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.
Queuing theory has been extensively used in the modelling and performance analysis of cloud computing systems. The phenomenon of the task (or request) reneging, that is, the dropping of requests from the request queue often occur in cloud computing systems, and it is important to consider it when developing performance evaluations models for cloud computing infrastructures. Majority of studies in the performance evaluation of cloud computing data centres with the use of queuing theory do not consider the fact that the tasks could be removed from queue without being serviced. The removal of tasks from the queue could be due to the user impatience, execution deadline expiration, security reasons, or as an active queue management strategy. The reneging could be correlated in nature, that is, if a request is dropped (or reneged) at any time epoch, and then there is a probability that a request may or may not be dropped at the next time epoch. This kind of dropping (or reneging) of requests is referred to as correlated request reneging. In this paper we have modelled a cloud computing infrastructure with correlated request reneging using queuing theory. An M/M/1/N queuing model with correlated reneging has been used to study the performance analysis of the load balancing server of a cloud computing system. The steady-state as well as the transient performance analyses have been carried out. Important measures of performance like average queue size, average delay, probability of task blocking, and the probability of no waiting in the queue are studied. Finally, some comparisons are performed which describe the effect of correlated task reneging over simple exponential reneging.
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 interreneging 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.
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