On the stochastic domination for batch-arrival, batch-service and assemble-transfer queueing networks

2003 ◽  
Vol 40 (4) ◽  
pp. 1103-1120 ◽  
Author(s):  
Antonis Economou
Keyword(s):  
Continuous Time ◽  
Queueing Networks ◽  
Stochastic Order ◽  
Batch Arrival ◽  
Batch Service ◽  
Stochastic Monotonicity ◽  
Stochastic Domination ◽  
Jackson Networks ◽  
Continuous Time Markov Chains

Stochastic monotonicity properties for various classes of queueing networks have been established in the literature mainly with the use of coupling constructions. Miyazawa and Taylor (1997) introduced a class of batch-arrival, batch-service and assemble-transfer queueing networks which can be thought of as generalized Jackson networks with batch movements. We study conditions for stochastic domination within this class of networks. The proofs are based on a certain characterization of the stochastic order for continuous-time Markov chains, written in terms of their associated intensity matrices.

On the stochastic domination for batch-arrival, batch-service and assemble-transfer queueing networks

2003 ◽  
Vol 40 (04) ◽  
pp. 1103-1120 ◽  
Author(s):  
Antonis Economou
Keyword(s):  
Continuous Time ◽  
Queueing Networks ◽  
Stochastic Order ◽  
Batch Arrival ◽  
Batch Service ◽  
Stochastic Monotonicity ◽  
Stochastic Domination ◽  
Jackson Networks ◽  
Continuous Time Markov Chains

Stochastic monotonicity properties for various classes of queueing networks have been established in the literature mainly with the use of coupling constructions. Miyazawa and Taylor (1997) introduced a class of batch-arrival, batch-service and assemble-transfer queueing networks which can be thought of as generalized Jackson networks with batch movements. We study conditions for stochastic domination within this class of networks. The proofs are based on a certain characterization of the stochastic order for continuous-time Markov chains, written in terms of their associated intensity matrices.

NECESSARY AND SUFFICIENT CONDITIONS FOR THE STOCHASTIC COMPARISON OF JACKSON NETWORKS

2003 ◽  
Vol 17 (1) ◽  
pp. 143-151 ◽  
Author(s):  
Antonis Economou
Keyword(s):  
Continuous Time ◽  
Stochastic Order ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Stochastic Comparison ◽  
Jackson Networks ◽  
Continuous Time Markov Chains ◽  
Service Rates ◽  
Necessary And Sufficient

External and internal monotonicity properties for Jackson networks have been established in the literature with the use of coupling constructions. Recently, Lopez et al. derived necessary and sufficient conditions for the (strong) stochastic comparison of two-station Jackson networks with increasing service rates, by constructing a certain Markovian coupling. In this article, we state necessary and sufficient conditions for the stochastic comparison of L-station Jackson networks in the general case. The proof is based on a certain characterization of the stochastic order for continuous-time Markov chains, written in terms of their associated intensity matrices.

Stochastic domination and Markovian couplings

2000 ◽  
Vol 32 (4) ◽  
pp. 1064-1076 ◽  
Author(s):  
F. Javier López ◽  
Servet Martínez ◽  
Gerardo Sanz
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Sufficient Conditions ◽  
Stochastic Domination ◽  
Jackson Networks ◽  
Partially Ordered ◽  
Continuous Time Markov Chains ◽  
Necessary And Sufficient

For continuous-time Markov chains with semigroups P, P' taking values in a partially ordered set, such that P ≤ stP', we show the existence of an order-preserving Markovian coupling and give a way to construct it. From our proof, we also obtain the conditions of Brandt and Last for stochastic domination in terms of the associated intensity matrices. Our result is applied to get necessary and sufficient conditions for the existence of Markovian couplings between two Jackson networks.

Stochastic monotonicity in general queueing networks

1989 ◽  
Vol 26 (02) ◽  
pp. 413-417 ◽  
Author(s):  
J. George Shanthikumar ◽  
David D. Yao
Keyword(s):  
Queueing Networks ◽  
Stochastic Monotonicity ◽  
Jackson Networks ◽  
Monotonicity Properties

We develop a representation for general queueing networks, and study some stochastic monotonicity properties that have been previously established in Jackson networks (e.g. Shanthikumar and Yao (1986), (1987)).

