Delays at signalised intersections with exhaustive traffic control
DDelays at signalised intersections with exhaustive traffic control ∗ M.A.A. Boon † [email protected] I.J.B.F. Adan † [email protected] E.M.M. Winands ‡ [email protected]. Down § [email protected], 2011 Abstract
In this paper we study a traffic intersection with vehicle-actuated traffic signal control.Traffic lights stay green until all lanes within a group are emptied. Assuming generalrenewal arrival processes, we derive exact limiting distributions of the delays under HeavyTraffic (HT) conditions. Furthermore, we derive the Light Traffic (LT) limit of the meandelays for intersections with Poisson arrivals, and develop a heuristic adaptation of thislimit to capture the LT behaviour for other interarrival-time distributions. We combine theLT and HT results to develop closed-form approximations for the mean delays of vehiclesin each lane. These closed-form approximations are quite accurate, very insightful andsimple to implement.
Keywords: delays, intersection, vehicle-actuated traffic signals, polling, light traffic,heavy traffic, approximation
Traffic signals play an important part in the infrastructure of towns and cities all over theworld, and waiting before a red traffic light has become an unavoidable nuisance in everydaylife. It is obvious that it is important to find optimal settings, i.e. green and red times,for signalised intersections. When evaluating the quality of traffic signal settings, the mostcommonly used criterion for optimality is a weighted average of the expected vehicle delays.Surprisingly, however, the most commonly used formulas for calculating the mean delays arebased on major simplifications of the actual situation encountered in reality (see, e.g., [8, 33]).On the other hand, for most realistic cases hardly any good alternatives exist. Typical traffic ∗ The research was done in the framework of the BSIK/BRICKS project, and of the European Network ofExcellence Euro-NF. † Eurandom and Department of Mathematics and Computer Science, Eindhoven University of Technology,P.O. Box 513, 5600MB Eindhoven, The Netherlands ‡ Department of Mathematics, Section Stochastics, VU University, De Boelelaan 1081a, 1081HV Amster-dam, The Netherlands § McMaster University, 1280 Main Street West, Hamilton, ON L8S 4L7, Canada. a r X i v : . [ m a t h . P R ] A ug ights settings can be divided into three strategies: fixed cycle, vehicle-actuated, and traffic-actuated signals. The fixed-cycle policy for signalised intersections is the oldest strategy,defining fixed red, amber and green times. Vehicle-actuated traffic signals have flexible greenphases with minimum and maximum green times. Detectors gather information about thepresence of vehicles at the different traffic flows and use this information to determine whetherthe best option is to stay green, or switch to an all-red phase. Traffic-actuated signals aremuch like vehicle-actuated signals, with the additional possibility that actual queue lengthsare determined. The combined information about all queue lengths can be used to create moresophisticated traffic signal settings. In the present paper we study a signalised intersectionwith vehicle-actuated, exhaustive traffic control. The policy is exhaustive, because a greenphase ends as soon as no vehicle is present in any flow that faces a green light. The mainadvantage of an exhaustive control policy is its efficiency, due to the fact that the trafficlights do not turn red until all vehicles in the corresponding flows have left. This implies thatthe exhaustive policy minimises the mean total amount of unprocessed work in the system,i.e., the total time required by all vehicles present at the intersection to discharge at thecorresponding saturation flow rates. Newell [26] argues that, for isolated intersections, anexhaustive control policy should be preferred over alternative strategies. Nevertheless, a well-known disadvantage of the exhaustive control policy is its unfairness with respect to vehiclesin flows where relatively few vehicles arrive. For this reason, in practice often time-limitedcontrol policies are used.Although the model in the present paper is vehicle-actuated, we will give a short literaturereview about fixed-cycle traffic signals first. Intersections based on a fixed cycle have beenstudied since a long time. One of the first, and perhaps still the most influential and practicallyapplied papers, is written by Webster [40] who analyses a fixed-cycle traffic-light queue.Although more sophisticated analyses have appeared throughout the years (cf. [18, 23, 24,37]), his formula for the mean delay is still used in most traffic engineering manuals. Thepresent paper does not focus on settings based on a fixed cycle or on traffic-actuated signals,but on vehicle-actuated traffic signal control. Considering this is the most commonly usedcontrol type nowadays, it is surprising how little mathematical literature is available to analysea typical, realistic vehicle-actuated signalised intersection. The earliest literature on vehicle-actuated systems dates from the early 1960s when Darroch et al. [12] analysed a systemconsisting of two intersecting Poisson traffic streams that are served exhaustively. A modelwith two lanes that does not assume Poisson input has been studied by Lehoczky [19], who usesan alternating priority queueing model. Newell [25] analyses an intersection with two one-waystreets using fluid and diffusion queueing approximations. In [27], Newell and Osuna study afour-lane intersection where two opposite flows face a green light simultaneously. A variationof the two-lane intersection is introduced by Greenberg et al. [16], who analyse mean delayson a single rail line that has to be shared by trains arriving from opposite directions. Thismodel is extended by Yamashita et al. [42], who study alternating traffic crossing a narrowone-lane bridge on a two-lane road. In many of the discussed papers traffic is modelled asfluid passing through the road. These types of approximations are fairly accurate when thetraffic intensity is relatively high, but do not perform very well if there is a lot of variationin the arrival or departure processes. Vlasiou and Yechiali [39] use a different approach,modelling a traffic intersection as a polling system, consisting of multiple queues with aninfinite number of servers visiting each queue simultaneously. A disadvantage that all of theaforementioned papers have in common, is that the methods can be applied only to situations2hat are significant simplifications of modern intersections encountered in practice. Most ofthe papers focus on two or four lanes only, and Newell and Osuna [27] have written one of thefew papers studying an intersection where multiple flows of vehicles can receive a green lightsimultaneously. Haijema and Van der Wal [17] study a model that allows for multiple groupsof different flows, but their approach uses a Markov Decision Problem (MDP) formulation,which does not lead to closed-form, transparent expressions for the mean delay that can beused for optimisation purposes. Summarising, in the literature on vehicle-actuated trafficsignals, either the models are very simplified versions of reality, or the resulting algorithmor expression to determine the mean delays is so complex that it cannot be implemented forreal-life intersections. Additional advantages, apart from the comprehensiveness, of a closed-form expression for the mean delay, is that it is perfectly suitable for optimisation purposes,and that it can easily be adapted to extensions of the model discussed in the present paper.The goal of this paper is to provide a comprehensive, novel analysis for traffic intersectionswith a vehicle-actuated, exhaustive control policy. The model in the present paper is more re-alistic than most vehicle-actuated models, in the sense that we allow combinations of multipleflows to receive a green light simultaneously, we take more types of randomness into account(e.g., in interarrival times and interdeparture times), and we do not limit ourselves to Poissonarrivals. The main contribution of the paper is twofold. Firstly, we capture the limitingbehaviour of the model under Light Traffic (LT) and Heavy Traffic (HT) conditions. TheHT results give insight into the system behaviour when the intersection becomes saturated,which makes them very usable by themselves. The LT results describe the system when it ishardly exposed to any traffic at all. These results are mostly interesting because they form anessential building block for the second main contribution of the paper, a closed-form approxi-mation for the mean delay. This approximation is created using an interpolation between theLT and HT limits, which results in a comprehensive expression that is very insightful, simpleto implement, and suitable for optimisation purposes.We use a polling model to describe the traffic intersection. A polling model is a queueingsystem with multiple queues of customers, served by a single server in a cyclic order. Theswitch of the server from one queue to the next requires some (possibly random) time, and iscalled a switch-over time. An advantage of using a polling model, with customers representingthe vehicles, is that we can model randomness in the interarrival times, but also in the servicetimes, corresponding to the times between two successive vehicles as they pass the stop line.When considering the features of a polling model, it seems like the natural way to modela traffic intersection. However, traffic intersections do exhibit features that have not beenstudied in the polling literature before, which impels us to considerably extend the queueinganalysis of these systems. In particular, the feature that multiple queues (correspondingto the different traffic flows) should be allowed to receive service simultaneously has notbeen investigated yet in the huge literature on polling systems (for good surveys on pollingsystems and their applications, see, for example, [3, 20, 31, 38]). The main reason why itis difficult to analyse a polling system with simultaneous service of multiple queues, is thatthe system loses the so-called branching-property . We do not discuss this property in moredetail here, but we just mention that Resing [30] and Fuhrmann [14] have shown that forpolling models satisfying this property, performance measures like cycle time distributionand waiting time distributions can be obtained. If this branching property is, however, notsatisfied, the corresponding polling model defies an exact analysis except for some special(two-queue or symmetric) cases. In the present paper we show that, despite the fact that3he branching property is not satisfied, the delays can still be studied under LT and HTconditions. Furthermore, using these LT and HT results we propose an accurate closed-form approximation of the mean delay for a signalised intersection with vehicle-actuated,exhaustive traffic control.The structure of the present paper is as follows. In the next section we present an outline ofthe model and describe in more detail how a traffic intersection can be modelled using a pollingmodel with simultaneous service of multiple queues. The notation required for the remainderof the paper is also introduced in Section 2. In Section 3 we study the distribution of thedelays under Heavy Traffic conditions. This means that we increase the arrival intensitiesand, hence, the load of the system until it reaches the point of saturation. In Section 4 westudy the behaviour of the system under Light Traffic conditions. It goes without sayingthat this situation is rather opposite to the previous section, and mean delays of vehicles inan (almost) empty system are analysed. Using the mean delays under LT conditions, andthe mean delays under HT conditions, we develop interpolations between these two limits inSection 5. These interpolations can be used as approximations for the mean delay under anysystem load. In Section 6 we discuss the accuracy of the approximations and show numericalresults for three intersections, located in The Netherlands. We finish with some conclusionsand topics for further research. We model the traffic intersection as a polling system. A typical polling system consists of N queues attended by a single server in a cyclic order. Each flow of vehicles, sometimes referredto as stream, corresponds to a queue in the polling system. Note that traffic approaching theintersection from the same direction, sharing one lane, but with different destinations (e.g.,flow 9 in Figure 1), is modelled as one single queue, whereas the same situation with two dif-ferent lanes (e.g., flows 1 and 2) is modelled as two separate queues. The main contribution
