Queues or waiting lines are an integral part of health care facilities such as hospitals, outpatient clinics, medical laboratories, and many other health facilities. Health care management must have waiting lines control strategies for smooth functioning. Due to the lack of proper queuing control and management, patients may become dissatisfied and may leave (renege) the health care facilities without getting service. But, the reneging of patients at two consecutive time marks may be correlated in the sense that if a patient reneges at the current time mark, then there is a probability that a patient may or may not renege at the next time mark. This kind of reneging is referred to as correlated reneging. In this paper, we have introduced the concept of correlated reneging in a finite capacity multi-server queuing model with balking with its application in health care. The steady-state as well as the transient analyses of the model are carried out. We have also derived an expression for the correlation coefficient between the inter-reneging times and for the rate at which the health facility is losing patients (patient loss probability) due to insufficient capacity, reneging, and balking. We have provided numerical examples in order to demonstrate the effect of balking and correlated reneging on performance measures such as the mean number of patients waiting to be serviced, mean waiting time of patients, and the probability of patient rejection. Further, the effect of the number of servers on performance measures is investigated. Finally, the effect of the correlation coefficient between the inter-reneging times on performance measures is studied. The queuing model discussed in this paper could be useful to the health care firms facing the problem of patient impatience and capacity constraints.

In this paper, the concept of correlated reneging is introduced in queuing theory. The reneging considered so far is dependent on system size, but there are many real life situations where customers may renege due to exogenous factors other than the state of the system. Further, the reneging of customer may induce the other customers to renege at two successive time points. Such reneging is called correlated reneging. An M/M/1/K queuing model with correlated reneging is studied. Runge-Kutta method of fourth order is presented to obtain the transient solution of the model. Some performance measures like expected system size and expected waiting time in the system are studied.

This chapter deals with a multi-server queuing system with exponentially distributed inter-arrival and service times, feedback of served customers and retention of reneging customers. Customers’ impatience is harmful to any business. The idea of retention of reneging (impatient) customers is based on the fact that if the firms providing service utilize some customer retention mechanism, then there is a probability that a reneging (impatient) customer may be retained in the system for his service. Moreover, the dissatisfied customers, after receiving service, may re-join the queue for further service; such customers are known as feedback customers. We derive the time-dependent solution of the queuing model by using the technique of probability generating function. Important queuing models are obtained as the particular cases of the model.

In this article, we consider a single server queuing model with correlated reneging and feedback of served customers. The steady-state solution of the model is obtained using the matrix-decomposition method. The transient analysis of the model is also performed with the help of the Runge-Kutta method. Important measures of performance like expected queue length and expected waiting time in the queue are computed. Some comparisons are performed, and finally, the sensitivity analysis of the model is also carried out.