The Shot Noise Representation for Noise for a Free Traffic StreamAuthor(s): George H. WeissSource: Advances in Applied Probability, Vol. 8, No. 2 (Jun., 1976), pp. 224-225Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1425872 .

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5TH CONFERENCE ON STOCHASTIC PROCESSES AND APPLICATIONS

The shot noise representation for noise for a free traffic stream

GEORGE H. WEISS, National Institutes of Health, Bethesda, Maryland

The theory of acoustic noise from a stream of traffic is of great interest at the

present time because of various efforts to decrease the effects of this noise. A deterministic theory of this noise was first given by Johnson and Saunders [1] and the first stochastic development by myself in 1970 [2]. The theory envisions a stream of traffic modelled as acoustic point sources which move along an infinite one-dimensional highway, the distance between successive cars being indepen- dent random variables. Let D be the distance of the measuring instrument from the highway and let x be a distance along the highway measured from the point of closest approach. Then the noise can be represented as

I(D)= Q 2+D2

where Q is the unit sound intensity (these are assumed constant) and dN(x) is the number of cars in (x, x + dx). Thus, the acoustic noise can be regarded as a

generalized shot noise process. When the distances between successive cars on the road have a negative exponential distribution all of the cumulants can be found in closed form, as can the characteristic function. This assumption is

appropriate for light traffic [3]-[5]. The detailed results indicate that the deterministic model of Johnson and Saunders greatly underestimates the variance of the noise. The asymptotic form of the probability distribution for small values of I(D) can be derived by Tauberian techniques. Surprisingly, data taken by Kurze [6] indicate that the distribution so developed gives an excellent fit to some data taken for purposes of comparison.

Marcus [7] has developed the theory to take into account variations in vehicle

types and different kinds of vehicles. He concludes that such variations consider-

ably enhance the variance of noise level. More recently we have investigated the effect of a non-exponential distribution of headways as would be appropriate for flows of more than 600 cars/hr/lane. This model only allows the calculation of low-order moments [8]. The variance is reduced under that predicted by the

original shot noise theory. A calculation of the spectral density for cars moving at uniform speed indicates that a qualitative difference between results for a

negative exponential headway distribution and a gamma distribution with

positive index. Marcus [9] has developed an analogous theory for atmospheric pollution

induced by a stream of vehicles, and Shaw and Olson [10], have described environmental noise pollution in Ottawa by a similar model. We are currently investigating the statistics of the noise in a fixed time interval.

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University of Maryland, 9-13 June 1975 University of Maryland, 9-13 June 1975

References

[1] JOHNSON, D. R. AND SAUNDERS, E. G. (1968) The evaluation of noise from freely flowing traffic. J. Sound Vib. 7, 287-309.

[2] WEISS, G. H. (1970) On the noise generated by a stream of vehicles. Transp. Res. 4, 229-233.

[3] WEISS, G. H. AND HERMAN, R. (1962) Statistical properties of low density traffic. Quart. Appl. Math. 20, 121-130.

[4] BREIMAN, L. (1963) The Poisson tendency in traffic distributions. Ann. Math. Statist. 34, 308-311.

[5] THED?EN, T. (1964) A note on the Poisson tendency in traffic. Ann. Math. Statist. 35, 1823-1824.

[6] KURZE, U. J. (1974) Frequency curves of road traffic noise. J. Sound Vib. 33, 171-185.

[7] MARCUS, A. H. (1973) Traffic noise as a filtered Markov renewal process. J. Appl. Prob. 10, 32-39.

[8] BLUMENFELD, D. E. AND WEISS G. H. (1975) Effects of headway distributions on second order properties of traffic noise. J. Sound Vib. 41, 93-102.

[9] MARCUS, A. H. (1973) A stochastic model of micro-scale air pollution from highway traffic. Technometrics 15, 353-363.

[10] SHAW, E. A. G. AND OLSON, N. (1972) Theory of steady state urban noise for an ideal

homogeneous city. J. Acoust. Soc. Amer. 51, 1781-1793.

II. CONTRIBUTED PAPERS

II A. Measures

Factorization identities for homogeneous random processes PRISCILLA GREENWOOD, University of British Columbia

A number of factorization identities for maxima and minima of homogeneous random processes are collected and related. Transform equations are inter-

preted as decompositions of time-changed processes. Discrete- and continuous- time versions are related. Factorizations in terms of generators, in terms of Levy measures and process decompositions are equivalent. The time-change view-

point raises questions for fluctuation theory.

Levy random measures

ALAN F. KARR, The Johns Hopkins University

A Levy random measure possesses a conditional independence structure reminiscent of the Markov property. Let M be a random measure on (E, E), let

References

[1] JOHNSON, D. R. AND SAUNDERS, E. G. (1968) The evaluation of noise from freely flowing traffic. J. Sound Vib. 7, 287-309.

[2] WEISS, G. H. (1970) On the noise generated by a stream of vehicles. Transp. Res. 4, 229-233.

[3] WEISS, G. H. AND HERMAN, R. (1962) Statistical properties of low density traffic. Quart. Appl. Math. 20, 121-130.

[4] BREIMAN, L. (1963) The Poisson tendency in traffic distributions. Ann. Math. Statist. 34, 308-311.

[5] THED?EN, T. (1964) A note on the Poisson tendency in traffic. Ann. Math. Statist. 35, 1823-1824.

[6] KURZE, U. J. (1974) Frequency curves of road traffic noise. J. Sound Vib. 33, 171-185.

[7] MARCUS, A. H. (1973) Traffic noise as a filtered Markov renewal process. J. Appl. Prob. 10, 32-39.

[8] BLUMENFELD, D. E. AND WEISS G. H. (1975) Effects of headway distributions on second order properties of traffic noise. J. Sound Vib. 41, 93-102.

[9] MARCUS, A. H. (1973) A stochastic model of micro-scale air pollution from highway traffic. Technometrics 15, 353-363.

[10] SHAW, E. A. G. AND OLSON, N. (1972) Theory of steady state urban noise for an ideal

homogeneous city. J. Acoust. Soc. Amer. 51, 1781-1793.

II. CONTRIBUTED PAPERS

II A. Measures

Factorization identities for homogeneous random processes PRISCILLA GREENWOOD, University of British Columbia

A number of factorization identities for maxima and minima of homogeneous random processes are collected and related. Transform equations are inter-

preted as decompositions of time-changed processes. Discrete- and continuous- time versions are related. Factorizations in terms of generators, in terms of Levy measures and process decompositions are equivalent. The time-change view-

point raises questions for fluctuation theory.

Levy random measures

ALAN F. KARR, The Johns Hopkins University

A Levy random measure possesses a conditional independence structure reminiscent of the Markov property. Let M be a random measure on (E, E), let

225 225

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