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Figure 1: An example of an intersection with traffic approaching from four directions.of the present paper is that we extend the standard polling model by dividing the queues intogroups that are served simultaneously, which turns out to complicate the queueing analysissignificantly. We impose the restriction that each flow can be part of only one group. Note4hat the Highway Capacity Manual 2000 (HCM, [33]) often does not use the terms “flow”or “stream”, but prefers lane group , defined as a set of lanes established at an intersectionapproach for separate capacity and level-of-service analysis. We will not adopt this termi-nology, except for a few times in this section, because it might be confused with a group(or combination) of non-conflicting flows that receive a green light simultaneously. A cycle consists of multiple phases, where groups of non-conflicting flows receive the right-of-way si-multaneously. The switch-over times in the polling model correspond to the all-red times ofa traffic intersection. Adopting the terminology used in the polling literature, we refer to thegreen periods as visit times , i.e., the times that the server visits a group of queues. Similarly,the red period of a group of flows may be referred to as intervisit time of that group. In thepresent paper we assume that the control policy, which is called service discipline in pollingliterature, is exhaustive service. Since multiple queues are served simultaneously, exhaustiveservice implies that a green period only ends when all queues in the corresponding groupare empty. The N flows of the intersection are divided into a number of groups, denotedby M . Each group, say group g = 1 , . . . , M , consists of N g nonconflicting flows of vehicles.We assume that each flow belongs to exactly one group, so (cid:80) Mg =1 N g = N . Denote by R g the all-red time starting at the end of the green period of group g , denoted by G g . Thetotal all-red time in a cycle, denoted by R , equals R = (cid:80) Mg =1 R g . Throughout this paper, weassume that the all-red times R g are independent random variables. In reality, all-red timesare generally deterministic, which simplifies many results obtained in this paper.The times between two consecutive vehicles (in the same flow) crossing the stop line aregenerally called departure headways and will be denoted by B i , i = 1 , . . . , N . In the pollingmodel, these headways correspond to service times of customers, but in the literature on trafficsignals it is more common to use the reciprocal, 1 / E [ B i ], which is referred to as dischargerate or saturation flow rate. An advantage of adopting a polling model, is that it allows forrandomness in the headways, which is generally ignored in the literature on traffic signals.This randomness enables us to distinguish between slow and fast, or between big and smallvehicles. An important aspect of our model, is that we make a distinction between theheadways of queued vehicles, and vehicles approaching the intersection without queue infront of them. If a queue clears before the green period terminates, all vehicles that arriveduring the remainder of this visit period pass through the system and experience no delaywhatsoever. This assumption is quite common in the literature on traffic light queues todistinguish between headways of cars that need to accelerate and cars that arrive at fullspeed (see, e.g., [34, 37]), but it is not common in the polling literature. Furthermore, weassume that the headways B i are independent of each other and of all other random variablesin the model. This assumption is generally not satisfied in practice, because the first fewvehicles might require a slightly longer time to accelerate. However, this can be circumventedby incorporating any systematic, additional delay of the first cars crossing the intersectionafter an all-red phase in the preceding all-red time ( R g − for vehicles belonging to group g = 1 , . . . , M ).We assume that the interarrival times of vehicles, corresponding to customers in the pollingsystem, are independent, generally distributed random variables A i , i = 1 , . . . , N . The arrival rates are denoted by λ i = 1 / E [ A i ], i = 1 , . . . , N . The main performance measures of interestin the present paper are the delays W i of vehicles in flow i . Note that for a queued vehicle,the delay is the waiting time plus B i , whereas vehicles approaching a clear intersection duringa green period experience no delay at all. We study the delay as a function of the total traffic5oad offered to the system, denoted by ρ . The load of a particular flow, say flow i , is theproduct of the arrival rate and the mean departure headway: ρ i = λ i E [ B i ]. The total loadis the sum of the loads of the different flows: ρ = (cid:80) Ni =1 ρ i . The HCM refers to ρ i as the flow ratio of lane group i , being the ratio of the actual flow rate ( λ i ) to the saturation flowrate (1 / E [ B i ]) for lane group i at an intersection. Since, starting from the next section, weconsider the delay as a function of ρ , the total amount of traffic offered to the system, wehave to specify in more detail how the scaling takes place. The traffic intensity is varied bykeeping the headways B i fixed, and scaling the interarrival times A i (or arrival rates λ i ). Wedenote unscaled quantities by putting a hat on the scaled quantities. The settings that wecall “unscaled” correspond to the situation where ˆ ρ = (cid:80) Ni =1 ˆ ρ i = 1, which has the advantagethat ˆ ρ i can be interpreted as the fraction of the total traffic load that is routed to flow i . Thismeans that, if the total load equals ρ , the actual flow ratio of flow i is ρ i = ˆ ρ i × ρ . Hence, theunscaled interarrival times ˆ A i are the interarrival times that lead to a system with load 1: N (cid:88) i =1 ˆ ρ i = N (cid:88) i =1 ˆ λ i E [ B i ] = N (cid:88) i =1 E [ B i ] E [ ˆ A i ] = 1 . The other scaled/unscaled quantities follow from the relation A i = ˆ A i /ρ . Another interestingquantity is the departure headway of an arbitrary vehicle (in any flow), denoted by B . Sincean arbitrary vehicle arrives in flow i with probability λ i / (cid:80) Nj =1 λ j , we have E [ B ] = 1 (cid:80) Nj =1 λ j N (cid:88) i =1 λ i E [ B i ] = ρ (cid:80) Nj =1 λ j = 1 (cid:80) Ni =1 ˆ λ i . In the remainder of the paper, we use the subscripts { g, j } for flow j within group g . Withoutloss of generality, we order the flows within a group according to their flow ratios: ρ g, >ρ g, ≥ · · · ≥ ρ g,N g . For example, supposing that in Figure 1 flows 4 and 9 are part of group 1,and flow 4 has a higher flow ratio than flow 9, we use the notation B , = B , and B , = B .The flows with the highest flow ratios, flows { g, } for g = 1 , . . . , M , are called dominant flows ,or critical lane groups in the HCM. Since we study the limiting behaviour of the intersectionas it becomes saturated, the stability condition is an important issue for our analysis. Theorem 2.1
The system is stable (i.e., the intersection is undersaturated) if and only if M (cid:88) g =1 ρ g, < . Proof:
The proof is provided in Appendix A. (cid:3)
Note that only the dominant flows play a role in this stability condition. We study thesteady-state of stable systems only, so throughout the paper we assume that0 ≤ Lρ < , where L = (cid:80) Mg =1 ˆ ρ g, , i.e., the total relative load of the dominant flows. The quantity Lρ iscalled the critical volume-to-capacity ratio in the HCM, defined as the proportion of available6ntersection capacity used by vehicles in critical lane groups. Sometimes it is more compre-hensive to use Lρ instead of ρ , because its value is always between 0 and 1, for all stableintersections. It turns out to be convenient to introduce ρ g, • = (cid:80) N g j =1 ρ g,j as the load of group g , and λ g, • = (cid:80) N g j =1 λ g,j as the total arrival rate of group g .Finally, the (equilibrium) residual length of a random variable X is denoted by X res , with E [ X res ] = E [ X ] / E [ X ]. See, e.g., [7], pp. 108 – 115, for more information. In the present section we study an intersection with exhaustive control policy under HeavyTraffic conditions. This means that we increase the load of the system until it reaches thepoint of saturation. From Theorem 2.1 we learn that the critical load for which the pointof saturation is reached, is completely determined by the dominant flows in each group: thesystem becomes saturated as (cid:80) Mg =1 ρ g, →
1, which is equivalent to Lρ →
1. As the totalload of the system, ρ , increases, the green times, the cycle times and waiting times becomelarger and will eventually grow to infinity. For this reason, we scale them appropriately andconsider the scaled versions. Since we consider finite all-red times, they become negligiblecompared to the waiting times as the load is increased. Polling systems under HT conditionshave been studied by Coffman et al. [5, 6], and by Olsen and Van der Mei [28, 29]. The keyobservation in these papers, is the occurrence of a so-called Heavy Traffic Averaging Principle(HTAP). When a polling system becomes saturated, two limiting processes take place. Let V denote the total workload of the system. As the load offered to the system, ρ , tends to 1, thescaled total workload (1 − ρ ) V tends to a Bessel-type diffusion. However, the work in eachqueue is emptied and refilled at a faster rate than the rate at which the total workload ischanging. This implies that during the course of a cycle, the total workload can be consideredas constant, while the loads of the individual queues fluctuate like a fluid model. The HTAPrelates these two limiting processes and provides expressions for the stationary distributionsof the scaled cycle times, switch-over times, and waiting times. In order to derive the HTlimits for traffic intersections, we introduce and analyse a novel fluid model . Subsequently, weadapt and extend the HTAP to relate the results of this fluid model with the original trafficintersection model.We introduce a fluid model, with work flowing in at constant rate lim ρ → /L ρ g,j = ˆ ρ g,j /L for flow { g, j } . The all-red times are not considered in the fluid model, because under HTconditions they become negligible. A graphical illustration of the fluid model is presentedin Figure 2. On the horizontal axis, the course of a cycle with length c is plotted. On thevertical axis, the scaled workloads in flows { g, } and { g, } are plotted. Because the lengthof the green periods is determined by the dominant flows, the system becomes saturated as (cid:80) Mg =1 ρ g, = Lρ →
1. A formal proof of this statement is provided in Appendix A. For thetotal load offered to the system, this translates to ρ → L . We (arbitrarily) start the cycleat the moment that the traffic lights for the flows in group g turn green. For now, duringthe first part of the analysis, we assume that the amount of work at the beginning of a cycleis fixed. This implies that the length of a cycle, denoted by c , is also fixed because it isdetermined by the amount of work present at the beginning of the cycle. Throughout thecycle, work arrives with intensity 1 /L and a fraction ˆ ρ g,j is directed to flow { g, j } . During7 ρ g, L − ˆ ρ g, L ˆ ρ g, L − ˆ ρ g, L ˆ ρ g, L (cid:0) − ˆ ρ g, L (cid:1) c ˆ ρ g, L (cid:0) − ˆ ρ g, L (cid:1) c ˆ ρ g, L c (cid:0) − ˆ ρ g, L (cid:1) c | {z } c Figure 2: Heavy traffic fluid limits.the green periods, work flows out of each stream at rate 1 as long as it is not empty. As soonas the stream is empty, it stays empty (hence, the work flows out at rate ˆ ρ g,j /L ) until theend of the green period. As a consequence, although the total amount of work in the systemat the end of a cycle is back to the level of the beginning of a cycle, it varies throughoutthe cycle. However, if we consider the workload in dominant flows only, the total workloadremains constant, because it flows out at rate 1, and flows in at rate (cid:80) g ˆ ρ g, /L = 1. Thisresult follows directly from the observation that in the fluid model, the non-dominant flowsdo not contribute to the cycle length and, hence, do not influence the total workload in thedominant flows.From the viewpoint of fluid in flow { g, j } , a cycle of length c consists of three parts. Duringthe first part, starting at the moment that the lights turn green, fluid starts to drain out ofthe system until flow { g, j } is empty. The second part is the time between the emptying offlow { g, j } and the moment that the dominant flow of the group, { g, } becomes empty andtraffic lights turn red. The third part is the red period of group g . It is easily seen that thelength of this third part is (cid:0) − ˆ ρ g, /L (cid:1) c , and the length of the first two parts together (thegreen period of group g ) is ˆ ρ g, L c . Using Figure 2 we see that the length of the first part,with flow { g, j } being non-empty, is ˆ ρ g,j L − ˆ ρ g, /L − ˆ ρ g,j /L c . We denote the lengths of the three partsrespectively by P j , P , and P R . The probability distribution of the delay of an arbitrary fluidparticle arriving in flow { g, j } , denoted by W fluid g,j , can now be computed. Theorem 3.1 W fluid g,j d = (cid:40) P /c,U × P R , w.p. ( P j + P R ) /c, (3.1)where U is a uniformly distributed random variable on the interval [0 , Proof:
We condition on the arrival epoch of an arbitrary fluid particle in flow { g, j } . With probability P R /c , the arrival takes place during the red period. The probability that the arrival takes8lace during the first part of the green period, when there is still other fluid present in flow { g, j } , is P j /c . If the particle arrives during the second part of the green period, when thereis no fluid present in flow { g, j } , its delay is 0. This happens with probability P /c . So wecan write: W fluid g,j d = P /c,W green g,j w.p. P j /c,W red g,j w.p. P R /c. A particle arriving during the first part of the green period (with length P j ) has to wait untilall the fluid in front of it has left the system. Let the uniform random variable U G denotethe fraction of P j that has elapsed at the arrival epoch of this particle. Then the amount offluid left in flow { g, j } is (1 − ˆ ρ g,j /L ) × (1 − U G ) P j = ˆ ρ g,j L (cid:0) − ˆ ρ g, /L (cid:1) (1 − U G ) c . Hence, W green g,j d = ˆ ρ g,j L (1 − U G ) (1 − ˆ ρ g, /L ) c. (3.2)The delay of a particle arriving during the red period can be analysed similarly. Let theuniform random variable U R denote the fraction of the red period that has elapsed at thearrival epoch of this particle. Then the amount of fluid present in flow { g, j } is ˆ ρ g,j L × U R P R .The arriving particle has to wait for this amount of fluid to drain, after it has waited for theresidual red period (1 − U R ) P R . Hence, we have W red g,j d = (cid:18) ˆ ρ g,j L U R + (1 − U R ) (cid:19) (cid:0) − ˆ ρ g, /L (cid:1) c. (3.3)If we study (3.2) and (3.3) more carefully, we see that W green g,j is uniformly distributed on theinterval [0 , ˆ ρ g,j L P R ] and W red g,j is uniformly distributed on the interval [ ˆ ρ g,j L P R , P R ]. Recall that W fluid g,j d = W green g,j with probability P j /c = ˆ ρ g,j /L − ˆ ρ g,j /L P R /c = ˆ ρ g,j L ( P j + P R ) /c , and W fluid g,j d = W red g,j with probability P R /c = (cid:16) − ˆ ρ g,j L (cid:17) ( P j + P R ) /c . This implies that W fluid g,j is uniformly dis-tributed on [0 , P R ] with probability ( P j + P R ) /c , and it is 0 otherwise. (cid:3) Now that we have established the distribution of the delay of a particle in the fluid model, wecan find the distribution of the scaled delay of a vehicle in the original model for the trafficintersection, under HT conditions.
