An inequality between two speed averages

Tranrpn. Rcs -8. Vol. 128. No Pnnwd in Great Bntam.

6. P,J Wi-175.

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11191.?6lSi&i PS.OlI c .W ,D 19Ui3 Petgamon Press plc

1988

NOTES

AND

RESEARCH

COMMUNICATIONS

AN INEQUALITY BETWEEN SPEED AVERAGES Department

of Mathematics.

TWO

S. K. STEIN University of California. Davis, CA 95616, U.S.A.

(Received 26 October 1987)

Abstract-In

this note we invoke Schwartz’s

inequality

to relate

two speed averages.

Consider n vehicles travelling around a circular track at various fixed speeds, uI, u?, . . . , u,. The average speed of these vehicles is

c

i=I u, n

(1)

This is the average that is found, say, by equipping each vehicle with a transmitter by which the driver communicates his/her speed. In practice, however, a device is installed on the track that records the speed of each vehicle as it passes. A fast vehicle is therefore recorded more often than a slow one. The average speed determined this way is

since the frequency at which a given speed is observed is proportional to that speed. (This average is proportional to total vehicle miles travelled during the period of observation.) As Haight and Mosher (1962) suggest, the second average is larger, since a higher speed has more chances to be recorded than does a lower speed. The intuition is correct and this inequality between the two speed averages is a consequence of Schwarz’s inequality from calculus, as we will show. We wish to prove that

or, equivalently,

NOW, the Schwarz inequality asserts that if pi and qi, 1 5 i I n, are numbers, then

(4) 469

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470

Notes

and Research

Communications

Moreover, equality holds in (4) only when the 4;‘s are proportional to the p,‘s. Letting p, = 1 and q, = u, in (4), we obtain (3). [Equality holds in (3) only when all the u,s are equal.] A similar inequality holds when we consider a single vehicle moving at a varying speed and compare its average speed with respect to time with its average speed with respect to distance. (The first is estimated by observing its speed, say, every second and averaging the results; the second, by observing its speed, say. every kilometer and averaging these results.) Intuition again suggests that the first average is smaller. The continu,ous version of Schwarz’s inequality shows that this is the case, as follows. Let the speed at time f be u(t) and let s(t) be the distance travelled up to time t. During the time interval from time a to timbe b the average of velocity with respect to time is defined as u(t) dt = r(b)

- s(a)

(5)

b-a

b-a

On the other hand, the average of speed with respect to distance is defined as

r

u(s) ds

s(b)

- s(a)’

s(b)

Jr(o)

(6)

where u(s) denotes the velocity when the distance covered iss. Changing the independent variable in the numerator of (6) from s to t by the relation ds = u(t)dt, we obtain r(b)

I

u(s) ds

s(b)

- s(a)

b

I

0

s(a)

Noting that s(b)

- s(a)

=

= jk u(t)dt

u(f) u(f) dt

s(b)

- s(a)

’

and b - a = Js 1 dt, we wish to show that

i

.” (u(t))? dr

5

u(f) dt

or, equivalently,

(ru(f)df)‘
(8)

Now, Schwarz’s inequality for integrals asserts that iff(r ) andg(t) are real functions, then

(9)

Choosingf(t) = 1 and g(t) = u(t) shows that (8) is a consequence of (9). Schwa& inequality shows up elsewhere in traffic theory. For instance, assume that n+ 1 buses on some municipal route are dispatched at time intervals between them a,,

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Communications

471

a?, . . . , a,. The average time interval between buses, as viewed by the dispatcher,

is

However, persons arriving at random at a bus stop experience a different average since they are more likely to arrive during a long interval between buses than during a short interval. Their average waiting time is half the average time interval between buses, as viewed by the prospective riders. The latter average is

pi

(11) i: a, 1=I

That (10) is not larger than (11) is, again, a consequence of Schwarz’s inequality. Similar inequalities between averages occur in other areas. For instance, the average class size at a university, as viewed by the registrar, is never larger than the average class size as viewed by the students. This, too, is an immediate consequence of Schwarz’s inequality. (Only if all the classes have the same size are the two averages equal.) For an elementary proof of Schwarz’s inequality see Stein (1987, p. 421); the index lists further applications. REFERENCES Haight E A. and Mosher. Jr. W. W. (1962) A practical method for improving the accuracy of vehicular distribution measurements. Highway Res. Board Bull. 341, 92-116. Stein S. K. (1987) Culculw and Analyfic Geomerry, 4th ed. McGraw-Hill, New York.

speed

A COMMENT ON THE NUMBER OF STOPS AND HEADWAY FOR A FIXED-ROUTE TRANSIT SYSTEM JUI-HSIEN LING and MICHAEL A. P. TAYLOR Department of Civil Engineering. Monash University. Clayton. 3168 (Received 5 November Abstract-In

this note we (1985) paper; (ii) offer two (iii) present the comparison of the revised models lead frequencies (headway) and

Australia

1987; in revised form 25 April 198S)

will (i) comment on three inappropriate assumptions in Kikuchi’s revised models and rework some findings of the original paper; and of the results between the revised original models. Our examinations to some new conclusions. particularly concerning optimum service number of stops.

The purpose of this note is to offer a revised version of the model described by Kikuchi (1985), in terms of a more realistic assumption about the vehicle travel time and the relationship between transit standing time and travel demand. We first revise Kikuchi’s model and rework the procedures in the original paper, and then compare the results between the revised original models. The model in the Kikuchi’s paper is incorrect in three respects. First, the author wrongly assumed that the same passenger boarding or alighting rate would apply for