Source: http://article.sapub.org/10.5923.j.ajor.20170701.01.html
Timestamp: 2019-04-21 02:06:54+00:00

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Correspondence to: Sushil Ghimire , Pulchowk Campus, Institute of Engineering, Tribhuvan University, Nepal.
Cite this paper: Sushil Ghimire , Gyan Bahadur Thapa , Ram Prasad Ghimire , Sergei Silvestrov , A Survey on Queueing Systems with Mathematical Models and Applications, American Journal of Operational Research, Vol. 7 No. 1, 2017, pp. 1-14. doi: 10.5923/j.ajor.20170701.01.
Queueing theory is one of the branches of applied mathematics which studies and models the waiting lines. Danish mathematician A. K. Erlang (1878–1929), who published his first paper entitled “The Theory of Probability and Conversations” in 1909 , is considered as the father of queueing theory. Further going back to the history, it can be observed that a viable queueing theory was developed by French Mathematician S. D. Poisson (1781–1840), who created a probability distribution function for the total outcomes of independent trials. He used statistical approach for these distributions which can be applied to the situations where excessive demands are to be fulfilled on a limited resource. During the late 1800s, all telephone calls used to be switched manually to the recipient by an operator. Each customer used to call the operator first and the operator used to fix the call for the customer. In this process, telephone companies were facing problem to appoint more operators. Callers who were unable to reach to an operator may simply hung up for several minutes with frustration and might think that it was a busy time for the operators. On the other hand, some would be waiting their turn to talk to the operator. And some others would call repeatedly thinking that the operator would be sufficiently annoyed by repeated calls to serve them next. These type of behaviour of the customers caused problems for traffic engineers because they affected the level of demand for service from an operator. A call which was not reached to the operator could be lost and could be effectively out of the system. To overcome this situation and to reduce the number of switchboards in an area, the most important application of queuing theory was developed. Those callers who repeatedly try for the operator increased demands on the system by appearing several requests. Poisson's formula was meant only for the repeated callers. Kendell  presented a paper that opened a general review of some points in congestion theory to enhance the study for a single server queue where input is Poisson and service time is generally distributed. In , he further extended the study of the stochastic processes for the theory of queues and their analysis by the method of the imbedded Markov chain. The study was carried out first reviewing on single-server queues and using the similar technique to the analysis of many server queuing system.
Stochastic process is a key factor to specify in queueing systems because it describes the arrival pattern as well as the structure and the discipline of the service facility. Queueing system deals with queue length and waiting times. The concept of queue is applied not only in the waiting system by the human beings but also in modern technology of computer and other service providers by the devices. In general, it is not necessary that service will be immediately available to address the demand of all the customers, so that they are forced to line up. In the queueing system, the one who demands the service is referred as customer, which may be a person, a task or a commodity. The other element of the queueing system is the one who provides the service with some defined discipline, called the server. It may be people, machine or objects. Some of the service disciplines are first come first served (FCFS), last come first served (LCFS), service in random order (SIRO), priority, processor sharing (PS), round-robin (RR). We study performance measures of a queueing system where only the limited number of customers are served and arrival or service or both occur in a batch. If any of the customers come after the prescribed quota has already been served, the server does not provide the service to the new comer.
The main goal of this study is to present and analyse measures of performance effectiveness for some specific queuing models. The models we investigate have important applications in the study of machine repair problem, tele-traffic, computer and flexible manufacturing systems, production processes, transportation, monitoring, controlling and managing complex engineering systems that have finite buffer system. Transient study makes the problem more realistic. The situations considered are also applicable to various day-to-day activities. It has been attracting the attention of mathematicians and operations research scientists in the field of social and liberal globalization of economy. The study provides a prospect to the development of research and design in related fields.
Rest of the paper is organized as follows: Section 2 presents the state of arts of the queueing system and associated theories developed by various researchers. Section 3 describes the different components of queueing system along with the standard notations used in queueing models. Some of the mathematical formulae for different queueing models and the derivation of simple Markovian queue are explained in Section 4. Section 5 includes Little’s law with its use for the calculation of performance measures in queueing systems. Some applications of queueing models in supply chain management are observed in Section 6. Finally, Section 7 concludes the paper.
