Discrete Time Markov Chains Competing over Resources: Product Form Steady-State Distribution

2008 ◽  
Author(s):  
Jean-Michel Fourneau
Keyword(s):  
Steady State ◽  
Markov Chains ◽  
Discrete Time ◽  
Product Form ◽  
Steady State Distribution ◽  
State Distribution

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.

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.

On steady-state queue size distributions of the discrete-time GI/G/1 queue

1996 ◽  
Vol 28 (04) ◽  
pp. 1177-1200 ◽  
Author(s):  
Tao Yang ◽  
M. L. Chaudhry
Keyword(s):  
Steady State ◽  
Discrete Time ◽  
Linear Transformation ◽  
State System ◽  
Interarrival Time ◽  
Steady State Distribution ◽  
State Distribution ◽  
The Matrix ◽  
Matrix Geometric Solution ◽  
System Length

In this paper, we present results for the steady-state system length distributions of the discrete-timeGI/G/1 queue. We examine the system at customer arrival epochs (customer departure epochs) and use the residual service time (residual interarrival time) as the supplementary variable. The embedded Markov chain is ofGI/M/1 type if the embedding points are arrival epochs and is ofM/G/1 type if the embedding points are departure epochs. Using the matrix analytic method, we identify the necessary and sufficient condition for both Markov chains to be positive recurrent. For theGI/M/1 type chain, we derive a matrix-geometric solution for its steady-state distribution and for theM/G/1 type chain, we develop a simple linear transformation that relates it to theGI/M/1 type chain and leads to a simple analytic solution for its steady-state distribution. We also show that the steady-state system length distribution at an arbitrary point in time can be obtained by a simple linear transformation of the matrix-geometric solution for theGI/M/1 type chain. A number of applications of the model to communication systems and numerical examples are also discussed.

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.

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).

On steady-state queue size distributions of the discrete-time GI/G/1 queue

1996 ◽  
Vol 28 (4) ◽  
pp. 1177-1200 ◽  
Author(s):  
Tao Yang ◽  
M. L. Chaudhry
Keyword(s):  
Steady State ◽  
Discrete Time ◽  
Linear Transformation ◽  
State System ◽  
Interarrival Time ◽  
Steady State Distribution ◽  
State Distribution ◽  
The Matrix ◽  
Matrix Geometric Solution ◽  
System Length

In this paper, we present results for the steady-state system length distributions of the discrete-time GI/G/1 queue. We examine the system at customer arrival epochs (customer departure epochs) and use the residual service time (residual interarrival time) as the supplementary variable. The embedded Markov chain is of GI/M/1 type if the embedding points are arrival epochs and is of M/G/1 type if the embedding points are departure epochs. Using the matrix analytic method, we identify the necessary and sufficient condition for both Markov chains to be positive recurrent. For the GI/M/1 type chain, we derive a matrix-geometric solution for its steady-state distribution and for the M/G/1 type chain, we develop a simple linear transformation that relates it to the GI/M/1 type chain and leads to a simple analytic solution for its steady-state distribution. We also show that the steady-state system length distribution at an arbitrary point in time can be obtained by a simple linear transformation of the matrix-geometric solution for the GI/M/1 type chain. A number of applications of the model to communication systems and numerical examples are also discussed.

ON THE STEADY-STATE DISTRIBUTION OF NUMBER IN THE QUEUE Geom/G/1

1999 ◽  
Vol 13 (1) ◽  
pp. 121-125 ◽  
Author(s):  
M. L. Chaudhry
Keyword(s):  
Steady State ◽  
Discrete Time ◽  
Continuous Time ◽  
Telecommunication Networks ◽  
Steady State Distribution ◽  
State Distribution ◽  
Discrete Time Systems ◽  
Discrete Time Queues ◽  
Time Queues ◽  
Time Systems

Discrete-time queues are frequently used in telecommunication networks. This paper proposes a new method which is based only on probabilistic arguments. It derives the distributions of numbers of customers in the discrete-time systems Geom/G/1 and Geom/G/1/M. Though the derivation does not involve use of the transforms, the transforms may be obtained, if desired. Another advantage of this is that the numerical results obtained are stable as there are no negative signs involved in summations. Further, the method can be easily used to solve more complex problems in discrete- and continuous-time queues.

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