Networks of Queues with Customers, Signals and Arbitrary Service Time Distributions

1995 ◽  
Vol 43 (3) ◽  
pp. 537-544 ◽  
Author(s):  
Xiuli Chao
Keyword(s):  
Service Time ◽  
Networks Of Queues ◽  
Arbitrary Service Time

On queueing systems by retrials

1983 ◽  
Vol 20 (02) ◽  
pp. 380-389 ◽  
Author(s):  
Vidyadhar G. Kulkarni
Keyword(s):  
General Result ◽  
Service Time ◽  
Queueing Systems ◽  
Single Server ◽  
Expected Number ◽  
Waiting Room ◽  
Second Moments ◽  
Different Types ◽  
General Distributions ◽  
Arbitrary Service Time

A general result for queueing systems with retrials is presented. This result relates the expected total number of retrials conducted by an arbitrary customer to the expected total number of retrials that take place during an arbitrary service time. This result is used in the analysis of a special system where two types of customer arrive in an independent Poisson fashion at a single-server service station with no waiting room. The service times of the two types of customer have independent general distributions with finite second moments. When the incoming customer finds the server busy he immediately leaves and tries his luck again after an exponential amount of time. The retrial rates are different for different types of customers. Expressions are derived for the expected number of retrial customers of each type.

An extension of Erlang's formulas which distinguishes individual servers

1972 ◽  
Vol 9 (1) ◽  
pp. 192-197 ◽  
Author(s):  
Jan M. Chaiken ◽  
Edward Ignall
Keyword(s):  
Service Time ◽  
Time Distribution ◽  
The State ◽  
Service Time Distribution ◽  
Loss System ◽  
Arbitrary Service Time

For a particular kind of finite-server loss system in which the number and identity of servers depends on the type of the arriving call and on the state of the system, the limits of the state probabilities (as t → ∞) are found for an arbitrary service-time distribution.

Insensitive bounds for the stationary distribution of non-reversible Markov chains

1988 ◽  
Vol 25 (1) ◽  
pp. 9-20 ◽  
Author(s):  
Arie Hordijk ◽  
AD Ridder
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Service Time ◽  
Form Solution ◽  
Tree Form ◽  
Standard Product ◽  
Reversible Markov Chains ◽  
Stationary Probabilities ◽  
Networks Of Queues ◽  
General Method

A general method is developed to compute easy bounds of the weighted stationary probabilities for networks of queues which do not satisfy the standard product form. The bounds are obtained by constructing approximating reversible Markov chains. Thus, the bounds are insensitive with respect to service-time distributions. A special representation, called the tree-form solution, of the stationary distribution is used to derive the bounds. The results are applied to an overflow model.

On queueing systems by retrials

1983 ◽  
Vol 20 (2) ◽  
pp. 380-389 ◽  
Author(s):  
Vidyadhar G. Kulkarni
Keyword(s):  
General Result ◽  
Service Time ◽  
Queueing Systems ◽  
Single Server ◽  
Expected Number ◽  
Waiting Room ◽  
Second Moments ◽  
Different Types ◽  
General Distributions ◽  
Arbitrary Service Time

A general result for queueing systems with retrials is presented. This result relates the expected total number of retrials conducted by an arbitrary customer to the expected total number of retrials that take place during an arbitrary service time. This result is used in the analysis of a special system where two types of customer arrive in an independent Poisson fashion at a single-server service station with no waiting room. The service times of the two types of customer have independent general distributions with finite second moments. When the incoming customer finds the server busy he immediately leaves and tries his luck again after an exponential amount of time. The retrial rates are different for different types of customers. Expressions are derived for the expected number of retrial customers of each type.

Networks of queues with customers of different types

1975 ◽  
Vol 12 (3) ◽  
pp. 542-554 ◽  
Author(s):  
F. P. Kelly
Keyword(s):  
Service Time ◽  
Time Distribution ◽  
Service Time Distribution ◽  
Basic Model ◽  
Open Case ◽  
Negative Exponential Distribution ◽  
Different Types ◽  
The Individual ◽  
Negative Exponential ◽  
Networks Of Queues

The behaviour in equilibrium of networks of queues in which customers may be of different types is studied. The type of a customer is allowed to influence his choice of path through the network and, under certain conditions, his service time distribution at each queue. The model assumed will usually cause each service time distribution to be of a form related to the negative exponential distribution.Theorems 1 and 2 establish the equilibrium distribution for the basic model in the closed and open cases; in the open case the individual queues are independent in equilibrium. In Section 4 similar results are obtained for other models, models which include processes better described as networks of colonies or as networks of stacks. In Section 5 the effect of time reversal upon certain processes is used to obtain further information about the equilibrium behaviour of those processes.

Insensitive bounds for the stationary distribution of non-reversible Markov chains

1988 ◽  
Vol 25 (01) ◽  
pp. 9-20 ◽  
Author(s):  
Arie Hordijk ◽  
AD Ridder
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Service Time ◽  
Form Solution ◽  
Tree Form ◽  
Standard Product ◽  
Reversible Markov Chains ◽  
Stationary Probabilities ◽  
Networks Of Queues ◽  
General Method

A general method is developed to compute easy bounds of the weighted stationary probabilities for networks of queues which do not satisfy the standard product form. The bounds are obtained by constructing approximating reversible Markov chains. Thus, the bounds are insensitive with respect to service-time distributions. A special representation, called the tree-form solution, of the stationary distribution is used to derive the bounds. The results are applied to an overflow model.

A Note on Queueing Networks with Signals and Random Triggering Times

1994 ◽  
Vol 8 (2) ◽  
pp. 213-219 ◽  
Author(s):  
Xiuli Chao
Keyword(s):  
Stationary Distribution ◽  
Service Time ◽  
Queueing Networks ◽  
Time Distribution ◽  
Network Models ◽  
Form Solution ◽  
Service Time Distribution ◽  
Signal Changes ◽  
Networks Of Queues ◽  
Random Amount

There is a growing interest in networks of queues with customers and signals. The signals in these models carry commands to the service nodes and trigger customers to move instantaneously within the network. In this note we consider networks of queues with signals and random triggering times; that is, when a signal arrives at a node, it takes a random amount of time to trigger a customer to move with distribution depending on the source of the signal. By appropriately choosing the triggering times, we can obtain network models such that a signal changes a customer's service time distribution – for example, the signal increases or decreases the service time of a customer. We show that the stationary distribution of this model has a product form solution.

An extension of Erlang's formulas which distinguishes individual servers

1972 ◽  
Vol 9 (01) ◽  
pp. 192-197 ◽  
Author(s):  
Jan M. Chaiken ◽  
Edward Ignall
Keyword(s):  
Service Time ◽  
Time Distribution ◽  
The State ◽  
Service Time Distribution ◽  
Loss System ◽  
Arbitrary Service Time

For a particular kind of finite-server loss system in which the number and identity of servers depends on the type of the arriving call and on the state of the system, the limits of the state probabilities (as t → ∞) are found for an arbitrary service-time distribution.

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