scholarly journals M(t)/M/1 Queueing System with Sinusoidal Arrival Rate

2016 ◽  
Vol 11 (1) ◽  
pp. 120-127
Author(s):  
A. P. Pant ◽  
R. P. Ghimire
Keyword(s):  
Mathematical Model ◽  
Waiting Time ◽  
Queueing System ◽  
Arrival Rate ◽  
Rate Function ◽  
Recursive Method ◽  
Mean Waiting Time ◽  
Number Of Customers ◽  
The Mathematical Model ◽  
Mean Time

This paper deals with the study of M (t)/M/1 queueing system with customers arrive to the system with sinusoidal arrival rate function λ (t) and are served exponentially with the rate μ. On formulating the mathematical model, we obtain the expressions for mean waiting time in the queue, mean time spent in the system, mean number of customers in the queue and in the system by using recursive method. Some numerical illustrations are also obtained by using computing software so as to show the applicability of the model under study.Journal of the Institute of Engineering, 2015, 11(1): 120-127

scholarly journals Mathematical Models of Mb/M/1 Bulk Arrival Queueing System

2014 ◽  
Vol 10 (1) ◽  
pp. 184-191 ◽  
Author(s):  
Sushil Ghimire ◽  
R. P. Ghimire ◽  
Gyan Bahadur Thapa
Keyword(s):  
Mathematical Model ◽  
Function Method ◽  
Queueing System ◽  
Batch Size ◽  
Queueing Model ◽  
Mean Waiting Time ◽  
Bulk Arrival ◽  
Number Of Customers ◽  
The Mathematical Model ◽  
Mean Time

 This paper deals with the study of bulk queueing model with the fixed batch size ‘b’ and customers arrive to the system with Poisson fashion with the rate λ and are severed exponentially with the rate μ. On formulating the mathematical model, we obtain the expressions for mean waiting time in the queue, mean time spent in the system, mean number of customers/work pieces in the queue and in the system by using generating function method. Some numerical illustrations are also obtained by using MATLAB-7 so as to show the applicability of the model under study.DOI: http://dx.doi.org/10.3126/jie.v10i1.10899Journal of the Institute of Engineering, Vol. 10, No. 1, 2014, pp. 184–191

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.

ESTIMATING WAITING TIMES WITH THE TIME-VARYING LITTLE'S LAW

2013 ◽  
Vol 27 (4) ◽  
pp. 471-506 ◽  
Author(s):  
Song-Hee Kim ◽  
Ward Whitt
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Waiting Times ◽  
Arrival Rate ◽  
Rate Function ◽  
Time Varying ◽  
Waiting Time Distribution ◽  
Mean Waiting Time ◽  
Little's Law ◽  
Little’S Law

When waiting times cannot be observed directly, Little's law can be applied to estimate the average waiting time by the average number in system divided by the average arrival rate, but that simple indirect estimator tends to be biased significantly when the arrival rates are time-varying and the service times are relatively long. Here it is shown that the bias in that indirect estimator can be estimated and reduced by applying the time-varying Little's law (TVLL). If there is appropriate time-varying staffing, then the waiting time distribution may not be time-varying even though the arrival rate is time varying. Given a fixed waiting time distribution with unknown mean, there is a unique mean consistent with the TVLL for each time t. Thus, under that condition, the TVLL provides an estimator for the unknown mean wait, given estimates of the average number in system over a subinterval and the arrival rate function. Useful variants of the TVLL estimator are obtained by fitting a linear or quadratic function to arrival data. When the arrival rate function is approximately linear (quadratic), the mean waiting time satisfies a quadratic (cubic) equation. The new estimator based on the TVLL is a positive real root of that equation. The new methods are shown to be effective in estimating the bias in the indirect estimator and reducing it, using simulations of multi-server queues and data from a call center.

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 A queueing model for error control of partial buffer sharing in ATM

1999 ◽  
Vol 5 (4) ◽  
pp. 329-348
Author(s):  
Boo Yong Ahn ◽  
Ho Woo Lee
Keyword(s):  
Error Control ◽  
Approximation Scheme ◽  
Queueing System ◽  
Recursive Method ◽  
Loss Probabilities ◽  
Mean Waiting Time ◽  
System Equations ◽  
The Mean ◽  
Buffer Sharing

We model the error control of the partial buffer sharing of ATM by a queueing systemM1,M2/G/1/K+1with threshold and instantaneous Bernoulli feedback. We first derive the system equations and develop a recursive method to compute the loss probabilities at an arbitrary time epoch. We then build an approximation scheme to compute the mean waiting time of each class of cells. An algorithm is developed for finding the optimal threshold and queue capacity for a given quality of service.

