A simple renewal model of throughput at an oversaturated signalized intersection

Trunspn Res. Vol. 1, pp. 57-65.

Pergamon Prew 1967. Printed in Great Britain

A SIMPLE RENEWAL MODEL OF THROUGHPUT AT AN OVERSATURATED SIGNALIZED INTERSECTIONt C. J. ANCICFB,JR. and A. V. GAF~N System Development Corporation, Santa Monica, California (Received 3 1 October 1966) 1. INTRODUCTION THE ~RATIONAL efficiency of an oversaturated signalized intersection has received some prior’attention, principally in connection with optimizing throughput (Gazis, 1964; Gazis and Potts, 1965). Much of this analysis assumes knowledge of the demand function and the service function. The service function, which is actually a sawtooth curve because of the succession of red and green phases, is usually drawn as a smooth curve for simplicity. The effects of randomness in the service function are not considered. In this paper, a first attempt is made to study the random fluctuations of the service curve during each go-phase. It will be seen that an appropriate mean value model very accurately portrays the average throughput characteristics. On the other hand, variability in headways does cause significant variations in throughput characteristics. We have underway, at present, a program to measure and analyze successive headways in an oversaturated situation. The results of this study will be reported in a later paper.

2. THE MODEL In this model we make the following assumptions: (1) An infinite queue of vehicles is stacked in a single lane at a signalized intersection. (2) The go-phase of the signal has a fixed time-length, T. (3) The interval of time between the initiation of the go-phase and the entry of the first car into the intersection is a positive random variable, X. (4) All subsequent entry time headways are assumed equal to a fixed minimum time, T, plus a random variable with the same distribution as X. (5) All random variables are assumed to be independent. (6) If a vehicle reaches the intersection before the end of a go-phase, it is committed to a passage through. This model might apply to a single lane or one-way traffic, such as a freeway off-ramp or to each interior lane of a multilane arterial. In this problem the number of vehicles that pass through the intersection during a single go-phase, T, is the fundamental random variable, N. This variable is equivalent to the basic variable of renewal theory, namely the number of renewals in time T (Cox, 1962). As defined, the process is clearly a modified renewal process, i.e. one in which the distribution of the time to the first renewal differs from the distribution of subsequent inter-renewal times. In particular, iff(t) is the density function for the time to the first renewal, the density function t Presented under the title “The effect of driver response time on throughput at a saturated sign&lized intersection” at the 28th National Meeting of the Operations Research Society of America, 4 November 1965, Houston, Texas. 57

58

C. J. ANCKER, JR. and A. V. GAFARL~N

for the subsequent inter-renewal times isf(f--‘F). However, the process may also be viewed as an ordinary renewal process, i.e. one in which all the distributions are the same: Let x,, &, ***,X, denote n independent values of the random variable X; then P[Nan] or P[N2n]

= P

2 &+(n-1)7QT [ 2-l

5(Xt+7) means the largest integer less than X. It can be shown (see Cox, 1962, Chapter 3) that iff*(s) is the Laplace transform of the density of the random variable X+ T, and c is any real number greater than the real part of any singularity of If*(s)]/s, then

n] = L

P[zv>

2ri

= P

1 1

c+tco _

f c ia,

ew MT+ 415 Lf*Wl”ds,

= 0, Then,p,=P[N=n]=P[N>n]-P[Nan+l]or pn = &

“_“*exp MT+ T)] i{ I~*(s)]~- If*(s)]“+‘} ds, s c cw C+kO

1 P+l =

G

s

c_-(to

n=O,l,...,

exp [s(T+ T)] f y’*(~)]<~/‘>+l ds

(2) n>

Pn = 0,

By using equation (2), the expected number of renewals is

/T\T >

T

i

07

+l

+1 pN=

F,

and the variance of the number of renewals is -+l

3. A SPECIFIC

EXAMPLE

In order to get some idea of the effect of randomness on throughput, distribution will be assumed for the random variable, X, namely the Erlangian -xXp-lexp(-bx)

UP)

b,x>O,

a specific

p = 1,2 ,...

with mean p = p/b, variance a2 = p/b2, third central moment b = 2p/bs, and Laplace transform

Simple renewal model of throughput at an oversaturated signalized intersection

59

Hence f*(s), which appears in equations (l-4) and which is the transform of the random variable X+T, is given by f*(s) = exp(--ST) (&j’

(7)

This assumption for the random variable Xmakes it convenient to use the theory of residues to evaluate the various integrals. An inspection of equations (l-4) shows that the only integral involved is of the form C+iCO

fC-iCOexp [s(T+ T)] 5 V’*(S)]~ ds,

n = 0, 1, . . ..

