scholarly journals An asymptotic property of branching-type overloaded polling networks

2019 ◽  
Vol 17 (1) ◽  
pp. 1476-1489 ◽  
Author(s):  
Yuejiao Wang ◽  
Zaiming Liu ◽  
Yuqing Chu ◽  
Yingqiu Li
Keyword(s):  
Asymptotic Property ◽  
Branching Processes ◽  
Heavy Traffic ◽  
Natural Extension ◽  
Service Discipline ◽  
Polling System ◽  
Fluid Limit ◽  
Polling Model ◽  
Routing Policy ◽  
Service Disciplines

Abstract Remerova et al. [Random fluid limit of an overloaded polling model, Adv. Appl. Probab., 2014, 46, 76–101] studied the fluid asymptotics of the joint queue length process for an overloaded cyclic polling system with multigated service discipline by exploiting the connection with multi-type branching processes. In contrast to the heavy traffic behaviors, the cycle time of the overloaded polling system increases by a deterministic times over times under passage to the fluid dynamics and the fluid limit preserves some randomness. The present paper aims to extend the overloaded asymptotics in Remerova et al. [Random fluid limit of an overloaded polling model, Adv. Appl. Probab., 2014, 46, 76–101] to the corresponding polling system with general branching-type service disciplines and customer re-routing policy. A unifying overloaded asymptotic property is derived. Due to the exhaustiveness, the property is a natural extension of the classical polling model with multigated service discipline in Remerova et al. [Random fluid limit of an overloaded polling model, Adv. Appl. Probab., 2014, 46, 76–101] and provides new exact results that have not been observed before for rerouting policy. Additionally, a stochastic simulation is undertaken for the validation of the fluid limit and the optimization of the gating indexes to minimize the total population is considered as an example to demonstrate the usefulness of the random fluid limit.

scholarly journals A Globally Gated Polling System with Server Interruptions, and Applications to the Repairman Problem

1993 ◽  
Vol 7 (2) ◽  
pp. 187-208 ◽  
Author(s):  
O. J. Boxma ◽  
J. A. Weststrate ◽  
U. Yechiali
Keyword(s):  
Preventive Maintenance ◽  
Waiting Times ◽  
Optimal Operation ◽  
Service Discipline ◽  
Single Server ◽  
Polling System ◽  
Corrective Maintenance ◽  
Polling Model ◽  
Real World Problem ◽  
First Moment

A repair crew is responsible for the maintenance and operation of N installations. The crew has to perform a collection of preventive maintenance tasks at the various installations. The installations may break down from time to time, generating corrective maintenance requests which have priority over the preventive maintenance tasks. We formulate and analyze this real-world problem as a single-server multi-queue polling model with Globally Gated service discipline and with server interruptions. We derive closed-form expressions for the Laplace-Stieltjes Transform and the first moment of the waiting time distributions of the preventive and corrective maintenance requests at the various installations, and obtain simple and easily implementable static and dynamic rules for optimal operation of the system. We further show that, for the socalled elevator-type polling scheme, mean waiting times of preventive maintenance jobs at all installations are equal.

Expected delay analysis of polling systems in heavy traffic

1998 ◽  
Vol 30 (02) ◽  
pp. 586-602 ◽  
Author(s):  
R. D. van der Mei ◽  
H. Levy
Keyword(s):  
Heavy Traffic ◽  
Waiting Times ◽  
Delay Analysis ◽  
Polling Systems ◽  
Heavy Load ◽  
Polling Model ◽  
Mean Delay ◽  
Service Disciplines ◽  
The Mean ◽  
Cyclic Polling

We study the expected delay in a cyclic polling model with mixtures of exhaustive and gated service in heavy traffic. We obtain closed-form expressions for the mean delay under standard heavy-traffic scalings, providing new insights into the behaviour of polling systems in heavy traffic. The results lead to excellent approximations of the expected waiting times in practical heavy-load scenarios and moreover, lead to new results for optimizing the system performance with respect to the service disciplines.

