Stochastic Monotonicities in Jackson Queueing Networks

1997 ◽  
Vol 11 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Torgny Lindvall
Keyword(s):  
Queueing Networks ◽  
Queueing Network ◽  
Zero Point ◽  
Monotonicity Property ◽  
Stochastic Monotonicity ◽  
Property A ◽  
Jackson Network ◽  
Queueing Process ◽  
Network Process ◽  
Number Of Customers

When starting from 0, a standard M/M/k queueing process has a second-order stochastic monotonicity property of a strong kind: its increments are stochastically decreasing (the SDI property). A first attempt to generalize this to the Jackson queueing network fails. This gives us reason to reexamine the underlying theory for stochastic monotonicity of Markov processes starting from a zero-point, in order to find a condition on a function of a Jackson network process to have the SDI property. It turns out that the total number of customers at time t has the desired property, if the network is idle at time O. We use couplings in our analysis; they are also of value in the comparison of two networks with different parameters.

Realization probability in closed Jackson queueing networks and its application

1987 ◽  
Vol 19 (03) ◽  
pp. 708-738 ◽  
Author(s):  
X. R. Cao
Keyword(s):  
Steady State ◽  
Perturbation Analysis ◽  
Queueing Networks ◽  
Sample Path ◽  
Queueing Network ◽  
Jackson Network ◽  
The Mean ◽  
Service Times ◽  
A New Technique ◽  
Number Of Customers

Perturbation analysis is a new technique which yields the sensitivities of system performance measures with respect to parameters based on one sample path of a system. This paper provides some theoretical analysis for this method. A new notion, the realization probability of a perturbation in a closed queueing network, is studied. The elasticity of the expected throughput in a closed Jackson network with respect to the mean service times can be expressed in terms of the steady-state probabilities and realization probabilities in a very simple way. The elasticity of the throughput with respect to the mean service times when the service distributions are perturbed to non-exponential distributions can also be obtained using these realization probabilities. It is proved that the sample elasticity of the throughput obtained by perturbation analysis converges to the elasticity of the expected throughput in steady-state both in mean and with probability 1 as the number of customers served goes to This justifies the existing algorithms based on perturbation analysis which efficiently provide the estimates of elasticities in practice.

Realization probability in closed Jackson queueing networks and its application

1987 ◽  
Vol 19 (3) ◽  
pp. 708-738 ◽  
Author(s):  
X. R. Cao
Keyword(s):  
Steady State ◽  
Perturbation Analysis ◽  
Queueing Networks ◽  
Sample Path ◽  
Queueing Network ◽  
Jackson Network ◽  
The Mean ◽  
Service Times ◽  
A New Technique ◽  
Number Of Customers

Perturbation analysis is a new technique which yields the sensitivities of system performance measures with respect to parameters based on one sample path of a system. This paper provides some theoretical analysis for this method. A new notion, the realization probability of a perturbation in a closed queueing network, is studied. The elasticity of the expected throughput in a closed Jackson network with respect to the mean service times can be expressed in terms of the steady-state probabilities and realization probabilities in a very simple way. The elasticity of the throughput with respect to the mean service times when the service distributions are perturbed to non-exponential distributions can also be obtained using these realization probabilities. It is proved that the sample elasticity of the throughput obtained by perturbation analysis converges to the elasticity of the expected throughput in steady-state both in mean and with probability 1 as the number of customers served goes to This justifies the existing algorithms based on perturbation analysis which efficiently provide the estimates of elasticities in practice.

scholarly journals Impact of Routeing on Correlation Strength in Stationary Queueing Network Processes

2008 ◽  
Vol 45 (03) ◽  
pp. 846-878 ◽  
Author(s):  
Hans Daduna ◽  
Ryszard Szekli
Keyword(s):  
Queueing Networks ◽  
Queueing Network ◽  
Dependence Structure ◽  
Spectral Gaps ◽  
Stochastic Monotonicity ◽  
Closed Queueing Networks ◽  
First Order ◽  
Correlation Strength ◽  
Concordance Order ◽  
Asymptotic Variances