Original model.
For ordinary polling models, the link between the fluid model and thepolling model under HT conditions is the Heavy Traffic Averaging Principle. This principlestates that the work in each queue is emptied and refilled at a rate that is so much faster thanthe rate at which the total workload is changing, that during the course of a cycle, the totalworkload can be considered as constant, while the loads of the individual queues fluctuatelike a fluid model. A novel contribution of the present paper is the adaptation and extensionof the HTAP for polling systems to the traffic intersection model. Also in this model, whenthe system becomes saturated, the diffusion limits of the total workload process and theworkload in the individual flows can be related using the HTAP. The main difference is thatin the fluid model for the traffic intersection, the total workload does not remain constant,but the total workload in the dominant flows does. From the fluid model it has become clear9hat the non-dominant flows do not play a role in the length of the green periods, because thedominant flow in each group will always be the last flow that becomes empty. For this reason,we can ignore the non-dominant flows temporarily and focus on the workload in the dominantflows only. This turns our traffic light model into an ordinary polling model with exhaustiveservice. First, we define a random variable with a Gamma distribution as a random variablewith probability density function f ( t ) = 1Γ( α ) e − µt µ α t α − , t ≥ , where Γ( α ) = (cid:82) ∞ e − t t α − d t . The positive parameters α and µ are respectively the scale andrate parameter.When the load ρ approaches 1 /L , the system becomes overloaded and the queue lengths andwaiting times tend to infinity. For this reason, we consider the scaled delay (1 − Lρ ) W g,j ,which stays finite for ρ → /L . Before we can formulate the main result of this section, thedistribution of the scaled delay as the HT limit is approached, we need some lemmas. Notethat the proofs of these lemmas all rely on the conjectures posed in [29]. Although theseconjectures are widely accepted to be true, they have only been proven for systems consistingof two queues (cf. [5, 6]), systems with Poisson arrivals (cf. [28]), or for the means ratherthan the complete distributions (cf. [36]). Lemma 3.2
Denote by V • , the amount of work in the dominant flows of the intersection,at the beginning of a cycle. For ρ → /L , (1 − Lρ ) V • , has a Gamma distribution withparameters α = 2 E [ R ] δ/σ + 1 and µ = 2 /σ , where δ = (cid:80) Mg =1 ˆ ρ g, L (1 − ˆ ρ g, L ) /
2, and σ = (cid:80) Mg =1 ˆ λ g, L (cid:16) V ar[ B g, ] + ˆ ρ g, V ar[ ˆ A g, ] (cid:17) . Proof:
By the HTAP, the total workload in the dominant flows of the system may be regarded asunchanged over the course of a cycle. So if we regard the system with dominant flows only,Lemma 3.2 follows directly from Olsen and Van der Mei [29], Conjecture 1. More specifically,we use the special case of cyclic, exhaustive service to obtain the distribution of the scaledamount of work in the dominant flows. (cid:3)
Before continuing, it is important to realise the interpretation of δ . Assume, just like inFigure 2, that c is the length of a cycle, starting at the beginning of green period G g . Thenthe workload in flow { g, } at the beginning of the cycle is ˆ ρ g, L (cid:0) − ˆ ρ g, L (cid:1) c . The flow is emptyat the moment that a fraction ˆ ρ g, L of the cycle has passed, and at the end of the cycle it hasreached the same level as at the beginning. The mean workload in flow { g, } during thiscycle is
12 ˆ ρ g, L (cid:0) − ˆ ρ g, L (cid:1) c . A summation over all dominant flows shows that δc is the meantotal workload of the dominant flows during the course of one cycle of length c Given the scaled amount of work in the dominant queues at the start of a green period ofgroup g , we can derive the distribution of the scaled cycle time (1 − Lρ ) C . In fact, we considerthe length-biased (or time-averaged) scaled cycle time (1 − Lρ ) C . If a random variable X has probability density function F X ( x ), then we define the length-biased random variable X as a random variable with probability density function f X ( x ) = xf X ( x ) / E [ X ]. From renewaltheory, we know that the length-biased cycle length accounts for the fact that an arbitraryarriving vehicle arrives with a higher probability during a long cycle, than during a short one.10he length of a cycle depends on the amount of work at the beginning of that cycle. Denoteby C ( x ) the length of a cycle, given that a total amount of x work is present in the dominantflows. We are now ready to formulate the second lemma, needed to find the distributions ofthe scaled delays under HT conditions. Lemma 3.3
Denote by I g the length-biased red-time (or intervisit time) of group g , g =1 , . . . , M . For ρ → /L , we find that (1 − Lρ ) I g converges in distribution to a random variablehaving a Gamma distribution with parameters α = 2 E [ R ] δ/σ + 1 and µ g := 2 δ/ (cid:0) σ (1 − ˆ ρ g, /L ) (cid:1) . Proof:
The proof proceeds along the same lines as the argument given to support Conjecture 2 in[29]. It uses Lemma 3.2 and the fact that, due to the averaging principle, the scaled workloadin the dominant flows remains effectively constant. In steady-state, we have the followingrelation: δC ( x ) = x. This relation can easily be verified graphically from Figure 2, because the total amount ofwork in the dominant queues remains constant throughout the course of a cycle. Given a cyclelength of c , the total amount of work in the dominant queues is (cid:80) Mg =1 ˆ ρ g, L (cid:0) − ˆ ρ g, L (cid:1) c/ δc .Hence, given an amount of work x in the dominant flows, the cycle time is C ( x ) = x/δ . Nowwe use Lemma 3.2, which states that, in the HT limit, the scaled workload in the dominantflows has a Gamma distribution with parameters α and µ . This implies that for ρ → /L ,the scaled, length-biased cycle time (1 − Lρ ) C follows a Gamma distribution with, again,scale parameter α , but with rate parameter µδ . The distributions of the scaled length-biasedintervisit times, denoted by (1 − Lρ ) I g for group g = 1 , . . . , M , can now be determined. Theintervisit time of group g is the time that the signals in group g are red. Given that x is theamount of work present at the dominant queues at the beginning of a cycle, the intervisittime conditioned on x is obviously I g ( x ) = C ( x ) (cid:0) − ˆ ρ g, L (cid:1) . The limiting distribution of (1 − Lρ ) I g now readily follows from the limiting distribution of(1 − Lρ ) C . (cid:3) Finally, we formulate the main result of the present section.
Theorem 3.4 As ρ ↑ /L , the scaled delay is 0 with probability ˆ ρ g, − ˆ ρ g,j L − ˆ ρ g,j , and it is theproduct of a uniformly distributed random variable on [0 , U , and a randomvariable Γ I having the same distribution as the limiting distribution of (1 − Lρ ) I g , withprobability − ˆ ρ g, /L − ˆ ρ g,j /L : (1 − Lρ ) W g,j d → (cid:40) ˆ ρ g, − ˆ ρ g,j L − ˆ ρ g,j ,U × Γ I , w.p. − ˆ ρ g, /L − ˆ ρ g,j /L , (3.4)for ρ → /L . Proof:
A combination of Theorem 3.1 and Lemma 3.3 yields the desired result. The scaled intervisit11ime (1 − Lρ ) I g converges in distribution to a random variable having a Gamma distributionwith parameters α and µ g . The HTAP states that we can simply replace P R , the deterministicred time in the fluid model, in (3.1) by the scaled, length-biased red time in the original model,(1 − Lρ ) I g , because the random variables U and I g are independent. (cid:3) For the approximations developed in Section 5, the mean scaled delay for ρ → /L will beused. Corollary 3.5 lim ρ → L (1 − Lρ ) E [ W g,j ] = (1 − ˆ ρ g, /L ) − ˆ ρ g,j /L (cid:18) E [ R ]2 + σ δ (cid:19) , (3.5)where δ = (cid:80) Mg =1 ˆ ρ g, L (1 − ˆ ρ g, L ) /
2, and σ = (cid:80) Mg =1 ˆ λ g, (cid:16) V ar[ B g, ] + ˆ ρ g, V ar[ ˆ A g, ] (cid:17) . Proof:
The result immediately follows from (3.4). (cid:3)
Note that any possible variations in the all-red times R g have no influence on the mean scaleddelay under HT conditions. Remark 3.6
For Poisson arrivals σ can be simplified to σ = E [ B • , ] E [ B • , ] , where B • , = (cid:80) Mg =1 ˆ λ g, B g, (cid:80) Mg =1 ˆ λ g, is the headway of an arbitrary vehicle arriving in a dominant flow. Remark 3.7 (Convergence to the HT limit)
Although the distribution of the scaled de-lay in the HT limit ρ ↑ /L is exact, it is interesting to know how fast this limiting distributionis approached. Unfortunately, the analysis does not provide any insight in the speed at whichthe scaled delay converges to this limiting distribution, so we have to resort to simulations.Denote by p g,j the steady-state probability that flow { g, j } is the flow from which the lastdeparture takes place before the corresponding green phase G g ends. This probability de-pends on ρ , the total load of the system. We have shown in this section that, in the limitingsituation, the dominant flow of each group is always the last flow to become empty in thisgroup. For ρ < /L , the fraction of green periods in which the last departure indeed takesplace from the dominant flow becomes smaller. In fact, in the LT limit one can easily calculatethe exact value of p g,j . In LT, the probability that the last vehicle in group g departs fromflow { g, j } is proportional to the arrival rates of the flows in this group. Summarising:lim ρ ↑ /L p g,j = (cid:40) j = 1 , j > , lim ρ ↓ p g,j = ˆ λ g,j ˆ λ g, • . This implies that we can regard p g,j as a measure for how close the distribution of the scaleddelay is to the limiting distribution. In Figure 3 we show an example that illustrates how p g,j might depend on ρ . The specific intersection chosen for this figure, has two groups ofthree flows each. The traffic intensities of the flows within each group are relatively closeto each other. In this situation, it takes rather long before the probabilities p g,j converge to12heir limiting values for ρ → /L . For example, in Figure 3 one can see that at Lρ = 0 . p , (corresponding to flow 6) is less than 0 .
