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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In classical queuing models it is assumed that customers are willing to wait until they get serviced. However in real life, customers may lose their patience due to the long service times or long queues and leave the system without having service.
Impatience behavior of customers can be experienced as balking and/or reneging. A customer may discouraged by the queue if the queue is too long and he may decide not to enter the system. If a customer decides not to enter the queue upon arrival, the customer is said to have balked. On the other hand, a customer may enter the queue, but after a time decide to leave. In this case, the customer is said to have reneged (Gross and Harris, 1998). The importance of queuing models with impatient customers appears in many real life problems such as in situations involving impatient telephone switchboard customers, hospital emergency rooms handling critical patients, inventory systems with storage of perishable goods, as well as in certain military problems (Stanford, 1979).
Queuing models with impatient customers, where the source of impatience is due to the long waiting times in the queue, have been studied extensively. Palm (1937), who is the notable pioneer of queuing theory, first mentioned about the reneging behavior. However, his results have apparently been reproduced or derived independently in studies beginning in 1957 (Stanford, 1979). An M/M/1 queue with reneging is proposed by Haight (1959), and it is assumed customers renege with a specified probability which is a function of the total number of customers in the system. Barrer (1957) considers M/M/1 queues with reneging. The reneging times are assumed to be fixed and the distribution of number of waiting customers are derived.
Ancker and Gafarian (1962) examine the Markovian queuing system with exponential reneging times and derived the waiting time distribution. Choudhury (2008) deals with a Markovian queuing system M/M/1 with reneging and it is assumed that customer's patience time is an exponential random variable. A derivation of the distribution of virtual waiting time in the system and in the queue is presented. Perel and Yechiali (2010) consider a Markovian queue with slow servers and impatient customers. It is assumed customers become impatient after a random amount of time because of the slow service rate. The balance equations and steady state solution are presented for single-server, multiple-server, and infinite-number-of-server cases.
Rao(1968) treats an M/G/1 queuing system with exponential reneging times, and derives the generating function for the Laplace transform of the joint distribution of the length of a busy period and the number served during a busy period. Martin and Artalejo (1995) examine M/G/1 type queuing system with two types of arrival stream; patient and impatient customers and developed the performance measures. De Kok and Tijms (1985) consider a single-server queuing system with Poisson input and general service times, where a customer becomes a lost customer when his service has not begun within fixed time after his arrival. Useful approximations for performance measures as the loss probability and the average delay in the queue are presented. Bacelli et. al (1984) study a single-server queue with general interarrival and service times with impatient customers. Bae et. al. (2001) consider an M/G/1 queue with reneging. Customers wait for service only for limited fixed time and leave the system if their service do not start during this period. They analyzed the virtual waiting time and the busy period of the system.