For exponential open and closed queueing networks, we investigate the internal dependence structure, compare the internal dependence for different networks, and discuss the relation of correlation formulae to the existence of spectral gaps and comparison of asymptotic variances. A central prerequisite for the derived theorems is stochastic monotonicity of the networks. The dependence structure of network processes is described by concordance order with respect to various classes of functions. Different networks with the same first-order characteristics are compared with respect to their second-order properties. If a network is perturbed by changing the routeing in a way which holds the routeing equilibrium fixed, the resulting perturbations of the network processes are evaluated.

scholarly journals Analysis of Closed Queueing Networks with Batch Service

2020 ◽  
Vol 20 (4) ◽  
pp. 527-533
Author(s):  
Elena P. Stankevich ◽  
◽  
Igor E. Tananko ◽  
Vitalii I. Dolgov ◽  
◽  
...  
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Queueing Networks ◽  
Manufacturing Systems ◽  
Queueing Network ◽  
Transport Systems ◽  
Batch Service ◽  
Single Server ◽  
Closed Queueing Networks ◽  
Number Of Customers

We consider a closed queuing network with batch service and movements of customers in continuous time. Each node in the queueing network is an infinite capacity single server queueing system under a RANDOM discipline. Customers move among the nodes following a routing matrix. Customers are served in batches of a fixed size. If a number of customers in a node is less than the size, the server of the system is idle until the required number of customers arrive at the node. An arriving at a node customer is placed in the queue if the server is busy. The batсh service time is exponentially distributed. After a batсh finishes its execution at a node, each customer of the batch, regardless of other customers of the batch, immediately moves to another node in accordance with the routing probability. This article presents an analysis of the queueing network using a Markov chain with continuous time. The qenerator matrix is constructed for the underlying Markov chain. We obtain expressions for the performance measures. Some numerical examples are provided. The results can be used for the performance analysis manufacturing systems, passenger and freight transport systems, as well as information and computing systems with parallel processing and transmission of information.

A Geometric Product-Form Distribution for a Queueing Network by Non-Standard Batch Arrivals and Batch Transfers

1997 ◽  
Vol 29 (02) ◽  
pp. 523-544 ◽  
Author(s):  
Masakiyo Miyazawa ◽  
Peter G. Taylor
Keyword(s):  
Stationary Distribution ◽  
Queueing Networks ◽  
Queueing Network ◽  
Product Form ◽  
Arrival Process ◽  
Service Discipline ◽  
Batch Service ◽  
Batch Arrivals ◽  
Geometric Product ◽  
Number Of Customers

We introduce a batch service discipline, called assemble-transfer batch service, for continuous-time open queueing networks with batch movements. Under this service discipline a requested number of customers is simultaneously served at a node, and transferred to another node as, possibly, a batch of different size, if there are sufficient customers there; the node is emptied otherwise. We assume a Markovian setting for the arrival process, service times and routing, where batch sizes are generally distributed. Under the assumption that extra batches arrive while nodes are empty, and under a stability condition, it is shown that the stationary distribution of the queue length has a geometric product form over the nodes if and only if certain conditions are satisfied for the extra arrivals. This gives a new class of queueing networks which have tractable stationary distributions, and simultaneously shows that the product form provides a stochastic upper bound for the stationary distribution of the corresponding queueing network without the extra arrivals.

scholarly journals Impact of Routeing on Correlation Strength in Stationary Queueing Network Processes

2008 ◽  
Vol 45 (3) ◽  
pp. 846-878 ◽  
Author(s):  
Hans Daduna ◽  
Ryszard Szekli
Keyword(s):  
Queueing Networks ◽  
Queueing Network ◽  
Dependence Structure ◽  
Spectral Gaps ◽  
Stochastic Monotonicity ◽  
Closed Queueing Networks ◽  
First Order ◽  
Correlation Strength ◽  
Concordance Order ◽  
Asymptotic Variances

For exponential open and closed queueing networks, we investigate the internal dependence structure, compare the internal dependence for different networks, and discuss the relation of correlation formulae to the existence of spectral gaps and comparison of asymptotic variances. A central prerequisite for the derived theorems is stochastic monotonicity of the networks. The dependence structure of network processes is described by concordance order with respect to various classes of functions. Different networks with the same first-order characteristics are compared with respect to their second-order properties. If a network is perturbed by changing the routeing in a way which holds the routeing equilibrium fixed, the resulting perturbations of the network processes are evaluated.