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 Probability of number of customers and mean number of customers in the system at time 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.
If any customer is willing to get a service, s/he should check whether a server is idle or not. If the server is vacant, customer gets the service immediately. However, if at least one customer is waiting for the service in front of each of the servers, then the new arrival should line up. Figure 1 represents the basic queueing model where the procedure of a simple queueing system is shown.
From the time someone starts standing in a queue until getting served, there are certain steps to follow. These steps are called the components of a queue which are characterized by the arrival process of customers, behaviour of customers, service times, service discipline and service capacity. These components are briefly described in the followings.
The inter-arrival times are assumed to be independent having a common distribution. In many practical situations, customers arrive according to a Poisson stream (i.e. exponential inter-arrival times). Customers may arrive one by one or in batches. An example of batch arrivals is the customs office at the border where travel documents of passengers are to be checked.
Customers may be patient and willing to wait for the service after some times or may be impatient and leave after a while. The behaviour of customers who leave the queue realizing that they have to wait longer than they have expected is called balking. There are some customers who leave the queue feeling that they are tired of waiting in the queue. This type of customers’ behaviour is called reneging. There is another behaviour of customers who re-join the queue which they had left earlier either by balking or by reneging, is called jockeying.
When a customer joins a queue, server takes a certain time to serve the customers. This time is called the service time which can be deterministic or exponentially distributed. It can also occur that service times are dependent of the queue length. For example, the processing rates of the machines in a production system can be increased once the number of jobs waiting to be processed becomes larger.
First Come First Served (FCFS): In this process, service is provided in the order of arrival. First comer gets the service first. Generally, this method is applied in the supermarket, billing counters and many other queuing systems. It is called First In First Out (FIFO) as well.
Service In Random Order (SIRO): This is the method in which service is provided without any fixed rule. People can be observed randomly in security checkpoints. Service provider collects data randomly for statistical studies.
Last Come First Served (LCFS): It is the another method of providing service to the customers in which the last arrival gets service at first. This method is applied in the production line where the products are kept one over the other. While selling these products, the item kept at the top is sold at first though it is placed there at last.
Priorities: There can be some customers who need the service immediately for many reasons. Those customers could be in a rush or may take shorter processing time. Some emergency cases could also be there in the doctor’s clinic and some would be ready to pay more for the quicker service.
Processor Sharing: Concept of processor sharing is specially applied in the communication system where computers divide their processing power equally over all the other computers. Number of customers can get access to the internet from a single router.
Round Robin (RR): This method can be seen mainly in cyclic queueing system in which either server or the customers move in a cycle. We have experienced a revolving dining table where customers are stationary but the server moves carrying different menu on the table.
There may be a single server or group of servers to help arrivals having limitations with respect to the number of customers. For example, in a data communication network, only finitely many cells can be buffered in a switch. The determination of good buffer size is an important issue in the design of these networks.
where denotes inter-arrival time distribution, is the service time distribution, represents the number of servers and denotes the maximum number of jobs that can be occupied in the system (waiting and in service) with infinite number of waiting positions for default. Likewise, indicates queueing discipline (FCFS, LCFS, SIRO, RR etc.) having FCFS for default, and the last notation is for population size from which customers rush to the system.
where P0 is the steady-state probability that the server is idle. and denote the mean busy period and mean idle period of the server respectively.
where and respectively denote the mean sojourn times of the chain in A and during a cycle.
where denotes service time; and are random variables and their means denoted by and , are appropriate for measuring the efficiency of the system. It is not easy in general to obtain their distribution function.
The primary aim is to get their distributions which always may not be possible. In many of the situations, we have only their mean values or their generating function.
With these, we describe some of the queueing models with their respective formulae in the following subsections.
A Birth-Death process is a Markov process in which states are numbered by an integer and transitions are only permitted between two neighbouring states. Births are the cases when state variables are increased by one and deaths are the cases when state variables are decreased by one. When birth occurs, the state N moves to state N + 1 and when the death occurs, state N changes to state N - 1.
