scholarly journals Jackson networks with unlimited supply of work

2005 ◽  
Vol 42 (03) ◽  
pp. 879-882 ◽  
Author(s):  
Gideon Weiss
Keyword(s):  
Joint Distribution ◽  
Product Form ◽  
Traffic Intensity ◽  
Stationary Distributions ◽  
Marginal Distributions ◽  
Flow Rates ◽  
Jackson Network ◽  
Jackson Networks ◽  
Unlimited Supply ◽  
Infinite Supply

We consider a Jackson network in which some of the nodes have an infinite supply of work: when all the customers queued at such a node have departed, the node will process a customer from this supply. Such nodes will be processing jobs all the time, so they will be fully utilized and experience a traffic intensity of 1. We calculate flow rates for such networks, obtain conditions for stability, and investigate the stationary distributions. Standard nodes in this network continue to have product-form distributions, while nodes with an infinite supply of work have geometric marginal distributions and Poisson inflows and outflows, but their joint distribution is not of product form.

scholarly journals Jackson networks with unlimited supply of work

2005 ◽  
Vol 42 (3) ◽  
pp. 879-882 ◽  
Author(s):  
Gideon Weiss
Keyword(s):  
Joint Distribution ◽  
Product Form ◽  
Traffic Intensity ◽  
Stationary Distributions ◽  
Marginal Distributions ◽  
Flow Rates ◽  
Jackson Network ◽  
Jackson Networks ◽  
Unlimited Supply ◽  
Infinite Supply

We consider a Jackson network in which some of the nodes have an infinite supply of work: when all the customers queued at such a node have departed, the node will process a customer from this supply. Such nodes will be processing jobs all the time, so they will be fully utilized and experience a traffic intensity of 1. We calculate flow rates for such networks, obtain conditions for stability, and investigate the stationary distributions. Standard nodes in this network continue to have product-form distributions, while nodes with an infinite supply of work have geometric marginal distributions and Poisson inflows and outflows, but their joint distribution is not of product form.

On ergodicity and recurrence properties of a Markov chain by an application to an open jackson network

1992 ◽  
Vol 24 (02) ◽  
pp. 343-376 ◽  
Author(s):  
Arie Hordijk ◽  
Flora Spieksma
Keyword(s):  
Markov Chain ◽  
Product Form ◽  
Geometric Ergodicity ◽  
Two Dimensional ◽  
Marginal Distributions ◽  
Jackson Network ◽  
Strong Ergodicity ◽  
Key Theorem ◽  
Stieltjes Transforms

This paper gives an overview of recurrence and ergodicity properties of a Markov chain. Two new notions for ergodicity and recurrence are introduced. They are calledμ-geometric ergodicity andμ-geometric recurrence respectively. The first condition generalises geometric as well as strong ergodicity. Our key theorem shows thatμ-geometric ergodicity is equivalent to weakμ-geometric recurrence. The latter condition is verified for the time-discretised two-centre open Jackson network. Hence, the corresponding two-dimensional Markov chain isμ-geometrically and geometrically ergodic, but not strongly ergodic. A consequence ofμ-geometric ergodicity withμof product-form is the convergence of the Laplace-Stieltjes transforms of the marginal distributions. Consequently all moments converge.

scholarly journals Nonergodic Jackson networks with infinite supply–local stabilization and local equilibrium analysis

2016 ◽  
Vol 53 (4) ◽  
pp. 1125-1142 ◽  
Author(s):  
Jennifer Sommer ◽  
Hans Daduna ◽  
Bernd Heidergott
Keyword(s):  
Performance Measures ◽  
Queue Length ◽  
Local Equilibrium ◽  
Length Distribution ◽  
Equilibrium Analysis ◽  
Product Form ◽  
Jackson Networks ◽  
Closed Form Solutions ◽  
Queue Length Distribution ◽  
Infinite Supply

Abstract Classical Jackson networks are a well-established tool for the analysis of complex systems. In this paper we analyze Jackson networks with the additional features that (i) nodes may have an infinite supply of low priority work and (ii) nodes may be unstable in the sense that the queue length at these nodes grows beyond any bound. We provide the limiting distribution of the queue length distribution at stable nodes, which turns out to be of product form. A key step in establishing this result is the development of a new algorithm based on adjusted traffic equations for detecting unstable nodes. Our results complement the results known in the literature for the subcases of Jackson networks with either infinite supply nodes or unstable nodes by providing an analysis of the significantly more challenging case of networks with both types of nonstandard node present. Building on our product-form results, we provide closed-form solutions for common customer and system oriented performance measures.