For n = 0, the only pole is at s = 0 with residue = 1; thus

g(O)= 1

(9)

For n 2 1, there is a pole of order np at s = -b as well as the pole of order 1 at s = 0. After much calculation, to determine the residue at s = -b, we have

= I[pFz,b++l)]

where y[ *, -1 is the incomplete gamma function and I[ -, -1 is the incomplete gamma function ratio. Now, by substitution of (9) and (10) into (l), (2), (3) and (4) the final results are P[N)n]

= 1,

n=O

= +,br[;-(n-l)]),

II = 1,2, . . .. (;)+l

=

(11)

0,

(12) ,,=$pn,b+n++-l[p(n+l),b+-n)],

P
n=1,2,...,(;)

3 b+-c))]

(13)

(14)

C. J. ANCKER,JR. and A. V. GNU&W

6Q

(16) Figures 1 and 2 are plots of the frequency function of N for various reasonable values of the parameters. These figures were computed using Molina’s (1942) tables in equations (12), (13) and (14). Figure 1 illustrates the fact that for small values of T (the go-phase),

P”30

30

P” .m

20

Jo

Jo 0

0

0123456 ;I

(c",

FIG. 1. Frequency function of N, go-phase, T, increasing.

there is very little spread around the mean, while for larger values of T the probability of less (or more) than the mean number getting through is appreciable. Figure 2 illustrates in a quantitative way how increasing the variance of the headway time increases the variance of N. Thus we can see that if the go-phase is reasonably long and/or the headway time has a moderately large variance, there will be a significant number who are delayed longer than the average. The next question to be examined is how well a mean value model of our process will predict the mean value of N. This is illustrated in Figs. 3-5. The solid curves are plots of ,_&Nfrom equation (15). They were computed on a Philco 2ooO machine using a series

Simple renewal model .of throughput at an oversaturated signalized intersection

61

expansion of (15). The step function is a plot of PNI=(($$)+l=(

(17)

@)+l

where ((x)) is the largest integer Zessthan or equal to x. This would seem to be a reasonable mean value model to fit the assumptions given earlier. That is, the first vehicle enters the

O"4

5

6

7

8

9

10 11 12

t:,

(a;

FIG. 2. Frequency function of N, variance of headway time increasing.

intersection at time p after the go-phase begins, and all subsequent vehicles have headway equal to T+P. The upper and lower envelopes (dashed lines) are given by TIL 7

clN*=-+l l+k and

respectively.

7

(18)

C. J. ANCYER,JR. and A. V. GAFARIAN

62

4

3

CN’ 2

/ /’

z ,’

0

/

/

1

/

/ /’

7 /’ / // 2

3

4

5

TIT

FIG. 3. Average throughput for large average headway time.

It is evident from these plots that a very good approximation halfway between the envelopes or T

for most regions is a line

u

In those cases where &T, U/T and T/T are small, equation (17) is the best approximation (these situations are of little or no practical significance).

Simple renewal model of throughput at an oversaturated signalized intersection

63

4

3

0

1

2

3

4

5

T/T FIG. 4.

Average throughput for moderate average headway time.

In renewal theory there are asymptotic results (good for large T’) for the mean number of renewals (see Cox, 1962, p. 47) and for the variance of the number of renewals (see Cox, 1962, p. 58), which in our case are

and

2-S-T 1 - CT *-- 1 PNHCL+7+Z ( /.L+r ) 2

a N *(T+T) uN - (jL++

+ [k+S(&)?&]

(21) (221

64

C. J. ANCKER,JR. and A. V. GAQNUAN

“N

2

39

38 BN 37

36

/ 35 36.

39

38 VT W

FIG. 5. Average throughput for small average headway time.

Simple renewal model of throughput at an oversaturated signalized intersection

where b is the third central moment off(t). then these approximations reduce to

65

If UQ p + T (which implies cog< p + r for p 2 4),

T+T 1 pN2i--LL+7 2

and (24) Equation (23) is identical with pN4 (equation 20) which we deduced heuristically. As an example of the use of these asymptotic equations, let T = 90, 7 = 1, p = 1.5, u2 = O-36; these values seem reasonable based on an inspection of some of our preliminary experimental results. Then from equations (23) and (24),

4. CONCLUSION

This study has been concerned with the effect of headway variability on the flow through an oversaturated signalized intersection. It seems clear that insofar as average throughput characteristics are concerned an appropriate mean value model is quite adequate. For this purpose, equation (20) is the best one. However, the effect of throughput variability from cycle to cycle may be large enough that it should seriously affect control policy. A thorough study of throughput is indicated. It is important to determine the actual nature of the headway distribution. An earlier study has shown how the mean value differs as a function of vehicle position in the queue (Greenshields et al., 1947). As stated before, an experimental study is presently underway to determine headway distributions as well as to test whether, in fact, successive headways are independent. The results of these experiments will be reported in a subsequent paper. REFERENCES COX D. R. (1962). Renewal Theory. Methuen, London. GAZIS D. C. (1964). Optimum control of a system of oversaturated intersections. Ops Res. 12,815-831. GA~IS D. C. and Pox-m R. B. (1965). The over-saturated intersection. Proc. 2nd Int. Symp. Theory of

Traffic Flow, pp. 221-237.

The Organisation for Economic Co-operation and Development,

Paris.

GREENWFJLDSB. D., SHAPIROD. and ERICKSENE. L. (1947). TruJic Performance at Urban Street Znfer-

sections. Yale Bureau of Highway Traffic, New Haven. MOLINA E. C. (1942). Poisson’s Exponential Binomial Limit.

5

Van Nostrand, Princeton.