Delay at controlled intersections: the old theory revised Henk J. van Zuylen, Francesco Viti
Abstract — The calculation of delays at an intersection is an old problem that has been studied and solved by many researchers in the last 50 years. This paper reanalyzes the problem by using a Markov chain model for the probability distribution of queue length. From the dynamics of the expectation value of the queue length, the authors derive a formula for the delay in fixed time traffic control. The effect of the overflow queue is made explicit. The result is a new formula, but even more important, a clear understanding of the role of the overflow queue.
I. INTRODUCTION
D
ELAYS of vehicles at a controlled intersection are an issue that has been studied by many authors. Beckman et al. [1] and Webster [2] have been early pioneers to derive and publish delay formulas, consisting of two main components: a uniform delay term, which is the delay in case of a uniform arrival process and a term that has to take into account that the arrival process is a random process, leading to overflow queues, i.e. queues that still remain at the end of a green phase. Several formulas have been derived; most of them are based partly on heuristics to calculate the effect of overflow queues [3]. The overflow queue itself has still some interesting aspects. It is obvious that in case of an oversaturated intersection, the length of the overflow queue will grow from cycle to cycle. The assumption in the older delay models is that the overflow queue is static, constant from cycle to cycle. The unrealistic character of this assumption becomes clear in the analysis of the transition between under saturated intersections and over saturated ones. If the degree of saturation comes close to 1 – a fully saturated intersection – the overflow queue becomes infinite according to formulas like Webster’s, while if it becomes larger than 1, the length of the overflow queue becomes finite and linear in time. It has been the merit of researchers like Kimber and Hollis [4] and Akcelik [5] to solve this anomaly by introducing a kind of transformation that gives a smooth transition for the delay expressions for undersaturated and oversaturated conditions. The approach that is followed is a heuristic one, i.e. the formula that is derived fits with the asymptotic situations for very low and very high saturation of the Manuscript received March 20, 2006. This work was supported in part by the Transport Research Centre Delft and the program Next Generation Infrastructure Both authors are with Delft University of Technology and TRAIL research school. P.O.Box 5048, 2600 GA Delft, The Netherlands (tel.: +31152782761, fax: +31152783179, e-mail
[email protected] and
[email protected] )
intersection and in the regime of nearly saturated or slightly oversaturated intersections the length of the overflow queue is obtained by some interpolation approach. Scope of this paper is to provide an exact probabilistic formulation of the queuing process within a cycle, which enables one to justify the dynamic and stochastic character of overflow queues, especially at signals that operate near capacity. This formulation is derived by assuming the queue evolution as a one-step Markov Chain process. This has been done in other studies, e.g. [6], [7], [8], but only to evaluate the queue length probability distribution cycle by cycle. A Markov Chain model formulation of expected queue lengths is derived here and evaluated numerically within a cycle, which, as a consequence, justifies the same cycle-by-cycle behavior and gives a clearer insight into the role of overflow queues in the delays travelers may experience. The paper is structured as follows. The next section introduces the Markov model formulation that computes the probability distribution of overflow queues cycle by cycle. The new model formulation for within-cycle dynamics is presented in section III, while exact formulations of the uniform and the stochastic delay components are derived later in section IV. Section V compares the result with the traditional random delay formulas. Finally section VI gives conclusions of this new approach. II. MARKOV CHAINS The length of the overflow queue can be derived in a fundamental way by describing the probability distribution of the queue length as a Markov chain process: the probability distribution of the queue length at the end of the green phase depends on the probability distribution of the queue length at the end of the previous green phase and the probability distribution of the arrivals during the cycle (figure 1). This has been proposed by van Zuylen [6] and Olszewski [7].
