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On the relative mix transition probabilities in repairman problem of two different types with batch
deterministic repairs
1Department of Mathematical Sciences, KolaDaisi University, Ibadan, Oyo State, Nigeria
2 Department of Mathematics, Obafemi Awolowo University, Ile –Ife, Nigeria
October 2018
ABSTRACT
In this paper, the relative mix in the distribution of each machine type and the link between the durability and maintenance of each type of machine, are presented. Compared to existing machine repair problem, this study considered the repairman problem with multiple batch deterministic repairs of two different machine types A_1 and A_2. Without loss of generality, it is assumed that, there are integer k_1=2 of type A_1 machines and integer k_2=2 of type A_2 machines in the system such that k_1+k_2=k. Each type A_1 (A_2) machine occasionally breaks down and moves into repair queue Q_1 (Q_2) at Poisson rate λ_1(λ_2). The Repair station has a capacity to repair at most two machines in a session. Flow balance equations are obtained for each of the 22 states of the system as defined by observing the system at repair time points. The resulting equations were solved for stationary probabilities at repair point and some server occupancy mode conditions using Gaussian elimination. Steady state repair completion mix probabilities (p_1,p_2,p_3,⋯,p_22 ) and steady state occupancy probabilities q_j:j=1,2,⋯,7 are obtained. Subsequently, the work are evaluated with adopted arrival rates λ_1= 0.25,0.50,0.75,1.00,⋯,2.50 , λ_2=2.00, and specific deterministic repair times D_st for s and t units of type A_1 and A_2 machines respectively. Using the Minkwoski distance of the various runs and system configuration as a guide, it was established that the closer the system is to a balanced situation the less is the discrepancy in the relative mix of the state probability distribution. We concluded that the relative mix can be rather different from each other depending on the appropriate parameter for the model.
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