0.1776 0.92378 4 0.5498 0.5348 36.6344 1.2036e-06 0.1872 0.92148 5 0.5462 0.5311 36.1087 1.3594e-06 0.1447 0.92898 6 0.5428 0.5275 35.6145 1.5257e-06 0.1627 0.92698 7 0.5497 0.5346 36.6159 1.2088e-06 0.1582 0.92778 8 0.5321 0.5165 34.1113 2.1793e-06 0.3014 0.8971
For the heterogeneous case, we have plotted the distancemax kj d Mkj ( L ) (for different M ) in Fig. 8. Our conclusionsare the same as in the homogeneous case. B. Distribution of Increments
We now focus on supporting the part of Claim 3 statingthat the increments of T L converge in distribution to anormal random variable. For this purpose, we divide[0 ,
1] into N intervals of equal length. For fixed N , wedenote the corresponding increments of T L by I Lk , andtheir standard deviations by σ Lk , where k ∈ { , . . . , N } .Denote by t n − the cumulative distribution function of a t -distribution with n − σ Lk ,and, therefore, do not have a complete description ofthe limiting distribution. Hence, we slightly modify thetwo methods by considering the random variables I Lk / ˆ σ Lk ,where ˆ σ Lk is the maximum-likelihood estimator for σ Lk , es-timated from a simulated sample of size n = 10 . Claim 3implies that, as L → ∞ , I Lk / ˆ σ Lk converges in distributionto a random variable that has distribution t n − , and it isthis implication that we will support.With our first experiment, we aim to show that, forevery k ∈ { , . . . , N } , d sup k ( L ) := (cid:107) ˆ F Lk − t n − (cid:107) ∞ → L → ∞ , where (cid:107)·(cid:107) ∞ denotes the supremum norm, andˆ F Lk denotes the empirical distribution function of I Lk / ˆ σ Lk .In our second experiment, we use the novel method thatwas explained in Section IV B. To be precise, we apply TABLE II. Results linear regression of L versus estimated E M ( L ) (heterogeneous case). N k
Rsq Rsq adj
F p intercept slope1 1 0.7541 0.7459 92.0002 1.1971e-10 0.7161 1.07022 1 0.6878 0.6773 66.0790 4.4923e-09 0.5324 0.97242 2 0.5639 0.5493 38.7879 7.3848e-07 2.0480 0.75804 1 0.7518 0.7436 90.8835 1.3761e-10 0.9568 0.82304 2 0.6827 0.6721 64.5423 5.7407e-09 0.3722 0.88144 3 0.6955 0.6854 68.5310 3.0624e-09 0.3669 0.89004 4 0.7479 0.7395 88.9982 1.7462e-10 0.9977 0.82288 1 0.6871 0.6767 65.8841 4.6332e-09 1.0612 0.68208 2 0.6306 0.6183 51.2062 5.8243e-08 0.8349 0.75628 3 0.4906 0.4737 28.8981 8.0635e-06 1.3538 0.61378 4 0.4476 0.4292 24.3093 2.8351e-05 1.6853 0.57148 5 0.4880 0.4709 28.5930 8.7378e-06 1.5911 0.58108 6 0.4839 0.4667 28.1235 9.8957e-06 1.4242 0.61848 7 0.6181 0.6053 48.5450 9.6884e-08 0.7438 0.75488 8 0.6540 0.6425 56.7122 2.1412e-08 0.6693 0.7848 the chi-squared goodness-of-fit test, with the hypotheses H ( L ) : I Lk / ˆ σ Lk d = t n − ; H ( L ) : I Lk / ˆ σ Lk d (cid:54) = t n − , to determine M ( L ). We estimate E M ( L ) by repeatingthe procedure 10 times, and aim to show that E M ( L )diverges as L → ∞ . Support (of Claim 3) . Consider the first experiment,and the homogeneous case. For N ∈ { , , , } , we haveplotted d sup1 ( L ) in Fig. 9. By symmetry, the results for k (cid:54) = 1 are similar. As the graphs are all linear in L ona log-log scale, the distance decreases in L like a powerlaw. This is in turn sufficient to conclude that for each1 ≤ k ≤ N , d sup k ( L ) → L → ∞ , and thus supportsClaim 3.For the heterogeneous case, Fig. 10 depicts the dis-tance max k d sup k ( L ) as a function of L . The results arein line with those of the homogeneous case. Hence, theexperiment supports convergence in distribution of theincrements of T L . Support (of Claim 3) . Now consider the second exper-iment. For N ∈ { , , , } and k ∈ { , . . . , N } we haveestimated the E M ( L ) for L ∈ { , , . . . , } . Then,we applied a log-transformation to L and E M ( L ), afterwhich we have applied linear regression to find the bestlinear fit. The idea is that if the linear fit on a log-log scaleis good and strictly increasing, then E M ( L ) is strictlyincreasing in L via a power law, i.e., E M ( L ) ∼ L β , where β is the slope of the linear fit found by the regression.The results of the linear regression are given in Table Ifor the homogeneous case, and in Table II for the het-erogeneous case. Here, N and k are as before, ‘Rsq’ and1 l og ( M ( L )) log ( L ) I. PLAATJES l og ( M ( L )) log ( L ) l og ( M ( L )) log ( L ) I. PLAATJES −0.8−0.400.40.81.2 −2 −1 0 1 2Theoretical quantile E m p i r i ca l qu a n til e −0.400.40.81.2 −2 −1 0 1 2Theoretical quantile E m p i r i ca l qu a n