A Geometric Product-Form Distribution for a Queueing Network by Non-Standard Batch Arrivals and Batch Transfers

1997 ◽  
Vol 29 (2) ◽  
pp. 523-544 ◽  
Author(s):  
Masakiyo Miyazawa ◽  
Peter G. Taylor
Keyword(s):  
Stationary Distribution ◽  
Queueing Networks ◽  
Queueing Network ◽  
Product Form ◽  
Arrival Process ◽  
Service Discipline ◽  
Batch Service ◽  
Batch Arrivals ◽  
Geometric Product ◽  
Number Of Customers

We introduce a batch service discipline, called assemble-transfer batch service, for continuous-time open queueing networks with batch movements. Under this service discipline a requested number of customers is simultaneously served at a node, and transferred to another node as, possibly, a batch of different size, if there are sufficient customers there; the node is emptied otherwise. We assume a Markovian setting for the arrival process, service times and routing, where batch sizes are generally distributed.Under the assumption that extra batches arrive while nodes are empty, and under a stability condition, it is shown that the stationary distribution of the queue length has a geometric product form over the nodes if and only if certain conditions are satisfied for the extra arrivals. This gives a new class of queueing networks which have tractable stationary distributions, and simultaneously shows that the product form provides a stochastic upper bound for the stationary distribution of the corresponding queueing network without the extra arrivals.

scholarly journals Characterising scoring rules by their solution in iteratively undominated strategies

2021 ◽  
Author(s):  
Christian Basteck
Keyword(s):  
Monotonicity Property ◽  
Scoring Rules ◽  
The Other ◽  
Approval Voting ◽  
Property A ◽  
Borda Rule ◽  
Scoring Rule ◽  
Direct Mechanism ◽  
Social Choice Correspondence ◽  
Choice Correspondence

AbstractWe characterize voting procedures according to the social choice correspondence they implement when voters cast ballots strategically, applying iteratively undominated strategies. In elections with three candidates, the Borda Rule is the unique positional scoring rule that satisfies unanimity (U) (i.e., elects a candidate whenever it is unanimously preferred) and is majoritarian after eliminating a worst candidate (MEW)(i.e., if there is a unanimously disliked candidate, the majority-preferred among the other two is elected). In a larger class of rules, Approval Voting is characterized by a single axiom that implies both U and MEW but is weaker than Condorcet-consistency (CON)—it is the only direct mechanism scoring rule that is majoritarian after eliminating a Pareto-dominated candidate (MEPD)(i.e., if there is a Pareto-dominated candidate, the majority-preferred among the other two is elected); among all finite scoring rules that satisfy MEPD, Approval Voting is the most decisive. However, it fails a desirable monotonicity property: a candidate that is elected for some preference profile, may lose the election once she gains further in popularity. In contrast, the Borda Rule is the unique direct mechanism scoring rule that satisfies U, MEW and monotonicity (MON). There exists no direct mechanism scoring rule that satisfies both MEPD and MON and no finite scoring rule satisfying CON.

scholarly journals Finite Queueing Modeling and Optimization: A Selected Review

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
F. R. B. Cruz ◽  
T. van Woensel
Keyword(s):  
Performance Evaluation ◽  
Queueing Networks ◽  
Queueing Network ◽  
Expansion Method ◽  
Network Analyzer ◽  
Modeling And Optimization ◽  
Product Engineering ◽  
Service Rates

This review provides an overview of the queueing modeling issues and the related performance evaluation and optimization approaches framed in a joined manufacturing and product engineering. Such networks are represented as queueing networks. The performance of the queueing networks is evaluated using an advanced queueing network analyzer: the generalized expansion method. Secondly, different model approaches are described and optimized with regard to the key parameters in the network (e.g., buffer and server sizes, service rates, and so on).

scholarly journals A M/M/1 Queueing Network with Classical Retrial Policy

2021 ◽  
Vol 12 (1S) ◽  
pp. 329-337
Author(s):  
S. Shanmugasundaram, Et. al.
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Queueing Network ◽  
State Probability ◽  
Queueing Model ◽  
Steady State Probability ◽  
Numerical Examples ◽  
System Length ◽  
Number Of Customers ◽  
Little's Formula

In this paper we study the M/M/1 queueing model with retrial on network. We derive the steady state probability of customers in the network, the average number of customers in the all the three nodes in the system, the queue length, system length using little’s formula. The particular case is derived (no retrial). The numerical examples are given to test the correctness of the model.

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