## A Survey on Queueing Systems with Mathematical Models and Applications

Read this paper on waiting line analysis and queues. It provides a good survey of the theory and uses of this type of analysis. Pay particular attention to sections 1 through 3. How might these models be used to balance firm costs with different levels of customer satisfaction?

### 2. Literature Review

In many practical situations, customers arrive following a Poisson stream which is an exponential inter-arrival times. Customers perform different nature coming alone or in a batch. Some are silent and wait in the queue for a long time whereas some are impatient and do not bother to leave after a while. We have noticed in our daily life also that customers wait even for a long time in the call centres until an operator is available. In spite of the different nature of customers, there are some common features on which a queue depends, namely service times, service discipline and the number of servers. These are the key factors to determine a queue. In many cases, we assume that there is an independent service time which is identically distributed with the provision of independent inter-arrival times. Among different types of queueing system, our focus is mainly on the finite capacity and batch queueing system. In finite capacity queueing system, a fixed number of customers are served and in batch queueing system, arrival and service can take place in a bulk. On top of this, we report some of the literatures in the following.

Kovalenko studied rare events during busy periods along with some useful rare event theorems and singular state aggregation theorems presenting some analysis and numerical methods. Cohen and Boxma gathered the information of queuing theory from its origin up to the maturity as a branch of mathematics in the field of Operations Research to calculate the performance measures. Hui investigated a survey in china for five main areas namely transient behaviour, classical problems, approximation theory, model structure and applications. Fomundam and Herrmann reported a survey of queuing theory application in healthcare focusing on the area of waiting time and utilization analysis, system design, and appointment systems. Lade et al. used simulation of queueing system in hospitals to predict the parameters like total waiting time, average waiting time of patients, average queue length and to decrease the waiting time of patients. Shanthikumar et al. carried out a survey paper on queueing theory which is applicable for semiconductor manufacturing systems. They put their efforts to improve the model assumptions and model input, mainly in averages and the variations of products. Jackson and Adelson dealt with the calculation of customer characteristics in some simple cases to explain the literature for complex and practical queueing systems.

Jouini and Dallery considered Markovian multi-server queue with a finite waiting line in which a customer may decide to give up for service if waiting time in queue exceeds its random deadline. They focused on the performance measures in terms of the probability of being served under both transient and stationary behaviours. Karaesmen et al. examined a finite buffered queue in which the queue length is controlled by low and high service rates with higher operating cost for faster service rate. Moreover, holding cost for waiting customers to proceed and setup costs for the change in service rate were included along with some numerical examples for the validity of the model. Laxmi and Suchitra studied finite buffer N-policy GI/M(n)/1 queue with Bernoulli-schedule vacation interruption, where server takes a vacation and works with a slower rate if there are less than N customers waiting for service. They used supplementary variable technique and recursive method to obtain the steady state system length distributions at pre-arrival and arbitrary epochs. Kwon dealt queueing network model for the performance analysis of a flexible manufacturing system composed of workstations with limited buffers. Performance measures were developed and numerical examples were presented to verify the effectiveness of the approximation algorithm used in the model. Chakravarthy illustrated numerical examples using matrix-analytic method of multi-server queueing model in which one sever is considered as the main server. The main server is connected to the other servers to provide the consultation on the FCFS basis. Ghimire and Basnet studied finite capacity queueing system under the provision of vacation and service breakdown for the calculation of queue length and waiting time distributions. Yang and Wu considered M/M/1/N queue with server subject to breakdowns and repairs during the time of operation where server works at a lower service rate even at the failure times. They computed transient state probabilities and some system performance measures using fourth-order Runge–Kutta method. Kaczynski et al. put their effort to study M/M/s queue with initially presented k customers for the exact distribution of nth customer's sojourn time. Algorithms for computing the covariance between sojourn times for an M/M/1 queue with k customers present at time zero was developed using maple computer code.