The queueing system M/M/1 is the simplest non-trivial queue where the customers arrive according to a Poisson process with rate , that is, the inter-arrival times are independent, exponentially distributed random variables with parameter . The service times are assumed to be independent and exponentially distributed with parameter . Furthermore, all the involved random variables are supposed to be independent of each other.
The queuing system M/M/c is the queueing discipline where c service channels are ready for the arriving customers following Poisson process. and have the usual meanings with all the random variables independent as described in the subsection 4.2. Followings are some of the formulae to for the performance measures of this model.
In this model, customers arrive according to a Poisson process describing independent and exponentially distributed inter-arrival times. The service times are also assumed to be independent and exponentially distributed. The difference of this model with the previous models is the only k customers who can get the service by fixed number of c servers. Followings are the formulae for some of the performance measures of this queueing system.
Each of the proposed modes should have some kind of applications in the real life. It is important to verify those results by means of some established tools. These results are called the performance measures and are calculated by using different techniques. One of the methods to calculate these performance measures is the Little’s Law. In this Section, we briefly describe about the performance measures and the Little’s Law.
(i) Distribution of the waiting and the sojourn times: Time spent by a customer in a queue is calculated in two categories. The first is the waiting time before starting to receive the service and the second is the sojourn time which includes the waiting time plus the service time.
(ii) Distribution of the number of customers: In a single server queueing system, there will be one customer receiving the service whereas in multiple server queueing system, there could be the customers equal to the number of servers receiving the service. Number of customers in a queueing system refers to the customers including or excluding the one or all the service.
(iii) Distribution of amount of work: It is the sum of service times of the waiting customers and the residual service time of the customers in service. Residual service time signifies the time that a new arrival waits until being served in a non-empty queue.
(iv) Distribution of busy period: When customers arrive to the server for the service, the server becomes busy. Busy period of a server is the time during which server is working continuously. While calculating the performance measures, we are interested in mean performance like the mean waiting time and the mean queue length.
It may be applied to separate parts of much larger queueing systems, such as subsystems in a queueing network. In such a case, L should be defined with respect to the number of customers in a subsystem and W with respect to the total time in that subsystem. Little’s law may also refer to a specific class of customer in a queueing system or to subgroups of customers, and so on. Its range of applicability is very wide indeed.
• It is widely applicable (it requires only very weak assumptions). It will be valuable to us in checking the consistency of measurement of data.
• It is the main task in the algorithms for evaluating several queueing network models.
• In studying computer system, we frequently find two of the quantities related by Little’s law (the average number of requests in a system and the throughput of that system) and desire to know the third (the average system residence time, in this case).
• On rainy days, streets and highways are more crowded.
• Fast food restaurants need a smaller dining room than regular restaurants with the same customer arrival rate.
• Large buffering together with large arrival rate cause large delays.
Theorem 2 : In a closed Gordon-Newell network with m queues, write for the state of network. For a customer in transit to state i, let denotes the probability that immediately before arrival the customer sees the state of the system is Then the probability is same as the steady state probability for state for a network of the same type with one customer less.
• If a customer changes queue, the one s/he has left will start to move faster than the one s/he is in.
• Customer feels that her/his queue always goes the slowest.
• Whatever queue a customer joins, no matter how short it looks, will always take the longest for her/him to get served.
Queueing theory is applied in many of the daily life activities including computer networks, telecommunication systems, traffic flow systems, airport scheduling systems, banking and logistic operations and so on. Besides all these, queuing system is applied in the manufacturing industries as well. Items produced by industries have to be delivered to the retailers and then to the customers. If there is the proper chain to deliver those items, it can save time and money. Products of the industries can be delivered together in numbers but one machine can produce only one item at a time following a sequential order. Those produced items should be supplied to the wholesalers and to the retailers turn by turn maintaining a proper queue. In this sense, we can observe a close relationship between queuing system and supply chain management, which is described in rest of this Section.
where is the probability of a customer being blocked for queueing model.