Sojourn times in closed queueing networks

1983 ◽  
Vol 15 (03) ◽  
pp. 638-656 ◽  
Author(s):  
F. P. Kelly ◽  
P. K. Pollett
Keyword(s):  
Free Path ◽  
Queueing Networks ◽  
Joint Distribution ◽  
The State ◽  
Product Form ◽  
Closed Queueing Networks ◽  
Jackson Network ◽  
Simple Representation ◽  
Sojourn Times

This paper obtains the stationary joint distribution of a customer's sojourn times along an overtake-free path in a closed multiclass Jackson network. The distribution has a simple representation in terms of the product form distribution for the state of the network at an arrival instant.

On ergodicity and recurrence properties of a Markov chain by an application to an open jackson network

1992 ◽  
Vol 24 (2) ◽  
pp. 343-376 ◽  
Author(s):  
Arie Hordijk ◽  
Flora Spieksma
Keyword(s):  
Markov Chain ◽  
Product Form ◽  
Geometric Ergodicity ◽  
Two Dimensional ◽  
Marginal Distributions ◽  
Jackson Network ◽  
Strong Ergodicity ◽  
Key Theorem ◽  
Stieltjes Transforms

This paper gives an overview of recurrence and ergodicity properties of a Markov chain. Two new notions for ergodicity and recurrence are introduced. They are called μ -geometric ergodicity and μ -geometric recurrence respectively. The first condition generalises geometric as well as strong ergodicity. Our key theorem shows that μ -geometric ergodicity is equivalent to weak μ -geometric recurrence. The latter condition is verified for the time-discretised two-centre open Jackson network. Hence, the corresponding two-dimensional Markov chain is μ -geometrically and geometrically ergodic, but not strongly ergodic. A consequence of μ -geometric ergodicity with μ of product-form is the convergence of the Laplace-Stieltjes transforms of the marginal distributions. Consequently all moments converge.

Sojourn times in closed queueing networks

1983 ◽  
Vol 15 (3) ◽  
pp. 638-656 ◽  
Author(s):  
F. P. Kelly ◽  
P. K. Pollett
Keyword(s):  
Free Path ◽  
Queueing Networks ◽  
Joint Distribution ◽  
The State ◽  
Product Form ◽  
Closed Queueing Networks ◽  
Jackson Network ◽  
Simple Representation ◽  
Sojourn Times

This paper obtains the stationary joint distribution of a customer's sojourn times along an overtake-free path in a closed multiclass Jackson network. The distribution has a simple representation in terms of the product form distribution for the state of the network at an arrival instant.

Dependent Random Variables with Independent Subsets - II

1990 ◽  
Vol 33 (1) ◽  
pp. 24-28 ◽  
Author(s):  
Y. H. Wang
Keyword(s):  
Joint Distribution ◽  
Random Variables ◽  
Marginal Distributions ◽  
Dependent Random Variables ◽  
One Dimensional ◽  
The One

AbstractIn this paper, we consolidate into one two separate problems - dependent random variables with independent subsets and construction of a joint distribution with given marginals. Let N = {1,2,3,...} and X = {Xn; n ∊ N} be a sequence of random variables with nondegenerate one-dimensional marginal distributions {Fn; n ∊ N}. An example is constructed to show that there exists a sequence of random variables Y = {Yn; n ∊ N} such that the components of a subset of Y are independent if and only if its size is ≦ k, where k ≧ 2 is a prefixed integer. Furthermore, the one-dimensional marginal distributions of Y are those of X.

Distributions of the Longest Excursions in a Tied Down Simple Random Walk and in a Brownian Bridge

2007 ◽  
Vol 44 (04) ◽  
pp. 1056-1067 ◽  
Author(s):  
Andreas Lindell ◽  
Lars Holst
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Weak Convergence ◽  
Joint Distribution ◽  
Marginal Distribution ◽  
Brownian Bridge ◽  
Simple Random Walk ◽  
Marginal Distributions

Expressions for the joint distribution of the longest and second longest excursions as well as the marginal distributions of the three longest excursions in the Brownian bridge are obtained. The method, which primarily makes use of the weak convergence of the random walk to the Brownian motion, principally gives the possibility to obtain any desired joint or marginal distribution. Numerical illustrations of the results are also given.

A heterogeneous blocking system in a random environment

1996 ◽  
Vol 33 (01) ◽  
pp. 211-216 ◽  
Author(s):  
G. Falin
Keyword(s):  
Random Environment ◽  
Joint Distribution ◽  
Service System ◽  
External Environment ◽  
The State ◽  
Product Form ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

We obtain a necessary and sufficient condition for the interaction between a service system and an external environment under which the stationary joint distribution of the set of busy channels and the state of the external environment is given by a product-form formula.

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