Queueing Systems – A Numerical Approach (original) (raw)
Calculation of delay characteristics for multiserver queues with constant service times
European Journal of Operational Research, 2009
We consider a discrete-time infinite-capacity queueing system with a general uncorrelated arrival process, constant-length service times of multiple slots, multiple servers and a first-come-first-served queueing discipline. Under the assumption that the queueing system can reach a steady state, we first establish a relationship between the steady-state probability distributions of the system content and the customer delay. Next, by means of this relationship, an explicit expression for the probability generating function of the customer delay is obtained from the known generating function of the system content, derived in previous work. In addition, several characteristics of the customer delay, namely the mean value, the variance and the tail distribution of the delay, are derived through some mathematical manipulations. The analysis is illustrated by means of some numerical examples.
THE ANALYSIS OF THE RESULTS OF SIMULATION MODELS OF LIMITED AND UNLIMITED QUEUING SERVICE SYSTEMS
The comparative analysis of the results of simulation models of limited and unlimited queuing service systems is reviewed. Here, when the restriction put on the time of presence of requests is violated, the loss of requests occurs. The analysis of possible situations leading to the loss in such systems is one of the important issues. These situations occur practically in service processes of different technical systems. An experiment was conducted; results were obtained and comparatively analyzed for both cases.
The-Application-of-Queuing-Analysis-in-modeling-Optimal-Service-level.pdf
Queues are common scenario faced in the modern day Banks and other financial Institutions. Queuing theory is the mathematical study of waiting lines; this can also be applicable queues in the banking system. This study examine the queuing system at Guarantee Trust Bank (GTB), putting into consideration the waiting time spend by Customers, Service time spend by a Customer and the average cost a customer loses while in queue and the service cost of each server in order to optimize the system. The First Come First Serve (FCFS) Multi-Server queuing model was used to model the queuing process. The waiting time was assumed to follow a Poisson distribution while the service rate follows an Exponential distribution. This study adopted a case study approach by randomly administering questionnaires, interviews and observation of the participants. The data were collected at the GTB cash deposit unit for four days period. The data collected were analyzed using TORA optimization window based software as well as standard queuing formula. The results of the analysis showed that the average queue length, waiting time of customer with a minimum Total Cost that utilize the system is by using five Servers against the present server level of Three Servers which incur a high total cost to both the Customers and the system.
Statistical techniques for a numerical evaluation of the proximity of and queueing systems
Computers & Mathematics with Applications, 2011
We study the strong stability of a G/M/1 queueing system after perturbation of the service times. We are interested in the determination of the proximity error between the corresponding service time distributions of G/G/1 and G/M/1 systems, the approximation error on the stationary distributions, and confidence intervals for the difference between the corresponding characteristics of the quoted systems in the stationary state, when the general distribution of service times G in the G/G/1 system is unknown and must be estimated by means of a nonparametric estimation method. We use the Student test to accept or reject the equality of the corresponding characteristics. The boundary effects are taken into account. Simulation studies are performed to support the results.
A NON LINEAR APPROACH TO QUEUEING SYSTEM MODELLING
2010
This paper seeks to establish queueing models that can help organizations to improve on their customer service within and outside their establishment. New Models were created using non linear regression analysis which is more convenient for the organizations to assess. It was found that The Coefficient of determination, R 2 value equals 1 and that the Degree of Correlation is 100% which indicates that 100% of the original uncertainty has been explained by the model.
Numerical method for the analysis of queuing models with priority jumps1
Cybernetics and Systems Analysis, 2013
An algorithmic approach to study queuing models with common finite and infinite buffer and jump priorities is developed. It is assumed that upon arrival of a low-priority call, one call of such kind can be transferred with some probability to the end of the queue of high-priority calls. The transition probability depends on the state of the queue of heterogeneous calls. The algorithms are proposed to calculate the quality of service metrics of such queuing models.
A model for waiting times for non-stationary queueing systems
Journal of Management and Science, 2016
Queueing models are stochastic models that represent the probability that a queueing system will be found in a particular configuration or state. Several interesting stationary queueing systems have been solved analytically; on the other hand, non-stationary queueing systems are relatively unexplored. The present study analyses the waiting times of a non-stationary M/M/1 queueing system using simulation methods.
