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.

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.

The general N server finite queue

1971 ◽  
Vol 8 (04) ◽  
pp. 828-834 ◽  
Author(s):  
Asha Seth Kapadia
Keyword(s):  
Waiting Time ◽  
Busy Period ◽  
Queueing System ◽  
List Type ◽  
Type Number ◽  
Stochastic Nature ◽  
Mean Waiting Time ◽  
Queue Discipline ◽  
Mean And Variance ◽  
The Mean

Kingman (1962) studied the effect of queue discipline on the mean and variance of the waiting time. He made no assumptions regarding the stochastic nature of the input and the service distributions, except that the input and service processes are independent of each other. When the following two conditions hold: (a) no server sits idle while there are customers waiting to be served; (b) the busy period is finite with probability one (i.e., the queue empties infinitely often with probability one); he has shown that 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 order of their arrival. Conditions (a) and (b) will henceforward be called Kingman conditions and a queueing system satisfying Kingman conditions will be referred to in the text as a Kingman queue.

The general N server finite queue

1971 ◽  
Vol 8 (4) ◽  
pp. 828-834 ◽  
Author(s):  
Asha Seth Kapadia
Keyword(s):  
Waiting Time ◽  
Busy Period ◽  
Queueing System ◽  
List Type ◽  
Type Number ◽  
Stochastic Nature ◽  
Mean Waiting Time ◽  
Queue Discipline ◽  
Mean And Variance ◽  
The Mean

Kingman (1962) studied the effect of queue discipline on the mean and variance of the waiting time. He made no assumptions regarding the stochastic nature of the input and the service distributions, except that the input and service processes are independent of each other. When the following two conditions hold: (a)no server sits idle while there are customers waiting to be served;(b)the busy period is finite with probability one (i.e., the queue empties infinitely often with probability one); he has shown that 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 order of their arrival. Conditions (a) and (b) will henceforward be called Kingman conditions and a queueing system satisfying Kingman conditions will be referred to in the text as a Kingman queue.

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 Delay Analysis of anM/G/1/KPriority Queueing System with Push-out Scheme

2007 ◽  
Vol 2007 ◽  
pp. 1-12 ◽  
Author(s):  
Yutae Lee ◽  
Bong Dae Choi ◽  
Bara Kim ◽  
Dan Keun Sung
Keyword(s):  
Communication Networks ◽  
Waiting Time ◽  
Queueing System ◽  
Waiting Times ◽  
Loss Probability ◽  
Mean Waiting Time ◽  
The Mean ◽  
Recursive Equations ◽  
Push Out ◽  
Stieltjes Transforms

This paper considers anM/G/1/Kqueueing system with push-out scheme which is one of the loss priority controls at a multiplexer in communication networks. The loss probability for the model with push-out scheme has been analyzed, but the waiting times are not available for the model. Using a set of recursive equations, this paper derives the Laplace-Stieltjes transforms (LSTs) of the waiting time and the push-out time of low-priority messages. These results are then utilized to derive the loss probability of each traffic type and the mean waiting time of high-priority messages. Finally, some numerical examples are provided.

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.

A relation between stationary queue and waiting time distributions

1971 ◽  
Vol 8 (03) ◽  
pp. 617-620 ◽  
Author(s):  
Rasoul Haji ◽  
Gordon F. Newell
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Arrival Process ◽  
Time Interval ◽  
Queue Size ◽  
Random Time Interval ◽  
Queue Discipline ◽  
Stationary Waiting Time ◽  
Number Of Customers ◽  
First In First Out

A theorem is proved which, in essence, says the following. If, for any queueing system, (i) the arrival process is stationary, (ii) the queue discipline is first-in-first-out (FIFO), and (iii) the waiting time of each customer is statistically independent of the number of arrivals during any time interval after his arrival, then the stationary random queue size has the same distribution as the number of customers who arrive during a random time interval distributed as the stationary waiting time.

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.

A relation between stationary queue and waiting time distributions

1971 ◽  
Vol 8 (3) ◽  
pp. 617-620 ◽  
Author(s):  
Rasoul Haji ◽  
Gordon F. Newell
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Arrival Process ◽  
Time Interval ◽  
Queue Size ◽  
Random Time Interval ◽  
Queue Discipline ◽  
Stationary Waiting Time ◽  
Number Of Customers ◽  
First In First Out

A theorem is proved which, in essence, says the following. If, for any queueing system, (i) the arrival process is stationary, (ii) the queue discipline is first-in-first-out (FIFO), and (iii) the waiting time of each customer is statistically independent of the number of arrivals during any time interval after his arrival, then the stationary random queue size has the same distribution as the number of customers who arrive during a random time interval distributed as the stationary waiting time.

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