Expected delay analysis of polling systems in heavy traffic

1998 ◽  
Vol 30 (2) ◽  
pp. 586-602 ◽  
Author(s):  
R. D. van der Mei ◽  
H. Levy
Keyword(s):  
Heavy Traffic ◽  
Waiting Times ◽  
Delay Analysis ◽  
Polling Systems ◽  
Heavy Load ◽  
Polling Model ◽  
Mean Delay ◽  
Service Disciplines ◽  
The Mean ◽  
Cyclic Polling

We study the expected delay in a cyclic polling model with mixtures of exhaustive and gated service in heavy traffic. We obtain closed-form expressions for the mean delay under standard heavy-traffic scalings, providing new insights into the behaviour of polling systems in heavy traffic. The results lead to excellent approximations of the expected waiting times in practical heavy-load scenarios and moreover, lead to new results for optimizing the system performance with respect to the service disciplines.

Heavy traffic analysis for continuous polling models

1997 ◽  
Vol 34 (03) ◽  
pp. 720-732 ◽  
Author(s):  
Dirk P. Kroese
Keyword(s):  
Steady State ◽  
Gamma Distribution ◽  
Branching Processes ◽  
Heavy Traffic ◽  
Traffic Analysis ◽  
Polling System ◽  
Polling Models ◽  
Heavy Traffic Analysis ◽  
Age Dependent ◽  
The Relationship

We consider a continuous polling system in heavy traffic. Using the relationship between such systems and age-dependent branching processes, we show that the steady-state number of waiting customers in heavy traffic has approximately a gamma distribution. Moreover, given their total number, the configuration of these customers is approximately deterministic.

POLLING SYSTEMS WITH TWO-PHASE GATED SERVICE

2008 ◽  
Vol 22 (4) ◽  
pp. 623-651 ◽  
Author(s):  
R. D. van der Mei ◽  
J. A. C. Resing
Keyword(s):  
Waiting Time ◽  
Branching Processes ◽  
Heavy Traffic ◽  
Polling Systems ◽  
Polling System ◽  
Two Phase ◽  
General Parameter ◽  
Multitype Branching Processes ◽  
Poisson Arrivals ◽  
Gated Service

We study an asymmetric cyclic polling system with Poisson arrivals, general service-time and switch-over time distributions, and so-called two-phase gated service at each queue, an interleaving scheme that aims to enforce some level of “fairness” among the different customer classes. For this model, we use the classical theory of multitype branching processes to derive closed-form expressions for the Laplace–Stieltjes transform of the waiting-time distributions when the load tends to 1, in a general parameter setting and under proper heavy-traffic scalings. This result is strikingly simple and provides new insights in the behavior of two-phase polling systems. In particular, the result provides insight in the waiting-time performance and the trade-off between efficiency and fairness of two-phase gated polling compared to the classical one-phase gated service policy.

scholarly journals Random Fluid Limit of an Overloaded Polling Model

2014 ◽  
Vol 46 (1) ◽  
pp. 76-101 ◽  
Author(s):  
Maria Remerova ◽  
Sergey Foss ◽  
Bert Zwart
Keyword(s):  
Queue Length ◽  
Busy Period ◽  
Branching Processes ◽  
Oscillatory Behavior ◽  
Time Interval ◽  
Finite Time Interval ◽  
Fluid Limit ◽  
Polling Model ◽  
Multitype Branching Processes ◽  
Cyclic Polling

In the present paper, we study the evolution of an overloaded cyclic polling model that starts empty. Exploiting a connection with multitype branching processes, we derive fluid asymptotics for the joint queue length process. Under passage to the fluid dynamics, the server switches between the queues infinitely many times in any finite time interval causing frequent oscillatory behavior of the fluid limit in the neighborhood of zero. Moreover, the fluid limit is random. In addition, we suggest a method that establishes finiteness of moments of the busy period in an M/G/1 queue.

scholarly journals Random Fluid Limit of an Overloaded Polling Model

2014 ◽  
Vol 46 (01) ◽  
pp. 76-101 ◽  
Author(s):  
Maria Remerova ◽  
Sergey Foss ◽  
Bert Zwart
Keyword(s):  
Queue Length ◽  
Busy Period ◽  
Branching Processes ◽  
Oscillatory Behavior ◽  
Time Interval ◽  
Finite Time Interval ◽  
Fluid Limit ◽  
Polling Model ◽  
Multitype Branching Processes ◽  
Cyclic Polling

In the present paper, we study the evolution of an overloaded cyclic polling model that starts empty. Exploiting a connection with multitype branching processes, we derive fluid asymptotics for the joint queue length process. Under passage to the fluid dynamics, the server switches between the queues infinitely many times in any finite time interval causing frequent oscillatory behavior of the fluid limit in the neighborhood of zero. Moreover, the fluid limit is random. In addition, we suggest a method that establishes finiteness of moments of the busy period in an M/G/1 queue.

