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.

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).

The steady-state distribution of spent service times present in the M/G/1 foreground–background processor-sharing queue

1988 ◽  
Vol 25 (1) ◽  
pp. 194-203 ◽  
Author(s):  
R. Schassberger
Keyword(s):  
Steady State ◽  
Random Measure ◽  
Processor Sharing ◽  
Steady State Distribution ◽  
State Distribution ◽  
Generating Functional ◽  
Service Times

The steady-state distribution of spent service times present in the M/G/1 foreground-background processor-sharing queue is described by a random measure, whose generating functional is obtained.

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.

scholarly journals G-NETWORKS OF UNRELIABLE NODES

2016 ◽  
Vol 30 (3) ◽  
pp. 361-378 ◽  
Author(s):  
Jean-Michel Fourneau
Keyword(s):  
Steady State ◽  
Customer Service ◽  
Product Form ◽  
Steady State Distribution ◽  
State Distribution ◽  
Negative Customers ◽  
Service Completion

We study G-networks with positive and negative customers and signals. We consider two types of signals: they can make a subnetwork of queues operational or down. As signals are sent by queues after a customer service completion, one can model the availability of a sub-network of queues controlled by another network of queues. We prove that under classical assumptions for G-networks and assumptions on the rerouting probabilities when a subnetwork is not operational, the steady-state distribution, if it exists, has a product form steady state distribution. Some examples are given.

Insensitive average residence times in generalized semi-Markov processes

1981 ◽  
Vol 13 (4) ◽  
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.

The steady-state distribution of spent service times present in the M/G/1 foreground–background processor-sharing queue

1988 ◽  
Vol 25 (01) ◽  
pp. 194-203
Author(s):  
R. Schassberger
Keyword(s):  
Steady State ◽  
Random Measure ◽  
Processor Sharing ◽  
Steady State Distribution ◽  
State Distribution ◽  
Generating Functional ◽  
Service Times

The steady-state distribution of spent service times present in the M/G/1 foreground-background processor-sharing queue is described by a random measure, whose generating functional is obtained.

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