the probability function of the queue length: Pn (t), the probability that a queue of n vehicles is waiting at time t. The dynamics of this distribution function during the red phase is given by:
probability probability
P(1,0
P(0,0 )
P(1,1 ) P(1,3 ) P(1,4
P(0,1 P(0,3
n
∑ P (t − ∆t ). p
Pn (t ) =
j
P(0,4
n− j
( ∆t )
(3)
j =0
0 1 2 3
Queue length this cycle
0 1 2 3
The expectation value of the queue is then given by:
Queue length previous cycle
Figure 1: The transition process of the probability distribution of the queue length at the end of the green phase from cycle to cycle
∞
E{Q (t )} = ∞
n
n=0
l =0
∑ n∑ P
n−l
∑ p P ( m − 1, j − l + s.t ) l
g
∞
(1)
E{Q(t )} =
l =0
( t − ∆t ) p l ( ∆t )
∞
∑∑ nP
n −l
(t − ∆t ) pl (∆t ) =
l = 0 n =l ∞
where: • P(m,j) is the probability of a queue length j at the end of the green phase of cycle m, • pl is the probability of l arrivals in a cycle, • tg is the length of the green phase, and • s is the saturation flow. This approach has been used in past studies to evaluate the dynamic and stochastic character of overflow queues, in particular in conditions near capacity [3]. The same approach can be applied for the queue length distribution at any moment during the cycle as it is shown in the next section. This allows one to calculate the expected value of the delay correctly. III. QUEUE CALCULATION The delay at an approach of a controlled intersection depends on the arrivals the length of the red and green phases and the initial queue. Delays will differ from cycle to cycle due to the stochastic character of the traffic. The expected delay is:
=
∞
∑∑
∞
(n − l ) Pn −l (t − ∆t ) pl (∆t ) +
l = 0 n =l
∑
∞
lpl (∆t )
l =0
∑P
n −l
(t − ∆t )
n =l
giving the simple and intuitive result:
E{Q(t )} = (5) = E{Q (t − ∆t )} + E{a(∆t )} = E{Q (t − ∆t )} + q∆t i.e. the expectation function of the queue length is a simple linear function of the time and grows proportional to the average arrival rate q. The expression (5) is derived without using any assumption about the arrival distribution pl apart from the assumption that this arrival distribution is uniform over the red phase. This justifies the simplified representation of the queue length as a linear function as is used in many estimations of the delay. If the same approach is used for the queue length in the green phase, the situation is considerably complicated. Formula (3) has to be modified in the following way: n + s ∆t
Pn (t ) =
∑P
n − l + s ∆t
(t − ∆t ). pl ( ∆t )
if n > 0
(6)
l =0
C
E{W } = ∫ E{Q(t )}dt
(4)
If this expression is rearranged it can be evaluated exactly:
j + s .tg
P ( m, j ) =
n
n=0
= If one assumes a constant, deterministic departure rate, while no specific distribution is assumed for the arrivals, the transition can be described with:
∑ n.P (t ) =
(2)
0
Where W is the total delay in a cycle C and Q(t) is the queue length at time t. The queue length is in one single cycle a step function that increases with one at the arrival of a vehicle. If we take the expectation value of the queue length it becomes a continuous function. The expectation value of the queue can be derived from
s ∆t
P0 (t ) =
s ∆t − l
∑ p ( ∆t ) ∑ P ( t − ∆t ) l
l =0
j
(7)
j =0
Expression (7) is just the consequence of the fact that the probability distribution function has only finite values for positive queue lengths. Figure 2 shows the evolution process of the distribution function:
Probability
Queue length
E{Q (t )} = E{Q (0)} + qt − s (t − t r , eff ) +
E{Q(t)}=E{Q(0)}+E{a(t)} [ s ( t − t r , eff )]
+
∑ l =0
time
Figure 2: The probability distribution of the queue length in the green phase. If the departure process in the green phase is assumed to be deterministic with constant headways, the dynamics of the queue in the green phase becomes: ∞
∞
n + s ∆t
n =1
n=0
l =0
E {Q (t )} = ∑ n ⋅ Pn (t ) = ∑ n ∑ Pn − l + s∆t (t − ∆t ) pl ( ∆t ) (8)
∞
n ⋅ Pn − l + s∆t (t − ∆t ) pl ( ∆t ) =
l = 0 n = max( 0, l − s ∆t ) ∞
=
∞
∑ p ( ∆t ) ∑
( n − l + s ∆t ) Pn − l + s∆t (t − ∆t )
l
l =0
n = max( 0, l − s ∆t )
∞
∑
l =0
n = max(0,l − s ∆t )
(9)
∞
+ ∑ lpl (∆t )
Pn −l + s∆t (t − ∆t )
∞
∑
l =0
n = max(0, l − s ∆t )
− s∆t ∑ pl ( ∆t )
( s (t − t r , eff ) − m − l )Pm (0)
m =0
One may easily verify that the expression approaches zero when t→∞. The calculation of the third term can be done numerically, where the initial queue length probability distribution can be calculated with the Markov chain model of van Zuylen [4] or Olszewski [5], as described in detail in [3]. If we assume that the cycle begins with a deterministic state E{Q(0)} = Q0, the correction term in eq. (11) becomes: [ s ( t − t r , eff )]
∑
pl (t )( s (t − t r , eff ) − Q0 − l )
(12)
Since the first terms of eq. (11), the deterministic queue length, continues also after the moment that the deterministic queue disappears, the correction term has to compensate for that. The deterministic queue is Qdet (t ) = Q0 + qt − s (t − t r , eff ) if t < Qdet (t ) = 0
∞
∑ ∑
∑
l =0
A numerical evaluation of formula (8) shows that the queue initially decreases linearly until the moment that the standard deviation of the queue length distribution becomes of the same order of magnitude as the expectation value of the queue, as shown in figure 2. Equation (8) can be rewritten by rearranging the summation: E {Q (t )} =
(11)
[ s ( t − t r , eff )] − l
pl ( t )
∞
Pn −l + s∆t (t − ∆t )
(Q0 + s ⋅ t r , eff ) s−q
(13)
otherwise
Since the first part of eq. (11) continues to decrease, the correction term has to compensate the negative part. Figure 3 shows the expectation of the queue length computed with formula (11) for different degrees of saturation if a Poisson distribution is assumed for the number of arrivals within the cycle. The results are computed using g=24, c=60, s=1800 veh/h and Q0=0. The behavior of the decreasing part of the queue length confirms the illustrative example drawn in figure 2.
