Product form in networks of queues with batch arrivals and batch services

1990 ◽  
Vol 6 (1) ◽  
pp. 71-87 ◽  
Author(s):  
W. Henderson ◽  
P. G. Taylor
Keyword(s):  
Product Form ◽  
Batch Arrivals ◽  
Batch Services ◽  
Networks Of Queues

Some new results on queueing networks with batch movement

1991 ◽  
Vol 28 (02) ◽  
pp. 409-421 ◽  
Author(s):  
W. Henderson ◽  
P. G. Taylor
Keyword(s):  
Continuous Time ◽  
Queueing Networks ◽  
Equilibrium Distribution ◽  
Product Form ◽  
Batch Arrivals ◽  
Before And After ◽  
Equilibrium Distributions ◽  
Batch Services ◽  
Independence Results ◽  
Networks Of Queues

Product-form equilibrium distributions in networks of queues in which customers move singly have been known since 1957, when Jackson derived some surprising independence results. A product-form equilibrium distribution has also recently been shown to be valid for certain queueing networks with batch arrivals, batch services and even correlated routing. This paper derives the joint equilibrium distribution of states immediately before and after a batch of customers is released into the network. The results are valid for either discrete- or continuous-time queueing networks: previously obtained results can be seen as marginal distributions within a more general framework. A generalisation of the classical ‘arrival theorem' for continuous-time networks is given, which compares the equilibrium distribution as seen by arrivals to the time-averaged equilibrium distribution.

Some new results on queueing networks with batch movement

1991 ◽  
Vol 28 (2) ◽  
pp. 409-421 ◽  
Author(s):  
W. Henderson ◽  
P. G. Taylor
Keyword(s):  
Continuous Time ◽  
Queueing Networks ◽  
Equilibrium Distribution ◽  
Product Form ◽  
Batch Arrivals ◽  
Before And After ◽  
Equilibrium Distributions ◽  
Batch Services ◽  
Independence Results ◽  
Networks Of Queues

Product-form equilibrium distributions in networks of queues in which customers move singly have been known since 1957, when Jackson derived some surprising independence results. A product-form equilibrium distribution has also recently been shown to be valid for certain queueing networks with batch arrivals, batch services and even correlated routing.This paper derives the joint equilibrium distribution of states immediately before and after a batch of customers is released into the network. The results are valid for either discrete- or continuous-time queueing networks: previously obtained results can be seen as marginal distributions within a more general framework. A generalisation of the classical ‘arrival theorem' for continuous-time networks is given, which compares the equilibrium distribution as seen by arrivals to the time-averaged equilibrium distribution.

Queueing networks with instantaneous movements: a unified approach by quasi-reversibility

2000 ◽  
Vol 32 (01) ◽  
pp. 284-313 ◽  
Author(s):  
Xiuli Chao ◽  
Masakiyo Miyazawa
Keyword(s):  
Queueing Networks ◽  
Product Form ◽  
Unified Approach ◽  
Stationary Distributions ◽  
Batch Arrivals ◽  
Unified View ◽  
Batch Services ◽  
Product Form Queueing Networks

In this paper we extend the notion of quasi-reversibility and apply it to the study of queueing networks with instantaneous movements and signals. The signals treated here are considerably more general than those in the existing literature. The approach not only provides a unified view for queueing networks with tractable stationary distributions, it also enables us to find several new classes of product form queueing networks, including networks with positive and negative signals that instantly add or remove customers from a sequence of nodes, networks with batch arrivals, batch services and assembly-transfer features, and models with concurrent batch additions and batch deletions along a fixed or a random route of the network.

Networks of queues with batch services and customer coalescence

1996 ◽  
Vol 33 (03) ◽  
pp. 858-869 ◽  
Author(s):  
Xiuli Chao ◽  
Michael Pinedo ◽  
Dequan Shaw
Keyword(s):  
Single Unit ◽  
Production Systems ◽  
Queueing Network ◽  
Form Solution ◽  
Product Form ◽  
Batch Size ◽  
Jackson Network ◽  
Batch Sizes ◽  
Batch Services ◽  
Networks Of Queues

Consider a queueing network with batch services at each node. The service time of a batch is exponential and the batch size at each node is arbitrarily distributed. At a service completion the entire batch coalesces into a single unit, and it either leaves the system or goes to another node according to given routing probabilities. When the batch sizes are identical to one, the network reduces to a classical Jackson network. Our main result is that this network possesses a product form solution with a special type of traffic equations which depend on the batch size distribution at each node. The product form solution satisfies a particular type of partial balance equation. The result is further generalized to the non-ergodic case. For this case the bottleneck nodes and the maximal subnetwork that achieves steady state are determined. The existence of a unique solution is shown and stability conditions are established. Our results can be used, for example, in the analysis of production systems with assembly and subassembly processes.

scholarly journals Spatial birth-death processes with multiple changes and applications to batch service networks and clustering processes

1990 ◽  
Vol 22 (2) ◽  
pp. 433-455 ◽  
Author(s):  
Richard J. Boucherie ◽  
Nico M. Van Dijk
Keyword(s):  
Form Solution ◽  
Product Form ◽  
Symmetry Condition ◽  
Batch Service ◽  
Transition Rates ◽  
Service Networks ◽  
Batch Services ◽  
Concentration Equation ◽  
Networks Of Queues ◽  
Birth Death

Reversible spatial birth-death processes are studied with simultaneous jumps of multi-components. A relationship is established between (i) a product-form solution, (ii) a partial symmetry condition on the jump rates and (iii) a solution of a deterministic concentration equation. Applications studied are reversible networks of queues with batch services and blocking and clustering processes such as those found in polymerization chemistry. As illustrated by examples, known results are hereby unified and extended. An expectation interpretation of the transition rates is included.

