Performance Analysis of M/M(a,b)/1 Queuing Model with Balking and State-dependent Reneging and Service (original) (raw)
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The paper investigates a M / M ( b , b ) /1 queuing model with bulk service. The server serves the customers in batches of fixed size b , and the service time is assumed to be exponentially distribution. Customers arrive to the system as a Poisson process and may renege after waiting in the queue for an exponentially distributed time. The reneging of a customer depends on the state of the system. The model is analyzed to find the different measures of effectiveness of the model. The approach adopted is based on embedded Markov chains.
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The ultimate objective of the analysis of queuing systems is to understand the behaviour of their underlying process so that informed and intelligent decisions can be made by the management. The application of queuing concepts is an attempt to minimize cost through minimization of inefficiency and delays in a system. Various methods of solving queuing problems have been proposed. In this study we have explored single –server Markovian queuing model with both interarrival and service times following exponential distribution with parameters and , respectively, and unlimited queue size with FIFO queuing discipline and unlimited customer population. We apply this model to catering data and estimate parameters for the same. A sensitivity analysis is the carried out to evaluate stability of the system.
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The phenomena are balking can be said to have been observed when a customer who has arrived into queuing system decides not to join it. Reverse balking is a particular type of balking wherein the probability that a customer will balk goes down as the system size goes up and vice versa. Such behavior can be observed in investment firms (insurance company, Mutual Fund Company, banks etc.). As the number of customers in the firm goes up, it creates trust among potential investors. Fewer customers would like to balk as the number of customers goes up. In this paper, we develop an M/M/1/k queuing system with reverse balking. The steady-state probabilities of the model are obtained and closed forms of expression of a number of performance measures are derived.
An M/M/1/N Queueing Model with Retention of Reneged Customers and Balking
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The concept of customer balking and reneging has been exploited to a great extent in recent past by the queuing modelers. Economically, if we see, the customer impatience (due to balking and reneging) leads to the loss of potential customers and thereby results into the loss in the total revenue. Taking into consideration this customers' loss due to impatience, a new queuing model has been developed that deals with retention of reneged customers. According to this model, a reneged customer can be convinced in many cases by employing certain convincing mechanism to stay in the queue for completion of his service. Thus, a reneged customer can be retained in the queuing system with some probability (say, q) and it may leave the queue without receiving service with probability p (=1-q). This process is referred to as customer retention. We consider a single server, finite capacity queuing system with customer retention and balking in which the interarrival and service times follow negative-exponential distribution. The reneging times are assumed to be exponentially distributed. An arriving customer may not join the queue if there is at least one customer in the system, i.e. the customer may balk. The steady state solution of the model has been obtained. Some performance measures have been computed. The sensitivity analysis of the model has been carried out. The effect of probability of retention on the average system size has been studied. The numerical results show that the average system size increases proportionately and steadily as the probability of retention increases. Some particular cases of the model have been derived and discussed.
Bulk Service Queuing System A Computational Approach
2016
The paper investigates a M/M/1 queuing model with bulk service customers in batches of fixed size b, and the service time is assumed to be exponentially distribution. Customers arrive to the system as a Poisson process and may renege after waiting in the queue for an exponentially distributed time the system. The model is analyzed to find the different measures of effectiveness of the model approach adopted is based on embedded Markov chains ______________________________
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We consider a queuing network with single-server nodes and heterogeneous customers. The number of customers, which can obtain service simultaneously, is restricted. Customers that cannot be admitted to the network upon arrival make repeated attempts to obtain service. The service time at the nodes is exponentially distributed. After service completion at a node, the serviced customer can transit to another node or leave the network forever. The main features of the model are the mutual dependence of processes of customer arrivals and retrials and the impatience and non-persistence of customers. Dynamics of the network are described by a multidimensional Markov chain with infinite state space, state inhomogeneous behavior and special structure of the infinitesimal generator. The explicit form of the generator is derived. An effective algorithm for computing the stationary distribution of this chain is recommended. The expressions for computation of the key performance measures of the...
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The paper studies a queuing model with Poisson arrival process and bulk service. The server serves the customers in batches of fixed size b, and the service time is assumed to be exponentially distribution. The model is analyzed to find the steady-state distribution of the number of customers stranded following each service. The approach adopted is based on discrete-time Markov chains, instead of Laplace transforms that is usually used in literature. A simulation study is carried out to estimate the expected number of stranded customers at any point of time, its variance and the downside risk for given values of the system parameters.
M/Ck/1 Queue with Impatient Customers
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A single-server queuing system with impatient customers and Coxian service is examined. It is assumed that arrivals are Poisson with a constant rate. When the server is busy upon an arrival, customer joins the queue and there is an infinite capacity of the queue. Since the variance of the service time is relatively high, the service time distribution is characterized by k-phase Cox distribution. Due to the high variability of service times and since some of the services take extremely long time, customers not only in the queue, but also in the service may become impatient. Each customer, upon arrival, activates an individual timer and starts his patience time. The patience time for each customer is a random variable which has exponential distribution. If the service does not completed before the customer's time expires, the customer abandons the queue never to return. The model is expressed as birth-and-death process and the balance equations are provided.
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Customers often get attracted by lucrative deals and discounts offered by firms. These, attracted customers are termed as encouraged arrivals. In this paper, we developed a multi-server Feedback Markovian queuing model with encouraged arrivals, customer impatience, and retention of impatient customers. The stationary system size probabilities are obtained recursively. Also, we presented the necessary measures of performance and gave numerical illustrations. Some particular, and special cases of the model are discussed.
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In this paper, a single server queueing model, wherein the units arrive in bulk with varying arrival rates in Poisson process, is considered. It is assumed that the service time of units is arbitrarily distributed. Also, we incorporate the optional deterministic vacations for the server. The server may take a vacation of a fixed duration at the completion of each service or may continue to be available in the system for the next service. At busy and vacation states, the customers may balk from the system with different balking probabilities. By using the basic assumptions of probability reasoning and supplementary variable technique, the steady state behaviour of the system is studied and various performance measures are obtained. In order to obtain the approximate values of the system state probabilities, the principle of maximum entropy is also employed. To verify the tractability of the performance measures obtained, the numerical illustrations are provided. Further, the sensitivity analysis is carried out to examine the system performance with respect to different parameters.