A note on the equilibrium M/G/1 queue length

1988 ◽  
Vol 25 (01) ◽  
pp. 228-231 ◽  
Author(s):  
Gordon E. Willmot
Keyword(s):  
Generating Function ◽  
Queue Length ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Infinite Divisibility ◽  
Queue Length Distribution ◽  
Service Distribution ◽  
Bulk Arrival ◽  
Finite Sum ◽  
The Right

This note concerns the distribution of the equilibrium M/G/1 queue length. A representation for the probability generating function is given which allows for an explicit finite sum representation of the associated probabilities. The radius of convergence of the probability generating function and an asymptotic formula for the right tail of the distribution also follow from this representation, as well as infinite divisibility of the queue-length distribution when the service distribution is infinitely divisible. Extension of these results to the bulk arrival case is straightforward.

A note on the equilibrium M/G/1 queue length

1988 ◽  
Vol 25 (1) ◽  
pp. 228-231 ◽  
Author(s):  
Gordon E. Willmot
Keyword(s):  
Generating Function ◽  
Queue Length ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Infinite Divisibility ◽  
Queue Length Distribution ◽  
Service Distribution ◽  
Bulk Arrival ◽  
Finite Sum ◽  
The Right

This note concerns the distribution of the equilibrium M/G/1 queue length. A representation for the probability generating function is given which allows for an explicit finite sum representation of the associated probabilities. The radius of convergence of the probability generating function and an asymptotic formula for the right tail of the distribution also follow from this representation, as well as infinite divisibility of the queue-length distribution when the service distribution is infinitely divisible. Extension of these results to the bulk arrival case is straightforward.

Traffic light queues as a generalization to queueing theory

1971 ◽  
Vol 8 (3) ◽  
pp. 480-493 ◽  
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.

On a non-Markovian time-dependent queueing system

1989 ◽  
Vol 26 (1) ◽  
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.

On a non-Markovian time-dependent queueing system

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.

Traffic light queues as a generalization to queueing theory

1971 ◽  
Vol 8 (03) ◽  
pp. 480-493 ◽  
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.

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

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.

Relationships and decomposition in the delayed bernoulli feedback queueing system

1988 ◽  
Vol 25 (1) ◽  
pp. 169-183 ◽  
Author(s):  
D. König ◽  
M. Miyazawa
Keyword(s):  
Generating Function ◽  
Queue Length ◽  
Queueing System ◽  
Length Distribution ◽  
Arrival Process ◽  
Stieltjes Transform ◽  
Bernoulli Feedback ◽  
Queue Length Distribution ◽  
Workload Distribution ◽  
Feedback Queue

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.

Relationships and decomposition in the delayed bernoulli feedback queueing system

1988 ◽  
Vol 25 (01) ◽  
pp. 169-183
Author(s):  
D. König ◽  
M. Miyazawa
Keyword(s):  
Generating Function ◽  
Queue Length ◽  
Queueing System ◽  
Length Distribution ◽  
Arrival Process ◽  
Stieltjes Transform ◽  
Bernoulli Feedback ◽  
Queue Length Distribution ◽  
Workload Distribution ◽  
Feedback Queue

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.

On the steady-state solution of the M/G/2 queue

1979 ◽  
Vol 11 (01) ◽  
pp. 240-255 ◽  
Author(s):  
Per Hokstad
Keyword(s):  
Steady State ◽  
General Solution ◽  
Laplace Transform ◽  
Queue Length ◽  
Joint Distribution ◽  
Length Distribution ◽  
Steady State Solution ◽  
Queue Length Distribution ◽  
The Difference ◽  
Number Of Customers

The asymptotic behaviour of the M/G/2 queue is studied. The difference-differential equations for the joint distribution of the number of customers present and of the remaining holding times for services in progress were obtained in Hokstad (1978a) (for M/G/m). In the present paper it is found that the general solution of these equations involves an arbitrary function. In order to decide which of the possible solutions is the answer to the queueing problem one has to consider the singularities of the Laplace transforms involved. When the service time has a rational Laplace transform, a method of obtaining the queue length distribution is outlined. For a couple of examples the explicit form of the generating function of the queue length is obtained.

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