The Intersection Delay Problem with Correlated Gap Acceptance

The Intersection Delay Problem with Correlated Gap AcceptanceAuthor(s): George H. WeissSource: Operations Research, Vol. 14, No. 4 (Jul. - Aug., 1966), pp. 614-618Published by: INFORMSStable URL: http://www.jstor.org/stable/168723 .

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THE INTERSECTION DELAY PROBLEM WITH

CORRELATED GAP ACCEPTANCE

George H. Weiss

National Cancer Institute, Bethesda, lMaryland

(Received December 14, 1965)

This paper considers a modification of the single-car intersection delay problem in which the waiting driver, upon finding a suitable gap for crossing also examines the succeeding one, and uses it if it is larger. It is shown that the additional delay time caused by this method of gap acceptance is generally quite small.

IN THE maniy mathematical treatments of the problem of delay at an uncontrolled intersection it is invariably assumed that a driver on the

minor road examines gaps in the major road traffic one at a time until a suitable one is selected for crossing.[1-7] At the recently held Third Inter- national Symposium on Traffic Flow DR. A. J. M\IILLER suggested that the process of selection can be slightly more complicated in that the minor road driver, finding an acceptable gap, iyight also examine the following gap. If that gap is larger the driver waits the additional time for the more desirable gap. Although there is no experimental information available on this subject one can still postulate a plausible model for this process to ascertain whether the additional delay resulting from the more complicated test for gap acceptance is significant. It is the purpose of this paper to present an analysis of this model. It will be shown that the effects of the modified gap acceptance procedure are generally negligible.

Many of the assumptions of this paper duplicate those of reference 5. In particular we will assume a single lane of traffic on the main road with independent, random headways (where headways will be measured in time) between successive cars. The headway density will be denoted by so(t). The gap aceeptance function will be denoted by a(t). However, in the present case, if an acceptable gap presents itself we will assume that the minor road driver examines one succeeding gap to see whether it is more acceptable. The examination is carried out with probability p(t), where t is the duration of the first headway. This probability is introduced because it is unreasonable to suppose tbat a very large first gap will be rejected even if the succeeding gap is larger. Finally, we let n(t) be the probability that an acceptable gap of t is followed by an even more desirable one. We can consider this probability to be made up of the probability

614



Intersection Delay Proble b 616 5

that the succeeding gap is x and the probability that a gap of x will be nmore desirable than one of t. If this latter probability is g(t, x) then

Xqt ~V(x) g(t, x) dx. (1)

The remainder of our notation is mainly that given in reference 5; for con- venience we provide a list:

P(t ) =fX <(x) dx,

A=f x<(x) dx = mean headway,

o(Pt)O (1/,u)b( t) = headway density for the first gap assuming that the minor road driver arrives at the intersection at t =0,

S2(t) =p.d.f. of delay,

t(t =s(t) [1 - a(t) ],

to (t) =woo(t) [I - a(t)]

a! = f c(x) s?(x) dx,

ao= a(x) poo(x) dx,

a (t)-So (t) a (t) p (t) ()

ao (t) wo (t) a (t) p (t) ?()

c(t) dt= probability that a car in the major stream passes the intersection at some time in (t, t+dt), conditional on the minor stream driver not having crossed because all previous gaps were unacceptable.

In addition to this notation we also use the convention that the Laplace transform of a function of time is that same functioii with an asterisk and arguments. Thus, we write Q*(s)=-2{Q(t)}.

We begin the analysis by decomposing Q(t) into two parts:

Q?(t)-Q1(t) +22(t), (2)

where Qi(t) is the delay density conditional on the first gap being the first acceptable gap, and 22(t) is the delay density for all other possibilities. An expression for p1(t) can immediately be written as



616 George H. Weiss

where b(t) is a Dirac delta function. The first term corresponds to zero delay and the second term to the situation in which the first gap is acceptable but the minor road driver waits for the succeeding gap. An expression for Q2(t) is

Q2(0 =W(t)J <(u) ax(u)[1-p(u) r7(u)] du+l w(u) a(t-u) du, (4)

in which the first term represents the possibility that the gap beginning at t is the first acceptable one, and the following gap is either not considered, or rejected. The second term represents the possibility that the gap immediately preceding t is the first acceptable one but the minor road driver waited for the next gap. The Laplace transforms of equations (2), (3), and (4) yield

