Analysis of an M^([X])/G(a,b)/1 Unreliable G-queue with Loss, Instantaneous Bernoulli Feedback, Vacation and Two Delays of Verification

This paper deals with a batch arrival that customers arrive to the system according to a compound Poisson process. The customer’s behavior is incorporated according to which loss with a certain probability and the server begins to provide a service only when a queue size minimum say ‘a’ and maximum service capacity is ‘b’. Once the server completes the service, the unsatisfied customers may get the same service under Bernoulli schedule is termed as instantaneous Bernoulli feedback. The occurrence of negative customer cause the server to fail and removes a group of customers or an amount of work if present upon its arrival. As soon as the failure instant, the service channel send to the two delays of verification, the first verification delay starts before the repair process and the second verification delay begins after the repair process. We use the generating function method to derive the stationary queue size distribution. Some important performance measures such as different states of the system and the expected length of the queue explicitly. Some important special cases and numerical examples are determined.

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R&D projects analyzed by semimartingale methods

Journal of Applied Probability ◽

10.1017/s0021900200037761 ◽

1985 ◽

Vol 22 (02) ◽

pp. 288-299 ◽

Cited By ~ 3

Author(s):

Knut K. Aase

Keyword(s):

Brownian Motion ◽

Stochastic Equation ◽

Compound Poisson Process ◽

General Nature ◽

Decision Variable ◽

Non Linear Equation ◽

Special Cases ◽

Dynamic Stochastic ◽

Present Setting

In this article we examine R&D projects where the project status changes according to a general dynamic stochastic equation. This allows for both continuous and jump behavior of the project status. The time parameter is continuous. The decision variable includes a non-stationary resource expenditure strategy and a stopping policy which determines when the project should be terminated. Characterization of stationary policies becomes straightforward in the present setting. A non-linear equation is determined for the expected discounted return from the project. This equation, which is of a very general nature, has been considered in certain special cases, where it becomes manageable. The examples include situations where the project status changes according to a compound Poisson process, a geometric Brownian motion, and a Brownian motion with drift. In those cases we demonstrate how the exact solution can be obtained and the optimal policy found.

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Finite Buffer GI/M(n)/1 Queue with Bernoulli-Schedule Vacation Interruption under N-Policy

International Scholarly Research Notices ◽

10.1155/2014/392317 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8

Author(s):

P. Vijaya Laxmi ◽

V. Suchitra

Keyword(s):

Performance Measures ◽

Finite Buffer ◽

Supplementary Variable ◽

Working Vacation ◽

Variable Technique ◽

Special Cases ◽

Vacation Interruption ◽

System Length ◽

Vacation Period ◽

Bernoulli Schedule

We study a finite buffer N-policy GI/M(n)/1 queue with Bernoulli-schedule vacation interruption. The server works with a slower rate during vacation period. At a service completion epoch during working vacation, if there are at least N customers present in the queue, the server interrupts vacation and otherwise continues the vacation. Using the supplementary variable technique and recursive method, we obtain the steady state system length distributions at prearrival and arbitrary epochs. Some special cases of the model, various performance measures, and cost analysis are discussed. Finally, parameter effect on the performance measures of the model is presented through numerical computations.

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An N-policy discrete-time Geo/G/1 queue with modified multiple server vacations and Bernoulli feedback*

RAIRO - Operations Research ◽

10.1051/ro/2017027 ◽

2019 ◽

Vol 53 (2) ◽

pp. 367-387

Author(s):

Shaojun Lan ◽

Yinghui Tang

Keyword(s):

Discrete Time ◽

Queue Length ◽

Length Distribution ◽

Probability Generating Function ◽

Renewal Theory ◽

Function Technique ◽

Single Server ◽

Bernoulli Feedback ◽

Queue Length Distribution ◽

Special Cases

This paper deals with a single-server discrete-time Geo/G/1 queueing model with Bernoulli feedback and N-policy where the server leaves for modified multiple vacations once the system becomes empty. Applying the law of probability decomposition, the renewal theory and the probability generating function technique, we explicitly derive the transient queue length distribution as well as the recursive expressions of the steady-state queue length distribution. Especially, some corresponding results under special cases are directly obtained. Furthermore, some numerical results are provided for illustrative purposes. Finally, a cost optimization problem is numerically analyzed under a given cost structure.

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Filtering of Markov renewal queues, I: Feedback queues

Advances in Applied Probability ◽

10.1017/s0001867800021212 ◽

1983 ◽

Vol 15 (02) ◽

pp. 349-375 ◽

Cited By ~ 3

Author(s):

Jeffrey J. Hunter

Keyword(s):

Queue Length ◽

Queueing Systems ◽

Renewal Processes ◽

Focus Attention ◽

Bernoulli Feedback ◽

State Dependent ◽

Markov Renewal Processes ◽

Special Cases ◽

Filtering Procedure ◽

Feedback Queues

Queueing systems which can be formulated as Markov renewal processes with basic transitions of three types, ‘arrivals', ‘departures' and ‘feedbacks' are examined. The filtering procedure developed for Markov renewal processes by Çinlar (1969) is applied to such queueing models to show that the queue-length processes embedded at any of the ‘arrival', ‘departure', ‘feedback', ‘input', ‘output' or ‘external' transition epochs are also Markov renewal. In this part we focus attention on the derivation of stationary and limiting distributions (when they exist) for each of the embedded discrete-time processes, the embedded Markov chains. These results are applied to birth–death queues with instantaneous state-dependent feedback including the special cases of M/M/1/N and M/M/1 queues with instantaneous Bernoulli feedback.