Arrival rate of forward flow of product p to bidirectional facility b.
where and are the recorder quantity and recorder point at distribution centre K. Wang et al.  collected a review defining supply chain and discussing literature in the areas, namely service supply management, service demand management and the coordination of service supply chains to observe the state in each area. He  derived supply risk sharing contracts for the equilibrium between the recycling price decision and the remanufacturing quantity decision using game theory illustrating some numerical examples for managerial results. Zhalechian et al.  studied environmental impact of sustainable closed-loop location - routing -inventory model using a stochastic-probabilistic programming approach presenting some real-world applications. Sadjadi et al.  derived optimization model using queuing approach for allocation of the retailers’ demands, and inventory replenishment decisions so as to minimize the total expected cost of location, transportation and inventory. Jin  formulated link transmission model and link queue model defining demand and supply to present queuing models for a point queue and their discrete versions.
Upon observing the contributions of the researchers in the field of queueing theory, it can be noticed that plenty of works have been done pointing out many extendable areas. Change of one parameter in any of the proposed models might cause a huge change in the result of performance measures. Small change in the arrival rate may create large queue or no queue, and small change in service rate may make the customers very happy for the quick service or may have to wait for a long time. For any of the queue, time plays a vital role. It is very important how long a customer waits in the queue to get the service and how fast a server provides the service. To make the service more effective, sometimes we need to add the servers and increase the efficiency. We have seen the proposed queueing models for finite and infinite capacities. Some of the queueing models have finite capacity and some are ready to serve for any number of customers. Some are time dependent studied under transient fashion and some are used for the optimization model. We have chosen Markovian queuing model with finite number of customers for a single or multiple servers. Besides the usual and standard mathematical modelling in queueing theory, consideration of customers’ behaviours, servers’ breakdown or vacation along with the limitation in arrivals can be introduced to make the model more realistic and challenging. On the other hand, suggesting some mathematical models considering those limitations may not always be reliable, so verifying those models in the real life situations with the use of computer simulation would be a remarkable contribution in the study of queueing theory.
All the studies carried out by the researchers have several applications in the real life. These applications are specially focused for making the life easier specially by saving time and money. In this process, number of models are proposed with applications in the different areas. The other motivation is to get the maximum profit with the minimum utilization of the limited resources, called the constraints. We intend to study the conditions for the optimal solution in order to maximize the production or the benefits using those limited constraints. These basic phenomena can be applied in telecommunication, traffic control, employee allocation, computer scheduling, supermarket, hospitals and many other fields. Variations of arrival and service disciplines in a queueing problem is the challenging work to tackle in the days to come. In addition, the simultaneous study of queueing operations with manufacturing and logistics may yield some interesting interlinks. Some of the literatures are described in the former Section. Such comparative study can be applied in the real problems of the industries to optimize production and distribution operations. The detail study of the application of queueing theory in supply chain networks will be our due course.
The first author is thankful to Erasmus Mundus SmartLink project for financial support as a PhD exchange student to carry out the work in Burgas Free University, Bulgaria from Sep 2016 – Sep 2017.
The second author is thankful to the Erasmus Mundus LEADERS Project for funding him as a Post Doc Research Fellow to carry out the work in Department of Mathematics, University of Evora, Portugal from Nov 2016 – Aug 2017.
 Kendall, D. G. (1951). Some Problems in the Theory of Queues, Journal of the Royal Statistical Society. Series B (Methodological), 13(2), 151-185.
 Kendall, D. G. (1953). Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain, The Annals of Mathematical Statistics, 24(3), 338-354.
 Kovalenko, I. N. (1994). Rare events in queueing system - A survey, Queueing System, 16, 1-49.
 Cohen, J.W. and Boxma O.J. (1985). A survey of the evolution of queueing theory, Statistica Neerlandica, 39, 143-158.
 Hui, G. HSU (1990). A survey of queueing theory, Annals of Operations Research, 24, 29-43.
 Fomundam, S. F. and Herrmann, J. W. (2007). A survey of queuing theory applications in healthcare, The Institute for Systems Research, 24, 1-23.
 Lade, I. P., Chowriwar, S. A. and Sawaitul, P. B. (2013). Simulation of queueing analysis in hospital, International Journal of Mechanical Engineering and Robotics Research, 2(3), 122-128.
 Shanthikumar, J. G., Ding, S. and Mike Tao Zhang, M. T. (2007). Queueing theory for semiconductor manufacturing systems: A survey and open problems, IEEE Transactions on Automation Science and Engineering, 4(4), 513-522.
