scholarly journals Transient solution of /G/1 with Second Optional Service Subject to Reneging during Vacation and Breakdown time

2015 ◽  
Vol 3 (1) ◽  
pp. 206-213
Author(s):  
Suganya S Vijay
Keyword(s):  
Arrival Rate ◽  
Queuing Model ◽  
Breakdown Time ◽  
Mean Waiting Time ◽  
Optional Service ◽  
Second Optional Service ◽  
Poisson Stream ◽  
The Mean ◽  
System Breakdown ◽  
Mean Queue Length

In this paper, we present a batch arrival non- Markovian queuing model with second optional service. Batches arrive in Poisson stream with mean arrival rate ?, such that all customers demand the first essential service, whereas only some of them demand the second "optional" service. We consider reneging to occur when the server is unavailable during the system breakdown or vacations periods. The time-dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been derived explicitly. Also the mean queue length and the mean waiting time have been found explicitly.

scholarly journals Transient Solution of M[X]=G=1 With Second Optional Service, Bernoulli Schedule Server Vacation and Random Break Downs

2013 ◽  
Vol 3 (3) ◽  
pp. 45-55
Author(s):  
S. SHYAMALA ◽  
Dr. G. AYYAPPAN
Keyword(s):  
Arbitrary Distribution ◽  
Breakdown Rate ◽  
Mean Waiting Time ◽  
Optional Service ◽  
Second Optional Service ◽  
Poisson Stream ◽  
The Mean ◽  
Mean Queue Length ◽  
Bernoulli Schedule ◽  
Essential Service

In this model, we present a batch arrival non- Markovian queueingmodel with second optional service, subject to random break downs andBernoulli vacation. Batches arrive in Poisson stream with mean arrivalrate (> 0), such that all customers demand the rst `essential' ser-vice, wherein only some of them demand the second `optional' service.The service times of the both rst essential service and the second op-tional service are assumed to follow general (arbitrary) distribution withdistribution function B1(v) and B2(v) respectively. The server may un-dergo breakdowns which occur according to Poisson process with breakdown rate . Once the system encounter break downs it enters the re-pair process and the repair time is followed by exponential distributionwith repair rate . Also the sever may opt for a vacation accordingto Bernoulli schedule. The vacation time follows general (arbitrary)distribution with distribution function v(s). The time-dependent prob-ability generating functions have been obtained in terms of their Laplacetransforms and the corresponding steady state results have been derivedexplicitly. Also the mean queue length and the mean waiting time havebeen found explicitly.

scholarly journals An M/M/1 Queue with Working Vacation and Vacation Interruption Under Bernoulli Schedule

2018 ◽  
Vol 7 (4.10) ◽  
pp. 448
Author(s):  
P. Manoharan ◽  
A. Ashok
Keyword(s):  
Geometric Approach ◽  
Stochastic Decomposition ◽  
Working Vacation ◽  
System Parameters ◽  
Mean Waiting Time ◽  
The Mean ◽  
Vacation Interruption ◽  
Mean Queue Length ◽  
Vacation Period ◽  
Bernoulli Schedule

This work deals with M/M/1 queue with Vacation and Vacation Interruption Under Bernoulli schedule. When there are no customers in the system, the server takes a classical vacation with probability p or a working vacation with probability 1-p, where . At the instants of service completion during the working vacation, either the server is supposed to interrupt the vacation and returns back to the non-vacation period with probability 1-q or the sever will carry on with the vacation with probability q. When the system is non empty after the end of vacation period, a new non vacation period begins. A matrix geometric approach is employed to obtain the stationary distribution for the mean queue length and the mean waiting time and their stochastic decomposition structures. Numerous graphical demonstrations are presented to show the effects of the system parameters on the performance measures.  

Queues with time-dependent arrival rates. III — A mild rush hour

1968 ◽  
Vol 5 (3) ◽  
pp. 591-606 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Queueing Theory ◽  
Queue Length ◽  
Arrival Rate ◽  
Time Dependent ◽  
Service Rate ◽  
Diffusion Approximations ◽  
Dimensionless Form ◽  
Queue Lengths ◽  
The Mean ◽  
Mean Queue Length

The arrival rate of customers to a service facility is assumed to have the form λ(t) = λ(0) — βt2 for some constant β. Diffusion approximations show that for λ(0) sufficiently close to the service rate μ, the mean queue length at time 0 is proportional to β–1/5. A dimensionless form of the diffusion equation is evaluated numerically from which queue lengths can be evaluated as a function of time for all λ(0) and β. Particular attention is given to those situations in which neither deterministic queueing theory nor equilibrium stochastic queueing theory apply.

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.

