On the M/G/1 queue by additional inputs

A single-server queueing system is studied, the input into which consists of the sum of two independent stochastic processes. One of these is an ‘M/G' type input process, the other a much more general process which need not be Markov. There are two types of busy period, depending on which arrival process started the busy period. Stochastic monotonicity results are derived and it is found that under a stationarity-like condition the probability of being in a busy period which started with an ‘M/G' arrival is independent of time and is the same it would be with the ‘M/G' process as only input process. Also, distributional results are obtained for the virtual waiting-time process, and these results are used to reduce the study of a single-server queueing system with as input the sum of independent ‘M/G' and ‘GI/G' input streams to the study of a related GI/G/1 queueing system.The purpose of this paper is to pave the way for a study of an M/G/1 queueing system with periodic arrivals of additional work, and for optimal scheduling of maintenance processes in certain real-time computer systems.

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On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals

Journal of Applied Probability ◽

10.2307/3212173 ◽

1971 ◽

Vol 8 (3) ◽

pp. 494-507 ◽

Cited By ~ 34

Author(s):

E. K. Kyprianou

Keyword(s):

Waiting Time ◽

Busy Period ◽

Queueing System ◽

Random Variable ◽

Single Server ◽

Virtual Waiting Time ◽

Waiting Time Process ◽

Poisson Arrivals ◽

Image Position ◽

Quasi Stationary Distribution

We consider a single server queueing system M/G/1 in which customers arrive in a Poisson process with mean λt, and the service time has distribution dB(t), 0 < t < ∞. Let W(t) be the virtual waiting time process, i.e., the time that a potential customer arriving at the queueing system at time t would have to wait before beginning his service. We also let the random variable denote the first busy period initiated by a waiting time u at time t = 0.

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On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals

Journal of Applied Probability ◽

10.1017/s0021900200035580 ◽

1971 ◽

Vol 8 (03) ◽

pp. 494-507 ◽

Cited By ~ 4

Author(s):

E. K. Kyprianou

Keyword(s):

Waiting Time ◽

Busy Period ◽

Queueing System ◽

Random Variable ◽

Single Server ◽

Virtual Waiting Time ◽

Waiting Time Process ◽

Poisson Arrivals ◽

Image Position ◽

Quasi Stationary Distribution

We consider a single server queueing system M/G/1 in which customers arrive in a Poisson process with mean λt, and the service time has distribution dB(t), 0 < t < ∞. Let W(t) be the virtual waiting time process, i.e., the time that a potential customer arriving at the queueing system at time t would have to wait before beginning his service. We also let the random variable denote the first busy period initiated by a waiting time u at time t = 0.

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A limit theorem for priority queues in heavy traffic

Journal of Applied Probability ◽

10.1017/s0021900200096133 ◽

1973 ◽

Vol 10 (04) ◽

pp. 907-912 ◽

Cited By ~ 2

Author(s):

J. Michael Harrison

Keyword(s):

Limit Theorem ◽

Queueing System ◽

Heavy Traffic ◽

Critical Value ◽

Single Server ◽

Traffic Condition ◽

Transient Distribution ◽

Virtual Waiting Time ◽

Waiting Time Process ◽

Priority Queueing

A single server, two priority queueing system is studied under the heavy traffic condition where the system traffic intensity is either at or near its critical value. An approximation is developed for the transient distribution of the low priority customers' virtual waiting time process. This result is stated formally as a limit theorem involving a sequence of systems whose traffic intensities approach the critical value.

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A limit theorem for priority queues in heavy traffic

Journal of Applied Probability ◽

10.2307/3212396 ◽

1973 ◽

Vol 10 (4) ◽

pp. 907-912 ◽

Cited By ~ 9

Author(s):

J. Michael Harrison

Keyword(s):

Limit Theorem ◽

Queueing System ◽

Heavy Traffic ◽

Critical Value ◽

Single Server ◽

Traffic Condition ◽

Transient Distribution ◽

Virtual Waiting Time ◽

Waiting Time Process ◽

Priority Queueing

A single server, two priority queueing system is studied under the heavy traffic condition where the system traffic intensity is either at or near its critical value. An approximation is developed for the transient distribution of the low priority customers' virtual waiting time process. This result is stated formally as a limit theorem involving a sequence of systems whose traffic intensities approach the critical value.