til e FIG. 11. Illustration results of linear regression. The upperfigures showing the regression and the lower figures showingthe corresponding QQ-plots, with the homogeneous case leftand the heterogeneous case right. ‘Rsq adj’ are, respectively, the ordinary and adjusted R from ordinary least squares, F is the F-statistic, and p isits corresponding p-value. The last two columns containthe intercept and slope of the regression line given byordinary least squares.The tables show that under the assumption of standardnormally distributed residuals, the fit for each pair of N and k is good, since R is large and the p-value from thecorresponding F-statistic is very small. Also, the slope isalways significantly positive. As explained above, we thusconclude that E M ( L ) diverges like a power law in L .In both the homogeneous and heterogeneous case, wehave to verify that the residuals of the regressions arenormally distributed, and that the conclusions we draware therefore valid. To do so, we made QQ-plots forevery pair of N and k ; the case N = 4 and k = 1 isgiven in Fig. 11 for illustration. The data from whichthese residuals stem is drawn on a log-log scale in Fig. 11together with the best-fit line. None of the QQ-plots givesrise to question the assumption of normally distributedresiduals, and hence our conclusions are valid. VII. SCALING LIMIT FOR QUEUES
We now focus on the behavior of the total queue lengthin a segment of the roundabout, as L → ∞ . We claim that,for every subdivision of the roundabout into segments,the sum of the queue lengths within these segments isPoisson distributed. Similar to our results for the cells,one could use these results for the queues in the design ofthe roundabout. For example, using the Poisson limit incombination with Little’s law we can approximate mean −13−11−9−7−5 5 6 7 8 9 10log ( L ) l og ( d M ( L )) M = 128M = 256M = 512M = 1024 −14−13−12−11−10−9−8 5 6 7 8 9 10log ( L ) l og ( m a x k j d k j M ( L )) M = 128M = 256M = 512M = 1024
FIG. 12. Graphs of d M ( L ) for the homogeneous case (left)and max kj d Mkj ( L ) for the heterogeneous case (right), for M ∈{ , , , } . waiting times; one could thus design the roundabout suchthat these delays remain within an acceptable bound.Before we formulate our claim, we introduce some no-tation. Recall that Q i denotes the length of queue i inequilibrium. Define P L = 0 and P Lk := Q + · · · + Q k , k ≥ . Furthermore, define P L : [0 , → N by P L ( u ) = P L (cid:100) uL (cid:101) .We now claim the following: Claim 4. As L → ∞ , P L converges in distribution to atime-inhomogeneous Poisson process P . The intuition for this claim primarily stems from study-ing the behavior of specific quantities in the round-about model, as L → ∞ . We have p i = O (1 /L ) and q ij = O (1 /L ), so that π i = O (1). We write σ ikl for thestationary probability of the event { C i = k, Q i = l } , andrecall that λ ik denotes the stationary probability that Q i = k . By considering what happens when we start theMarkov chain from the stationary distribution, and let ittake one step, one can derive the identities π i +1 , = L (cid:88) j =1 π ij q ij + σ i (1 − p i ); (5) λ i = λ i (1 − p i ) + σ i (1 − p i ) + σ i p i . (6)Furthermore, a calculation shows that (1) and (2) imply L (cid:88) j =1 π ij (1 − q ij ) = 1 − π i +1 , − p i . (7)Combining (7) with (5) and (2) yields σ i = π i − p i − p i , implying that σ i = O (1). Using that σ i ≥
0, it thenfollows from (6) that λ i = O (1) as well.This line of reasoning fails to determine the order of λ ik ,but it is conceivable that λ ik = O (1 /L k ). The argumentbehind this is as follows. Since π i = O (1), the time wehave to wait for an empty cell is of constant order. For2 L d s up ( L ) N = 1N = 2N = 4N = 8
FIG. 13. Graph of d sup1 ( L ), for N ∈ { , , , } (homogeneouscase). L S a m p l e M ea n k = 1k = 2k = 3k = 4 L S a m p l e V a r i a n ce k = 1k = 2k = 3k = 4 L D i s p e r s i on k = 1k = 2k = 3k = 4 L S ca l e d M ea n k = 1k = 2k = 3k = 4 FIG. 14. Poisson characteristics for the increments (cid:80) ki =1 J Li ,for k ∈ { , , , } and N = 4, in the homogeneous case.With the figures showing the means (upper left), the variances(upper right), the dispersions (lower left), and the scaled means(lower right). a queue of length k to build up from an empty queue,we need to have at least k arrivals within this constanttime. The probability that this happens is of order 1 /L k ,because p i = O (1 /L ).Under the proviso that λ ik = O (1 /L k ), it follows thatthe functions P L behave asymptotically as counting pro-cesses. For convergence to a Poisson process, it thensuffices that the finite-dimensional distributions convergeto those of a Poisson process (see, e.g., [16, Theorem 12.6]).To support Claim 4, we therefore verify below (1) thatthe increments of P L become independent as L → ∞ ,and (2) that they converge in distribution to a Poissonrandom variable. A. Independence of Increments
