Queueing networks by negative customers and negative queue lengths

1993 ◽  
Vol 30 (4) ◽  
pp. 931-942 ◽  
Author(s):  
W. Henderson
Keyword(s):  
Queueing Networks ◽  
Queue Length ◽  
Equilibrium Distribution ◽  
State Dependence ◽  
Balance Equations ◽  
Negative Customers ◽  
Jackson Networks ◽  
Geometric Product ◽  
Queue Lengths ◽  
Number Of Customers

A number of papers have recently appeared in the literature in which customers, in moving from node to node in the network arrive as either a positive customer or as a batch of negative customers. A positive customer joining its queue increases the number of customers at the queue by 1 and each negative customer decreases the queue length by 1, if possible. It has been shown that the equilibrium distribution for these networks assumes a geometric product form, that certain partial balance equations prevail and that the parameters of the geometric distributions are, as in Jackson networks, the service facility throughputs of customers. In this paper the previous work is generalised by allowing state dependence into both the service and routing intensities and by allowing the possibility, although not the necessity, for negative customers to build up at the nodes.

Queueing networks by negative customers and negative queue lengths

1993 ◽  
Vol 30 (04) ◽  
pp. 931-942 ◽  
Author(s):  
W. Henderson
Keyword(s):  
Queueing Networks ◽  
Queue Length ◽  
Equilibrium Distribution ◽  
State Dependence ◽  
Balance Equations ◽  
Negative Customers ◽  
Jackson Networks ◽  
Geometric Product ◽  
Queue Lengths ◽  
Number Of Customers

A number of papers have recently appeared in the literature in which customers, in moving from node to node in the network arrive as either a positive customer or as a batch of negative customers. A positive customer joining its queue increases the number of customers at the queue by 1 and each negative customer decreases the queue length by 1, if possible. It has been shown that the equilibrium distribution for these networks assumes a geometric product form, that certain partial balance equations prevail and that the parameters of the geometric distributions are, as in Jackson networks, the service facility throughputs of customers. In this paper the previous work is generalised by allowing state dependence into both the service and routing intensities and by allowing the possibility, although not the necessity, for negative customers to build up at the nodes.

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.

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.

Strong Approximations of Irreducible Closed Queueing Networks

1997 ◽  
Vol 29 (2) ◽  
pp. 498-522 ◽  
Author(s):  
Hanqin Zhang
Keyword(s):  
Queueing Networks ◽  
Queue Length ◽  
Rates Of Convergence ◽  
Closed Queueing Networks ◽  
Brownian Motions ◽  
Strong Approximations ◽  
Queue Lengths

A sequence of irreducible closed queueing networks is considered in this paper. We obtain that the queue length processes can be approximated by reflected Brownian motions. Using these approximations, we get rates of convergence of the distributions of queue lengths.

Jackson networks in nonautonomous random environments

2016 ◽  
Vol 48 (2) ◽  
pp. 315-331 ◽  
Author(s):  
Ruslan Krenzler ◽  
Hans Daduna ◽  
Sonja Otten
Keyword(s):  
Explicit Expression ◽  
Random Environment ◽  
Queueing Networks ◽  
The Other ◽  
Random Environments ◽  
Jackson Networks ◽  
Random Direction ◽  
Queue Lengths ◽  
Joint Stationary Distribution ◽  
The Impact

Abstract We investigate queueing networks in a random environment. The impact of the evolving environment on the network is by changing service capacities (upgrading and/or degrading, breakdown, repair) when the environment changes its state. On the other side, customers departing from the network may enforce the environment to jump immediately. This means that the environment is nonautonomous and therefore results in a rather complex two-way interaction, especially if the environment is not itself Markov. To react to the changes of the capacities we implement randomised versions of the well-known deterministic rerouteing schemes 'skipping' (jump-over protocol) and `reflection' (repeated service, random direction). Our main result is an explicit expression for the joint stationary distribution of the queue-lengths vector and the environment which is of product form.

On normalization constants for closed queueing networks with finite local buffers

1998 ◽  
Vol 35 (3) ◽  
pp. 600-607
Author(s):  
Ulrich A. W. Tetzlaff
Keyword(s):  
Queueing Networks ◽  
Delta Function ◽  
Partition Functions ◽  
Balance Equations ◽  
Steady State Flow ◽  
Closed Queueing Networks ◽  
Single Class ◽  
Closed Form Solutions ◽  
Flow Balance ◽  
Number Of Customers

We present new closed form solutions for partition functions used to normalize the steady-state flow balance equations of certain Markovian type queueing networks. The results focus on single class closed product form networks with state space constraints at the queueing stations. They are achieved by combining the partition function of the open network, having finite local buffers with a delta function in order to fix the number of customers in the system.