Q*(s) = &o-ao*(O) +ao*(s)+w*(s)[&-a*(O) +a*(s)] (5)

in terms of c*(s). However, this last quantity is known from reference 5 in the context of the present problem; it is

@*(s)-to*(s)/[1l-1,*(s)], (6)

and so the formal calculation of Q(t) can be considered to be accomplished. Most forms of the relevant functions are sufficiently complicated so that

the expression for Q(t) cannot be obtained through inversion of Q*(s). Nevertheless Q*(s) can be used as a moment generating function. If t denotes the mean delay, and to denotes the mean delay as calculated for the rule of always accepting the first acceptable gap, then a differentiation of equation (5) followed by setting s = 0 yields

t=|ft[ao(t)+1 aO a(t)] dt+to-ia+10 (7)

where we have used the relation Q*(s) = ao?&w*(s) proved in reference 5. Hence Ta represents the additional delay time caused by considering an additional gap after the first acceptable one.

Let us now consider a plausible specialization of the general model. If H(x) denotes the Heaviside step function, i.e., H(x) =0, x<O, H(x) = 1, x>0 then we choose

a(t) =H(t- T), p(t) =eX (t-T) (t_ T) (8)

that is to say, a step gap acceptance function, and an exponentially de- creasing choice function. If X = O, p( t) = 1, and the gap following an acceptable one is invariably considered. The case X = oo implies that the succeeding gap is never considered. We assume that the headway density is negative exponential

-;O(O =wP(t =ae`(9)



Inttersection Delay Problem 617

and that a following gap will be more desirable than a present (acceptable) one provided that it is larger. This implies that q(t) is given by

e-x dx=e-t (10)

ADDITIONAL MEAN DELAY(ta)AS A FUNCTION OF to FOR T=5 secs., X=0,0.1,0.5secs.7

2

30

A =0.1

1 2 3 4 5 6 7 8 9 10 15 20

t. ( seconds )

Figure I

Hence the additional mean delay is

t. = [ae" /(2o-+ X)2][ I + (2o +S X) T], ( 11) as contrasted to

to- ?T1 --T) lo-. ( 12)

For large o- aiid oaT, corresponding to heavy traffic, it is clearly the case



618 George H. Weiss

that the contribution from Ta is negligible in comparison to that from to. In Fig. 1 T, is plotted as a function of to for T =5 sec and various values of X. It can be seen that the only time T, takes on measureable values is at very low traffic densities and very low values of X. This can be under- stood in the following terms: Low X implies that the second gap is nearly always considered. If the second acceptable gap is the one actually taken, the resulting wait can be quite long because the mean headway is small. However, application of the theory in its present form to the low (a, X) region is absurd because few drivers will look for a longer gap than say 10 seconds, in which case the form of p (t) given in equation (8) is qualitatively incorrect.

These calculations strongly suggest that the present formulation of the problem of single-car delay at an intersection will give quantitatively reasonable results without further modification. Although the functions I have chosen are arbitrary, I believe that this conclusion is insensitive to their particular form. There still remains the possibility that results for a queue of cars at an intersection may be more strongly dependent on the mode of gap acceptance. I conjecture that this would be true only for heavy traffic in the minor stream. It is not clear how the N-lane gap acceptance function would be modified by a rule analogous to that considered in the present paper.

REFERENCES

1. F. GARWOOD, "An Application of the Theory of Probability to the Operation of Vehicular-Controlled Traffic Signals," J. Roy. Stat. Soc. B 7, 65 (1940).

2. M. S. RAFF, "The Distribution of Blocks in an Uncongested Stream of Auto- mobile Traffic," J. Am. Stat. Soc. 46, 114 (1951).

3. J. C. TANNER, "The Delay to Pedestrians Crossing a Road," Biometrika 38, 383 (1951).

4. A. J. MAYNE, "Some Further Results in the Theory of Pedestrians and Road Traffic," Biometrika 41, 375 (1954).

5. G. H. WEISS AND A. A. MARADUDIN, "Some Problems in Traffic Delay," Opns. Res. 10, 74-104 (1962)

6. R. HERMAN AND G. H. WEISS, "Some Comments on the Highway Crossing Problem," Opns. Res. 9, 828-840 (1961).

7. D. C. GAZIS, G. F. NEWELL, P. WARREN, AND G. H. WEISS, "The Delay Prob- lem for Crossing an N-Lane Highway," Proc. Third International Symposium on Traffic Flow (to appear).



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The Intersection Delay Problem with Correlated Gap Acceptance