7. The fact that this value isstill quite different from its limiting value 1, has a direct impact on the distribution of thescaled delay. The (simulated) mean scaled delay (1 − Lρ ) E [ W , ] at Lρ = 0 . .
5, whereaslim Lρ ↑ (1 − Lρ ) E [ W , ] = 3 .
5. For this reason, one has to be careful when applying the resultsfor ρ ≈ /L to situations with smaller loads. In Section 6, we continue the discussion on thenumerical accuracy. For more information about the intersection which is used for Figure 3,see Scenario V in Example 1. Flow 1Flow 2Flow 30.0 0.2 0.4 0.6 0.8 1.0 L Ρ Group 1
Flow 4Flow 5Flow 60.0 0.2 0.4 0.6 0.8 1.0 L Ρ Group 2Figure 3: Simulated fractions of the green periods where the corresponding flows have beenthe last in their groups from which a vehicle departs before the traffic lights turn red.
In the present section we study the delay of vehicles arriving at the traffic intersection underLight Traffic conditions, i.e., for ρ ↓
0. In the first part of the section, we find an expressionfor the mean delay of vehicles, assuming Poisson arrivals. Subsequently, we heuristicallyadapt this expression to find an approximation for the mean delay for intersections withgeneral renewal arrivals. The analysis of the present section follows the lines of [4], wherethe derivation of the LT limit of mean waiting times for polling systems is discussed, withmodifications for the specific traffic intersection model.For the first part in this section we assume that the arrival processes in all flows are Poisson.Under this assumption, the LT limit of the mean delay can be determined by conditioning onthe arrival epoch of an arbitrary vehicle in, say, flow { g, j } . By the LT limit of E [ W g,j ], wemean the expression for E [ W g,j ] as a function of the load in the system, ρ , up to O ( ρ ) termsfor ρ ↓
0. In the following theorem we formulate the main result for the system with Poissonarrivals.
Theorem 4.1
Under the assumption of Poisson arrivals, the LT limit of the mean delay of13n arbitrary type { g, j } vehicle is E [ W LT,Poisson g,j ] = ρ g,j (cid:0) E [ B res g,j ] + E [ B g,j ] (cid:1) + g − (cid:88) m = g − M +1 N m (cid:88) k =1 ρ m,k (cid:32) E [ B res m,k ] + g − (cid:88) l = m E [ R l ] + E [ B g,j ] (cid:33) + g − (cid:88) m = g − M E [ R m ] E [ R ] (cid:32) E [ R res m ] (cid:16) − ρ + 2 g − (cid:88) k = m +1 ρ k, • + ρ g,j (cid:17) + m − (cid:88) k = g − M E [ R k ] (cid:16) g − (cid:88) l = m +1 ρ l, • + ρ g,j (cid:17) + g − (cid:88) k = m +1 E [ R k ] (cid:16) − ρ + g − (cid:88) l = m +1 ρ l, • (cid:17) + (1 − ρ ) E [ B g,j ] + O ( ρ ) . (4.1) Proof:
The proof is essentially a Mean Value Analysis (MVA) that ignores all O ( ρ ) terms, andfocusses mainly on the amount of work instead of number of vehicles. The first step is tocondition on the arrival epoch of a vehicle in flow { g, j } . A cycle consists of the green phases G g and all-red phases R g , for g = 1 , . . . , M . At the beginning of green time G g , the probabilitythat one vehicle has arrived in any of the flows in group g during the preceding red time (or:intervisit time) I g , is λ g, • E [ I g ] + O ( λ g, • ). The probability that more than one vehicle hasarrived in any flow of group g is O ( λ g, • ) and therefore negligible in LT analysis. Therefore,we have that E [ G g ] = N g (cid:88) j =1 ρ g,j E [ I g ] + O ( ρ g,j ) = N g (cid:88) j =1 ρ g,j E [ R ] + O ( ρ ) . (4.2)The mean cycle time is E [ C ] = M (cid:88) g =1 E [ G g ] + E [ R ] = E [ R ] (1 + ρ ) + O ( ρ ) . (4.3)Now we can find the LT limit of E [ W g,j ] by conditioning on the arrival epoch of an arbitrarycustomer. The probability that an arbitrary vehicle arrives during the all-red phase R k , for k = 1 , . . . , M , is P (arrival during R k ) = E [ R k ] E [ C ] = E [ R k ] E [ R ] (1 − ρ ) + O ( ρ ) . The probability that an arbitrary vehicle arrives during the green phase G k , for k = 1 , . . . , M ,is P (arrival during G k ) = E [ G k ] E [ C ] = E [ G k ] E [ R ] (1 − ρ ) + O ( ρ ) = N k (cid:88) l =1 ρ k,l + O ( ρ ) . If a vehicle arrives during G k , the vehicle crossing the intersection at that moment is a type { k, l } vehicle with probability ρ k,l ρ k, • , for l = 1 , . . . , N k . This means that we can formulate themean delay of a type { g, j } vehicle as follows. E [ W g,j ] = M (cid:88) k =1 (cid:32) E [ R k ] E [ R ] (1 − ρ ) E [ W ( R k ) g,j ] + N k (cid:88) l =1 ρ k,l E [ W ( G k,l ) g,j ] (cid:33) + O ( ρ ) , (4.4)14here E [ W ( R k ) g,j ] is the mean delay of a { g, j } vehicle that arrives during R k , and E [ W ( G k,l ) g,j ] isthe mean delay of a { g, j } vehicle that arrives during the service of a { k, l } vehicle. We willstudy these conditional mean delays in more detail.Firstly, we note that the mean delay of a { g, j } vehicle arriving while another vehicle of thesame type is crossing the intersection, is simply the residual headway of the crossing vehicle,plus the departure headway of the vehicle itself: E [ W ( G g,j ) g,j ] = E [ B res g,j ] + E [ B g,j ] + O ( ρ ) , g = 1 , . . . , M ; j = 1 , . . . , N g . (4.5)The O ( ρ ) terms are of no interest in (4.5), because the probability of an arrival during a greenperiod is an O ( ρ ) term itself. Secondly, a { g, j } vehicle arriving while another vehicle of thesame group, but not of the same flow, is crossing the intersection experiences no delay at all,because the probability that another vehicle of the same type is present at the intersection is O ( ρ ): E [ W ( G g,l ) g,j ] = 0 + O ( ρ ) , g = 1 , . . . , M ; j = 1 , . . . , N g ; l (cid:54) = j. (4.6)The mean delay of a { g, j } vehicle arriving while a vehicle of another group is crossingthe intersection, is composed of the mean residual headway of that other vehicle, plus allsubsequent all-red times until the vehicle itself can start crossing the intersection, plus itsown headway. Note that the intermediate green times are negligible, because their totallength is O ( ρ ). E [ W ( G k,l ) g,j ] = E [ B res k,l ] + g − (cid:88) i = k E [ R i ] + E [ B g,j ] + O ( ρ ) , g (cid:54) = k. (4.7)Note that the sum in (4.7) has to be taken cyclic over the all-red periods between the greentimes of groups k and g . The mean delay of vehicles arriving during all-red times is slightlymore complicated. We have to include all O ( ρ ) terms now, because the probability of anarrival during an all-red period is not O ( ρ ). The mean delay of a { g, j } vehicle arrivingduring all-red period R m consists of the residual all-red period R res m , plus the green and all-red periods between group m and group g . In more detail, the mean delay is composedof: 1. The departure headways of all vehicles that arrive in groups m + 1 , . . . , g − and in flow { g, j } during the all-red times R g , . . . , R m − , and the elapsed part of R m at the arrivalepoch, denoted by R past m ;2. The residual red time R m , denoted by R res m , plus the headways of all vehicles arrivingin groups m + 1 , . . . , g − R m +1 , . . . , R g − plus the headways of the vehicles that arrive duringthese all-red times and will be served between R m and the green period of group g ;4. The headway of the arriving vehicle itself.Note that R past m is the same, in distribution, as R res m . This leads to the following expression15or E [ W ( R m ) g,j ], where the terms (4.8) − (4.11) correspond to items 1 − E [ W ( R m ) g,j ] = m − (cid:88) k = g − M E [ R k ] + E [ R past m ] (cid:32) g − (cid:88) l = m +1 ρ l, • + ρ g,j (cid:33) (4.8)+ E [ R res m ] (cid:32) g − (cid:88) l = m +1 ρ l, • (cid:33) (4.9)+ g − (cid:88) k = m +1 E [ R k ] (cid:16) g − (cid:88) l = m +1 ρ l, • (cid:17) (4.10)+ E [ B g,j ] . (4.11)Combining Equations (4.2)–(4.11) completes the proof of Theorem 4.1. (cid:3) Remark 4.2
Equation (4.6) will turn out to be the only place in the LT analysis wherethe assumption that an empty flow stays empty during the remainder of the green periodplays a role. Without this assumption, we would have that E [ W ( G g,l ) g,j ] = E [ B g,j ] + O ( ρ ) for l (cid:54) = j . The HT analysis in the previous section would not change at all. This means thatthe assumption, perhaps surprisingly, does not have much impact at all on the real and theapproximated mean delays. In Section 6, Example 3, we discuss in more detail the impact ofthe assumption that vehicles arriving at an empty flow during a green phase do not experienceany delay.The main result of this section is an adaptation of (4.1), which can be used as an approxi-mation for the LT limit of the mean delay for general renewal arrivals. Corollary 4.3