A functional central limit theorem for the jump counts of Markov processes with an application to Jackson networks

1995 ◽  
Vol 27 (2) ◽  
pp. 476-509
Author(s):  
Venkat Anantharam ◽  
Takis Konstantopoulos
Keyword(s):  
Central Limit ◽  
Queueing Networks ◽  
Weighted Sums ◽  
Counting Processes ◽  
Transition Diagram ◽  
Jackson Networks ◽  
Number Of Customers ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit ◽  
Traffic Processes

Each feasible transition between two distinct states i and j of a continuous-time, uniform, ergodic, countable-state Markov process gives a counting process counting the number of such transitions executed by the process. Traffic processes in Markovian queueing networks can, for instance, be represented as sums of such counting processes. We prove joint functional central limit theorems for the family of counting processes generated by all feasible transitions. We characterize which weighted sums of counts have zero covariance in the limit in terms of balance equations in the transition diagram of the process. Finally, we apply our results to traffic processes in a Jackson network. In particular, we derive simple formulas for the asymptotic covariances between the processes counting the number of customers moving between pairs of nodes in such a network.

scholarly journals Product-form queueing networks with negative and positive customers

1991 ◽  
Vol 28 (3) ◽  
pp. 656-663 ◽  
Author(s):  
Erol Gelenbe
Keyword(s):  
Queueing Networks ◽  
Queue Length ◽  
Queueing Network ◽  
Form Solution ◽  
Product Form ◽  
Usual Sense ◽  
Negative Customers ◽  
Flow Equations ◽  
New Class ◽  
Service Times

We introduce a new class of queueing networks in which customers are either ‘negative' or ‘positive'. A negative customer arriving to a queue reduces the total customer count in that queue by 1 if the queue length is positive; it has no effect at all if the queue length is empty. Negative customers do not receive service. Customers leaving a queue for another one can either become negative or remain positive. Positive customers behave as ordinary queueing network customers and receive service. We show that this model with exponential service times, Poisson external arrivals, with the usual independence assumptions for service times, and Markovian customer movements between queues, has product form. It is quasi-reversible in the usual sense, but not in a broader sense which includes all destructions of customers in the set of departures. The existence and uniqueness of the solutions to the (nonlinear) customer flow equations, and hence of the product form solution, is discussed.

On normalization constants for closed queueing networks with finite local buffers

1998 ◽  
Vol 35 (03) ◽  
pp. 600-607
Author(s):  
Ulrich A. W. Tetzlaff
Keyword(s):  
Queueing Networks ◽  
Delta Function ◽  
Partition Functions ◽  
Balance Equations ◽  
Steady State Flow ◽  
Closed Queueing Networks ◽  
Single Class ◽  
Closed Form Solutions ◽  
Flow Balance ◽  
Number Of Customers

We present new closed form solutions for partition functions used to normalize the steady-state flow balance equations of certain Markovian type queueing networks. The results focus on single class closed product form networks with state space constraints at the queueing stations. They are achieved by combining the partition function of the open network, having finite local buffers with a delta function in order to fix the number of customers in the system.

A functional central limit theorem for the jump counts of Markov processes with an application to Jackson networks

1995 ◽  
Vol 27 (02) ◽  
pp. 476-509
Author(s):  
Venkat Anantharam ◽  
Takis Konstantopoulos
Keyword(s):  
Central Limit ◽  
Queueing Networks ◽  
Weighted Sums ◽  
Counting Processes ◽  
Transition Diagram ◽  
Jackson Networks ◽  
Number Of Customers ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit ◽  
Traffic Processes

Each feasible transition between two distinct states i and j of a continuous-time, uniform, ergodic, countable-state Markov process gives a counting process counting the number of such transitions executed by the process. Traffic processes in Markovian queueing networks can, for instance, be represented as sums of such counting processes. We prove joint functional central limit theorems for the family of counting processes generated by all feasible transitions. We characterize which weighted sums of counts have zero covariance in the limit in terms of balance equations in the transition diagram of the process. Finally, we apply our results to traffic processes in a Jackson network. In particular, we derive simple formulas for the asymptotic covariances between the processes counting the number of customers moving between pairs of nodes in such a network.

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