This results in E{Q (t )} = E{Q (t − ∆t )} + q∆t − s ∆t + [ s ∆t ]
[ s ∆t ] − l
l =0
m= 0
+ ∑ pl ( ∆t ) ∑ ( s∆t − m − l )Pm (t − ∆t )
(10)
The square brackets around s∆t mean the integer value of that expression. The first three terms of eq. (10) are simply the expression of the linearly decreasing queue, the last term is the effect of the stochastic character of the arrivals. Eq. (10) can simply be rewritten in terms of the probability distribution function of the queue at t=0, the beginning of the cycle – i.e. at the start of the red phase. Figure 3: Expected queue length within a cycle The fundamental role of the compensation term is clearly visualized at every demand condition. For very low degrees
of saturation (as x=0.5 in the figure) it is small error to consider no expected overflow queue at the end of the cycle. The larger this value, the larger positive expectation value results at the end of the cycle. The compensation term allows one to compute the expectation value also when x=1, what cannot be calculated with static models. The new formula (11) enables one to calculate a finite value for the overflow queue at the end of each cycle. This value is nonetheless dependent on the assumed probability distribution of the arrivals.
IV. DELAY CALCULATION From equations (2), the definition of the delay in terms of the queue as a function of the time and the expressions (5) and (10) the delay can be calculated according to eq. (2). If we take just the deterministic part of the queue we get the uniform delay with a modification for the initial queue (which might be the result of the stochastic arrival process in previous cycles):
E{Wuniform } =
(Q0 + s.t r , eff )(Q0 + q.t r , eff )
(14)
2( s − q )
In many models the term with Q0 is also denoted as a random delay term. Expression (14) reduces to the first term in Webster’s delay formula if Q0 = 0. The random delay component due to stochastic arrivals in the cycle itself becomes
E{Wrandom } = C [ s ( t − t r ,eff )]
∫ ∑
[ s ( t − t r ,eff )] − l
pl ( t )
∑
l=0
t r ,eff
( s (t − t r , eff ) − m − l )Pm (0) dt − (15)
m=0
C
Figure 4: Expected queue at x=1
∫
−
{Q0 + qt − s (t − t r , eff ) /( s − q )}dt
( Q0 + s . t r ,eff ) /( s − q )
Figure 4 gives also insight of how different cycle lengths affect the expectation value of the queue during the green phase and at the end of the cycle. Although the increase is relatively small in the example, the larger the cycle length, the higher overflow queue is observed at the end of the cycle. On the other hand, the compensation term does not have a particular role when the initial queue is large, as one can expect, since the probability of having zero queue at the end of the cycle is negligible (figure 5). The expected value at the end of the cycle is very well represented by the deterministic term only.