An approximation for the LT limit of the mean delay for vehicles in flow { g, j } for a traffic intersection with general renewal arrivals and deterministic all-red times is: E [ W LT g,j ] ≈ ρ g,j (cid:16) E [ ˆ A g,j ]ˆ g g,j (0) − (cid:17) E [ B res g,j ]) + ρ E [ B res ] + E [ B g,j ] − (cid:88) k (cid:54) = j ρ g,k (cid:0) E [ B res g,k ] + E [ B g,j ] (cid:1) + 12 (1 + ρ + ρ g,j − ρ g, • ) R. (4.12)We derive approximation (4.12) by adapting (4.1) to create an LT approximation for the caseof general renewal arrivals. This adaptation is similar as in [4], based on the observation thatthe first term in (4.1), ρ g,j E [ B res g,j ], is the LT limit of the mean waiting time (excluding theservice time) in an M/G/ E [ W M/G/ ] = ρ E [ B res ] + O ( ρ ) . (4.13)We obtain an approximation for the mean waiting time for general renewal arrivals by replac-ing this term with the LT limit for the GI/G/ ρ ↓ E [ W GI/G/ ] ρ = 1 + cv B E [ ˆ A ]ˆ g (0) E [ B ] , (4.14)16here cv B is the Squared Coefficient of Variation (SCV) of the service times, and ˆ g ( t ) is thedensity of the interarrival times ˆ A at ρ = 1. Our approximation is based on the approxi-mative assumption that the Fuhrmann-Cooper decomposition (cf. [15]) holds for our systemwith general renewal arrivals. The Fuhrmann-Cooper decomposition for the waiting times ofcustomers in queue i of a polling system with Poisson arrivals and exhaustive service in queue i states that the waiting time of an arbitrary type i customer is the sum of two independentrandom variables. One of these random variables is the waiting time of a customer in the cor-responding M/G/ i : W i d = W i,M/G/ + I res i . If we combine this Fuhrmann-Cooper decomposition with (4.1), (4.13), and (4.14), we obtainthe following approximation for the mean delay (i.e., waiting time plus headway) of a vehiclein flow { g, j } for an intersection with general renewal arrivals: E [ W LT g,j ] ≈ ρ g,j (cid:16) E [ ˆ A g,j ]ˆ g g,j (0) E [ B res g,j ] + E [ B g,j ] (cid:17) + g − (cid:88) m = g − M +1 N m (cid:88) k =1 ρ m,k (cid:32) E [ B res m,k ] + g − (cid:88) l = m E [ R l ] + E [ B g,j ] (cid:33) + g − (cid:88) m = g − M E [ R m ] E [ R ] (cid:32) E [ R res m ] (cid:16) − ρ + 2 g − (cid:88) k = m +1 ρ k, • + ρ g,j (cid:17) + m − (cid:88) k = g − M E [ R k ] (cid:16) g − (cid:88) l = m +1 ρ l, • + ρ g,j (cid:17) + g − (cid:88) k = m +1 E [ R k ] (cid:16) − ρ + g − (cid:88) l = m +1 ρ l, • (cid:17) + (1 − ρ ) E [ B g,j ] . (4.15)This expression can be written into a more compact form. After some straightforward (buttedious) rewriting, it can be shown that (4.15) reduces to: E [ W LT g,j ] ≈ ρ g,j (cid:16) E [ ˆ A g,j ]ˆ g g,j (0) − (cid:17) E [ B res g,j ] + ρ E [ B res ] + E [ B g,j ] − (cid:88) k (cid:54) = j ρ g,k (cid:0) E [ B res g,k ] + E [ B g,j ] (cid:1) + (1 − ρ + ρ g,j ) E [ R res ] + ( ρ − ρ g, • ) E [ R ] + 1 E [ R ] g − (cid:88) m = g − M (cid:32) g − (cid:88) k = m +1 ρ k, • (cid:33) V ar[ R m ] . (4.16)For deterministic all-red times, we have E [ R res ] = R/ V ar[ R g ] = 0 for g = 1 , . . . , M .Carrying out these substitutions in (4.16) leads to expression (4.12). Remark 4.4
A simplification can be made to replace the expression E [ ˆ A g,j ]ˆ g g,j (0). In [4, 41]it is shown that a good approximation for this term is E [ ˆ A i ]ˆ g i (0) ≈ cv Ai cv Ai +1 if cv A i > , (cid:0) cv A i (cid:1) if cv A i ≤ , where cv A i is the squared coefficient of variation of A i (and, hence, also of ˆ A i ). Note that thissimplification results in an approximation that requires only the first two moments of eachinput variable (i.e., headways, all-red times, and interarrival times).17 emark 4.5 (Convergence to the LT limit) In this paragraph we study the convergenceof the mean delays to their LT limiting values, similarly to what we have done for the HTlimit in the previous section. Although the LT limit of the mean delay (4.15) has been ob-tained in a rather heuristical way, it turns out to be very accurate. In contrast to the HTlimit, expression (4.15) is not exact when the arrival processes are not Poisson, because thecorrection term ρ g,j (cid:16) E [ ˆ A g,j ]ˆ g g,j (0) − (cid:17) E [ B res g,j ]) is based on a decomposition of the meandelay (see [4] for more details) which is known not be true in general. Nevertheless, whencomparing the approximated delays with simulated delays, results show that the approxima-tion is very accurate (see Section 6). We have also been able to test the accuracy of the LTlimit for small intersections (4 flows) by comparing it to exact results under the assumptionthat all the departure headways, interarrival times and switch-over times are exponentiallydistributed. This has provided much insight in the behaviour of the delay as a function of theload ρ . Expression (4.15) captures the LT behaviour very well, except for some cases wherethe behaviour exposes more curvature. To illustrate this, we consider a simple intersectionwith four flows, divided into two groups, each having a saturation flow rate of 1800 vehiclesper hour. The mean all-red times are 6 seconds each. The relative loads are ˆ ρ = ˆ ρ = 0 . ρ = ˆ ρ = 0 .
4. We assume exponentially distributed headways, interarrival times andall-red times, because this enables us to model this system as a Markov chain. In order tosolve this system numerically, we need a finite state space, so we take a maximum queuelength of 6 vehicles per flow. This is sufficient since we are only focussing on LT behaviour.Figure 4 shows the mean delays for flows 1 and 2 (and, because of symmetry, also of flows3 and 4) as a function of ρ , the total load in the system. These pictures illustrate that thederivative at ρ = 0 of the mean delay indeed equals the LT expression. But Figure 4(a) alsoillustrates that the actual behaviour may be non-linear, which can lead to deviations betweenthe approximation, developed in the next section, and the actual mean delay. Ρ W (a) Light TrafficExact Ρ W (b)Figure 4: The exact mean delays of flow 1 (left) and flow 2 (right) and the LT limits of theexample in Section 4. In the previous two sections we have given expressions for the mean delay in the extreme caseswhere the system is either hardly exposed to traffic, or completely saturated. In the presentsection we use these results to develop interpolations between these two cases that give an18pproximation for the mean delay for any load, as long as the system is not oversaturated(cf. [4]). In fact, we develop two different interpolations. At the end of this section we discussthe conditions under which each interpolation should be preferred. E [ W app1 g,j ] = K (cid:48) ,g,j + K (cid:48) ,g,j ρ − Lρ , g = 1 , . . . , M ; j = 1 , . . . , N g . (5.1) E [ W app2 g,j ] = K ,g,j + K ,g,j ρ + K ,g,j ρ − Lρ , g = 1 , . . . , M ; j = 1 , . . . , N g . (5.2)De denominator 1 − Lρ is required to capture the HT behaviour of the system, as discussedin Section 3. The numerator is a polynomial of first or second degree. The constants K (cid:48) ,g,j and K ,g,j (which will turn out to be the same) follow from the requirement that (5.1) and(5.2) should result in the same mean delay for ρ = 0 as the LT limit (4.15). We find K (cid:48) ,g,j and K ,g,j by imposing that the HT limit of the approximation is equal to the HT limit ofthe exact mean delay. Finally, the constant K ,g,j is found by adding the requirement thatalso the derivative with respect to ρ , taken at ρ = 0, of our approximation is equal to thederivative of the LT limit. A more formal definition of these requirements is presented below: E [ W app1 g,j ] (cid:12)(cid:12) ρ =0 = E [ W LT g,j ] (cid:12)(cid:12) ρ =0 , lim ρ ↑ /L (1 − Lρ ) E [ W app1 g,j ] = lim ρ → /L (1 − Lρ ) E [ W g,j ] , and for approximation (5.2), E [ W app2 g,j ] (cid:12)(cid:12) ρ =0 = E [ W LT g,j ] (cid:12)(cid:12) ρ =0 , dd ρ E [ W app2 g,j ] (cid:12)(cid:12) ρ =0 = dd ρ E [ W LT g,j ] (cid:12)(cid:12) ρ =0 , lim ρ ↑ /L (1 − Lρ ) E [ W app2 g,j ] = lim ρ → /L (1 − Lρ ) E [ W g,j ] , using (4.12) for E [ W LT g,j ] and (3.5) for lim ρ → /L (1 − Lρ ) E [ W g,j ]. This leads to the followingconstants: K (cid:48) ,g,j = R E [ B g,j ] , (5.3) K (cid:48) ,g,j = L (cid:32) (1 − ˆ ρ g, /L ) − ˆ ρ g,j /L (cid:18) R σ δ (cid:19) − K (cid:48) ,g,j (cid:33) , (5.4) K ,g,j = R E [ B g,j ] , (5.5) K ,g,j = ˆ ρ g,j (cid:16) E [ ˆ A g,j ]ˆ g g,j (0) − (cid:17) E [ B res g,j ] + E [ B res ] − L E [ B g,j ] − (cid:88) k (cid:54) = j ˆ ρ g,k (cid:0) E [ B res g,k ] + E [ B g,j ] (cid:1) + 12 (1 − L + ˆ ρ g,j − ρ g, • ) R, (5.6) K ,g,j = L (cid:32) (1 − ˆ ρ g, /L ) − ˆ ρ g,j /L (cid:18) R σ δ (cid:19) − K ,g,j (cid:33) − L K ,g,j , (5.7)19here δ = (cid:80) Mg =1 ˆ ρ g, L (1 − ˆ ρ g, L ) /
2, and σ = (cid:80) Mg =1 ˆ λ g, (cid:16) V ar[ B g, ] + ˆ ρ g, V ar[ ˆ A g, ] (cid:17) . In theabove expressions, we have assumed that the all-red times are deterministic, as is usually thecase. Sections 3 and 4 give insight in how to adapt these constants if randomness is involvedin (some of) the all-red times.An obvious question that arises now, is which of the two approximations, (5.1) or (5.2), shouldbe preferred. We can give some (heuristic) arguments, obtained by studying many numericalexamples. It can be shown that the two approximations, (5.1) or (5.2), are both exact for thelimiting cases ρ → ρ → /L and E [ R ] → ∞ . The difference between the two interpolationfunctions, respectively having a first and second order polynomial in the numerator, is theadditional requirement that the derivative in ρ = 0 of the approximated mean waiting timeshould be equal to the derivative of the LT expression (4.15). Since the LT expression is veryaccurate, using this additional information generally leads to more accurate approximations.For this reason, (5.2) should be preferred in most practical cases. However, there are somecircumstances under which (5.1) should be preferred, despite resulting in a derivative at ρ = 0which is not exact. In more detail, if the actual delay displays a strong non-linear behaviourfor small loads, using the second-order interpolation leads to bigger inaccuracies for loadsaround Lρ = 0 .
