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.

Geometric product form queueing networks with concurrent batch movements

1998 ◽  
Vol 30 (4) ◽  
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.

Overall station balance and decomposability for non-Markovian queueing networks

1998 ◽  
Vol 30 (03) ◽  
pp. 870-887 ◽  
Author(s):  
D. Fakinos ◽  
A. Economou
Keyword(s):  
Queueing Networks ◽  
Sufficient Conditions ◽  
Queueing Network ◽  
Necessary And Sufficient Conditions ◽  
Jackson Type ◽  
Necessary And Sufficient ◽  
Markovian Queueing Networks

Introducing the concept of overall station balance which extends the notion of station balance to non-Markovian queueing networks, several necessary and sufficient conditions are given for overall station balance to hold. Next the concept of decomposability is introduced and it is related to overall station balance. A particular case corresponding to a Jackson-type queueing network is considered in some more detail.

Overall station balance and decomposability for non-Markovian queueing networks

1998 ◽  
Vol 30 (3) ◽  
pp. 870-887 ◽  
Author(s):  
D. Fakinos ◽  
A. Economou
Keyword(s):  
Queueing Networks ◽  
Sufficient Conditions ◽  
Queueing Network ◽  
Necessary And Sufficient Conditions ◽  
Jackson Type ◽  
Necessary And Sufficient ◽  
Markovian Queueing Networks

Introducing the concept of overall station balance which extends the notion of station balance to non-Markovian queueing networks, several necessary and sufficient conditions are given for overall station balance to hold. Next the concept of decomposability is introduced and it is related to overall station balance. A particular case corresponding to a Jackson-type queueing network is considered in some more detail.

scholarly journals STRUCTURE-REVERSIBILITY OF A TWO-DIMENSIONAL REFLECTING RANDOM WALK AND ITS APPLICATION TO QUEUEING NETWORK

2014 ◽  
Vol 29 (1) ◽  
pp. 1-25 ◽  
Author(s):  
Masahiro Kobayashi ◽  
Masakiyo Miyazawa ◽  
Hiroshi Shimizu
Keyword(s):  
Random Walk ◽  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Queueing Network ◽  
Necessary And Sufficient Conditions ◽  
Negative Integer ◽  
Product Form ◽  
Two Dimensional ◽  
Necessary And Sufficient

We consider a two-dimensional reflecting random walk on the non-negative integer quadrant. It is assumed that this reflecting random walk has skip-free transitions. We are concerned with its time-reversed process assuming that the stationary distribution exists. In general, the time-reversed process may not be a reflecting random walk. In this paper, we derive necessary and sufficient conditions for the time-reversed process also to be a reflecting random walk. These conditions are different from but closely related to the product form of the stationary distribution.

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.

Stability of Product Form G-Networks

1992 ◽  
Vol 6 (3) ◽  
pp. 271-276 ◽  
Author(s):  
Erol Gelenbe ◽  
Rolf Schassberger
Keyword(s):  
Fixed Point ◽  
Queueing Networks ◽  
Existence And Uniqueness ◽  
Queueing Systems ◽  
Sufficient Conditions ◽  
Product Form ◽  
Load Factor ◽  
Flow Equations ◽  
General Stability ◽  
Necessary And Sufficient

We prove necessary and sufficient conditions for the existence and uniqueness of the stationary solution of the queueing networks (G-networks) with negative and positive customers introduced in Gelenbe [3], which have been shown to have product form. First, the existence of the solution of the nonlinear customer flow equations is established using Brouwer's fixed-point theorem; this result is valid for stable and unstable systems, as well as for certain networks that may not have product form. Then, the result is used to establish general stability related to the usual “load factor less than 1” criterion of queueing systems for G-networks with product form.

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.

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

1991 ◽  
Vol 23 (01) ◽  
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.

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.

PRODUCT-FORM MARKOVIAN QUEUEING SYSTEMS WITH MULTIPLE RESOURCES

2019 ◽  
pp. 1-9 ◽  
Author(s):  
Valeriy Naumov ◽  
Konstantin Samouylov
Keyword(s):  
Queueing Systems ◽  
Sufficient Conditions ◽  
Jump Process ◽  
Product Form ◽  
Temporary Increase ◽  
Stationary Probability Distribution ◽  
Markov Jump ◽  
Multiple Resources ◽  
Resource Requirements ◽  
Necessary And Sufficient

In the paper, we study general Markovian models of loss systems with random resource requirements, in which customers at arrival occupy random quantities of various resources and release them at departure. Customers may request negative quantities of resources, but total amount of resources allocated to customers should be nonnegative and cannot exceed predefined maximum levels. Allocating a negative volume of a resource to a customer leads to a temporary increase in its volume in the system. We derive necessary and sufficient conditions for the product-form of the stationary probability distribution of the Markov jump process describing the system.

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