To verify that the increments of P L become indepen-dent, we use the same experiment as the one used for L m a x k d k s up ( L ) N = 1N = 2N = 4N = 8
FIG. 15. Graph of max k d sup k ( L ), for N ∈ { , , , } (hetero-geneous case). L S a m p l e M ea n k = 1k = 2k = 3k = 4 L S a m p l e V a r i a n ce k = 1k = 2k = 3k = 4 L D i s p e r s i on k = 1k = 2k = 3k = 4 L S ca l e d M ea n k = 1k = 2k = 3k = 4 FIG. 16. Poisson characteristics for the increments (cid:80) ki =1 J Li ,for k ∈ { , , , } and N = 4, in the heterogeneous case.With the figures showing the means (upper left), the variances(upper right), the dispersions (lower left), and the scaled means(lower right). the Gaussian scaling limit. For completeness, we recallits main ingredients, and introduce some notation. Wedivide the roundabout into four segments, and denotethe increments of P L on these segments by J Lk , where k ∈ { , , , } . We use the metric defined in (4), where f kj is now the joint density of J Lk and J Lj , and where f k and f j are their respective marginal densities. For M ∈ { , , , } , we aim to show that for k (cid:54) = j , d Mkj ( L ) → L → ∞ . Support (of Claim 4) . In the left plot of Fig. 12 weshow the graph of the estimates of d M ( L ) as a functionof L . By symmetry, and because neighboring incrementshave the strongest dependence, it is enough to consider k = 1 and j = 2 in the homogeneous case. First, fromthe figure we establish that our estimate is the samefor each M , which is due to the small support of theempirical distributions. From the linearity of the plotthat d M ( L ) is decreasing according to a power law, whichis sufficient for d M ( L ) → L → ∞ . Finally, we also3see that d M ( L ) is already small for L = 32 and quitequickly becomes too small to estimate accurately withour sample size, meaning that the effect of the variancekicks in quite quickly. Rather than negating our findings,this actually makes our conclusion stronger, since thequeues are already only weakly dependent for small L .For the heterogeneous case, we plotted max k,j d Mkj ( L )as a function of L in the right panel of Fig. 12. Again,the function does not depend on M . The dependenciesare systematically small, so that we cannot show thatmax k,j d Mkj ( L ) → L → ∞ . However, the results stillsupport independence of the increments of P L in the limit,since the dependence is already negligible for L = 32. B. Distribution of Increments
To verify that the increments of P L are Poisson dis-tributed, we use an analogous experiment to the one usedin supporting Claim 3. Because we do not have a statisti-cal test with enough power to apply the second methodfrom Section IV B, we can only use the first method here,which looks at the distance between the empirical distri-bution function and a Poisson distribution. We dividethe roundabout into N segments of equal length, where N ∈ { , , , } . Each of these segments corresponds to anincrement of P L which, for fixed N , we denote by J Lk with k ∈ { , . . . , N } . Our claim is that in the limit L → ∞ , J Lk has a Poisson distribution with some parameter ν .For the homogeneous case, we estimate ν by the maxi-mum likelihood estimator ˆ ν = ¯ P (1), the bar denotingthe sample mean. We set ˆ ν k = ˆ ν/N for each k . In theheterogeneous case, we estimate the parameter separatelyfor each increment, as we do not expect a homogeneousPoisson process; so in this case, we have ˆ ν k = ¯ J k .The experiment is designed to support that d sup k ( L ) := (cid:107) ˆ G Lk − Ps(ˆ ν k ) (cid:107) ∞ → , for 1 ≤ k ≤ N , as L → ∞ . Here, ˆ G Lk denotes theempirical distribution function of J Lk , and Ps(ˆ ν k ) denotesa Poisson distribution with parameter ˆ ν k . To justify thatwe use ˆ ν k as the parameter, we estimate E (cid:80) ki =1 J Li , foreach L ∈ { , , . . . , } via the sample mean, andnumerically verify that the sample mean converges in L . Support (of Claim 