scholarly journals Renewal theory in two dimensions: asymptotic results

1974 ◽  
Vol 6 (3) ◽  
pp. 546-562 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Joint Distribution ◽  
Counting Process ◽  
Renewal Theory ◽  
Two Dimensions ◽  
Renewal Processes ◽  
Unified Theory ◽  
Two Dimensional ◽  
Renewal Function ◽  
Asymptotic Results ◽  
Renewal Counting Process

In an earlier paper (Renewal theory in two dimensions: Basic results) the author developed a unified theory for the study of bivariate renewal processes. In contrast to this aforementioned work where explicit expressions were obtained, we develop some asymptotic results concerning the joint distribution of the bivariate renewal counting process (Nx(1), Ny(2)), the distribution of the two-dimensional renewal counting process Nx,y and the two-dimensional renewal function &Nx,y. A by-product of the investigation is the study of the distribution and moments of the minimum of two correlated normal random variables. A comprehensive bibliography on multi-dimensional renewal theory is also appended.

scholarly journals Renewal theory in two dimensions: asymptotic results

1974 ◽  
Vol 6 (03) ◽  
pp. 546-562 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Joint Distribution ◽  
Counting Process ◽  
Renewal Theory ◽  
Two Dimensions ◽  
Renewal Processes ◽  
Unified Theory ◽  
Two Dimensional ◽  
Renewal Function ◽  
Asymptotic Results ◽  
Renewal Counting Process

In an earlier paper (Renewal theory in two dimensions: Basic results) the author developed a unified theory for the study of bivariate renewal processes. In contrast to this aforementioned work where explicit expressions were obtained, we develop some asymptotic results concerning the joint distribution of the bivariate renewal counting process (N x(1), N y(2)), the distribution of the two-dimensional renewal counting process N x,y and the two-dimensional renewal function &N x,y. A by-product of the investigation is the study of the distribution and moments of the minimum of two correlated normal random variables. A comprehensive bibliography on multi-dimensional renewal theory is also appended.

scholarly journals Renewal theory in two dimensions: Basic results

1974 ◽  
Vol 6 (02) ◽  
pp. 376-391 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Renewal Theory ◽  
Two Dimensions ◽  
Standard Theory ◽  
Renewal Processes ◽  
Counting Processes ◽  
Unified Theory ◽  
Two Dimensional ◽  
Renewal Function ◽  
Bivariate Exponential Distribution ◽  
Bivariate Exponential

In this paper a unified theory for studying renewal processes in two dimensions is developed. Bivariate generating functions and bivariate Laplace transforms are the basic tools used in generalizing the standard theory of univariate renewal processes. An example involving a bivariate exponential distribution is presented. This is used to illustrate the general theory and explicit expressions for the two-dimensional renewal density, the two-dimensional renewal function, the correlation between the marginal univariate renewal counting processes, and other related quantities are derived.

scholarly journals Renewal theory in two dimensions: Basic results

1974 ◽  
Vol 6 (2) ◽  
pp. 376-391 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Renewal Theory ◽  
Two Dimensions ◽  
Standard Theory ◽  
Renewal Processes ◽  
Counting Processes ◽  
Unified Theory ◽  
Two Dimensional ◽  
Renewal Function ◽  
Bivariate Exponential Distribution ◽  
Bivariate Exponential

In this paper a unified theory for studying renewal processes in two dimensions is developed. Bivariate generating functions and bivariate Laplace transforms are the basic tools used in generalizing the standard theory of univariate renewal processes. An example involving a bivariate exponential distribution is presented. This is used to illustrate the general theory and explicit expressions for the two-dimensional renewal density, the two-dimensional renewal function, the correlation between the marginal univariate renewal counting processes, and other related quantities are derived.

Renewal theory in two dimensions: bounds on the renewal function

1977 ◽  
Vol 9 (03) ◽  
pp. 527-541 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Renewal Theory ◽  
Random Variable ◽  
Two Dimensions ◽  
Upper And Lower Bounds ◽  
Renewal Processes ◽  
Renewal Function ◽  
Poisson Random Variable ◽  
Joint Distributions ◽  
The Family ◽  
Fréchet Bounds

In two earlier papers [6], [7] the properties of bivariate renewal processes and their associated two-dimensional renewal functions, H(x, y) were examined. By utilising the Fréchet bounds for joint distributions and the properties of univariate renewal processes, a collection of upper and lower bounds for H(x, y) are constructed. The evaluation of these bounds is carried out for the case of the family of bivariate Poisson processes. An interesting by-product of this investigation leads to a new inequality for the median of a Poisson random variable.

