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.

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.

Batch Routing Queueing Networks with Jump-Over Blocking

1996 ◽  
Vol 10 (2) ◽  
pp. 287-297 ◽  
Author(s):  
Richard J. Boucherie
Keyword(s):  
Queueing Networks ◽  
Equilibrium Distribution ◽  
Queueing Network ◽  
Capacity Constraints ◽  
Product Form

This paper shows that the equilibrium distribution of a queueing network with batch routing continues to be of product form if a batch that cannot enter its destination — for example, as a consequence of capacity constraints — immediately triggers a new transition that takes up the whole batch.

scholarly journals PRODUCT-FORM IN G-NETWORKS

2016 ◽  
Vol 30 (3) ◽  
pp. 345-360 ◽  
Author(s):  
Andrea Marin
Keyword(s):  
Stochastic Modeling ◽  
Queueing Networks ◽  
Equilibrium Distribution ◽  
Queueing Network ◽  
Point Of View ◽  
Product Form ◽  
Theoretical Point ◽  
Non Linear ◽  
Modular Analysis ◽  
Product Forms

The introduction of the class of queueing networks called G-networks by Gelenbe has been a breakthrough in the field of stochastic modeling since it has largely expanded the class of models which are analytically or numerically tractable. From a theoretical point of view, the introduction of the G-networks has lead to very important considerations: first, a product-form queueing network may have non-linear traffic equations; secondly, we can have a product-form equilibrium distribution even if the customer routing is defined in such a way that more than two queues can change their states at the same time epoch. In this work, we review some of the classes of product-forms introduced for the analysis of the G-networks with special attention to these two aspects. We propose a methodology that, coherently with the product-form result, allows for a modular analysis of the G-queues to derive the equilibrium distribution of the network.

scholarly journals Utility Optimization in Congested Queueing Networks

2011 ◽  
Vol 48 (1) ◽  
pp. 68-89 ◽  
Author(s):  
N. S. Walton
Keyword(s):  
Queueing Networks ◽  
Packet Switching ◽  
Queueing Network ◽  
Capacity Constraints ◽  
Switching Network ◽  
Single Server ◽  
Service Capacity ◽  
Asymptotic Regime ◽  
Utility Optimization ◽  
Aggregate Utility

We consider a multiclass single-server queueing network as a model of a packet switching network. The rates packets are sent into this network are controlled by queues which act as congestion windows. By considering a sequence of congestion controls, we analyse a sequence of stationary queueing networks. In this asymptotic regime, the service capacity of the network remains constant and the sequence of congestion controllers act to exploit the network's capacity by increasing the number of packets within the network. We show that the stationary throughput of routes on this sequence of networks converges to an allocation that maximises aggregate utility subject to the network's capacity constraints. To perform this analysis, we require that our utility functions satisfy an exponential concavity condition. This family of utilities includes weighted α-fair utilities for α > 1.

scholarly journals Product forms for queueing networks with state-dependent multiple job transitions

1991 ◽  
Vol 23 (1) ◽  
pp. 152-187 ◽  
Author(s):  
Richard J. Boucherie ◽  
Nico M. Van DIJK
Keyword(s):  
Continuous Time ◽  
Queueing Networks ◽  
Sufficient Conditions ◽  
Decomposition Theorem ◽  
Product Form ◽  
Job Transitions ◽  
State Dependent ◽  
Local Solutions ◽  
Necessary And Sufficient ◽  
Product Forms

A general framework of continuous-time queueing networks is studied with simultaneous state dependent service completions such as due to concurrent servicing or discrete-time slotting and with state dependent batch routings such as most typically modelling blocking. By using a key notion of group-local-balance, necessary and sufficient conditions are given for the stationary distribution to be of product form. These conditions and a constructive computation of the product form are based upon merely local solutions of the group-local-balance equations which can usually be solved explicitly for concrete networks. Moreover, a decomposition theorem is presented to separate service and routing conditions. General batch service and batch routing examples yielding a product form are hereby concluded. As illustrated by various examples, known results on both discrete- and continuous-time queueing networks are unified and extended.

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.

scholarly journals Utility Optimization in Congested Queueing Networks

2011 ◽  
Vol 48 (01) ◽  
pp. 68-89
Author(s):  
N. S. Walton
Keyword(s):  
Queueing Networks ◽  
Packet Switching ◽  
Queueing Network ◽  
Capacity Constraints ◽  
Switching Network ◽  
Single Server ◽  
Service Capacity ◽  
Asymptotic Regime ◽  
Utility Optimization ◽  
Aggregate Utility

We consider a multiclass single-server queueing network as a model of a packet switching network. The rates packets are sent into this network are controlled by queues which act as congestion windows. By considering a sequence of congestion controls, we analyse a sequence of stationary queueing networks. In this asymptotic regime, the service capacity of the network remains constant and the sequence of congestion controllers act to exploit the network's capacity by increasing the number of packets within the network. We show that the stationary throughput of routes on this sequence of networks converges to an allocation that maximises aggregate utility subject to the network's capacity constraints. To perform this analysis, we require that our utility functions satisfy an exponential concavity condition. This family of utilities includes weighted α-fair utilities for α > 1.

A Geometric Product-Form Distribution for a Queueing Network by Non-Standard Batch Arrivals and Batch Transfers

1997 ◽  
Vol 29 (02) ◽  
pp. 523-544 ◽  
Author(s):  
Masakiyo Miyazawa ◽  
Peter G. Taylor
Keyword(s):  
Stationary Distribution ◽  
Queueing Networks ◽  
Queueing Network ◽  
Product Form ◽  
Arrival Process ◽  
Service Discipline ◽  
Batch Service ◽  
Batch Arrivals ◽  
Geometric Product ◽  
Number Of Customers

We introduce a batch service discipline, called assemble-transfer batch service, for continuous-time open queueing networks with batch movements. Under this service discipline a requested number of customers is simultaneously served at a node, and transferred to another node as, possibly, a batch of different size, if there are sufficient customers there; the node is emptied otherwise. We assume a Markovian setting for the arrival process, service times and routing, where batch sizes are generally distributed. Under the assumption that extra batches arrive while nodes are empty, and under a stability condition, it is shown that the stationary distribution of the queue length has a geometric product form over the nodes if and only if certain conditions are satisfied for the extra arrivals. This gives a new class of queueing networks which have tractable stationary distributions, and simultaneously shows that the product form provides a stochastic upper bound for the stationary distribution of the corresponding queueing network without the extra arrivals.

Geometric product form queueing networks with concurrent batch movements

1998 ◽  
Vol 30 (04) ◽  
pp. 1111-1129 ◽  
Author(s):  
Hideaki Yamashita ◽  
Masakiyo Miyazawa
Keyword(s):  
Queueing Networks ◽  
Sufficient Conditions ◽  
Queueing Network ◽  
Product Form ◽  
Regularity Conditions ◽  
Geometric Product ◽  
Batch Sizes ◽  
Network States ◽  
The One ◽  
Necessary And Sufficient

Queueing networks have been rather restricted in order to have product form distributions for network states. Recently, several new models have appeared and enlarged this class of product form networks. In this paper, we consider another new type of queueing network with concurrent batch movements in terms of such product form results. A joint distribution of the requested batch sizes for departures and the batch sizes of the corresponding arrivals may be arbitrary. Under a certain modification of the network and mild regularity conditions, we give necessary and sufficient conditions for the network state to have the product form distribution, which is shown to provide an upper bound for the one in the original network. It is shown that two special settings satisfy these conditions. Algorithms to calculate their stationary distributions are considered, with numerical examples.

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