7. In most cases, this happens if the derivative in ρ = 0 is very small, oreven negative (see also Figure 4(a)). Studying the LT expression (4.15) shows that the mostnatural way for a negative derivative to appear, is when the combined load in all other groupsis smaller than the load of the other flows within the group. In terms of the model parameters:if, for a certain flow { g, i } , the criterion (cid:88) m (cid:54) = g ˆ ρ m, • − (cid:88) j (cid:54) = i ˆ ρ g,j < , (5.8)is satisfied, the first-order interpolation (5.1) is preferred over the second-order interpolation(5.2). Note that this is just a rule of thumb. We show the effect of choosing the wronginterpolation at the end of Example 2 in the next section. Remark 5.1
A negative slope at ρ = 0 is possible because of the assumption that the delay ofvehicles approaching an empty flow during a green period experience no delay at all. Undersome circumstances, mostly when Condition 5.8 is satisfied, an increase in traffic may bebeneficial for certain flows that hardly receive any traffic at all. The increase in traffic resultsin an increase of the mean green period, which results in a larger number of vehicles thatbenefit from the green light while their flow is empty. This might have a positive effect on themean delay for vehicles in this flow, although the effect disappears when the load increasesfurther. In the present section, we study typical features of the approximations and assess their accu-racies. In Example 1 we do this by taking an imaginary intersection and simulating severalscenarios. For each scenario we compare the simulated delays to the approximated delays. InExample 2 we take three real, existing intersections.20 xample 1: Accuracy of the approximations
In this example, we analyse an (imaginary) intersection with 6 traffic flows. The ratios ofthe arrival rates of the six flows are 1 : 2 : 3 : 4 : 5 : 6. The discharge rates are allequal, 1800 vehicles per hour, hence E [ B i ] = 2 seconds. For now, we assume that the SCVof the departure headways is 1. In our simulation we have used exponentially distributedrandom variables to achieve this. We will have a short discussion at the end of this exampleabout other probability distributions. Note that ˆ ρ i = i , because the mean headways are allequal. The total all-red time R in a cycle is 12 seconds, divided equally among the individualall-red times. We compare several different scenarios to get insight into the accuracy of theapproximations. The first seven scenarios, shown in Table 1, differ in the choice of which flowsto combine in one group. For each scenario, the mean delays for all queues have been obtainedby simulation, and are being compared to the approximated mean delays (5.1) or (5.2), usingCriterion (5.8) to decide which of the two interpolations is used. The mean delay of each queue,in each scenario, is obtained for 11 different loads: Lρ = { . , . , . , . . . , . , . , . } .Note that it is convenient to use Lρ instead of ρ , because Lρ takes values between 0 and1. Furthermore, Lρ can be interpreted as the level of saturation of the intersection. Bycomparing all approximated mean delays to the corresponding simulated values, the relativeerrors are calculated: e i ( Lρ ) = (cid:12)(cid:12)(cid:12)(cid:12) W app i − W sim i W sim i (cid:12)(cid:12)(cid:12)(cid:12) × , i = 1 , . . . , . In order to compare the scenarios, we introduce two quality measures. The first qualitymeasure,
QM1 , which gives insight in the worst performance, is the largest relative error (andthe corresponding queue, and the load at which this error is obtained). The second qualitymeasure,
QM2 , which provides a better insight in the overall performance, is the weightedmean relative error. This is computed by averaging the 11 relative errors for each flow, e i , andsubsequently taking the weighted average proportional to the arrival rates. In more detail: QM1 = max i,Lρ e i ( Lρ ) , QM2 = (cid:88) i =1 ˆ λ i (cid:80) j =1 ˆ λ j e j . Table 1 displays these two quality measures for scenarios I – VII .We discuss the scenarios I – VII first, later we add some extra scenarios. Scenario I dividesthe six signals into six separate groups. This reduces the model to an ordinary polling modelwith exhaustive service and Poisson arrivals. Exact results are known for this model, and theapproximation developed in the present paper simplifies to the expression found in [4]. In[4], this approximation is discovered to perform very well, especially for systems with morethan two queues. Therefore, it is no surprise that scenario I results in the best approximationaccuracy: an average mean error of only 0.06%, and the worst relative error, 0.3%, is obtainedfor flow 1 (with the smallest load) at Lρ = 0 .
7. The next scenario, Scenario II , has the lowestaccuracy, illustrating the greatest drawbacks of the approximation. The six flows are dividedinto three groups. The reason why this scenario performs so poorly, is that the flows in each21cenario Groups cv A i cv B Interpolation
QM1 QM2 orders error flow Lρ I { } , { } , { } , { } , { } , { } II { , } , { , } , { , } III { , } , { , } , { , } IV { , } , { , } , { , } V { , , } , { , , } VI { , , } , { , , } VII { , , } , { , , } I – VII used in Example 1, and the corresponding quality measures
QM1 and
QM2 .group have rather similar loads. When the HT limit is reached, the system behaviour is suchthat the heaviest loaded flows dominate the groups and determine the lengths of the greenperiods. As a result, the approximation converges to the HT limit only very slowly. For Lρ = 0 . e i are approximately 21% for all flows, whereas the errors areback to 7% for Lρ = 0 .
99. The overall mean relative error of 8.17% is reasonable, but in therange 0 . ≤ Lρ ≤ .
97 all relative errors are greater than 15%. We will have some furtherdiscussion on this issue later in this paper. In Scenario
III , the flows are also divided intothree groups, but now the best possible choice (for our approximation) is made, avoidinggroups with two flows having similar loads. Therefore, the results are excellent now, with
QM1 = 4 .
4% and
QM2 = 1 . IV might be the most interesting from a practicalpoint of view, combining good and bad combinations of flows in the groups. As expected, theperformance is better than Scenario II , and worse than Scenario III , with
QM1 = 10 .
3% and
QM2 = 3 . Lρ = 0 .
9, ranging from 7 . − II . We will return to Scenario IV later in this section, when we vary severalother parameters of the model.Now, we discuss Scenarios V - VII briefly. These scenarios have only two groups, each con-taining three flows. Scenario V can be compared to Scenario II , having the worst possibledivision into groups. An interesting aspect of this scenario, is that the mean delays of allflows in group 1 are approximated using the second-order approximation (5.2), whereas themean delays in group 2 use the first-order approximation (5.1). The reason to do this, isthat the Criterion (5.8) is satisfied for the flows in group 2, because this group contains thethree heaviest loaded flows of the intersection. This implies that the total relative load ofgroup 1, ˆ ρ , • , is much smaller than the relative load of any pair of flows in group 2, (cid:80) j (cid:54) = i ˆ ρ ,j ,for i = 1 , ,
3. Since Criterion (5.8) is satisfied, second order interpolations would severelyunderestimate the mean delays in group 2. In fact, if one would take (5.2) instead of (5.1)for group 2, the mean relative error for flows in this group would be increased from 4 .
07% to15 . Lρ . It can be seen that the derivative in Lρ = 0 of the approximated mean delay for flow 1 is not correct (it is slightly greater thanthe actual value), but it does lead to an overall better accuracy. In Scenarios VI and VII ,22 .0 0.2 0.4 0.6 0.8 1.0 L Ρ W L Ρ W ApproximationSimulation
Figure 5: Simulated and approximated mean delays for flow 1 (left) and flow 6 (right) inScenario V.the division into groups is better, resulting in more favourable results.Scenarios I − VII illustrate how the accuracy of the approximation depends on the distributionof the loads among the groups. Numerical experiments indicate that this is the biggest sourceof variation in the accuracy. For completeness, we show the effects of different interarrival-time distributions, and different distributions of the departure headways in Scenario IV . Inour simulations we have fitted phase-type distributions, as suggested in [32], matching thespecified SCVs. We can be brief in discussing the results, displayed in Table 2. The variationin the interarrival times and headways does not have a major impact on the accuracy of theapproximation, but in general it can be concluded that an increase (decrease) in variationresults in a decrease (increase) in the accuracy.Scenario Groups cv A i cv B QM1 QM2 error flow Lρ IV { , } , { , } , { , } VIII { , } , { , } , { , } IX { , } , { , } , { , } X { , } , { , } , { , } XI { , } , { , } , { , } XII { , } , { , } , { , } IV , VIII – XII used in Example 1, and the corresponding quality measures
QM1 and
QM2 . Example 2: Real-life examples
In the present subsection we test the approximation on three real-life situations. We take threeintersections, graphically displayed in Figure 6, and compare approximated mean delays withthe simulated values. The first two intersections are located in Eindhoven, The Netherlands,23
43 7 1258 6
754 10 6 3 89 12
Intersection 1 Intersection 2 Intersection 3
Motorised vehiclesBicycles
Figure 6: The intersections discussed in Example 2.and data have been obtained from the local city council. The data for the third intersectionare taken from the Dutch manual for configuration of traffic intersections [8]. Each of theseintersections contains several flows for motorised vehicles, and four bicycle lanes. The exactsettings for each intersection can be found in Appendix B. Table 3 shows the numericalresults of the approximated and simulated delays for the three different intersections. ForIntersection
QM1 QM2 error flow Lρ QM1 and
QM2 for the intersections in Section 6, Example2.all of the intersections, but especially for Intersections 1 and 2, the overall accuracy is quitegood, considering the overall mean relative error percentage
QM2 . But at the same time, oneof the drawbacks becomes apparent. In practice, these intersections have maximum greentimes, which are chosen such that the cycle time cannot exceed a specified value. In orderto minimise the maximum cycle time, flows are generally divided into groups such that thebusiest streams are put together in the same group, as long as no conflicts arise. As discussedin the first example, our approximation gives very accurate results, as long the loads of thebusiest flows in a group are not too close to each other. However, some of these exampleshave at least one group with two busy flows with similar relative loads. In Intersection 1, ingroup 4, we have ˆ ρ , = 0 .
12 and ˆ ρ , = 0 .
11. In Intersection 3, the highest two relative loadsin group 3 are ˆ ρ , = 0 .
15 and ˆ ρ , = 0 .
14. This means that the mean waiting time converges24o its HT limit (3.5) only very slowly. This explains that for these two intersections, theapproximation gives rather high relative errors (sometimes more than 20%) for high loads( Lρ = 0 . , . . . , . − Lρ ) W i , also for the simulateddelay and the two approximations. Clearly, the first-order interpolation should not be usedhere. For flow 3 in Intersection 1, Criterion (5.8) is satisfied. In Figure 7(c) and 7(d) wesee that the non-linearity for small values of Lρ results in an underestimation of the actualdelay by the second-order interpolation. The first-order interpolation is more suitable here.Finally, Figures 7(e) and 7(f) show why it is sometimes impossible to find a simple polynomialthat describes the behaviour of the scaled delay well. These figures, taken from Intersection3, show that the HT limit is reached only extremely late ( Lρ > .
99) for flow 2. In fact,for Lρ = 0 .
99 there is still a gap between the simulated value and the HT limit. Due tothe fact that the HT limit is being approached so slowly, the approximations are not veryaccurate in the range 0 . < Lρ < .