Heavy traffic analysis for continuous polling models

1997 ◽  
Vol 34 (3) ◽  
pp. 720-732 ◽  
Author(s):  
Dirk P. Kroese
Keyword(s):  
Steady State ◽  
Gamma Distribution ◽  
Branching Processes ◽  
Heavy Traffic ◽  
Traffic Analysis ◽  
Polling System ◽  
Polling Models ◽  
Heavy Traffic Analysis ◽  
Age Dependent ◽  
The Relationship

We consider a continuous polling system in heavy traffic. Using the relationship between such systems and age-dependent branching processes, we show that the steady-state number of waiting customers in heavy traffic has approximately a gamma distribution. Moreover, given their total number, the configuration of these customers is approximately deterministic.

Analysis of Single Buffer Random Polling System With State-Dependent Input Process and Server/Station Breakdowns

2018 ◽  
Vol 9 (1) ◽  
pp. 22-50 ◽  
Author(s):  
Thomas Y.S. Lee
Keyword(s):  
Steady State ◽  
Linear Model ◽  
Repair Process ◽  
Polling System ◽  
Steady State Analysis ◽  
Relaxation System ◽  
Polling Model ◽  
State Dependent ◽  
Non Linear ◽  
State Analysis

Models and analytical techniques are developed to evaluate the performance of two variations of single buffers (conventional and buffer relaxation system) multiple queues system. In the conventional system, each queue can have at most one customer at any time and newly arriving customers find the buffer full are lost. In the buffer relaxation system, the queue being served may have two customers, while each of the other queues may have at most one customer. Thomas Y.S. Lee developed a state-dependent non-linear model of uncertainty for analyzing a random polling system with server breakdown/repair, multi-phase service, correlated input processes, and single buffers. The state-dependent non-linear model of uncertainty introduced in this paper allows us to incorporate correlated arrival processes where the customer arrival rate depends on the location of the server and/or the server's mode of operation into the polling model. The author allows the possibility that the server is unreliable. Specifically, when the server visits a queue, Lee assumes that the system is subject to two types of failures: queue-dependent, and general. General failures are observed upon server arrival at a queue. But there are two possibilities that a queue-dependent breakdown (if occurs) can be observed; (i) is observed immediately when it occurs and (ii) is observed only at the end of the current service. In both cases, a repair process is initiated immediately after the queue-dependent breakdown is observed. The author's model allows the possibility of the server breakdowns/repair process to be non-stationary in the number of breakdowns/repairs to reflect that breakdowns/repairs or customer processing may be progressively easier or harder, or that they follow a more general learning curve. Thomas Y.S. Lee will show that his model encompasses a variety of examples. He was able to perform both transient and steady state analysis. The steady state analysis allows us to compute several performance measures including the average customer waiting time, loss probability, throughput and mean cycle time.

On a relationship between processor-sharing queues and Crump–Mode–Jagers branching processes

1992 ◽  
Vol 24 (3) ◽  
pp. 653-698 ◽  
Author(s):  
Sergei Grishechkin
Keyword(s):  
Processing Time ◽  
Branching Processes ◽  
Heavy Traffic ◽  
Processor Sharing ◽  
Batch Arrivals ◽  
Processor Sharing Queues ◽  
Residual Processing

The M/G/1 queue with batch arrivals and a queueing discipline which is a generalization of processor sharing is studied by means of Crump–Mode–Jagers branching processes. A number of theorems are proved, including investigation of heavy traffic and overloaded queues. Most of the results obtained are also new for the M/G/1 queue with processor sharing. By use of a limiting procedure we also derive new results concerning M/G/1 queues with shortest residual processing time discipline.

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