Exact Product Form Solution for Queueing Networks with Blocking

1987 ◽  
Vol C-36 (1) ◽  
pp. 122-125 ◽  
Author(s):  
I. F. Akyildiz
Keyword(s):  
Queueing Networks ◽  
Form Solution ◽  
Product Form ◽  
Exact Product

scholarly journals Product forms based on backward traffic equations

1995 ◽  
Vol 32 (02) ◽  
pp. 508-518
Author(s):  
Richard J. Boucherie
Keyword(s):  
Queueing Networks ◽  
Queueing Network ◽  
New Product ◽  
Capacity Constraints ◽  
Product Form ◽  
New Form ◽  
Product Forms ◽  
Exact Product

This paper introduces a new form of local balance and the corresponding product-form results. It is shown that these new product-form results allow capacity constraints at the stations of a queueing network without reversibility assumptions and without special blocking protocols. In particular, exact product-form results for heavily loaded queueing networks are obtained.

A PRODUCT FORM SOLUTION FOR TREE NETWORKS WITH DIVISIBLE LOADS

2011 ◽  
Vol 21 (01) ◽  
pp. 13-20 ◽  
Author(s):  
THOMAS G. ROBERTAZZI
Keyword(s):  
Parallel Processing ◽  
Equilibrium State ◽  
Queueing Networks ◽  
Form Solution ◽  
Product Form ◽  
Minimal Amount ◽  
Divisible Loads ◽  
Divisible Load ◽  
Multi Level ◽  
Open Question

A product form solution for the optimal fractions of divisible load to distribute to processors in a multi-level tree network is described. Here optimality involves parallel processing the load in a minimal amount of time. This tractable solution is similar to the product form solution for equilibrium state probabilities arising in Markovian queueing networks. The existence of this product form solution answers a long standing open question for divisible load scheduling.

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.

scholarly journals Product forms based on backward traffic equations

1995 ◽  
Vol 32 (2) ◽  
pp. 508-518 ◽  
Author(s):  
Richard J. Boucherie
Keyword(s):  
Queueing Networks ◽  
Queueing Network ◽  
New Product ◽  
Capacity Constraints ◽  
Product Form ◽  
New Form ◽  
Product Forms ◽  
Exact Product

This paper introduces a new form of local balance and the corresponding product-form results. It is shown that these new product-form results allow capacity constraints at the stations of a queueing network without reversibility assumptions and without special blocking protocols. In particular, exact product-form results for heavily loaded queueing networks are obtained.

scholarly journals Product-form queueing networks with negative and positive customers

1991 ◽  
Vol 28 (3) ◽  
pp. 656-663 ◽  
Author(s):  
Erol Gelenbe
Keyword(s):  
Queueing Networks ◽  
Queue Length ◽  
Queueing Network ◽  
Form Solution ◽  
Product Form ◽  
Usual Sense ◽  
Negative Customers ◽  
Flow Equations ◽  
New Class ◽  
Service Times

We introduce a new class of queueing networks in which customers are either ‘negative' or ‘positive'. A negative customer arriving to a queue reduces the total customer count in that queue by 1 if the queue length is positive; it has no effect at all if the queue length is empty. Negative customers do not receive service. Customers leaving a queue for another one can either become negative or remain positive. Positive customers behave as ordinary queueing network customers and receive service. We show that this model with exponential service times, Poisson external arrivals, with the usual independence assumptions for service times, and Markovian customer movements between queues, has product form. It is quasi-reversible in the usual sense, but not in a broader sense which includes all destructions of customers in the set of departures. The existence and uniqueness of the solutions to the (nonlinear) customer flow equations, and hence of the product form solution, is discussed.

Partial balances in batch arrival batch service and assemble-transfer queueing networks

1997 ◽  
Vol 34 (3) ◽  
pp. 745-752 ◽  
Author(s):  
Xiuli Chao
Keyword(s):  
Queueing Networks ◽  
Form Solution ◽  
Product Form ◽  
Batch Arrival ◽  
Arrival Process ◽  
Batch Service ◽  
Balance Equations ◽  
Steady State Probability ◽  
Simple Product ◽  
Necessary And Sufficient

Recently Miyazawa and Taylor (1997) proposed a new class of queueing networks with batch arrival batch service and assemble-transfer features. In such networks customers arrive and are served in batches, and may change size when a batch transfers from one node to another. With the assumption of an additional arrival process at each node when it is empty, they obtain a simple product-form steady-state probability distribution, which is a (stochastic) upper bound for the original network. This paper shows that this class of network possesses a set of non-standard partial balance equations, and it is demonstrated that the condition of the additional arrival process introduced by Miyazawa and Taylor is there precisely to satisfy the partial balance equations, i.e. it is necessary and sufficient not only for having a product form solution, but also for the partial balance equations to hold.

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