The mean waiting time in a G/G/m/∞ queue with the LCFS-P service discipline

1991 ◽  
Vol 23 (2) ◽  
pp. 406-428 ◽  
Author(s):  
Gunter Ritter ◽  
Ulrich Wacker
Keyword(s):  
Waiting Time ◽  
Service Discipline ◽  
Mixing Condition ◽  
Input Stream ◽  
Light Traffic ◽  
Preemptive Resume ◽  
Mean Waiting Time ◽  
The Mean ◽  
Number Of Customers ◽  
Busy Cycle

A single- or multiserver queue with work-conserving service discipline and a stationary and ergodic input stream with bounded service times and arbitrarily light traffic intensity may have infinite mean waiting time. We give an example of this paradox and we also give a mixing condition which, in the case of the preemptive-resume LCFS discipline, excludes this phenomenon. Furthermore, the same methods allow to estimate the durations of the first busy period and cycle and the number of customers served in the first busy cycle of a work-conserving queue.

The mean waiting time in a G/G/m/∞ queue with the LCFS-P service discipline

1991 ◽  
Vol 23 (02) ◽  
pp. 406-428
Author(s):  
Gunter Ritter ◽  
Ulrich Wacker
Keyword(s):  
Waiting Time ◽  
Service Discipline ◽  
Mixing Condition ◽  
Input Stream ◽  
Light Traffic ◽  
Preemptive Resume ◽  
Mean Waiting Time ◽  
The Mean ◽  
Number Of Customers ◽  
Busy Cycle

A single- or multiserver queue with work-conserving service discipline and a stationary and ergodic input stream with bounded service times and arbitrarily light traffic intensity may have infinite mean waiting time. We give an example of this paradox and we also give a mixing condition which, in the case of the preemptive-resume LCFS discipline, excludes this phenomenon. Furthermore, the same methods allow to estimate the durations of the first busy period and cycle and the number of customers served in the first busy cycle of a work-conserving queue.

On a property of the variance of the waiting time of a queue

1968 ◽  
Vol 5 (3) ◽  
pp. 702-703 ◽  
Author(s):  
D. G. Tambouratzis
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Mean Waiting Time ◽  
Queue Discipline ◽  
The Mean ◽  
Number Of Customers

In this note, we consider a queueing system under any discipline which does not affect the distribution of the number of customers in the queue at any time. We shall show that the variance of the waiting time is a maximum when the queue discipline is “last come, first served”. This result complements that of Kingman [1] who showed that, under the same assumptions, the mean waiting time is independent of the queue discipline and the variance of the waiting time is a minimum when the customers are served in the order of their arrival.

The mean waiting time of a GI/G/1 queue in light traffic via random thinning

1995 ◽  
Vol 32 (01) ◽  
pp. 256-266
Author(s):  
Soracha Nananukul ◽  
Wei-Bo Gong
Keyword(s):  
Waiting Time ◽  
Padé Approximation ◽  
Radius Of Convergence ◽  
Arrival Process ◽  
Maclaurin Series ◽  
Light Traffic ◽  
Numerical Examples ◽  
Mean Waiting Time ◽  
The Mean ◽  
Random Thinning

In this paper, we derive the MacLaurin series of the mean waiting time in light traffic for a GI/G/1 queue. The light traffic is defined by random thinning of the arrival process. The MacLaurin series is derived with respect to the admission probability, and we prove that it has a positive radius of convergence. In the numerical examples, we use the MacLaurin series to approximate the mean waiting time beyond light traffic by means of Padé approximation.

The mean waiting time of a GI/G/1 queue in light traffic via random thinning

1995 ◽  
Vol 32 (1) ◽  
pp. 256-266 ◽  
Author(s):  
Soracha Nananukul ◽  
Wei-Bo Gong
Keyword(s):  
Waiting Time ◽  
Padé Approximation ◽  
Radius Of Convergence ◽  
Arrival Process ◽  
Maclaurin Series ◽  
Light Traffic ◽  
Numerical Examples ◽  
Mean Waiting Time ◽  
The Mean ◽  
Random Thinning

In this paper, we derive the MacLaurin series of the mean waiting time in light traffic for a GI/G/1 queue. The light traffic is defined by random thinning of the arrival process. The MacLaurin series is derived with respect to the admission probability, and we prove that it has a positive radius of convergence. In the numerical examples, we use the MacLaurin series to approximate the mean waiting time beyond light traffic by means of Padé approximation.

