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).

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).

PROCESSOR SHARING G-QUEUES WITH INERT CUSTOMERS AND CATASTROPHES: A MODEL FOR SERVER AGING AND REJUVENATION

2017 ◽  
Vol 31 (4) ◽  
pp. 420-435 ◽  
Author(s):  
J.-M. Fourneau ◽  
Y. Ait El Majhoub
Keyword(s):  
Steady State ◽  
Product Form ◽  
Processor Sharing ◽  
Service Capacity ◽  
Steady State Distribution ◽  
State Distribution ◽  
Signal Arrival ◽  
Open Networks ◽  
Networks Of Queues

We consider open networks of queues with Processor-Sharing discipline and signals. The signals deletes all the customers present in the queues and vanish instantaneously. The customers may be usual customers or inert customers. Inert customers do not receive service but the servers still try to share the service capacity between all the customers (inert or usual). Thus a part of the service capacity is wasted. We prove that such a model has a product-form steady-state distribution when the signal arrival rates are positive.

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.

The solution of certain two-dimensional Markov models

1982 ◽  
Vol 14 (02) ◽  
pp. 295-308 ◽  
Author(s):  
G. Fayolle ◽  
P. J. B. King ◽  
I. Mitrani
Keyword(s):  
Steady State ◽  
Solution Method ◽  
Markov Models ◽  
Transition Rates ◽  
Two Dimensional ◽  
Steady State Distribution ◽  
State Distribution ◽  
Birth And Death Processes ◽  
State Dependent ◽  
Modelling Problems

A class of two-dimensional birth-and-death processes, with applications in many modelling problems, is defined and analysed in the steady state. These are processes whose instantaneous transition rates are state-dependent in a restricted way. Generating functions for the steady-state distribution are obtained by solving a functional equation in two variables. That solution method lends itself readily to numerical implementation. Some aspects of the numerical solution are discussed, using a particular model as an example.

The solution of certain two-dimensional Markov models

1982 ◽  
Vol 14 (2) ◽  
pp. 295-308 ◽  
Author(s):  
G. Fayolle ◽  
P. J. B. King ◽  
I. Mitrani
Keyword(s):  
Steady State ◽  
Solution Method ◽  
Markov Models ◽  
Transition Rates ◽  
Two Dimensional ◽  
Steady State Distribution ◽  
State Distribution ◽  
Birth And Death Processes ◽  
State Dependent ◽  
Modelling Problems

A class of two-dimensional birth-and-death processes, with applications in many modelling problems, is defined and analysed in the steady state. These are processes whose instantaneous transition rates are state-dependent in a restricted way. Generating functions for the steady-state distribution are obtained by solving a functional equation in two variables. That solution method lends itself readily to numerical implementation. Some aspects of the numerical solution are discussed, using a particular model as an example.

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.

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.

Insensitive average residence times in generalized semi-Markov processes

1981 ◽  
Vol 13 (04) ◽  
pp. 720-735 ◽  
Author(s):  
A. D. Barbour ◽  
R. Schassberger
Keyword(s):  
Steady State ◽  
Markov Processes ◽  
Broad Class ◽  
Residence Times ◽  
Steady State Distribution ◽  
State Distribution ◽  
Lifetime Distributions ◽  
Definition Of ◽  
Semi Markov Processes ◽  
Networks Of Queues

For a broad class of stochastic processes, the generalized semi-Markov processes, conditions are known which imply that the steady state distribution of the process, when it exists, depends only on the means, and not the exact shapes, of certain lifetime distributions entering the definition of the process. It is shown in the present paper that this insensitivity extends to certain average and conditional average residence times. Particularly interesting applications can be found in the field of networks of queues.

scholarly journals A TWO-ECHELON SPARE PARTS NETWORK WITH LATERAL AND EMERGENCY SHIPMENTS: A PRODUCT-FORM APPROXIMATION

2017 ◽  
Vol 32 (4) ◽  
pp. 536-555 ◽  
Author(s):  
Richard J. Boucherie ◽  
Geert-Jan van Houtum ◽  
Judith Timmer ◽  
Jan-Kees van Ommeren
Keyword(s):  
Steady State ◽  
Spare Parts ◽  
Inventory System ◽  
Product Form ◽  
Steady State Distribution ◽  
State Distribution ◽  
Poisson Demand ◽  
Lateral Transshipment ◽  
Repair Facility ◽  
Erlang Loss

We consider a single-item, two-echelon spare parts inventory model for repairable parts for capital goods with high downtime costs. The inventory system consists of multiple local warehouses, a central warehouse, and a central repair facility. When a part at a customer fails, if possible his request for a ready-for-use part is fulfilled by his local warehouse. Also, the failed part is sent to the central repair facility for repair. If the local warehouse is out of stock, then, via an emergency shipment, a ready-for-use part is sent from the central warehouse if it has a part in stock. Otherwise, it is sent via a lateral transshipment from another local warehouse, or via an emergency shipment from the external supplier. We assume Poisson demand processes, generally distributed leadtimes for replenishments, repairs, and emergency shipments, and a basestock policy for the inventory control.Our inventory system is too complex to solve for a steady-state distribution in closed form. We approximate it by a network of Erlang loss queues with hierarchical jump-over blocking. We show that this network has a product-form steady-state distribution. This enables an efficient heuristic for the optimization of basestock levels, resulting in good approximations of the optimal costs.

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