 Jackson, R. R. P. and Adelson, R. M. (1962). A critical survey of queueing theory, Part 1, Operational Research Society, 13(1), 13-22.
 Jouini, O. and Dallery, Y. (2007). Monotonicity properties for multi-server queues with reneging and finite waiting lines, Probability in the Engineering and Informational Sciences, 21, 335-360.
 Karaesmen, F., Avram, F. and Gupta, S. M. (1997). Service control in a finite buffered queue with multiple service rates, 1-26.
 Laxmi, P. V. and Suchitra, V. (2014), Finite buffer GI/M(n)/1 queue with Bernoulli-schedule vacation interruption under -policy, International Scholarly Research Notices, 2014, Article ID 392317.
 Kwon, S. T., Performance analysis of an FMS with finite capacity buffers and single AGV"
 Chakravarthy, S. R (2014). A multi-server queueing model with server consultations, European Journal of Operational Research, 233, 625-639.
 Ghimire, R. P. and Basnet, R. (2011). Finite capacity queueing system with vacation and service breakdown, International Journal of Engineering, 24(4), 387-394.
 Yang, D. Y. and Wu, Y. Y. (2014). Transient behavior analysis of a finite capacity queue march with working breakdowns and server vacations. Proceedings of the International Multi-Conference of Engineers and Computer Scientists IMECS, Vol. II, 12-14.
 Kaczynski, W. H., Leemis, L. M. and Drew, J. H. (2012). Transient queueing analysis. INFORMS Journal on Computing, 24(1), 10-28.
 Smith, J. M. G. (2010). Properties and performance modelling of finite buffer M/G/1/K networks, Computers & Operations Research, doi:10.1016/j.cor.2010.08.014.
 Cruz, F. R. B. and Woensel, T. V. (2014). Finite queueing modelling and optimization: A selected review, Journal of Applied Mathematics, 2014, Article ID 374962.
 Yadin, M. and Naor, P. (1963). Queueing system with a removal service station, Institute of Statistics, Mimeo Series No. 353.
 Schrage, L. (1969). Analysis and optimization of a queueing model of a real-time computer control system. IEEE Transactions on Computers, C-18(11), 997-1003.
 Akyildiz, I. F. and Liu, W. (1990). Performance optimization of distributed system models with unreliable servers. IEEE Transactions on Reliability, 39(2), 236-243.
 Krivulin, N. K. (1994). Recursive equations based models of queueing systems, Proc. European Simulation Symp., Istanbul, Turkey, 252-256.
 Bertsimas, D. and Mora, J. N. (1999). Optimization of multiclass queueing networks with changeover times via the achievable region approach: Part II, The multi-station case, Mathematics of Operations Research, 24(2), 331-361.
 Mishra, S. S. and Yadav, D. K. (2010). Computational approach to profit optimization of a loss-queueing system, Journal of Applied Computer Science & Mathematics, 9(4), 78-82.
 Brito, N. L. C., Duarte, A. R., Ferreira, J. H. and Cruz, F. R. B. (2012). Multi-objective optimization of finite queueing networks, International Conference on Engineering Optimization, Rio de Janeiro, Brazil, 1-5 July 2012.
 Pesu, T. and Knottenbelt, W. J. (2015). Dynamic subtask dispersion reduction in heterogeneous parallel queueing systems, Electronic Notes in Theoretical Computer Science, 318, 129-142.
 Shin, Y. W. and Choo, T. S. (2009). M/M/s queue with impatient customers and retrials, Applied Mathematical Modelling, 33, 2596-2606.
 Al-Seedy, R. O., El-Sherbiny, A. A., El-Shehawy, S. A. and Ammar, S. I. (2009). Transient solution of the M/M/c queue with balking and reneging, Computers and Mathematics with Applications, 57, 1280-1285.
 Ayyappan, G., Devipriya, G. and Subramanian, A. M. G. (2013). Transient analysis of single server queueing system with batch service under catastrophe, International Journal of Mathematical Archive, 4(5), 26-32.
 Choudhury, A. and Medhi, P. (2013). Performance evaluation of a finite buffer system with varying rates of impatience, Journal of the Turkish Statistical Association, 6(1), 42-55.