On a property of the variance of the waiting time of a queue

1968 ◽  
Vol 5 (03) ◽  
pp. 702-703 ◽  
Author(s):  
D. G. Tambouratzis
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Mean Waiting Time ◽  
Queue Discipline ◽  
The Mean ◽  
Number Of Customers

In this note, we consider a queueing system under any discipline which does not affect the distribution of the number of customers in the queue at any time. We shall show that the variance of the waiting time is a maximum when the queue discipline is “last come, first served”. This result complements that of Kingman [1] who showed that, under the same assumptions, the mean waiting time is independent of the queue discipline and the variance of the waiting time is a minimum when the customers are served in the order of their arrival.

scholarly journals Analysis of MAP/PH(1), PH(2)/2 Queue with Bernoulli Vacations

2008 ◽  
Vol 2008 ◽  
pp. 1-20 ◽  
Author(s):  
B. Krishna Kumar ◽  
R. Rukmani ◽  
V. Thangaraj
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Queueing System ◽  
Arrival Process ◽  
Waiting Time Distribution ◽  
Steady State Probability ◽  
Matrix Geometric Method ◽  
Mean Waiting Time ◽  
The Mean ◽  
Number Of Customers

We consider a two-heterogeneous-server queueing system with Bernoulli vacation in which customers arrive according to a Markovian arrival process (MAP). Servers returning from vacation immediately take another vacation if no customer is waiting. Using matrix-geometric method, the steady-state probability of the number of customers in the system is investigated. Some important performance measures are obtained. The waiting time distribution and the mean waiting time are also discussed. Finally, some numerical illustrations are provided.

scholarly journals Some Time-Dependent Properties of Symmetric M/G/1 Queues

2005 ◽  
Vol 42 (01) ◽  
pp. 223-234 ◽  
Author(s):  
Offer Kella ◽  
Bert Zwart ◽  
Onno Boxma
Keyword(s):  
Queue Length ◽  
Marginal Distribution ◽  
Geometric Distribution ◽  
Time Dependent ◽  
Service Discipline ◽  
Processor Sharing ◽  
Tail Behavior ◽  
Exponential Time ◽  
Preemptive Resume ◽  
Number Of Customers

We consider an M/G/1 queue that is idle at time 0. The number of customers sampled at an independent exponential time is shown to have the same geometric distribution under the preemptive-resume last-in-first-out and the processor-sharing disciplines. Hence, the marginal distribution of the queue length at any time is identical for both disciplines. We then give a detailed analysis of the time until the first departure for any symmetric queueing discipline. We characterize its distribution and show that it is insensitive to the service discipline. Finally, we study the tail behavior of this distribution.

scholarly journals Estimation of the mean waiting time of a customer subject to balking: a simulation study

2007 ◽  
Vol 19 (1) ◽  
pp. 63-63
Author(s):  
Jaejin Jang ◽  
Jaewoo Chung ◽  
Jungdae Suh ◽  
Jongtae Rhee
Keyword(s):  
Waiting Time ◽  
Simulation Study ◽  
Mean Waiting Time ◽  
The Mean

scholarly journals Waiting Time Problems for Patterns in a Sequence of Multi-State Trials

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1893
Author(s):  
Bara Kim ◽  
Jeongsim Kim ◽  
Jerim Kim
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Probability Generating Function ◽  
Finite Collection ◽  
Numerical Inversion ◽  
Analytic Method ◽  
Waiting Time Distribution ◽  
Mean Waiting Time ◽  
The Matrix ◽  
The Mean

In this paper, we investigate waiting time problems for a finite collection of patterns in a sequence of independent multi-state trials. By constructing a finite GI/M/1-type Markov chain with a disaster and then using the matrix analytic method, we can obtain the probability generating function of the waiting time. From this, we can obtain the stopping probabilities and the mean waiting time, but it also enables us to compute the waiting time distribution by a numerical inversion.

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