Relationships and decomposition in the delayed bernoulli feedback queueing system

For the delayed Bernoulli feedback queue with first come–first served discipline under weak assumptions a relationship for the generating functions of the joint queue-length distribution at various points in time is given. A decomposition for the generating function of the stationary total queue length distribution has been proven. The Laplace-Stieltjes transform of the stationary joint workload distribution function is represented by its marginal distributions. The arrival process is Poisson, renewal or arbitrary stationary, respectively. The service times can form an i.i.d. sequence at each queue. Different kinds of product form of the generating function of the joint queue-length distribution are discussed.

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On a non-Markovian time-dependent queueing system

Journal of Applied Probability ◽

10.2307/3214324 ◽

1989 ◽

Vol 26 (1) ◽

pp. 142-151 ◽

Cited By ~ 2

Author(s):

S. D. Sharma

Keyword(s):

Steady State ◽

Generating Function ◽

Queue Length ◽

Queueing System ◽

Length Distribution ◽

Probability Generating Function ◽

Transient State ◽

Time Dependent ◽

Queue Length Distribution ◽

State Behaviour

This paper studies the transient and steady-state behaviour of a continuous and discrete-time queueing system with non-Markovian type of departure mechanism. The Laplace transforms of the probability generating function of the time-dependent queue length distribution in the transient state are obtained and the probability generating function of the queue length distribution in the steady state is derived therefrom. Finally, some particular cases are discussed.

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On a non-Markovian time-dependent queueing system

Journal of Applied Probability ◽

10.1017/s0021900200041875 ◽

1989 ◽

Vol 26 (01) ◽

pp. 142-151

Author(s):

S. D. Sharma

Keyword(s):

Steady State ◽

Generating Function ◽

Queue Length ◽

Queueing System ◽

Length Distribution ◽

Probability Generating Function ◽

Transient State ◽

Time Dependent ◽

Queue Length Distribution ◽

State Behaviour

This paper studies the transient and steady-state behaviour of a continuous and discrete-time queueing system with non-Markovian type of departure mechanism. The Laplace transforms of the probability generating function of the time-dependent queue length distribution in the transient state are obtained and the probability generating function of the queue length distribution in the steady state is derived therefrom. Finally, some particular cases are discussed.

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Upper bound for the decay rate of the joint queue-length distribution in a two-node Markovian queueing system

Queueing Systems ◽

10.1007/s11134-008-9066-9 ◽

2008 ◽

Vol 58 (3) ◽

pp. 161-189 ◽

Cited By ~ 4

Author(s):

Ken’ichi Katou ◽

Naoki Makimoto ◽

Yukio Takahashi

Keyword(s):

Decay Rate ◽

Upper Bound ◽

Queue Length ◽

Queueing System ◽

Length Distribution ◽

Queue Length Distribution

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Transient solutions for denumerable-state Markov processes

Journal of Applied Probability ◽

10.1017/s0021900200045228 ◽

1994 ◽

Vol 31 (03) ◽

pp. 635-645

Author(s):

Guang-Hui Hsu ◽

Xue-Ming Yuan

Keyword(s):

Markov Process ◽

Markov Processes ◽

Queue Length ◽

Queueing System ◽

Length Distribution ◽

Initial Distribution ◽

Numerical Results ◽

Transient Solution ◽

Queue Length Distribution ◽

Transient Solutions

The algorithm for the transient solution for the denumerable state Markov process with an arbitrary initial distribution is given in this paper. The transient queue length distribution for a general Markovian queueing system can be obtained by this algorithm. As examples, some numerical results are presented.

