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

1997 ◽  
Vol 34 (1) ◽  
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.

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.

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.

State-dependent signalling in queueing networks

1994 ◽  
Vol 26 (02) ◽  
pp. 436-455 ◽  
Author(s):  
W. Henderson ◽  
B. S. Northcote ◽  
P. G. Taylor
Keyword(s):  
Queueing Networks ◽  
State Dependence ◽  
Product Form ◽  
Batch Size ◽  
Single Server ◽  
Negative Customers ◽  
Affect Control ◽  
State Dependent ◽  
Special Cases ◽  
Equilibrium Distributions

It has recently been shown that networks of queues with state-dependent movement of negative customers, and with state-independent triggering of customer movement have product-form equilibrium distributions. Triggers and negative customers are entities which, when arriving to a queue, force a single customer to be routed through the network or leave the network respectively. They are ‘signals' which affect/control network behaviour. The provision of state-dependent intensities introduces queues other than single-server queues into the network. This paper considers networks with state-dependent intensities in which signals can be either a trigger or a batch of negative customers (the batch size being determined by an arbitrary probability distribution). It is shown that such networks still have a product-form equilibrium distribution. Natural methods for state space truncation and for the inclusion of multiple customer types in the network can be viewed as special cases of this state dependence. A further generalisation allows for the possibility of signals building up at nodes.

A heterogeneous blocking system in a random environment

1996 ◽  
Vol 33 (01) ◽  
pp. 211-216 ◽  
Author(s):  
G. Falin
Keyword(s):  
Random Environment ◽  
Joint Distribution ◽  
Service System ◽  
External Environment ◽  
The State ◽  
Product Form ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

We obtain a necessary and sufficient condition for the interaction between a service system and an external environment under which the stationary joint distribution of the set of busy channels and the state of the external environment is given by a product-form formula.

State-dependent signalling in queueing networks

1994 ◽  
Vol 26 (2) ◽  
pp. 436-455 ◽  
Author(s):  
W. Henderson ◽  
B. S. Northcote ◽  
P. G. Taylor
Keyword(s):  
Queueing Networks ◽  
State Dependence ◽  
Product Form ◽  
Batch Size ◽  
Single Server ◽  
Negative Customers ◽  
Affect Control ◽  
State Dependent ◽  
Special Cases ◽  
Equilibrium Distributions

It has recently been shown that networks of queues with state-dependent movement of negative customers, and with state-independent triggering of customer movement have product-form equilibrium distributions. Triggers and negative customers are entities which, when arriving to a queue, force a single customer to be routed through the network or leave the network respectively. They are ‘signals' which affect/control network behaviour. The provision of state-dependent intensities introduces queues other than single-server queues into the network.This paper considers networks with state-dependent intensities in which signals can be either a trigger or a batch of negative customers (the batch size being determined by an arbitrary probability distribution). It is shown that such networks still have a product-form equilibrium distribution. Natural methods for state space truncation and for the inclusion of multiple customer types in the network can be viewed as special cases of this state dependence. A further generalisation allows for the possibility of signals building up at nodes.

Batch Routing Queueing Networks with Jump-Over Blocking

1996 ◽  
Vol 10 (2) ◽  
pp. 287-297 ◽  
Author(s):  
Richard J. Boucherie
Keyword(s):  
Queueing Networks ◽  
Equilibrium Distribution ◽  
Queueing Network ◽  
Capacity Constraints ◽  
Product Form

This paper shows that the equilibrium distribution of a queueing network with batch routing continues to be of product form if a batch that cannot enter its destination — for example, as a consequence of capacity constraints — immediately triggers a new transition that takes up the whole batch.

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.

Markovian couplings staying in arbitrary subsets of the state space

2002 ◽  
Vol 39 (01) ◽  
pp. 197-212 ◽  
Author(s):  
F. Javier López ◽  
Gerardo Sanz
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Transition Rates ◽  
Arbitrary Subset ◽  
State Spaces ◽  
Continuous Time Markov Chains ◽  
Countable State ◽  
Necessary And Sufficient

Let (X t ) and (Y t ) be continuous-time Markov chains with countable state spaces E and F and let K be an arbitrary subset of E x F. We give necessary and sufficient conditions on the transition rates of (X t ) and (Y t ) for the existence of a coupling which stays in K. We also show that when such a coupling exists, it can be chosen to be Markovian and give a way to construct it. In the case E=F and K ⊆ E x E, we see how the problem of construction of the coupling can be simplified. We give some examples of use and application of our results, including a new concept of lumpability in Markov chains.

Equilibrium distributions for an open migration process

1968 ◽  
Vol 5 (3) ◽  
pp. 567-571 ◽  
Author(s):  
P. Whittle
Keyword(s):  
Equilibrium Distribution ◽  
Sufficient Conditions ◽  
General Rule ◽  
Necessary And Sufficient Conditions ◽  
Migration Process ◽  
Immigration And Emigration ◽  
Equilibrium Distributions ◽  
Necessary And Sufficient

SummaryWe consider a system of colonies in which migration, immigration and emigration take place according to a fairly general rule. Necessary and sufficient conditions for equilibrium are given in the theorem of Section (3). The equilibrium distribution of colony numbers is determined by Equations (7) and (9); the colony sizes are thus statistically independent in equilibrium.Attention is also paid to the case where one colony becomes a “bottleneck” to circulation, when equilibrium changes character

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