Queues with time-dependent arrival rates I—the transition through saturation

1968 ◽  
Vol 5 (2) ◽  
pp. 436-451 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Equilibrium Distribution ◽  
Planck Equation ◽  
Arrival Rate ◽  
Service Capacity ◽  
Constant Rate ◽  
Mean And Variance ◽  
The Mean ◽  
Mean Queue Length ◽  
Pass Through ◽  
Stochastic Properties

Suppose that the arrival rate λ(t) of customers to a service facility increases with time at a nearly constant rate, dλ(t)/dt = a, so as to pass through the saturation condition, λ(t) = μ = service capacity, at some time which we label as t = 0. The stochastic properties of the queue are investigated here through use of the diffusion approximation (Fokker-Planck equation). It is shown that there is a characteristic time Tproportional to α–2/3 such that if , then the queue distribution stays close to the prevailing equilibrium distribution associated with the λ(t) and μ, evaluated at time t. For |t| = O(T), however, the mean queue length is much less than the equilibrium mean, and is measured in units of some characteristic length L which is proportional to α–1/3. For , the queue is approximately normally distributed with a mean of the order L larger than that predicted by deterministic queueing models. Numerical estimates are given for the mean and variance of the distribution for all t. The queue distributions are also evaluated in non-dimensional units.

Queues with time-dependent arrival rates. III — A mild rush hour

1968 ◽  
Vol 5 (03) ◽  
pp. 591-606 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Queueing Theory ◽  
Queue Length ◽  
Arrival Rate ◽  
Time Dependent ◽  
Service Rate ◽  
Diffusion Approximations ◽  
Dimensionless Form ◽  
Queue Lengths ◽  
The Mean ◽  
Mean Queue Length

The arrival rate of customers to a service facility is assumed to have the formλ(t) =λ(0) —βt2for some constantβ.Diffusion approximations show that forλ(0) sufficiently close to the service rateμ, the mean queue length at time 0 is proportional toβ–1/5. A dimensionless form of the diffusion equation is evaluated numerically from which queue lengths can be evaluated as a function of time for allλ(0) andβ.Particular attention is given to those situations in which neither deterministic queueing theory nor equilibrium stochastic queueing theory apply.

The Asymmetric PCF Control Technology of Gated Service Based on Continuous-Time

2013 ◽  
Vol 432 ◽  
pp. 582-586
Author(s):  
Guang Lan Zhao ◽  
Hong Wei Ding ◽  
Ying Ying Guo ◽  
Yi Fan Zhao ◽  
Ya Nan Hao
Keyword(s):  
Probability Generating Function ◽  
System Model ◽  
Embedded Markov Chain ◽  
Closed Expression ◽  
System Parameters ◽  
Mean Waiting Time ◽  
Defining System ◽  
The Mean ◽  
Mean Queue Length ◽  
And Performance

The paper starts with building system mathematical models and defining system parameters and working conditions to analyze the system model and performance, using the embedded Markov chain and probability generating function. The paper also deduces the accurate closed expression of the mean queue length and cycle time, gets approximate analysis expression of the mean waiting time well, and also verifies the correct theoretical analysis by simulation.

scholarly journals Performance Analysis of Heterogeneous Data Centers in Cloud Computing Using a Complex Queuing Model

2015 ◽  
Vol 2015 ◽  
pp. 1-15 ◽  
Author(s):  
Wei-Hua Bai ◽  
Jian-Qing Xi ◽  
Jia-Xian Zhu ◽  
Shao-Wei Huang
Keyword(s):  
Analytical Model ◽  
Data Center ◽  
Queuing Theory ◽  
Data Centers ◽  
Heterogeneous Data ◽  
Queuing Model ◽  
Research Attention ◽  
Cloud Data ◽  
Mean Waiting Time ◽  
The Mean

Performance evaluation of modern cloud data centers has attracted considerable research attention among both cloud providers and cloud customers. In this paper, we investigate the heterogeneity of modern data centers and the service process used in these heterogeneous data centers. Using queuing theory, we construct a complex queuing model composed of two concatenated queuing systems and present this as an analytical model for evaluating the performance of heterogeneous data centers. Based on this complex queuing model, we analyze the mean response time, the mean waiting time, and other important performance indicators. We also conduct simulation experiments to confirm the validity of the complex queuing model. We further conduct numerical experiments to demonstrate that the traffic intensity (or utilization) of each execution server, as well as the configuration of server clusters, in a heterogeneous data center will impact the performance of the system. Our results indicate that our analytical model is effective in accurately estimating the performance of the heterogeneous data center.

scholarly journals An Improved Queuing Model for Reducing Average Waiting Time of Emergency Surgical Patients Using Preemptive Priority Scheduling

2021 ◽  
Vol 9 (6) ◽  
pp. 827-840
Keyword(s):  
Waiting Time ◽  
Time Management ◽  
Arrival Rate ◽  
Preemptive Priority ◽  
Queuing Model ◽  
Priority Scheduling ◽  
Average Waiting Time ◽  
Public Sector Hospital ◽  
Medical Facilities ◽  
The Mean

Our research objective is to reduce the Average Waiting Time for patients in an Emergency Department of public sector hospital. We have based our model on M/M/s Queuing System, our study revealssignificant findings on arrival rate of patients. During this simulation, we have used a preemptive priority scheduling model. In our practice, the arrival rate followed a Poisson distribution, averaging 30 patients per hour, with the Mean Service time of1.5 hours and Average Waiting Time recorded around 12.13 minutes. This research offersvaluable help to achieve better time management in emergency departments of high-density medical facilities.

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