scholarly journals Necessary and sufficient ergodicity condition for open synchronized queueing networks

1989 ◽  
Vol 15 (4) ◽  
pp. 367-380 ◽  
Author(s):  
G. Florin ◽  
S. Natkin
Keyword(s):  
Queueing Networks ◽  
Ergodicity Condition ◽  
Necessary And Sufficient

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.

On stability of state-dependent queues and acyclic queueing networks

1989 ◽  
Vol 21 (3) ◽  
pp. 681-701 ◽  
Author(s):  
Nicholas Bambos ◽  
Jean Walrand
Keyword(s):  
Queueing Networks ◽  
Sufficient Conditions ◽  
Service Rate ◽  
Single Server ◽  
General Function ◽  
State Dependent ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Periodic Inputs ◽  
Networks Of Queues

We consider a single server first-come-first-served queue with a stationary and ergodic input. The service rate is a general function of the workload in the queue. We provide the necessary and sufficient conditions for the stability of the system and the asymptotic convergence of the workload process to a finite stationary process at large times. Then, we consider acyclic networks of queues in which the service rate of any queue is a function of the workloads of this and of all the preceding queues. The stability problem is again studied. The results are then extended to analogous systems with periodic inputs.

Local conditions for the stochastic comparison of particle systems

2004 ◽  
Vol 36 (4) ◽  
pp. 1252-1277 ◽  
Author(s):  
Rosario Delgado ◽  
F. Javier López ◽  
Gerardo Sanz
Keyword(s):  
Queueing Networks ◽  
Interacting Particle Systems ◽  
Sufficient Conditions ◽  
Explicit Construction ◽  
Particle Systems ◽  
Stochastic Comparison ◽  
Local Conditions ◽  
Jackson Networks ◽  
Finite Set ◽  
Necessary And Sufficient

We study the stochastic comparison of interacting particle systems where the state space of each particle is a finite set endowed with a partial order, and several particles may change their value at a time. For these processes we give local conditions, on the rates of change, that assure their comparability. We also analyze the case where one of the processes does not have any changes that involve several particles, and obtain necessary and sufficient conditions for their comparability. The proofs are based on the explicit construction of an order-preserving Markovian coupling. We show the applicability of our results by studying the stochastic comparison of infinite-station Jackson networks and batch-arrival, batch-service, and assemble-transfer queueing networks.

On stability of state-dependent queues and acyclic queueing networks

1989 ◽  
Vol 21 (03) ◽  
pp. 681-701 ◽  
Author(s):  
Nicholas Bambos ◽  
Jean Walrand
Keyword(s):  
Queueing Networks ◽  
Sufficient Conditions ◽  
Service Rate ◽  
Single Server ◽  
General Function ◽  
State Dependent ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Periodic Inputs ◽  
Networks Of Queues

We consider a single server first-come-first-served queue with a stationary and ergodic input. The service rate is a general function of the workload in the queue. We provide the necessary and sufficient conditions for the stability of the system and the asymptotic convergence of the workload process to a finite stationary process at large times. Then, we consider acyclic networks of queues in which the service rate of any queue is a function of the workloads of this and of all the preceding queues. The stability problem is again studied. The results are then extended to analogous systems with periodic inputs.

On closed support T-Invariants and the traffic equations

1998 ◽  
Vol 35 (2) ◽  
pp. 473-481 ◽  
Author(s):  
Richard J. Boucherie ◽  
Matteo Sereno
Keyword(s):  
Petri Nets ◽  
Queueing Networks ◽  
Stochastic Petri Nets ◽  
Product Form ◽  
Approximate Analysis ◽  
Existence Of A Solution ◽  
Exact Analysis ◽  
Necessary And Sufficient ◽  
Product Form Queueing Networks

The traffic equations are the basis for the exact analysis of product form queueing networks, and the approximate analysis of non-product form queueing networks. Conditions characterising the structure of the network that guarantees the existence of a solution for the traffic equations are therefore of great importance. This note shows that the new condition stating that each transition is covered by a minimal closed support T-invariant, is necessary and sufficient for the existence of a solution for the traffic equations for batch routing queueing networks and stochastic Petri nets.

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.

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.

A correspondence between product-form batch-movement queueing networks and single-movement networks

1997 ◽  
Vol 34 (01) ◽  
pp. 160-175
Author(s):  
J. L. Coleman ◽  
W. Henderson ◽  
C. E. M. Pearce ◽  
P. G. Taylor
Keyword(s):  
State Space ◽  
Queueing Networks ◽  
Equilibrium Distribution ◽  
The State ◽  
Product Form ◽  
Transition Rates ◽  
Equilibrium Distributions ◽  
Necessary And Sufficient

A number of recent papers have exhibited classes of queueing networks, with batches of customers served and routed through the network, which have generalised product-form equilibrium distributions. In this paper we look at these from a new viewpoint. In particular we show that, under standard assumptions, for a network to possess an equilibrium distribution that factorises into a product form over the nodes of the network for all possible transition rates, it is necessary and sufficient that it be equivalent to a suitably-defined single-movement network. We consider also the form of the state space for such networks.

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