Some researchers have studied and proposed some mathematical models for the nature of customers as well. Shin and Choo considered an M/M/s queue in which balking and reneging customers join the virtual pool of customers called orbit. Probabilities of joining the orbit by balking and reneging customers was determined by the number of customers in the service facility. Some numerical examples were also presented to validate the results. Al-Seedy et al. applied generating function technique for transient solution of system in an M/M/c queue having fixed probability for balking customers and a negative exponential distribution for reneging customers. Ayyappan et al. studied single server batch service of size k considering Poisson arrival rate λ, exponential service distribution μ and Poisson catastrophe rate v to calculate the mean and variance of all the parameters described in the model. Choudhury and Medhi analysed reneging behaviour where each customer is assumed to follow identical distribution of patience time ignoring the real life situations. They attempted to model a reneging property along with balking behaviour. Ghimire and Ghimire dealt M/M/1 queue with heterogeneous arrival and departure with the provision of server vacations and breakdowns to evaluate some performance measures using generating function method. Atencia and Moreno analysed a discrete-time Geo/G/1 retrial and without retrial queue to calculate the measure of the proximity between the system size and marginal distributions when the server is idle, busy or down. Ammar obtained explicit solution of multi-server transient queue with balking and reneging behaviour of customers using similar technique of along with the calculation of steady-state probabilities and some important performance measures. Gong and Li developed a maximum system utility optimization model considering customer's psychology to study the impact on their patience and rejection behaviour in a queue. They used a probability function to describe the change of customer's psychology and rejection behaviour. Li and Cheng dealt with infinite capacity queueing systems with two parallel servers having generally distributed service times where customer joined the shortest queue in the Poisson fashion. For both the queues of equal length, new arrival could join any of them with equal probability without jockeying. Shanmugasundaram and Banumathi analysed multi-server queueing system using Monte-Carlo simulation to find the future behaviour of railways to reduce the queue length and system length, queue time and system time with some numerical examples.

There are some queues for which arrival and service depend on time. Those queues are known as transient queues. Singla and Garg studied a feedback queueing system with correlated transient departures and calculated the transient-state queue length probabilities using Laplace Transform of the generating function. Zeng et al. investigated transient M/Ek/n queuing model for the evaluation of queue length and the average waiting time of the railway container terminal gate system, as well as the optimal number of service channels during the different time period. Chan et al. applied iterative and Crank–Nicolson pre-conditioner method to solve the system of linear equations for the transient solutions of M/M/2 queueing systems with two heterogeneous servers under a variant vacation policy. Jiang et al. proposed free flow, slow flow and jam flow vehicle velocities in which the transitions between slow and jam flow are controlled by the duration of slow flow queues. They revealed the fact that convective instability of queueing model could generate oscillation features. Ghimire et al. calculated the performance evaluations of multi-server M(t)/M(t)/n/n queuing system subject to breakdowns under transient frame work without accepting the queue of the waiting customers and verified the results using simulation. Tan et al. studied transient arrival finite capacity queue where arrival rate slowly varies with time for the large capacity $\mathrm{K}$. Probability of $\mathrm{n}$ number of customers and mean number of customers in the system at time $t$ was calculated using asymptotic approximations approach. Kempa derived explicit formulae for the queue size distribution of a finite-buffer GI/M/1/N transient queueing model. He calculated transient queue-size distribution convergence rate to the stationary distribution for the constant value given explicitly. Malligarjunan evaluated performance measures and total expected cost rate for a single server queueing system under transient behaviour and entropy measures on an inventory system with two demand classes. Selinka et al. developed a stationary backlog-carryover approach to compare the numerical results with simulations applicability in an analytical solution for a time-dependent performance evaluation of truck handling operations at an air cargo terminal. Ausina et al. chose a single server queueing system in which Bayesian inference for the transient behaviour and duration of a busy period with general, unknown distributions for the inter-arrival and service times has been investigated.

Batch or bulk arrival and service facility is the another area of research in queueing theory. Chang and Choi studied finite-buffer discrete-time GeoX/GY/1/K+B queue with multiple vacations that has a wide range of applications including high-speed digital telecommunication system and various related areas presenting some performance analysis. Goswami and Laxmi dealt a single server infinite or finite buffer bulk-service queues considering arbitrary inter-arrival and exponential service time distribution. The customers were served by a single server in accessible or non-accessible batches of maximum size 'b' with a minimum threshold value 'a'. Ghimire et al. established a bulk queueing model with the fixed batch size 'b' and obtained 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. Singh et al. investigated retrial queue with bulk arrivals and unreliable servers providing m-optional services to observe the validity of performance measures and the effect of parameters for the queue size distribution. Banergee et al. studied a single server bulk service finite capacity queue for the calculation of joint distribution of the random variables at various epochs in which service times depend on the batch size customers following Markovian arrival process. Luo et al. dealt with a ﬁnite buffer GeoX/G//1/N queue for the observation of queue-length distributions at departure, arbitrary, pre-arrival epochs with single working vacation and different input rates combining two techniques of supplementary variable and embedded Markov chain method. Claeys et al. analysed a versatile batch-service queueing model with correlation in the arrival process along with some performance evaluation and buffer management.