THE MEAN WAITING TIME FOR A PATTERN

1999 ◽  
Vol 13 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Sheldon M. Ross
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Waiting Time ◽  
Random Variables ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Mean Waiting Time ◽  
The Mean ◽  
Mean Time

Consider a sequence of independent and identically distributed random variables along with a specified set of k-vectors. We present an expression for E [T], the mean time until the last k observed random variables fall within this set. Not only can this expression often be used to obtain bounds on E[T], it also gives rise to an efficient way of approximating E[T] by a simulation. Specific lower and upper bounds for E[T] are also derived. These latter bounds are given in terms of a parameter, and a Markov chain Monte Carlo approach to approximate this parameter by a simulation is indicated. The results of this paper are illustrated by considering the problem of determining the mean time until a sequence of k-valued random variables has a run of size k that encompasses each value.

Fault Forecast Technology of Machine Based on Grey Theory for Aero-Engine Product Line

2011 ◽  
Vol 338 ◽  
pp. 718-722 ◽  
Author(s):  
Bing Chen ◽  
Ting Yang ◽  
Jun De Qi
Keyword(s):  
Mathematical Model ◽  
High Accuracy ◽  
Product Line ◽  
Grey Theory ◽  
Mean Time Between Failures ◽  
Time Between Failures ◽  
Aero Engine ◽  
Machine Fault ◽  
The Mathematical Model ◽  
Mean Time

Facing to the dynamic process of machine running, grey theory is introduced to increase the accuracy of forecast on machine fault. Firstly the mathematical model of the machine fault is constructed according to the life cycle of machine. Mean time between failures is defined as a tool to describe the fault on the machine. Moreover the fault is predicted respectively according to amount of data sample. And the produce to build the grey information model is given in this paper in detail. Finally an actual aero-engine casing production line is presented as an example to validate the algorithms in this paper. The results show that the fault forecast based on grey theory has high accuracy.

scholarly journals Mean Waiting Time in Large-Scale and Critically Loaded Power of d Load Balancing Systems

2021 ◽  
Vol 5 (2) ◽  
pp. 1-34
Author(s):  
Tim Hellemans ◽  
Benny Van Houdt
Keyword(s):  
Load Balancing ◽  
Waiting Time ◽  
Large Scale ◽  
Mean Field ◽  
Arrival Rate ◽  
Mean Field Limit ◽  
Mean Waiting Time ◽  
Constant Rate ◽  
Finite Set ◽  
The Mean

Mean field models are a popular tool used to analyse load balancing policies. In some exceptional cases the waiting time distribution of the mean field limit has an explicit form. In other cases it can be computed as the solution of a set of differential equations. In this paper we study the limit of the mean waiting time E[Wλ] as the arrival rate λ approaches 1 for a number of load balancing policies in a large-scale system of homogeneous servers which finish work at a constant rate equal to one and exponential job sizes with mean 1 (i.e. when the system gets close to instability). As E[Wλ] diverges to infinity, we scale with -log(1-λ) and present a method to compute the limit limλ-> 1- -E[Wλ]/l(1-λ). We show that this limit has a surprisingly simple form for the load balancing algorithms considered. More specifically, we present a general result that holds for any policy for which the associated differential equation satisfies a list of assumptions. For the well-known LL(d) policy which assigns an incoming job to a server with the least work left among d randomly selected servers these assumptions are trivially verified. For this policy we prove the limit is given by 1/d-1. We further show that the LL(d,K) policy, which assigns batches of K jobs to the K least loaded servers among d randomly selected servers, satisfies the assumptions and the limit is equal to K/d-K. For a policy which applies LL(di) with probability pi, we show that the limit is given by 1/ ∑i pi di - 1. We further indicate that our main result can also be used for load balancers with redundancy or memory. In addition, we propose an alternate scaling -l(pλ) instead of -l(1-λ), where pλ is adapted to the policy at hand, such that limλ-> 1- -E[Wλ]/l(1-λ)=limλ-> 1- -E[Wλ]/l(pλ), where the limit limλ-> 0+ -E[Wλ]/l(pλ) is well defined and non-zero (contrary to limλ-> 0+ -E[Wλ]/l(1-λ)). This allows to obtain relatively flat curves for -E[Wλ]/l(pλ) for λ ∈ [0,1] which indicates that the low and high load limits can be used as an approximation when λ is close to one or zero. Our results rely on the earlier proven ansatz which asserts that for certain load balancing policies the workload distribution of any finite set of queues becomes independent of one another as the number of servers tends to infinity.

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.

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