This can be simplified to: E{Wrandom } = C − t r ,eff
=
[ st ']
∫ ∑ p (t '+ t l
0t
l =0
(16)
[ s . t '] − l r , eff
)
∑ ( st '− m − l )P (0)dt '− 0.5( s − q)t
2
m
m=0
With t = C − (Q0 + s.t r , eff ) ( s − q ) the unsaturated green time. Formulas (14) and (16) represent exact formulations of uniform and random delays within a cycle. The calculation of the initial queue Q0 is not further discussed in this paper. One can still use in formula (11) one of the initial queue delay models available in literature - e.g. the HCM 2000 formula (TRB, 2000) [10], Akcelik et al. (1997) [11]. As an alternative, one can compute the initial queue delay in a stochastic fashion by using again a Markov chain approach ([7], [3]). In that approach the dynamics of the initial queue is calculated from a cycle to cycle Markov chain process. V. COMPARISON WITH OTHER RANDOM DELAY FORMULAS
Figure 5: Expected queue for Qo=20
If the probability distribution of the initial queue length P(0) is known and the arrival statistics p(t), e.g. a Poisson distribution, the random delay can easily be calculated numerically. Comparisons can be made with other expressions for the random delay, often chosen on a heuristic
basis or derived from some assumptions about the queuing process. The advantage of expression (16) is that it is ‘clean’, not influenced by any approximation. The expression (16) is not simple to evaluate by hand and the relationship with easily obtainable traffic characteristics is absent. The degree of saturation is not directly visible in the expression for the random delay. However, the calculation can very easily be executed by a rather simple computer program. Figure 6 shows that expression (16) is consistent with Webster’s delay formula for low degrees of saturation (x < 0.96)
Two options are possible: to use formula (16) as a validation for existing formulas of random delay or to search for a simple formula that approximates this expression. This last approach has been followed by Viti and van Zuylen [8] for the evolution of Q0 , the initial queue at the start of a red phase. The dynamics of the probability distribution of this queue length can be calculated with a Markov process similar to the one described in this paper. The dynamics of the expectation value and its standard deviations have been approximated very accurately by a mathematical function [3]. We propose to follow the other approach, to use formula (16) with the different random delay functions in the literature. For instance the well-known delay functions of [2] and [9] as representatives of the equilibrium delay models and [5] for the time-dependent models can be compared with the new expression for the delay. This comparison will be subject of further study.
VI. CONCLUSIONS
Figure 6 Comparison of the results from eq. (16) with the Webster formula [2]. Figure 7 shows a similar comparison with Ackcelik’s heuristic model. The compatibility between both models remains good up to x < 1.15.
Figure 7 The comparison of eq. (16) with Akcelik’s model [5]
Delay at controlled intersections has been a study subject for many years already. Several mathematical expressions have been derived to represent the so-called random delay component, the delay caused by the stochastic character of arriving traffic. The new formulation of this random delay component in this paper is derived without any assumptions about the statistical properties of the arrival process, apart from the assumption that the arrival distribution is uniform over the whole cycle. It fits for Poisson, binomial, normal distributed arrivals. The numerical value of the new random delay component can easily be computed and compared with approximated expressions of other authors. More important than the possibility to calculate the random delay with a more general model than the existing ones, is the insight that the derivation gives in the process that causes the random delay. The assumption used by some authors that the queue should be represented by a step function, appears to be superfluous. The stepwise character of the delay is transformed to a smooth character of the expected delay, linear increasing in the red-phase and the first part of the green phase. The expectation value of the queue in the green phase shows a non linear character as soon as the tail of the probability distribution comes close to zero. This phenomenon causes the overflow delay. Although fixed time controllers are considered as oldfashioned, while most intersections are controlled by closed loop controllers, the traffic dependent control becomes nearly fixed time during peak hours. That makes a good model for delays of fixed time controlled intersections still important.
VII. ACKNOWLEDGEMENT This research has been supported by Rijkswaterstaat and the Transport Research Centre Delft.
REFERENCES [1]
Beckmann, M.J., McGuire C.B., Winsten C.B., 1956. Studies in the Economics of Transportation. New Haven, Yale University Press [2] Webster F.V., 1958. Traffic Signal Settings. Road Research Lab, Technical Paper No. 39. Her Majesty Stationary Office, London, England. [3] Viti, F. (2006) thesis, TRAIL research school Delft. [4] Kimber and Hollis, 1979. Traffic queues and delay at road junctions, TRL LR 909. [5] Akçelik R., 1980. Time-Dependent Expressions for Delay, Stop Rate and Queue Length at Traffic Signals. Australian Road Research Board, Internal Report, AIR 367-1. [6] Van Zuylen, H.J. (1985) De dynamiek van wachtrijen voor een verkeerslicht (The dynamics of queues at traffic lights), Dutch lecture notes VB41, Verkeersakademie, Tilburg. [7] Olszewski P., (1990). Modeling of queue probability distribution at traffic signals. In Koshi M. ed. Transportation and Traffic Theory, Elsevier, New York. [8] Viti F. and van Zuylen H. J., 2004. Modeling queues at signalized intersections. Transportation Research Record 1883, pp. 68-77. National Research Council, Washington D.C. [9] McNeil D.R., 1968. A solution to the fixed cycle traffic light problem for compound Poisson arrivals. Journal of Applied Probability, 5, pp. 624-635. [10] TRB, 2000. Highway Capacity Manual, special report 209, National Research Council, Washington D.C., TRB. [11] Ackcelik, R., 1994. Akcelik, R., Chung, E. and Besley M. 1997. Recent research on actuated signal timing and performance evaluation and its application in SIDRA 5. Compendium of Technical Papers (CD). 67th Annual Meeting of the Institution of Transportation Engineers, Boston, USA.