99, with relative errors greater than 20% for all flows.Neither the first-order interpolation, nor the second-order interpolation is complex enoughto describe the behaviour of (1 − Lρ ) W i . These pictures also indicate that a fitting functionmore sophisticated that a first or second order polynomial is required if one wants to obtaina more accurate closed-form approximation. Example 3: The impact of the stay-empty assumption
Throughout the paper we have assumed that vehicles arriving during a green period whiletheir flow is already empty, experience no delay at all because they do not have to accelerateand cross the intersection at normal speed. This assumption, which we refer to as the “stay-empty assumption”, has been made because it makes the model more realistic than a standardqueueing model with queues emptying and possibly refilling during the same green period.In this example we study the impact of this assumption by comparing delays found in theprevious example to delays of vehicles in the same intersections, but assuming that queues donot stay empty. Before we carry out the numerical analysis, we discuss the model with refillingqueues in more detail. The derivation of the LT limit (4.15) uses the stay-empty assumptiononly in Equation (4.6). The LT limit of the model with refilling queues is obtained by replacing(4.6) with E [ W ( G g,l ) g,j ] = E [ B g,j ] + O ( ρ ) for l (cid:54) = j . This would, by the way, slightly simplifythe LT limit (4.15) and, hence, also simplify the second-order interpolation (5.2). Anotherinteresting observation is that the distributions of the scaled delays under HT conditionsdo not change at all. Without providing a rigorous proof here, we argue that the HT fluidlimits remain exactly the same if the stay-empty assumption is abandoned. Consequently,the stability condition (cid:80) Mg =1 ρ g, < .0 0.2 0.4 0.6 0.8 1.0 L Ρ W (a) L Ρ H - L Ρ L W (b) L Ρ W (c) L Ρ H - L Ρ L W (d) L Ρ W (e) L Ρ H - L Ρ L W (f) Second order interpolationFirst order interpolationSimulation
Figure 7: Three plots of mean waiting times W i and scaled mean waiting times (1 − Lρ ) W i ,taken from Example 2 in Section 6. Figures (a) and (b) correspond to flow 8 of Intersection2. Figures (c) and (d) correspond to flow 3 of Intersection 1. Figures (e) and (f) correspondto flow 2 of Intersection 3. 26he stay-empty assumption, perhaps surprisingly, does not have much impact at all on thesimulated and the approximated mean delays. We show one example of how close the meandelays for the two different models are. In Figure 8 we study the mean delays of vehicles inflow 3 of Intersection 1 again, just like in Figure 7(c). In Figure 8 the mean delays are plottedagainst Lρ for the model with and without the stay-empty assumption. As a reference, theapproximated mean values are also shown in the same figure. Since we used a first-orderinterpolation for this example, the approximations for the model with and without the stay-empty assumption are the same. The relative difference between the simulated values for thetwo models is at most 7%, for Lρ = 0 .
98. Summarising, the approximation for the meandelay can be used for models with and without the stay-empty assumption, although it isrecommended to adapt the LT limit slightly as stated before. However, one should keep inmind that scaled mean delay for the model with refilling queues converges to the HT limitslower than for the model with flows that stay empty. L Ρ W W ApproximationSimulation H stay empty L Simulation H do not stay empty L Figure 8: The mean delays for flow 3 of Intersection 1, for the model with and without thestay-empty assumption, compared to the approximated mean delay.
Under LT and HT conditions, we managed to accurately describe the behaviour of delayof vehicles approaching a traffic intersection with an exhaustive control policy. Based onthese limiting situations, mean delay approximations have been established for any workload.These approximations are easy to implement, and have been tested on real-life situations.The performance of the approximations is good, except when the two greatest flow ratioswithin a group are very close.Several extensions of the model considered in the present paper are interesting to study. • In order to build a better model for intersections that are part of an arterial system,it would be interesting to allow correlated batch arrivals: groups of vehicles arrivingsimultaneously. The studies by Levy and Sidi [21] and Van der Mei [35] on pollingsystems with correlated batch arrivals might prove useful in this respect. • Furthermore, using results on polling systems with priorities (see, e.g., [1, 2]), one couldalso analyse conflicting flows receiving a green light simultaneously. The conflicting27ows should be placed in the same lane group and by assigning priority levels to each ofthe conflicting flows, the right of way rules can be implemented. Only minor adaptationsto the LT and HT limits are required to introduce the priority levels. • It may also be possible to extend the distributional approximations for polling systemsdeveloped by Dorsman et al. [13] to traffic intersections. The two required ingredientsof such a distributional approximation are both derived in the present paper. That is,we need an HT approximation for the waiting-time distribution (as derived in Section 3)in conjunction with a mean waiting-time approximation for general load (as obtained inSection 5). Subsequently, the HT distributional approximation should be refined suchthat its mean coincides with the mean waiting-time approximation, while the resultingdistributional approximation keeps the correct limiting behavior in HT after refinement. • From a practical point of view, it would be desirable to have a model that allows flowsto be part of multiple groups, instead of just one. However, this would complicatethe analysis considerably at some points, because it is not straightforward anymore todetermine which flows are dominant in each group. • Finally, possibly the greatest challenge, is the analysis of intersections with time-limitedservice. Although these are by far the most commonly used intersections in practice,hardly any detailed results are known to reliably estimate the expected delays.
Acknowledgements
The authors wish to thank Onno Boxma for valuable discussions and for useful comments onearlier drafts of the present paper.
AppendixA Proof of stability
In this appendix, we provide a rigorous proof of Theorem 2.1. This theorem states that thestability condition of an intersection with exhaustive traffic control only depends on the flowratios of the dominant flows in each group.
A.1 Model
For each flow { g, j } , g = 1 , . . . , M and j = 1 , . . . , N g , there is an i.i.d. sequence of interarrivaltimes { A g,j,k } and an i.i.d. sequence of headways { B g,j,k } . All sequences are mutually inde-pendent. We assume that each sequence has a finite first moment and define the appropriaterates: λ g,j = 1 / E [ A g,j, ] ,µ g,j = 1 / E [ B g,j, ] .
28e assume that the interarrival time distribution is unbounded and spread-out, in otherwords P ( A g,j, > T ) > T >
0, and for some integer (cid:96) , P ( A g,j, + · · · + A g,j,(cid:96) ∈ d x ) ≥ q ( x ) d x where (cid:90) ∞ q ( x ) d x > . The control policy is to allow flows { , } , . . . , { , N } to operate until all of them are empty,then allow flows { , } , . . . , { , N } to operate until they are all empty and so on, until flows { M, } , . . . , { M, N M } have operated. Then, this cycle is repeated. We assume that whenswitching for the m th time from group 1 to group 2, there is an all-red period of R ,m . Ingeneral, for the m th switch from group g to group g + 1 (all group indices are modulo M ), theall-red time is given by R g,m . We assume that for g = 1 , . . . , M , { R g,m } are i.i.d. sequenceswith E [ R g, ] < ∞ .Let E g,j ( t ) be the number of vehicles arriving to flow { g, j } in (0 , t ]. Let S g,j ( t ) be the potential number of service completions by flow { g, j } in (0 , t ], i.e. S g,j ( t ) is the number of vehicles thatwould have departed from flow { g, j } between times 0 and t if there were no idling of server { g, j } . Let A res g,j ( t ) be the residual interarrival time at time t for stream { g, j } and let B res g,j ( t )be the residual headway of the vehicle crossing at flow { g, j } . We assume that A res g,j ( t ) and B res g,j ( t ) are right continuous. Thus we have E g,j ( t ) = max { n ≥ A res g,j (0) + A g,j, + · · · + A g,j,n − ≤ t } ,S g,j ( t ) = max { n ≥ B res g,j (0) + B g,j, + · · · + B g,j,n − ≤ t } . where the maximum of an empty set is defined to be zero. Let T g,j ( t ) be the cumulative busytime for server { g, j } in (0 , t ]. Then the number at server { g, j } , Q g,j ( t ) at time t , is Q g,j ( t ) = Q g,j (0) + E g,j ( t ) − S g,j ( T g,j ( t )) , (A.1)where T g,j ( t ) is determined by the control policy. Define X ( t ) = ( Q g,j ( t ) , A g,j ( t ) , B g,j ( t ) , I ( t ) : g = 1 , . . . , M, j = 1 , . . . , N g ) , where I ( t ) is the group number (1 , . . . , M ) that receives a green light at time t (it can be set toan arbitrary value during the all-red times). Then it is not difficult to see that X = { X ( t ) } isa Markov process and has the strong Markov property. The assumption that the interarrivaltime distribution is unbounded and spread-out allows us to conclude that the states where Q g,j ( t ) = 0 are reachable and to show the existence of a continuous component for X , seeMeyn and Down [22]. A.2 Stability of fluid models