 Ghimire, R. P. and Ghimire, S. G. (2011). Heterogeneous arrival and departure M/M/1 queue with vacation and service breakdown, Management Science and Engineering, 5(3), 61-67.
 Atencia, I. and Moreno, P. (2006). A discrete-time Geo/G/1 retrial queue with the server subject to starting failures, Ann Oper Res, 141, 85-107.
 Ammar, S. I. (2013). Transient analysis of a two-heterogeneous servers queue with impatient behavior, Journal of the Egyptian Mathematical Society, 28 June, 2013.
 Gong, J. and Li, M. (2014). Queuing time decision model with the consideration on call center customer abandonment behavior, Journal of Networks, 9(9), 2441-2447.
 Li, M. and Cheng, J. (2011). Transient state analysis of the shortest queueing system, IEEE, 2011.
 Shanmugasundaram, S. and Banumathi, P. (2016). A simulation study on M/M/C queueing models, International Journal for Research in Mathematics and Mathematical Sciences, 2(2), 52-61.
 Singla, N. and Garg, P. C. (2014). Transient and numerical solution of a feedback queueing system with correlated departures, American Journal of Numerical Analysis, 2(1), 20-28.
 Zeng, M., Cheng, W. and Guo, P. (2014). A transient queuing model for analyzing and optimizing gate congestion of railway container terminals, School of Mechanical Engineering, Hindawi Publishing Corporation, Mathematical Problems in Engineering, Vol. 2014, Article ID 914706.
 Chan, R. H., Lee, S. T. and Sun, H. W. (2012). Boundary value methods for transient solutions of queueing networks with variant vacation policy, Journal of Computational and Applied Mathematics, 236, 3948-3955.
 Jiang, H., Li Z., Jiang, R., Song, J. and Li, L. (2013). Three-velocity queueing model for congested traffic flow simulation, Procedia - Social and Behavioral Sciences, 96, 1389-1401.
 Ghimire, S., Ghimire, R. P. and Thapa, G. B. (2015). Performance evaluation of unreliable M(t)/M(t)/n/n queueing system, British Journal of Applied Science & Technology, 7(4), 412-422.
 Tan, X., Knessl, C., Yang, Y. (P.) (2013). On finite capacity queues with time dependent arrival rates, Stochastic Processes and their Applications, 123, 2175-2227.
 Kempa, W. M. (2017). A comprehensive study on the queue-size distribution in a finite-buffer system with a general independent input flow, Performance Evaluation, 108, 1-15.
 Malligarjunan, R. (2014). Transient behavior and entropy measures on an inventory system with two demand classes, Applied Mathematics and Computation, 226, 738-753.
 Selinka, G., Franz, A. and Stolletz, R. (2016). Time-dependent performance approximation of truck handling operations at an air cargo terminal, Computers & Operations Research, 65, 164-173.
 Ausina, M. C., Wiperb, M. P and Lillob, R. E. (2014). Bayesian prediction of the transient behavior and busy period in short and long tailed GI/G/1 queueing systems, Applied Mathematical Modelling, 38, 5870-5882.
 Chang, S. H. and Choi, D. W. (2005). Performance analysis of a finite-buffer discrete-time queue with bulk arrival, bulk service and vacations, Computers & Operations Research, 32, 2213-2234.
 Goswami, V. and Laxmi, P. V. (2011). Performance analysis of a renewal input bulk service queue with accessible and non-accessible batches, Quality Technology & Quantitative Management, 8(2), 87-100.
 Ghimire, S., Ghimire, R. P. and Thapa, G. B. (2014). Mathematical models of Mb/M/1 bulk arrival queueing system, Journal of the Institute of Engineering, 10(1), 184-191.
 Singh, C. J., Jain, M. and Kumar, B. (2016). MX/G/1 unreliable retrial queue with option of additional service and Bernoulli vacation, Ain Shams Engineering Journal, 7, 415-429.
 Banergee, A., Gupta, U.C. and, Chakravarthy, S.R. (2015). Analysis of a finite-buffer bulk-service queue under Markovian arrival process with batch-size-dependent service, Computers & Operations Research, 60, 138-149.