Performance Analysis of A Queueing System with Server Arrival and Departure
ACM SIGMETRICS Performance Evaluation Review
In many systems, in order to fulfill demand (computing or other services) that varies over time, service capacities often change accordingly. In this paper, we analyze a simple two dimensional Markov chain model of a queueing system in which multiple servers can arrive to increase service capacity, and depart if a server has been idle for too long. It is well known that multi-dimensional Markov chains are in general difficult to analyze. Our focus is on an approximation method of stationary performance of the system via the Stein method. For this purpose, innovative methods are developed to estimate the moments of the Markov chain, as well as the solution to the Poisson equation with a partial differential operator.
Modeling queues with simulation versus M/M/C models
Journal of Service Science Research, 2014
This paper examines the performance of single-queue service systems using a combination of computer simulation and M/M/C queuing models. Our results show that the accuracy of M/M/C models is significantly affected by the assumptions supporting the models. Managers should therefore exercise caution in using the M/M/C models for designing queuing systems when the models' assumptions are violated. Our results show that cost-centric and servicecentric firms should manage their queues differently. While cost-centric firms should target higher arrival load, single service session, and front-loaded arrival pattern for higher efficiency, service-centric firms should strive for lower arrival load, multiple short sessions and even arrival pattern for better service. In addition, both cost-centric and service-centric firms can consider pooling servers together and reducing the variability of inter-arrival and service times to improve both cost and service simultaneously.
Analysis and Comparison of Queues with Different Levels of Delay Information
Management Science, 2007
Information about delays can enhance service quality in many industries. Delay information can take many forms, with different degrees of precision. Different levels of information have different effects on customers and so on the overall system. The goal of this research is to explore these effects. We first consider a queue with balking under three levels of delay information: No information, partial information (the system occupancy) and full information (the exact waiting time). We assume Poisson arrivals, independent, exponential service times, and a single server. Customers decide whether to stay or balk based on their expected waiting costs, conditional on the information provided. By comparing the three systems, we identify some important cases where more accurate delay information improves performance. In other cases, however, information can actually hurt the provider or the customers.
A numerically stable algorithm for two server queue models
Queueing Systems, 1991
In this paper, we consider a queueing system in which there are two exponential servers, each having his own queue, and arriving customers will join the shorter queue. Based on the results given in Flatto and McKean, we rewrite the formula for the probability that there are exactly k customers in each queue, where k = 0, 1 ..... This enables us to present an algorithm for ~omputing these probabilities and then to find the joint distribution of the queue lengths in the system. A program and numerical examples are given.
Delay analysis of a discrete-time single-server queue with an occasional extra server
Annals of Operations Research, 2020
In this work we look at the delay analysis of a customer in a discrete-time queueing system with one permanent server and one occasional extra server. The arrival process is assumed to be general independent, the buffer size infinite and the service times deterministically equal to one slot. The system is assumed to be in one of two different states; during the UP-state 2 servers are available and during the DOWN-state 1 server is available. State changes can only occur at slot boundaries and mark the beginnings and ends of UP-periods and DOWN-periods. The lengths of the UP-periods, expressed in their number of slots, are assumed to follow a geometric distribution, while the lengths of the DOWN-periods follow a general distribution with rational probability generating function. We provide a method to compute the tail characteristics of the delay of an arbitrary customer based on the theory of the dominant singularity. We illustrate the developed method with several numerical examples.
Iterative Method of Analysis of Single Queue, Multi-Server with Limited System Capacity
International Journal of Science and Technology (STECH)., 2017
In this paper, analysis of single queue, multi-server with limited system capacity under first come first served discipline was carried out using iterative method. The arrivals of customers and service times of customers are assumed poisson and exponential distributions respectively. This queuing model is an extension of single queue, single server with limited system capacity. Performance measures of the model, such as the expected number of customers in the queue and in the system, the expected waiting times of customers in the queue and in the system respectively were derived. The performance measures so derived were compared with that of single queue, single server with limited capacity {M/M/1:(N/FCFS)} model. The numerical illustration indicates that single queue, multi-server with limited capacity {M/M/c:(N/FCFS)} model is more effective and efficient in handling congestions.
