M/M/1/N Queuing System with Retention of Reneged Customers

In this paper, we study an infinite capacity multi-server Markovian queuing system with balking and retention of reneging customers. The transient analysis of the model is performed. The probability generating function technique along with Bessel function properties is used to derive the time-dependent state probabilities explicitly.

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.

Customer impatience has a very negative impact on the queuing system under investigation. If we talk from business point of view, the firms lose their potential customers due to customer impatience, which affects their business as a whole. If the firms employ certain customer retention strategies, then there are chances that a certain fraction of impatient customers can be retained in the queuing system. A reneged customer may be convinced to stay in the queuing system for his further service with some probability, say q and he may abandon the queue without receiving the service with a probability p(=1? q). A finite waiting space Markovian single-server queuing model with discouraged arrivals, reneging and retention of reneged customers is studied. The steady state solution of the model is derived iteratively. The measures of effectiveness of the queuing model are also obtained. Some important queuing models are derived as special cases of this model.