4) . We present the homogeneouscase first. In Fig. 13 we show the graph of d sup1 ( L ) for N ∈ { , , , } , which supports our claim that d sup1 ( L )tends to zero. For k (cid:54) = 1 the results are equivalent dueto symmetry. In Fig. 14 we show the behavior of thesample means, sample variances and sample dispersionsof (cid:80) ki =1 J Li , and the scaled sample means (cid:80) ki =1 ¯ J Li / ( LkN ),for k ∈ { , , , } and N = 4. Observe from the first set of graphs that the sample means converge, so thatwe can indeed use ˆ ν as an estimate of the true Poissonparameter. Furthermore, the variances also converge. Thecorresponding dispersions tend to one, which is indicativeof the underlying random variable being Poisson, thusproviding additional support for our claim. Finally, thegraph of the scaled means shows that the infinitesimalcontribution of each queue goes to zero, but is equal forevery sub-division of N increments. Hence, even for L relatively small, P L behaves like a Poisson process.For the heterogeneous case, for N ∈ { , , , } , we haveplotted max k d sup k ( L ) as a function of L in Fig. 15. Fig. 16shows the sample means, variances, dispersions and scaledmeans, for N = 4. We see that the conclusions fromthe homogeneous case carry over to the heterogeneouscounterpart. VIII. CONCLUSION
Existing analytical papers on roundabout modelingtend to leave out relevant model features (on-/off-ramps,entry behavior, etc.), to facilitate the derivation of closed-form expressions. The obvious alternative is to realisti-cally model the underlying dynamics, but to resort tosimulation. The primary objective of our paper was todevelop a roundabout model that included relevant (ge-ometric) properties, while still allowing mathematicalanalysis.We have proposed a new roundabout model that modelsthe cars’ circulating behavior and has queueing at theon-ramps. The model is highly flexible; its parameterscan be directly calibrated to measurements. We find anexplicit expression for the marginal stationary distributionof the cells that the roundabout consists of. As it turnsout, the cells and the queues are dependent, so thatobtaining a joint stationary distribution remains out ofreach. The experiments, however, show that dependenciesare typically small, thus leading to various approximations.These approximations are tested in depth, and supportedby numerical evidence. They can be used when designingthe roundabout in such a way that delay or occupationmeasures are kept below a maximum allowable level.Our model includes many features that were not in-corporated in previously studied models. Nonetheless,various extensions can be thought of. One could, for in-stance, make the entry behavior and congestion on thecirculating ring more realistic (so as to capture the effectthat cars stop moving when cells in front of them areoccupied). Importantly, we do believe that, while theirfunctional forms might change, our findings generalize tomore realistic models; the underlying arguments and/ortechniques are not affected when one includes these fea-tures. In addition, a challenging research direction couldrelate to modeling roundabouts in networks.4 [1] S. Maerivoet and B. De Moor, arXiv:physics/0507126(2008).[2] F. van Wageningen-Kessels, H. Van Lint, K. Vuik, andS. Hoogendoorn, EURO Journal on Transportation andLogistics , 445 (2015).[3] S. Maerivoet and B. De Moor, Physics Reports , 1(2005).[4] M. E. Fouladvand, Z. Sadjadi, and M. R. Shaebani,Physical Review E , 046132 (2004).[5] R. Wang and H. J. Ruskin, Computer Physics Communi-cations , 570 (2002).[6] R. Wang and M. Liu, in International Conference onComputational Science (Springer, 2005) pp. 420–427.[7] N. P. Belz, L. Aultman-Hall, and J. Montague, Trans-portation research part C: emerging technologies , 134(2016).[8] A. Flannery, J. P. Kharoufeh, N. Gautam, and L. Elefte-riadou, in TRB Annual Conference Proceedings (2000). [9] A. Flannery, J. P. Kharoufeh, N. Gautam, and L. Eleft-eriadou, Civil Engineering and Environmental Systems , 133 (2005).[10] J. Tanner, Biometrika , 163 (1962).[11] D. Heidemann and H. Wegmann, Transportation researchpart B: methodological , 239 (1997).[12] M. E. Foulaadvand and P. Maass, Physical Review E ,012304 (2016).[13] HCM2010, Highway Capacity Manual Volumes 1-4. (Na-tional Research Council (U. S. ). Transportation ResearchBoard., 2010).[14] R. Ak¸celik, in
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