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Single-server queues with impatient customers

Advances in Applied Probability ◽

10.2307/1427345 ◽

1984 ◽

Vol 16 (4) ◽

pp. 887-905 ◽

Cited By ~ 123

Author(s):

F. Baccelli ◽

P. Boyer ◽

G. Hebuterne

Keyword(s):

Waiting Time ◽

Queueing System ◽

Distribution Functions ◽

Single Server ◽

Waiting Time Distribution ◽

Input Process ◽

Poisson Input ◽

Virtual Waiting Time ◽

Random Threshold ◽

The Stability

We consider a single-server queueing system in which a customer gives up whenever his waiting time is larger than a random threshold, his patience time. In the case of a GI/GI/1 queue with i.i.d. patience times, we establish the extensions of the classical GI/GI/1 formulae concerning the stability condition and the relation between actual and virtual waiting-time distribution functions. We also prove that these last two distribution functions coincide in the case of a Poisson input process and determine their common law.

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Single-server queues with impatient customers

Advances in Applied Probability ◽

10.1017/s0001867800022989 ◽

1984 ◽

Vol 16 (04) ◽

pp. 887-905 ◽

Cited By ~ 14

Author(s):

F. Baccelli ◽

P. Boyer ◽

G. Hebuterne

Keyword(s):

Waiting Time ◽

Queueing System ◽

Distribution Functions ◽

Single Server ◽

Waiting Time Distribution ◽

Input Process ◽

Poisson Input ◽

Virtual Waiting Time ◽

Random Threshold ◽

The Stability

We consider a single-server queueing system in which a customer gives up whenever his waiting time is larger than a random threshold, his patience time. In the case of a GI/GI/1 queue with i.i.d. patience times, we establish the extensions of the classical GI/GI/1 formulae concerning the stability condition and the relation between actual and virtual waiting-time distribution functions. We also prove that these last two distribution functions coincide in the case of a Poisson input process and determine their common law.

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Single server queues with modified service mechanisms

Journal of the Australian Mathematical Society ◽

10.1017/s144678870002749x ◽

1962 ◽

Vol 2 (4) ◽

pp. 499-507 ◽

Cited By ~ 32

Author(s):

G. F. Yeo

Keyword(s):

Time Distribution ◽

Busy Period ◽

Queueing System ◽

Probability Generating Function ◽

Single Server ◽

Waiting Time Distribution ◽

Period Distribution ◽

Time Dependent Problem ◽

Stationary Waiting Time ◽

Special Case

SummaryThis paper considers a generalisation of the queueing system M/G/I, where customers arriving at empty and non-empty queues have different service time distributions. The characteristic function (c.f.) of the stationary waiting time distribution and the probability generating function (p.g.f.) of the queue size are obtained. The busy period distribution is found; the results are generalised to an Erlangian inter-arrival distribution; the time-dependent problem is considered, and finally a special case of server absenteeism is discussed.

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An Uncertain Single-Server Queueing Model

Journal of Uncertain Systems ◽

10.1142/s175289092150001x ◽

2021 ◽

pp. 2150001

Author(s):

Kai Yao

Keyword(s):

Queueing Theory ◽

Busy Period ◽

Queueing System ◽

Single Server ◽

Queueing Model ◽

Uncertainty Distribution ◽

Uncertain Variables ◽

Service Efficiency ◽

Service Times ◽

Interarrival Times

In the queueing theory, the interarrival times between customers and the service times for customers are usually regarded as random variables. This paper considers human uncertainty in a queueing system, and proposes an uncertain queueing model in which the interarrival times and the service times are regarded as uncertain variables. The busyness index is derived analytically which indicates the service efficiency of a queueing system. Besides, the uncertainty distribution of the busy period is obtained.

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On a property of a refusals stream

Journal of Applied Probability ◽

10.1017/s0021900200101469 ◽

1997 ◽

Vol 34 (03) ◽

pp. 800-805 ◽

Cited By ~ 5

Author(s):

Vyacheslav M. Abramov

Keyword(s):

Waiting Time ◽

Service Time ◽

Busy Period ◽

Elementary Proof ◽

Queueing System ◽

Queueing Systems ◽

Single Server ◽

Wide Family

This paper consists of two parts. The first part provides a more elementary proof of the asymptotic theorem of the refusals stream for an M/GI/1/n queueing system discussed in Abramov (1991a). The central property of the refusals stream discussed in the second part of this paper is that, if the expectations of interarrival and service time of an M/GI/1/n queueing system are equal to each other, then the expectation of the number of refusals during a busy period is equal to 1. This property is extended for a wide family of single-server queueing systems with refusals including, for example, queueing systems with bounded waiting time.

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