A Result on Networks of Queues with Customer Coalescence and State-Dependent Signaling

1998 ◽  
Vol 35 (01) ◽  
pp. 151-164
Author(s):  
Xiuli Chao ◽  
Shaohui Zheng
Keyword(s):  
Steady State ◽  
Single Unit ◽  
Form Solution ◽  
Product Form ◽  
Transition Rates ◽  
Steady State Distribution ◽  
State Distribution ◽  
State Dependent ◽  
Batch Services ◽  
Networks Of Queues

In this paper we consider a network of queues with batch services, customer coalescence and state-dependent signaling. That is, customers are served in batches at each node, and coalesce into a single unit upon service completion. There are signals circulating in the network and, when a signal arrives at a node, a batch of customers is either deleted or triggered to move as a single unit within the network. The transition rates for both customers and signals are quite general and can depend on the state of the whole system. We show that this network possesses a product form solution. The existence of a steady state distribution is also discussed. This result generalizes some recent results of Hendersonet al. (1994), as well as those of Chaoet al. (1996).

A Result on Networks of Queues with Customer Coalescence and State-Dependent Signaling

1998 ◽  
Vol 35 (1) ◽  
pp. 151-164 ◽  
Author(s):  
Xiuli Chao ◽  
Shaohui Zheng
Keyword(s):  
Steady State ◽  
Single Unit ◽  
Form Solution ◽  
Product Form ◽  
Transition Rates ◽  
Steady State Distribution ◽  
State Distribution ◽  
State Dependent ◽  
Batch Services ◽  
Networks Of Queues

In this paper we consider a network of queues with batch services, customer coalescence and state-dependent signaling. That is, customers are served in batches at each node, and coalesce into a single unit upon service completion. There are signals circulating in the network and, when a signal arrives at a node, a batch of customers is either deleted or triggered to move as a single unit within the network. The transition rates for both customers and signals are quite general and can depend on the state of the whole system. We show that this network possesses a product form solution. The existence of a steady state distribution is also discussed. This result generalizes some recent results of Henderson et al. (1994), as well as those of Chao et al. (1996).

Networks of queues with batch services and customer coalescence

1996 ◽  
Vol 33 (3) ◽  
pp. 858-869 ◽  
Author(s):  
Xiuli Chao ◽  
Michael Pinedo ◽  
Dequan Shaw
Keyword(s):  
Single Unit ◽  
Production Systems ◽  
Queueing Network ◽  
Form Solution ◽  
Product Form ◽  
Batch Size ◽  
Jackson Network ◽  
Batch Sizes ◽  
Batch Services ◽  
Networks Of Queues

Consider a queueing network with batch services at each node. The service time of a batch is exponential and the batch size at each node is arbitrarily distributed. At a service completion the entire batch coalesces into a single unit, and it either leaves the system or goes to another node according to given routing probabilities. When the batch sizes are identical to one, the network reduces to a classical Jackson network. Our main result is that this network possesses a product form solution with a special type of traffic equations which depend on the batch size distribution at each node. The product form solution satisfies a particular type of partial balance equation. The result is further generalized to the non-ergodic case. For this case the bottleneck nodes and the maximal subnetwork that achieves steady state are determined. The existence of a unique solution is shown and stability conditions are established. Our results can be used, for example, in the analysis of production systems with assembly and subassembly processes.

scholarly journals Spatial birth-death processes with multiple changes and applications to batch service networks and clustering processes

1990 ◽  
Vol 22 (02) ◽  
pp. 433-455 ◽  
Author(s):  
Richard J. Boucherie ◽  
Nico M. Van Dijk
Keyword(s):  
Form Solution ◽  
Product Form ◽  
Symmetry Condition ◽  
Batch Service ◽  
Transition Rates ◽  
Service Networks ◽  
Batch Services ◽  
Concentration Equation ◽  
Networks Of Queues ◽  
Birth Death

Reversible spatial birth-death processes are studied with simultaneous jumps of multi-components. A relationship is established between (i) a product-form solution, (ii) a partial symmetry condition on the jump rates and (iii) a solution of a deterministic concentration equation. Applications studied are reversible networks of queues with batch services and blocking and clustering processes such as those found in polymerization chemistry. As illustrated by examples, known results are hereby unified and extended. An expectation interpretation of the transition rates is included.

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