 Luo, C., Li, W., Yu, K. and Ding, C. (2016). The matrix-form solution for GeoX/G/1/N working vacation queue and its application to state-dependent cost control, Computers & Operations Research, 67, 63-74.
 Claeys, D., Steyaert, B., Walraevens, J., Laevens, K. and Bruneel, H. (2013). Analysis of a versatile batch-service queueing model with correlation in the arrival process, Performance Evaluation, 70, 300-316.
 Sztrik, J., Basic queueing theory, University of Debrecen, Faculty of Informatics, 12-14.
 Sztrik, J. (2010). Queueing theory and its applications-a personal view, Proceedings of the 8th International Conference on Applied Informatics, January 27-30, 1, 9-30.
 Bhaskar, V. and Lallement, P. (2010). Modelling a supply chain using a network of queues, Applied Mathematical Modelling, 34, 2074-2088.
 Bhaskar, V. and Lallement, P. (2011). Queuing network model of uniformly distributed arrivals in a distributed supply chain using subcontracting, Decision Support Systems, 51, 65-76.
 Boulaksil, Y. (2016). Safety stock placement in supply chains with demand forecast updates, Operations Research Perspectives, 3, 27-31.
 Teimoury, E., Mazlomi, A., Nadafioun, R., Khondabi, I. G. and Fathi, M. (2011). A queuing-inventory model in multiproduct supply chains, Proceeding of the International Multi-Conference of Engineers and Computer Scientists, 3, March 16-18, 2011.
 Liu, L., Liu, X. and Yao, D. D. (2003). Analysis and optimization of a multi-stage inventory-queue system. November 10, 2003.
 Sivakumar, B., Jayaraman, R. and Arivarignan, G. (2012). A discrete time inventory system with postponed demands. Journal of Computational and Applied Mathematics, 236, 3073-3083.
 Andriansyah, R., Woensel, T. V., Cruz, F. R. B. and Duczmal, L. (2010). Performance optimization of open zero-buffer multi-server queueing networks, Computers & Operations Research, 37, 1472-1487.
 Mishra, S. S. and Yadav, D. K. (2010). Computational approach to cost and profit analysis of clocked queueing networks, Contemporary Engineering Sciences, 3(8), 365-370.
 Mary, K. J. R. and Christina, G. M. (2015). Analysis of total average cost for with fuzzy parameters using robust ranking technique, International Journal of Computer Applications, 121(24), 1-4.
 Smith, J. M. G. (2015). Queue decomposition & finite closed queueing network models, Computers & Operations Research, 53, 176-193.
 Babadi, A. Y., Moghaddam, R. T., Amiri, A. B., Seifi, S. (2017). Designing a reliable multi-Objective Queuing model of a petrochemical supply chain network under uncertainty: A case study, Computers and Chemical Engineering, 100, 177-197.
 Vahdani, B. and Mohammadi, M. (2015). A bi-objective interval-stochastic robust optimization model for designing closed loop supply chain network with multi-priority queuing system, Int. J. Production Economics, 170, 67-87.
 Diabat, A., Dehghani, E. and Jabbarzadeh, A. (2017). Incorporating location and inventory decisions into a supply chain design problem with uncertain demands and lead times, Journal of Manufacturing Systems, 43, 139–149.
 Wang, Y., Wallace, S. W., Shen, B. and Choi, T. M. (2015). Service supply chain management: A review of operational models, European Journal of Operational Research, 247, 685-698.
 He, Y. (2017). Supply risk sharing in a closed-loop supply chain, Int. J. Production Economics, 183, 39–52.
 Zhalechian, M., Moghaddam, R. T., Zahiri, B. and Mohammadi M. (2016). Sustainable design of a closed-loop location-routing-inventory supply chain network under mixed uncertainty, Transportation Research Part E, 89, 182–214.
 Sadjadi, S. J., Makui, A., Dehghani, E. and Pourmohammad, M. (2016). Applying queuing approach for a stochastic location-inventory problem with two different mean inventory considerations, Applied Mathematical Modelling, 40, 578–596.
 Jin, W. L. (2015). Point queue models: A uniﬁed approach, Transportation Research Part B, 77, 1–16.

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