## 4 Systems of Parallel Heterogeneous Servers with Recirculation  
Next, we considered the case when the rejected arrival rate recirculated and merged with the offered input flow into the system instead of blocking and direct loss. The system can represent a conveyor with recirculation or another system with retrial. The recirculating rate had priority over the external input and the system remained a loss system. In general, it is known that the merging input rate will not have a Poisson distribution [36]. Models B and E demonstrated their accuracy when evaluating the overflow of a parallel system of limited capacity. We modified model E to include a recirculation flow, as represented in Figure 9. When the conveyor recirculates the non-served arrivals, the offered load increases to include the flow recirculated in the time percentage the system is full (\(p_{f}\)), and thus, the expected recirculated rate is \(\lambda p_{f}\), and the net offered load to the system becomes \(\lambda_{r}=\lambda(1+p_{f})\) instead of \(\lambda\). Thus, considering the findings from model B, the loss of this net offered load will be \(\lambda_{r}\) (\(p_{f}+p_{b}\)) = \(\lambda(1+p_{f})\) (\(p_{f}+p_{b}\)), but in fact, the loss of the recirculating fraction \(\lambda\)\(p_{f}\) (\(p_{f}+p_{b}\)) does not exit the system, but it remains in recirculation; therefore, the net loss out of the  
Figure 8: Model E (m = 3) arrival rate \(\lambda\) vs. sojourn time CT for different service rate patterns (\(\mu_{1}\), \(\mu_{2}\), \(\mu_{3}\)) with average service rate 1 and buffer size k.  
system will be \(\lambda(I+p_{f})(p_{f}+p_{b})-\lambda\ p_{f}\ (p_{f}+p_{b})=\lambda\ (p_{f}+p_{b})\), which is consistent with the net balance with the external offered load \(\lambda\), which was already verified in models B and E. In consequence, the output rate is expected to be \(\lambda_{r}(I-p_{f}-p_{b})=\lambda(I+p_{f})\) (\(I-p_{f}-p_{b}\)). The recirculation does not generate any extra state in the transition state diagram of servers in model E (see Figure 4), but increases the offered load to the servers. In general, the probabilities \(p_{f}\) and \(p_{b}\) with recirculation will reach different values than those without recirculation. We formulated the recirculation model through the transition state matrix as a product form problem with the same states as model E; therefore, the size of the system provided a complete description of states with the probability summing to 1, but with the correction of the offered load at the entrance of the conveyor. Thus, the resulting matrix for the conveyor of maximum capacity 3, with three servers and ordered entry service was also (6), but the offered load was \(\lambda_{r}=\lambda(I+p_{f})\) instead of \(\lambda\). The state probability was the solution of \(A\cdot p=c\), where \(A=A(\lambda,\ \mu_{i},\ p_{f})\). The solution was obtained from the iteration on \(p_{f}\) as a fixed-point problem that converged very quickly. In general, the expression of the effective rate offered and the recirculating rate through the conveyor involved conditional probabilities, but the proposed transition state diagram was completed by incorporating the loss server state, and the iteration as a fixed-point problem provided the state probabilities solution. In Table 4, the results for the system of three servers with a queue (conveyor) of maximum capacity 3 are included, with and without recirculation. The output rate was evaluated using the average result of simulation over 600,000 recirculating cycles with a warm-up of 60,000 cycles, with 10 independent replications. Model E with recirculation produced simulation results within 1% in most of the ranges of operation, with a maximum below 1.4%.