A stochastic lower bound for assemble-transfer batch service queueing networks

2000 ◽  
Vol 37 (3) ◽  
pp. 881-889 ◽  
Author(s):  
Antonis Economou
Keyword(s):  
Lower Bound ◽  
Stationary Distribution ◽  
Upper Bound ◽  
Queueing Networks ◽  
Product Form ◽  
Arrival Process ◽  
Batch Service ◽  
Original Network ◽  
Departure Process ◽  
Geometric Product

Miyazawa and Taylor (1997) introduced a class of assemble-transfer batch service queueing networks which do not have tractable stationary distribution. However by assuming a certain additional arrival process at each node when it is empty, they obtain a geometric product-form stationary distribution which is a stochastic upper bound for the stationary distribution of the original network. In this paper we develop a stochastic lower bound for the original network by introducing an additional departure process at each node which tends to remove all the customers present in it. This model in combination with the aforementioned upper bound model gives a better sense for the properties of the original network.

A stochastic lower bound for assemble-transfer batch service queueing networks

2000 ◽  
Vol 37 (03) ◽  
pp. 881-889 ◽  
Author(s):  
Antonis Economou
Keyword(s):  
Lower Bound ◽  
Stationary Distribution ◽  
Upper Bound ◽  
Queueing Networks ◽  
Product Form ◽  
Arrival Process ◽  
Batch Service ◽  
Original Network ◽  
Departure Process ◽  
Geometric Product

Miyazawa and Taylor (1997) introduced a class of assemble-transfer batch service queueing networks which do not have tractable stationary distribution. However by assuming a certain additional arrival process at each node when it is empty, they obtain a geometric product-form stationary distribution which is a stochastic upper bound for the stationary distribution of the original network. In this paper we develop a stochastic lower bound for the original network by introducing an additional departure process at each node which tends to remove all the customers present in it. This model in combination with the aforementioned upper bound model gives a better sense for the properties of the original network.

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.

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

1997 ◽  
Vol 29 (2) ◽  
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.

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.

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

1997 ◽  
Vol 34 (03) ◽  
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.

ON THE TIME-DEPENDENT BEHAVIOR OF A MARKOVIAN REENTRANT-LINE MODEL

2018 ◽  
Vol 33 (3) ◽  
pp. 367-386
Author(s):  
Brian Fralix
Keyword(s):  
Stationary Distribution ◽  
Time Dependent ◽  
Product Form ◽  
Line Model ◽  
Transition Functions ◽  
Geometric Product ◽  
Dependent Behavior ◽  
Product Technique ◽  
Form Representation ◽  
Similar Product

We use the random-product technique from [5] to study both the steady-state and time-dependent behavior of a Markovian reentrant-line model, which is a generalization of the preemptive reentrant-line model studied in the work of Adan and Weiss [2]. Our results/observations yield additional insight into why the stationary distribution of the reentrant-line model from [2] exhibits an almost-geometric product-form structure: indeed, our generalized reentrant-line model, when stable, admits a stationary distribution with a similar product-form representation as well. Not only that, the Laplace transforms of the transition functions of our reentrant-line model also have a product-form structure if it is further assumed that both Buffers 2 and 3 are empty at time zero.

scholarly journals On the quasi-stationary distribution for queueing networks with defective routing

1997 ◽  
Vol 38 (4) ◽  
pp. 454-463 ◽  
Author(s):  
Richard J. Boucherie
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Discrete Time ◽  
Queueing Networks ◽  
Product Form ◽  
Stationary Distributions ◽  
Absorbing States ◽  
Quasi Stationary Distribution

AbstractThis note introduces quasi-local-balance for discrete-time Markov chains with absorbing states. From quasi-local-balance product-form quasi-stationary distributions are derived by analogy with product-form stationary distributions for Markov chains that satisfy local balance.

Non-product form of two-dimensional fluid networks with dependent Lévy inputs

2000 ◽  
Vol 37 (04) ◽  
pp. 1117-1122 ◽  
Author(s):  
Offer Kella
Keyword(s):  
Stationary Distribution ◽  
Queueing Networks ◽  
Product Form ◽  
Two Dimensional ◽  
Fluid Network ◽  
Fluid Networks ◽  
Stochastic Fluid

We show that the stationary distribution of a two-dimensional stochastic fluid network with (possibly dependent) Lévy inputs does not have product form other than in truly obvious cases. This is in contrast to queueing networks, where product form exists for non-obvious situations in which the inputs are independent, and for Brownian networks, where it typically exists for cases where the driving processes are actually dependent.

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.

scholarly journals On Two Interacting Markovian Queueing Systems

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 799
Author(s):  
Valeriy A. Naumov ◽  
Yuliya V. Gaidamaka ◽  
Konstantin E. Samouylov
Keyword(s):  
Stationary Distribution ◽  
Queueing Systems ◽  
Sufficient Conditions ◽  
Queuing System ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Product Form ◽  
Arrival Process ◽  
Arbitrary Structure ◽  
Necessary And Sufficient

In this paper, we study a Markovian queuing system consisting of two subsystems of an arbitrary structure. Each subsystem generates a multi-class Markovian arrival process of customers arriving to the other subsystem. We derive the necessary and sufficient conditions for the stationary distribution to be of product form and consider some particular cases of the subsystem interaction for which these conditions can be easily verified.

Sign in / Sign up

Export Citation Format

Share Document