NJPAS, 2019
Queues form an indispensible part of our everyday lives. They occur whenever there is competition for limited resources. For example, we line up in queues outside doctors' offices and supermarket checkout counters; airplanes queue along airport runways. Therefore, our ability to model and analyse systems of queues helps to minimize their inconveniences and maximize the use of the limited resources. In this paper, a queueing system comprising of two-phase hyper-exponential inter-arrival time, Erlang-3 distributed service time, single server and infinite number of arrivals has been investigated in order to provide some insight into the analysis of phase type distributions queues. This is carried out by using procedure for solving phase-type queueing systems by means of the matrix-geometric approach namely, construction of the block sub-matrices, forming the Neuts R matrix, solving the boundary equations and generating the successive components of the solution. Performance Measures for phase type Queueing system, Ph/Ph/1, are obtained as follows, the probability that there are customers present in the queueing system, for ≥ 1, is = ‖ ‖ = ‖ ‖ , the probability that the system is empty is given by = ‖ ‖ , The probability that the system is busy is given by 1 − = 1 − ‖ ‖ = 1 − , The probability that there or more customers present is, Prob [ ≥ ] = ∑ = ∑ = ∑ = (1 −). By Little's law and PASTA property, the average number of customers in phase type ℎ/ ℎ/1, Mean number of customers waiting in the queue, the average response time, the average time spent waiting in the queue were obtained. The practical examples were then considered for certain given values of ℷ, μ, and γ to obtained = (0.013 0.0356 0.0336) ∀ = 0, 1, 2.
Simulation : Analysis of Single Server Queuing Model
International Journal on Information Theory, 2014
A queue is a line of people or things to be handled in a sequential order. It is a sequence of objects that are waiting to be processed. Queuing theory is the study of queues for managing process and objects. Simulation has been applied successfully for modeling small and large complex systems and understanding queuing behavior. Analysis of the models helps to increases the performance of the system. In this paper we analyze various models of the Single server queuing system with necessaryimplementation using Matlab Software.
Mathematical Methods of Operations Research, 2010
Whereas the buffer content of batch-service queueing systems has been studied extensively, the customer delay has only occasionally been studied. The few papers concerning the customer delay share the common feature that only the moments are calculated explicitly. In addition, none of these surveys consider models including the combination of batch arrivals and a server operating under the full-batch service policy (the server waits to initiate service until he can serve at full capacity). In this paper, we aim for a complete characterisation -i.e., moments and tail probabilities -of the customer delay in a discrete-time queueing system with batch arrivals and a batch server adopting the full-batch service policy. In addition, we demonstrate that the distribution of the number of customer arrivals in an arbitrary slot has a significant impact on the moments and the tail probabilities of the customer delay.
Performance analysis approximation in a queueing system of type M/G/1
Mathematical Methods of Operations Research, 2006
In this work, we apply the strong stability method to obtain an estimate for the proximity of the performance measures in the M/G/1 queueing system to the same performance measures in the M/M/1 system under the assumption that the distributions of the service time are close and the arrival flows coincide. In addition to the proof of the stability fact for the perturbed M/M/1 queueing system, we obtain the inequalities of the stability. These results give with precision the error, on the queue size stationary distribution, due to the approximation. For this, we elaborate from the obtained theoretical results, the STR-STAB algorithm which we execute for a determined queueing system: M/Coxian − 2/1. The accuracy of the approach is evaluated by comparison with simulation results.
A queue is a line of people or things to be handled in a sequential order. It is a sequence of objects that are waiting to be processed. Queuing theory is the study of queues for managing process and objects. Simulation has been applied successfully for modelling small and large complex systems and understanding queuing behaviour. Analysis of the models helps to increases the performance of the system. In this paper we analyze various models of the Single server queuing system with necessary implementation using Microsoft Excel and Matlab Software. For better understanding we have considered a Virtual Telecommunication System, is presented with help of Microsoft Excel. Empirical distribution of Service Time and Inter-arrival Time of call has an impact on the performance of Queuing models. Examples include: A Telecommunication System staffing a call centre and measuring system parameters like average time that a call spend in the system ,average waiting time for those calls which are waiting. Numerous problems and more motivate the development of a queuing theory. Queuing theory as discussed in this paper is organized and a simulation of queuing system and the necessary mathematical tools are developed to analyze them. Finally, we illustrate the use of these models through various communication applications.
Queues with waiting time dependent service
Queueing Systems, 2011
Motivated by service levels in terms of the waiting-time distribution seen in e.g. call centers, we consider two models for systems with a service discipline that depends on the waiting time. The first model deals with a single server that continuously adapts its service rate based on the waiting time of the first customer in line. In the second model, one queue is served by a primary server which is supplemented by a secondary server when the waiting of the first customer in line exceeds a threshold. Using level crossings for the waiting-time process of the first customer in line, we derive steadystate waiting-time distributions for both models. The results are illustrated with numerical examples.