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A non-exponential queueing system with independent arrivals and batch servicing

Journal of Applied Probability ◽

10.1017/s0021900200038857 ◽

1990 ◽

Vol 27 (02) ◽

pp. 401-408

Author(s):

Nico M. Van Dijk ◽

Eric Smeitink

Keyword(s):

Queue Length ◽

Queueing System ◽

Length Distribution ◽

Service Systems ◽

Batch Service ◽

Repair System ◽

Theoretical Interest ◽

Form Expression ◽

Queue Length Distribution ◽

Service Times

We study a queueing system with a finite number of input sources. Jobs are individually generated by a source but wait to be served in batches, during which the input of that source is stopped. The service speed of a server depends on the mode of other sources and thus includes interdependencies. The input and service times are allowed to be generally distributed. A classical example is a machine repair system where the machines are subject to shocks causing cumulative damage. A product-form expression is obtained for the steady state joint queue length distribution and shown to be insensitive (i.e. to depend on only mean input and service times). The result is of both practical and theoretical interest as an extension of more standard batch service systems.

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Traffic light queues as a generalization to queueing theory

Journal of Applied Probability ◽

10.2307/3212172 ◽

1971 ◽

Vol 8 (3) ◽

pp. 480-493 ◽

Cited By ~ 1

Author(s):

Hisashi Mine ◽

Katsuhisa Ohno

Keyword(s):

Generating Function ◽

Discrete Time ◽

Queue Length ◽

Length Distribution ◽

Probability Generating Function ◽

Traffic Light ◽

Queue Length Distribution ◽

Mean Delay ◽

Stationary Queue Length ◽

Stationary Queue Length Distribution

Fixed-cycle traffic light queues have been investigated by probabilistic methods by many authors. Beckmann, McGuire and Winsten (1956) considered a discrete time queueing model with binomial arrivals and regular departure headways and derived a relation between the stationary mean delay per vehicle and the stationary mean queue-length at the beginning of a red period of the traffic light. Haight (1959) and Buckley and Wheeler (1964) considered models with Poisson arrivals and regular departure headways and investigated certain properties of the queue-length. Newell (1960) dealt with the model proposed by the first authors and obtained the probability generating function of the stationary queue-length distribution. Darroch (1964) discussed a more general discrete time model with stationary, independent arrivals and regular departure headways and derived a necessary and sufficient condition for the stationary queue-length distribution to exist and obtained its probability generating function. The above two authors, Little (1961), Miller (1963), Newell (1965), McNeil (1968), Siskind (1970) and others gave approximate expressions for the stationary mean delay per vehicle for fixed-cycle traffic light queues of various types. All of the authors mentioned above dealt with the queue-length.

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The BMAP/G/1 vacation queue with queue-length dependent vacation schedule

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s0334270000012479 ◽

1998 ◽

Vol 40 (2) ◽

pp. 207-221 ◽

Cited By ~ 7

Author(s):

Yang Woo Shin ◽

Chareles E. M. Pearce

Keyword(s):

Queue Length ◽

Length Distribution ◽

Arrival Process ◽

Service Discipline ◽

Single Server ◽

Server Vacation ◽

Batch Markovian Arrival Process ◽

Queue Length Distribution ◽

Special Cases ◽

Number Of Customers

AbstractWe treat a single-server vacation queue with queue-length dependent vacation schedules. This subsumes the single-server vacation queue with exhaustive service discipline and the vacation queue with Bernoulli schedule as special cases. The lengths of vacation times depend on the number of customers in the system at the beginning of a vacation. The arrival process is a batch-Markovian arrival process (BMAP). We derive the queue-length distribution at departure epochs. By using a semi-Markov process technique, we obtain the Laplace-Stieltjes transform of the transient queue-length distribution at an arbitrary time point and its limiting distribution

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Transient solutions for denumerable-state Markov processes

Journal of Applied Probability ◽

10.2307/3215144 ◽

1994 ◽

Vol 31 (3) ◽

pp. 635-645 ◽

Cited By ~ 5

Author(s):

Guang-Hui Hsu ◽

Xue-Ming Yuan

Keyword(s):

Markov Process ◽

Markov Processes ◽

Queue Length ◽

Queueing System ◽

Length Distribution ◽

Initial Distribution ◽

Numerical Results ◽

Transient Solution ◽

Queue Length Distribution ◽

Transient Solutions

The algorithm for the transient solution for the denumerable state Markov process with an arbitrary initial distribution is given in this paper. The transient queue length distribution for a general Markovian queueing system can be obtained by this algorithm. As examples, some numerical results are presented.

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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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