Let q = (cid:80) Mg =1 (cid:80) N g j =1 Q g,j (0). Suppose that the function ( ¯ Q g,j ( · ) , ¯ T g,j ( · ) : g = 1 , . . . , M, j =1 , . . . , N g ) is a limit point of( q − Q g,j ( qt ) , q − T g,j ( qt ) : g = 1 , . . . , M, j = 1 , . . . , N g ) , q → ∞ . When it exists ( ¯ Q g,j ( t ) , ¯ T g,j ( t ) : g = 1 , . . . , M, j = 1 , . . . , N g ) is called a fluid limit of the system. Since we have been able to describe the system dynamics in theform (A.1), Dai [10, Theorem 2.3.2] yields that a fluid limit exists (it may not be unique).Furthermore, each of the fluid limits is a solution of the following set of conditions, known asthe fluid model . ¯ Q g,j ( t ) = ¯ Q g,j (0) + λ g,j ( t ) − µ g,j ¯ T g,j ( t )¯ Q g,j ( t ) ≥ T g,j (0) = 0 and ¯ T g,j ( · ) is non-decreasingplus additional conditions on ¯ T g,j ( · ) due to the control policy. This last condition is usuallythe most important, even though it is vague at this point.The fluid model is said to be stable if there exists a fixed time t such that (cid:80) Mg =1 (cid:80) N g j =1 ¯ Q g,j ( t ) =0, for all t ≥ t , for any solution of the fluid model. The fluid model is said to be unstable if for every solution of the fluid model with (cid:80) Mg =1 (cid:80) N g j =1 ¯ Q g,j (0) = 0, there exists a δ > (cid:80) Mg =1 (cid:80) N g j =1 ¯ Q g,j ( δ ) (cid:54) = 0. Thus, stability of the fluid model states that eventuallyall flows will drain, and once drained, they remain empty.The connections between the Markov process and the fluid model are as follows: if the fluidmodel is stable, there exists a unique invariant probability for X (Theorem 4.2 of Dai [9]). Ifthe fluid model is unstable, then X is transient (Theorem 2.5.1 of Dai [10]). If one is interestedin finiteness of moments, then under corresponding assumptions on the underlying randomvariables, stability of the fluid model yields finiteness of moments, see Dai and Meyn [11]. Forexample, assuming finite second moments on the underlying random variables, stability of thefluid model implies the existence of an invariant probability and finite mean queue lengths.For the model under consideration, let ρ g,j = λ g,j /µ g,j and assume without loss of generalitythat ρ g, > ρ g, > · · · > ρ g,N g for all g = 1 , . . . , M . Then we have the following result. Proposition A.1 (i) If (cid:80) Mg =1 ρ g, <
1, the fluid model is stable and, thus, an invariantprobability ϕ exists for X .(ii) If (cid:80) Mg =1 ρ g, >
1, then the fluid model is unstable and X is transient. Proof: (i) First, we show that the all-red periods can be ignored in the stability analysis. Supposethat at time 0, group g has just completed and group g (cid:48) = ( g + 1) mod M + 1 begins service,where g (cid:48) is such that ¯ Q g (cid:48) ,j (0) > j ∈ { , . . . , N g (cid:48) } . We then have that there exists δ > Q g (cid:48) ,j ( s ) > s ∈ [0 , δ ]. In this case, we havelim q →∞ T g,j ( δq ) q = lim q →∞ max( δq − R g , q = lim q →∞ (cid:18) max (cid:18) δ − R g q , (cid:19)(cid:19) = δ from which we have ddt ¯ T g,j ( t ) | t =0 = 1 and thus without loss of generality we can assume thatthe all-red times are zero. 30ext, we show that if we are serving group g at time t and ¯ Q g,j ( t ) = 0 for all j ∈ { , . . . , N g } ,then for any fluid limit, we immediately switch to a new group (unless for all g, j , ¯ Q g,j ( t ) = 0).Note that for group g , queues 1 , . . . , N g operate as N g stable queues in parallel (until all queuesare simultaneously empty), which is a special case of a stable generalized Jackson network(as ρ g,j < j ). Denote by T ( n ,...,n Ng )0 the time to reach the state when all queues areempty, starting from the initial condition is that there are n i jobs at queue i . Theorem 3.8 of[22] implies that all queues being empty is reachable and as Q g,j ( qt ) q → , then T ( Q g, ( qt ) ,Q g, ( qt ) ,...,Q g,Ng ( qt ))0 q → g is immediate when all of the queuesare empty (on the fluid scale).Next, we show that from an arbitrary initial condition for the fluid model (i.e. ¯ Q g,j (0) = x g,j ≥
0) and with the server initially at group 1, we have that group i is switched away fromfor the first time at time t i , where t i is given by: t = max ≤ j ≤ N (cid:26) x ,j µ ,j − λ ,j (cid:27) t = t + max ≤ j ≤ N (cid:26) x ,j + λ ,j t µ ,j − λ ,j (cid:27) ... t M = t M − + max ≤ j ≤ N M (cid:26) x M,j + λ M,j t M − µ M,j − λ M,j (cid:27) . Clearly, if λ g,j /µ g,j < g, j , then t M < ∞ . However, at this point, by definition, it isclear that for t ≥ t M , ¯ Q g, ( t ) µ g, ≥ ¯ Q g,j ( t ) µ g,j (A.2)for all g and j ∈ { , . . . , N g } .Now, consider V ( t ) = M (cid:88) g =1 ¯ Q g, ( t ) µ g, . We trivially see that (cid:80) Mg =1 ddt ¯ T g, ( t ) = 1 if V ( t ) >
0, and thus for t ≥ t M and if V ( t ) > ddt V ( t ) = (cid:80) Mg =1 ρ g, − <
0. Therefore, there exists a
T < ∞ such that V ( t ) = 0for all t ≥ T and again using (A.2), we conclude that for all g, j ¯ Q g,j ( t ) = 0 for all t ≥ T .This completes the proof of (i).To show (ii), as (cid:80) Mg =1 ddt ¯ T g, ( t ) ≤
1, we see that ddt V ( t ) ≥ (cid:80) Mg =1 ρ g, −
1, which is strictlypositive and thus the fluid model is unstable. (cid:3) Input settings for Example 2
In this appendix we give the input settings for the intersections discussed in Section 6, Ex-ample 2. Intersection 1 N , . . . , , . . . , cv B i = 1 for cars ( i ≤ cv B i = 0 for bikes ( i ≥ { , , , } , { } , { , } , { , } All-red times 2 , , , N , . . . , , . . . , cv B i = 1 for cars ( i ≤ cv B i = 0 for bikes ( i ≥ { , , , } , { , } , { , } , { , , } All-red times 8 , , , N , . . . , , . . . , cv B i = 1 for cars ( i ≤ cv B i = 0 for bikes ( i ≥ { , , , } , { , } , { , , , } All-red times 4 , , eferences [1] M. A. A. Boon, I. J. B. F. Adan, and O. J. Boxma. A two-queue polling model with twopriority levels in the first queue. Discrete Event Dynamic Systems , 20(4):511–536, 2010.[2] M. A. A. Boon, I. J. B. F. Adan, and O. J. Boxma. A polling model with multiplepriority levels.
Performance Evaluation , 67:468–484, 2010.[3] M. A. A. Boon, R. D. van der Mei, and E. M. M. Winands. Applications of pollingsystems.
Surveys in Operations Research and Management Science , 16:67–82, 2011.[4] M. A. A. Boon, E. M. M. Winands, I. J. B. F. Adan, and A. C. C. van Wijk. Closed-formwaiting time approximations for polling systems.
Performance Evaluation , 68:290–306,2011.[5] E. G. Coffman, Jr., A. A. Puhalskii, and M. I. Reiman. Polling systems with zeroswitchover times: A heavy-traffic averaging principle.
The Annals of Applied Probability ,5(3):681–719, 1995.[6] E. G. Coffman, Jr., A. A. Puhalskii, and M. I. Reiman. Polling systems in heavy-traffic:A Bessel process limit.
Mathematics of Operations Research , 23:257–304, 1998.[7] J. W. Cohen.
The Single Server Queue . North-Holland, Amsterdam, revised edition,1982.[8] CROW.
Handboek verkeerslichtenregelingen . Publicatie 213. CROW kenniscentrum voorverkeer, vervoer en infrastructuur, Ede, 2006. ISBN 90 6628 444 7.[9] J. G. Dai. On positive Harris recurrence of multiclass queueing networks: a unified queuevia fluid limit models.
Annals of Applied Probability , 5:49–77, 1995.[10] J. G. Dai. Stability of fluid and stochastic processing networks. , 1999.[11] J. G. Dai and S. P. Meyn. Stability and convergence of moments for multiclass queueingnetworks via fluid models.
IEEE Transactions on Automatic Control , 40:1889–1904,1995.[12] J. N. Darroch, G. F. Newell, and R. W. J. Morris. Queues for a vehicle-actuated trafficlight.
Operations Research , 12(6):882–895, 1964.[13] J. L. Dorsman, R. D. van der Mei, and E. M. M. Winands. A new method for derivingwaiting-time approximations in polling systems with renewal arrivals.
Stochastic Models ,27(2):318–332, 2011.[14] S. W. Fuhrmann. Performance analysis of a class of cyclic schedules. Technical memo-randum 81-59531-1, Bell Laboratories, March 1981.[15] S. W. Fuhrmann and R. B. Cooper. Stochastic decompositions in the
M/G/
Operations Research , 33(5):1117–1129, 1985.3316] B. S. Greenberg, R. C. Leachman, and R. W. Wolff. Predicting dispatching delays on alow speed, single track railroad.
Transportation Science , 22:31–38, 1988.[17] R. Haijema and J. van der Wal. An MDP decomposition approach for traffic controlat isolated signalized intersections.
Probability in the Engineering and InformationalSciences , 22:587–602, 2007.[18] D. Heidemann. Queue length and delay distributions at traffic signals.
TransportationResearch Part B , 28(5):377–389, 1994.[19] J. P. Lehoczky. Traffic intersection control and zero-switch queues under conditions ofMarkov chain dependence input.
Journal of Applied Probability , 9(2):382–395, 1972.[20] H. Levy and M. Sidi. Polling systems: applications, modeling, and optimization.
IEEETransactions on Communications , 38:1750–1760, 1990.[21] H. Levy and M. Sidi. Polling systems with simultaneous arrivals.
IEEE Transactions onCommunications , 39:823–827, 1991.[22] S. P. Meyn and D. Down. Stability of generalized jackson networks.
Annals of AppliedProbability , 4:124–148, 1994.[23] A. J. Miller. Settings for fixed-cycle traffic signals.
Operational Research Quarterly , 14:373–386, 1963.[24] G. F. Newell. Approximation methods for queues with application to the fixed-cycletraffic light.
SIAM Review , 7(2):223–240, 1965.[25] G. F. Newell. Properties of vehicle-actuated signals: I. one-way streets.
TransportationScience , 3(2):31–52, 1969.[26] G. F. Newell. The rolling horizon scheme of traffic signal control.
Transportation ResearchPart A , 32(1):39–44, 1998.[27] G. F. Newell and E. E. Osuna. Properties of vehicle-actuated signals: II. two-way streets.
Transportation Science , 3(2):99–125, 1969.[28] T. L. Olsen and R. D. van der Mei. Polling systems with periodic server routeing inheavy traffic: distribution of the delay.
Journal of Applied Probability , 40:305–326, 2003.[29] T. L. Olsen and R. D. van der Mei. Polling systems with periodic server routing in heavytraffic: renewal arrivals.
Operations Research Letters , 33:17–25, 2005.[30] J. A. C. Resing. Polling systems and multitype branching processes.
Queueing Systems ,13:409 – 426, 1993.[31] H. Takagi. Queuing analysis of polling models.
ACM Computing Surveys (CSUR) , 20:5–28, 1988.[32] H. C. Tijms.
Stochastic models: an algorithmic approach . Wiley, Chichester, 1994.[33] TRB.
Highway Capacity Manual 2000 . Transportation Research Board, 2000.3434] M. S. van den Broek, J. S. H. van Leeuwaarden, I. J. B. F. Adan, and O. J. Boxma.Bounds and approximations for the fixed-cycle traffic-light queue.
Transportation Sci-ence , 40(4):484–496, 2006.[35] R. D. van der Mei. Polling systems with simultaneous batch arrivals.
Stochastic Models ,17(3):271–292, 2001.[36] R. D. van der Mei and E. M. M. Winands. A note on polling models with renewal arrivalsand nonzero switch-over times.
Operations Research Letters , 36:500–505, 2008.[37] J. S. H. van Leeuwaarden. Delay analysis for the fixed-cycle traffic-light queue.
Trans-portation Science , 40(2):189–199, 2006.[38] V. M. Vishnevskii and O. V. Semenova. Mathematical methods to study the pollingsystems.
Automation and Remote Control , 67(2):173–220, 2006.[39] M. Vlasiou and U. Yechiali.
M/G/ ∞ polling systems with random visit times. Probabilityin the Engineering and Informational Sciences , 22:81–106, 2008.[40] F. V. Webster. Traffic signal settings. Technical Paper 39, Road Research Laboratory,1958.[41] W. Whitt. An interpolation approximation for the mean workload in a
GI/G/
Operations Research , 37(6):936–952, 1989.[42] H. Yamashita, Y. Ishizuka, and S. Suzuki. Mean and variance of waiting time and theiroptimization for alternating traffic control systems.