Zero-sum games for continuous-time Markov chains with unbounded transition and average payoff rates

2003 ◽  
Vol 40 (02) ◽  
pp. 327-345 ◽  
Author(s):  
Xianping Guo ◽  
Onésimo Hernández-Lerma
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Transition Rates ◽  
Zero Sum Games ◽  
Stationary Strategies ◽  
Continuous Time Markov Chains ◽  
Average Payoff ◽  
Optimal Stationary Strategies ◽  
Zero Sum

This paper is a first study of two-person zero-sum games for denumerable continuous-time Markov chains determined by given transition rates, with an average payoff criterion. The transition rates are allowed to be unbounded, and the payoff rates may have neither upper nor lower bounds. In the spirit of the ‘drift and monotonicity’ conditions for continuous-time Markov processes, we give conditions on the controlled system's primitive data under which the existence of the value of the game and a pair of strong optimal stationary strategies is ensured by using the Shapley equations. Also, we present a ‘martingale characterization’ of a pair of strong optimal stationary strategies. Our results are illustrated with a birth-and-death game.

ON TRUNCATION PROPERTIES OF FINITE-BUFFER QUEUES AND QUEUEING NETWORKS

2000 ◽  
Vol 14 (4) ◽  
pp. 409-423 ◽  
Author(s):  
Xiuli Chao ◽  
Masakiyo Miyazawa
Keyword(s):  
Stochastic Process ◽  
Continuous Time ◽  
Queueing Networks ◽  
Additional Condition ◽  
Queueing Systems ◽  
Batch Service ◽  
Single Server ◽  
Finite Buffer ◽  
Finite Buffers ◽  
Batch Service Queues

We show that several truncation properties of queueing systems are consequences of a simple property of censored stochastic processes. We first consider a discrete-time stochastic process and show that its censored process has a truncated stationary distribution. When the stochastic process has continuous time, we present a similar result under the additional condition that the process is locally balanced. We apply these results to single-server batch arrival batch service queues with finite buffers and queueing networks with finite buffers and batch movements, and extend the well-known results on truncation properties of the MX/G/1/k queues and queueing networks with jump-over blocking.

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.

scholarly journals Networks—B

2021 ◽  
pp. 93-113
Author(s):  
Jean Walrand
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Time Reversal ◽  
Continuous Time ◽  
Queueing Networks ◽  
Product Form ◽  
Continuous Time Markov Chains

AbstractThis chapter provides the derivations of the results in the previous chapter. It also develops the theory of continuous-time Markov chains.Section 6.1 proves the results on the spreading of rumors. Section 6.2 presents the theory of continuous-time Markov chains that are used to model queueing networks, among many other applications. That section explains the relationships between continuous-time and related discrete-time Markov chains. Sections 6.3 and 6.4 prove the results about product-form networks by using a time-reversal argument.

Stochastic monotonicity in general queueing networks

1989 ◽  
Vol 26 (2) ◽  
pp. 413-417 ◽  
Author(s):  
J. George Shanthikumar ◽  
David D. Yao
Keyword(s):  
Queueing Networks ◽  
Stochastic Monotonicity ◽  
Jackson Networks ◽  
Monotonicity Properties

We develop a representation for general queueing networks, and study some stochastic monotonicity properties that have been previously established in Jackson networks (e.g. Shanthikumar and Yao (1986), (1987)).

Zero-sum games for continuous-time Markov chains with unbounded transition and average payoff rates

2003 ◽  
Vol 40 (2) ◽  
pp. 327-345 ◽  
Author(s):  
Xianping Guo ◽  
Onésimo Hernández-Lerma
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Transition Rates ◽  
Zero Sum Games ◽  
Stationary Strategies ◽  
Continuous Time Markov Chains ◽  
Average Payoff ◽  
Optimal Stationary Strategies ◽  
Zero Sum

This paper is a first study of two-person zero-sum games for denumerable continuous-time Markov chains determined by given transition rates, with an average payoff criterion. The transition rates are allowed to be unbounded, and the payoff rates may have neither upper nor lower bounds. In the spirit of the ‘drift and monotonicity’ conditions for continuous-time Markov processes, we give conditions on the controlled system's primitive data under which the existence of the value of the game and a pair of strong optimal stationary strategies is ensured by using the Shapley equations. Also, we present a ‘martingale characterization’ of a pair of strong optimal stationary strategies. Our results are illustrated with a birth-and-death game.

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