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DYNAMIC VISIT-ORDER RULES FOR BATCH-SERVICE POLLING

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964803173044 ◽

2003 ◽

Vol 17 (3) ◽

pp. 351-367 ◽

Cited By ~ 23

Author(s):

Jan van der Wal ◽

Uri Yechiali

Keyword(s):

Dynamic Control ◽

Performance Measure ◽

Polling Systems ◽

Current State ◽

Special Cases ◽

Expected Length ◽

Nonhomogeneous System ◽

Hard Problems ◽

Fixed Order ◽

Cyclic Type

We explore visit-order policies in nonsymmetric polling systems with switch-in and switch-out times, where service is in batches of unlimited size. We concentrate on so-called “Hamiltonian tour” policies in which, in order to give a fair treatment to the various users, the server attends every nonempty queue exactly once during each round of visits (cycle). The server dynamically generates a new visit schedule at the start of each round, depending on the current state of the system (number of jobs in each queue) and on the various nonhomogeneous system parameters. We consider three service regimes, globally gated, (locally) gated, and exhaustive, and study three different performance measures: (1) minimizing the expected weighted sum of all sojourn times of jobs within a cycle; (2) minimizing the expected length of the next cycle, and (3) maximizing the expected weighted throughput in a cycle. For each combination of performance measure and service regime, we derive characteristics of the optimal Hamiltonian tour. Some of the resulting optimal policies are shown to be elegant index-type rules. Others are the solutions of deterministic NP-hard problems. Special cases are reduced to assignment problems with specific cost matrices. The index-type rules can further be used to construct fixed-order, cyclic-type polling tables in cases where dynamic control is not applicable.

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Filtering of Markov renewal queues, III: semi-Markov processes embedded in feedback queues

Advances in Applied Probability ◽

10.2307/1427077 ◽

1984 ◽

Vol 16 (2) ◽

pp. 422-436 ◽

Cited By ~ 6

Author(s):

Jeffrey J. Hunter

Keyword(s):

Markov Processes ◽

Queue Length ◽

Markov Renewal Process ◽

Bernoulli Feedback ◽

State Dependent ◽

Special Cases ◽

Transition Points ◽

Semi Markov Processes ◽

Feedback Queues ◽

Birth Death

In Part I (Hunter) a study of feedback queueing models was initiated. For such models the queue-length process embedded at all transition points was formulated as a Markov renewal process (MRP). This led to the observation that the queue-length processes embedded at any of the ‘arrival', ‘departure', ‘feedback', ‘input', ‘output' or ‘external' transition epochs are also MRP. Part I concentrated on the properties of the embedded discrete-time Markov chains. In this part we examine the semi-Markov processes associated with each of these embedded MRP and derive expressions for the stationary distributions associated with their irreducible subspaces. The special cases of birth-death queues with instantaneous state-dependent feedback, M/M/1/N and M/M/1 queues with instantaneous Bernoulli feedback are considered in detail. The results obtained complement those derived in Part II (Hunter) for birth-death queues without feedback.

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Compound Poisson Process with a Poisson Subordinator

Journal of Applied Probability ◽

10.1239/jap/1437658603 ◽

2015 ◽

Vol 52 (2) ◽

pp. 360-374 ◽

Cited By ~ 11

Author(s):

Antonio Di Crescenzo ◽

Barbara Martinucci ◽

Shelemyahu Zacks

Keyword(s):

Poisson Process ◽

Compound Poisson Process ◽

Bell Polynomials ◽

Compound Poisson ◽

Convergence Results ◽

Special Cases ◽

Time Problem ◽

Crossing Time ◽

First Crossing Time ◽

Time Density

A compound Poisson process whose randomized time is an independent Poisson process is called a compound Poisson process with Poisson subordinator. We provide its probability distribution, which is expressed in terms of the Bell polynomials, and investigate in detail both the special cases in which the compound Poisson process has exponential jumps and normal jumps. Then for the iterated Poisson process we discuss some properties and provide convergence results to a Poisson process. The first-crossing time problem for the iterated Poisson process is finally tackled in the cases of (i) a decreasing and constant boundary, where we provide some closed-form results, and (ii) a linearly increasing boundary, where we propose an iterative procedure to compute the first-crossing time density and survival functions.

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A Batch Arrival Single Server Queue with Server Providing General Service in Two Fluctuating Modes and Reneging during Vacation and Breakdowns

Journal of Probability and Statistics ◽

10.1155/2014/319318 ◽

2014 ◽

Vol 2014 ◽

pp. 1-12

Author(s):

Monita Baruah ◽

Kailash C. Madan ◽

Tillal Eldabi

Keyword(s):

Repair Process ◽

Batch Arrival ◽

Single Server ◽

Single Server Queue ◽

Numerical Illustration ◽

Random Breakdown ◽

Special Cases ◽

Server Queue ◽

Service Rates ◽

General Service

We study the behavior of a batch arrival queuing system equipped with a single server providing general arbitrary service to customers with different service rates in two fluctuating modes of service. In addition, the server is subject to random breakdown. As soon as the server faces breakdown, the customer whose service is interrupted comes back to the head of the queue. As soon as repair process of the server is complete, the server immediately starts providing service in mode 1. Also customers waiting for service may renege (leave the queue) when there is breakdown or when server takes vacation. The system provides service with complete or reduced efficiency due to the fluctuating rates of service. We derive the steady state queue size distribution. Some special cases are discussed and numerical illustration is provided to see the effect and validity of the results.

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