On Poisson Type Stationary Mass Distributions in RnAuthor(s): J. H. B. KempermanSource: Advances in Applied Probability, Vol. 5, No. 1 (Apr., 1973), p. 16Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1425950 .

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J. H. B. KEMPERMAN J. H. B. KEMPERMAN

On Poisson type stationary mass distributions in R"

J. H. B. KEMPERMAN, University of Rochester

We will present a number of results concerning the distribution of the coverages of one or more straight line segments in R" by a random mass distribution in Rn. It is shown that the central limit theorem plays an important role in several ways when one superimposes many such random mass distri- butions.

Explicit formulae can often be given for the following mass distribution. Let {Pi} be a denumerable collection of points in R" which is random and has a Poisson distribution with constant density. Let further {A(W, * )} be a given ensemble (W random) of n-dimensional density functions and consider the random mass distribution with density d(P) = ]i A(Wi,P-Pi). Here, the Wi denote observations on W which are independent of each other and the Pi.

Other results concern the random set S = Ui (B(W) + Pi). Here, the B(Wi) are chosen from a given ensemble {B(W)} of subsets of Rn. Let L4 denote a given line segment in R" of length t > 0 and let C(t) denote the linear measure of S n Lt. A simple and useful explicit formula is given for the Laplace transform of E(e-"C(t)), which is valid provided the sets B(W) are all convex. It turns out that in this case the system of intervals S n Lt behaves as an alternating renewal model.

An algebraic approach to some waiting time problems

JACQUELINE LORIS-TEGHEM, Universitd Libre de Bruxelles

The algebraic approach to the waiting time process in GI/G/1, due essen- tially to Wendel (1958) and synthesized by Kingman (1966), is extended to a generalized GI/G/1 queueing system and to the queue GI/M/S.

1. A generalized GI/G/1 queueing system

The model considered is a generalized GI/G/1 queueing system with (N + 2) types of triplets (additional waiting time, service time, probability of joining the queue). It may be used in particular to describe a queueing system with rather general closing-down and setting-up processes for the service station.

On Poisson type stationary mass distributions in R"

J. H. B. KEMPERMAN, University of Rochester

We will present a number of results concerning the distribution of the coverages of one or more straight line segments in R" by a random mass distribution in Rn. It is shown that the central limit theorem plays an important role in several ways when one superimposes many such random mass distri- butions.

Explicit formulae can often be given for the following mass distribution. Let {Pi} be a denumerable collection of points in R" which is random and has a Poisson distribution with constant density. Let further {A(W, * )} be a given ensemble (W random) of n-dimensional density functions and consider the random mass distribution with density d(P) = ]i A(Wi,P-Pi). Here, the Wi denote observations on W which are independent of each other and the Pi.

Other results concern the random set S = Ui (B(W) + Pi). Here, the B(Wi) are chosen from a given ensemble {B(W)} of subsets of Rn. Let L4 denote a given line segment in R" of length t > 0 and let C(t) denote the linear measure of S n Lt. A simple and useful explicit formula is given for the Laplace transform of E(e-"C(t)), which is valid provided the sets B(W) are all convex. It turns out that in this case the system of intervals S n Lt behaves as an alternating renewal model.

An algebraic approach to some waiting time problems

JACQUELINE LORIS-TEGHEM, Universitd Libre de Bruxelles

The algebraic approach to the waiting time process in GI/G/1, due essen- tially to Wendel (1958) and synthesized by Kingman (1966), is extended to a generalized GI/G/1 queueing system and to the queue GI/M/S.

1. A generalized GI/G/1 queueing system

The model considered is a generalized GI/G/1 queueing system with (N + 2) types of triplets (additional waiting time, service time, probability of joining the queue). It may be used in particular to describe a queueing system with rather general closing-down and setting-up processes for the service station.

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