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.

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.

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.

Conditional job-observer property for multitype closed queueing networks

2002 ◽  
Vol 39 (4) ◽  
pp. 865-881 ◽  
Author(s):  
Hans Daduna ◽  
Ryszard Szekli
Keyword(s):  
Cycle Time ◽  
Queueing Networks ◽  
Closed Queueing Networks ◽  
Basic Formula ◽  
Sojourn Times ◽  
Number Of Customers ◽  
Insight Into

For functionals of multitype closed queueing networks, a conditional job-observer property is shown which provides more insight into the classical job-observer property. Applications and examples are given, including the classical job-observer property for the number of customers in a network, a representation of cycle time distributions and a basic formula for sojourn times.

Stationary state probabilities at arrival instants for closed queueing networks with multiple types of customers

1980 ◽  
Vol 17 (04) ◽  
pp. 1048-1061 ◽  
Author(s):  
S. S. Lavenberg ◽  
M. Reiser
Keyword(s):  
Stationary State ◽  
Queueing Networks ◽  
Product Form ◽  
Service Center ◽  
Closed Queueing Networks ◽  
Closed Networks

We consider closed networks of interconnected service centers with multiple types of customers and multiple classes, whose stationary state probabilities at arbitrary times have a product form. A customer can change its class but not its type as it traverses the network. We show that the stationary state probabilities at instants at which customers of a particular type arrive at a particular service center and enter a particular class are equal to the stationary state probabilities at arbitrary times for the network with one less customer of that type. Applications of this result are given.

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.

On First-Come First-Served Versus Random Service Discipline in Multiclass Closed Queueing Networks

1997 ◽  
Vol 11 (3) ◽  
pp. 313-326 ◽  
Author(s):  
Ronald Buitenhek ◽  
Geert-Jan van Houtum ◽  
Jan-Kees van Ommeren
Keyword(s):  
Steady State ◽  
Queueing Networks ◽  
Form Solution ◽  
Product Form ◽  
Service Discipline ◽  
Single Server ◽  
Service Time Distribution ◽  
Closed Queueing Networks ◽  
Steady State Probabilities ◽  
Closed Network

We consider multiclass closed queueing networks. For these networks, a lot of work has been devoted to characterizing and weakening the conditions under which a product-form solution is obtained for the steady-state distribution. From this work, it is known that, under certain conditions, all networks in which each of the stations has either the first-come first-served or the random service discipline lead to the same (product-form expressions for the) steady-state probabilities of the (aggregated) states that for each station and each job class denote the number of jobs in service and the number of jobs in the queue. As a consequence, all these situations also lead to the same throughputs for the different job classes. One of the conditions under which these equivalence results hold states that at each station all job classes must have the same exponential service time distribution. In this paper, it is shown that these equivalence results can be extended to the case with different exponential service times for jobs of different classes, if the network consists of only one single-server or multiserver station. This extension can be made despite of the fact that the network is not a product-form network anymore in that case. The proof is based on the reversibility of the Markov process that is obtained under the random service discipline. By means of a counterexample, it is shown that the extension cannot be made for closed network with two or more stations.

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