Analytical study of a complex feed back semi bi-tandem queue system differ in transition probabilities

In this paper we consider a complex queue network comprised of bi-serial and parallel service channels. The model consists of three subsystems in which one subsystem has two bi-serial sub channels and other consisting two parallel sub channels. The subsystem is a single service channel which is commonly connected in series with these two subsystems. The concept of feedback is introduced that means a customer can revisit the system at most once if he is unsatisfied. The queue model is analyzed in steady state under Poisson stream with the help of differential difference equations, probability generating function technique, L'Hospital rule and law of calculus. Partial queue lengths and mean queue length is derived by using classical formula under steady state behaviour of the model. Behavioral analysis is done graphically by giving particular values to all variables. A particular case is also discussed to check the validity of the model. The model is applicable to different real life situations like banking service system, supermarkets, business managements, industries etc.