Renewal theory in two dimensions: bounds on the renewal function

1977 ◽  
Vol 9 (3) ◽  
pp. 527-541 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Renewal Theory ◽  
Random Variable ◽  
Two Dimensions ◽  
Upper And Lower Bounds ◽  
Renewal Processes ◽  
Renewal Function ◽  
Poisson Random Variable ◽  
Joint Distributions ◽  
The Family ◽  
Fréchet Bounds

In two earlier papers [6], [7] the properties of bivariate renewal processes and their associated two-dimensional renewal functions, H(x, y) were examined. By utilising the Fréchet bounds for joint distributions and the properties of univariate renewal processes, a collection of upper and lower bounds for H(x, y) are constructed. The evaluation of these bounds is carried out for the case of the family of bivariate Poisson processes. An interesting by-product of this investigation leads to a new inequality for the median of a Poisson random variable.

Invariance principles in queueing theory

1989 ◽  
Vol 26 (04) ◽  
pp. 845-857
Author(s):  
Michael Alex ◽  
Josef Steinebach
Keyword(s):  
Convergence Rate ◽  
Stochastic Processes ◽  
Queueing Theory ◽  
Counting Process ◽  
Heavy Traffic ◽  
Renewal Processes ◽  
Light Traffic ◽  
Invariance Principles ◽  
Renewal Counting Process ◽  
Compound Renewal Processes

Several stochastic processes in queueing theory are based upon compound renewal processes . For queues in light traffic, however, the summands {Xk }and the renewal counting process {N(t)} are typically dependent on each other. Making use of recent invariance principles for such situations, we present some weak and strong approximations for the GI/G/1 queues in light and heavy traffic. Some applications are discussed including convergence rate statements or Darling–Erdös-type extreme value theorems for the processes under consideration.

Invariance principles in queueing theory

1989 ◽  
Vol 26 (4) ◽  
pp. 845-857 ◽  
Author(s):  
Michael Alex ◽  
Josef Steinebach
Keyword(s):  
Convergence Rate ◽  
Stochastic Processes ◽  
Queueing Theory ◽  
Counting Process ◽  
Heavy Traffic ◽  
Renewal Processes ◽  
Light Traffic ◽  
Invariance Principles ◽  
Renewal Counting Process ◽  
Compound Renewal Processes

Several stochastic processes in queueing theory are based upon compound renewal processes . For queues in light traffic, however, the summands {Xk}and the renewal counting process {N(t)} are typically dependent on each other. Making use of recent invariance principles for such situations, we present some weak and strong approximations for the GI/G/1 queues in light and heavy traffic. Some applications are discussed including convergence rate statements or Darling–Erdös-type extreme value theorems for the processes under consideration.

Classes of life distributions and renewal counting process

1994 ◽  
Vol 31 (4) ◽  
pp. 1110-1115 ◽  
Author(s):  
Yi-Hau Chen
Keyword(s):  
Exponential Distribution ◽  
Counting Process ◽  
Renewal Function ◽  
Life Distribution ◽  
Renewal Counting Process ◽  
Life Distributions

We prove that if the renewal function M(t) corresponding to a life distribution F is convex (concave) then F is NBU (NWU), and hence answer two questions posed by Shaked and Zhu (1992). Moreover, based-on the renewal function, some characterizations of the exponential distribution within certain classes of life distributions are given.

Classes of life distributions and renewal counting process

1994 ◽  
Vol 31 (04) ◽  
pp. 1110-1115 ◽  
Author(s):  
Yi-Hau Chen
Keyword(s):  
Exponential Distribution ◽  
Counting Process ◽  
Renewal Function ◽  
Life Distribution ◽  
Renewal Counting Process ◽  
Life Distributions

We prove that if the renewal function M(t) corresponding to a life distribution F is convex (concave) then F is NBU (NWU), and hence answer two questions posed by Shaked and Zhu (1992). Moreover, based-on the renewal function, some characterizations of the exponential distribution within certain classes of life distributions are given.

Urbanicity, City Delegations, and Two-Dimensional Liberalism

2018 ◽  
Author(s):  
Thomas K. Ogorzalek
Keyword(s):  
National Level ◽  
Two Dimensions ◽  
Political Order ◽  
Two Dimensional ◽  
Local Institutions ◽  
Horizontal Integration ◽  
National Politics ◽  
United Front ◽  
Strategies For Success

This theoretical chapter develops the argument that the conditions of cities—large, densely populated, heterogeneous communities—generate distinctive governance demands supporting (1) market interventions and (2) group pluralism. Together, these positions constitute the two dimensions of progressive liberalism. Because of the nature of federalism, such policies are often best pursued at higher levels of government, which means that cities must present a united front in support of city-friendly politics. Such unity is far from assured on the national level, however, because of deep divisions between and within cities that undermine cohesive representation. Strategies for success are enhanced by local institutions of horizontal integration developed to address the governance demands of urbanicity, the effects of which are felt both locally and nationally in the development of cohesive city delegations and a unified urban political order capable of contending with other interests and geographical constituencies in national politics.

Sign in / Sign up

Export Citation Format

Share Document