# STOCHASTIC DECOMPOSITION PROPERTY

For the system under consideration, we establish the decomposition of the random variable representing the system size in two different ways.

**Theorem 2**

- 1. The number of customers
*L,*in the system can be written as*L = L' + M',*where*L'*is the number of customers in “the standard*Geo IG1*1 queue with impatient customers and starting failures” and*M'*is the number of retrial customers when the server is free. - 2. The system size
*L*can also be rewritten as*L = L" + M",*where*L"*is the system size in “the*Geo*/*G*/1 queue with impatient customers” and*M"*is the number of customers retrying for service when the server is free or under repair.

**Proof. **We can easily rewrite equation (6.32) in the following two different ways.

Here, the first term is the PGF of the system size in “Geo */ G /*1 queue with starting failure” and the second term of the product is the PGF of the number of retrial customers when the server is free.

In this representation of

Geo IG11 standard queue and the other term is the PGF of the number of customers retrying their service when the server is idle or under repair.

# NUMERICAL ILLUSTRATION

Exhaustive numerical results are presented in this section to describe the behavior of the queueing systems under study. We examine the effect of the system parameters such as arrival probability *(a),* balking probability *(b).* feedback probability (v) and the probability that the server is started (0) to serve the customer successfully on some of the crucial performance measures of our model. The values of the parameters are chosen appropriately and the numerical results are obtained by using MATLAB software.

Figure 6.1 *(a)-(l)* depicts the impact of *a,b,%* and v on P(0). As expected intuitively, P(0), the probability that the system is empty, decreases with the increase in *a* for any values of *b.d* and v (Figure 6.1(a)-(c)). Further, P(0) is almost stable for increasing values of *b.* We can see from Figure 6.1 *(d)-(f),* P(0) remains unaltered as for as *b* is concerned but it is higher for higher values of *a,Q* and also v. Figure 6.1 *(g)-(i)* shows an increasing trend for P(0) for increasing values of 0. Figure 6.1 (*h*) confirms the observation that P(0) is not much affected by *b.* P(0) is plotted against v for different values of *a,b* and 0 in Figure 6.1 *(j)-(l).* We can see from these Figure 6.1 that P(0) decreases with the increment in v. Further, we can see it increases with increase in *b* and 0 but decreases with increase in *a.*

The effect of *a,b,%* and v on *E(L*) is plotted in Figure *6.2(a)-(l).* The behavior of E(L) with respect to *a* is described in Figure 6.2*(a)-(c).* It is noticed that the

FIGURE 6.1 Impact of and v on P(0).

FIGURE 6.1 *(Continued)*

expected system size *E(L*) increases for increasing values of *a* which matches with our intuition. Figure *6.2(d)-(f)* showcases the trend of *E(L)* for increasing values of *b. E(L)* decreases as *b* increases. Also it is observed that *E(L)* is higher for higher values of *a* and v but it is the other way for 0. *E(L)* is drawn for increasing values of 0 for different values of *a,b* and v in Figure *6.2(g)-(i).* As is expected, *E(L) *decreases for increasing values of 0 in all the three different cases. Also, *E(L)* is higher for the higher values of *a* and v but is very low for higher values of *b.* Finally, in Figure 6.2*(j)-(l), E(L)* is depicted for increasing values of v for various values of *a,b* and 0. We observe in Figure 6.2 an increasing trend for *E(L)* for increasing v for *a,b* and 0. Further, *E(L)* is higher for the higher values of *a* but it is on the other way for *b* and 0. Also, we see a different pattern when *a* is very small *(a =* 0.05) and *b* is more *(b =* 0.6) which may be the combined effect of v and *a* (Figure 6.2*(j))* and v and *b* (Figure 6.2(k)). The numerical results established above demonstrates that our analytical results are valid. Further, they can provide some information required for the concerned system designers for making optimal designs.

# CONCLUSIONS

In the present study, we considered a *Geo IG */1 queue with retrial customers. We assume that the server is prone to starting failure and the customer may rejoin the system for another service if his service is not satisfactory. Further, we have taken into consideration, the customer’s dilemma whether to join the system or leave if he or she anticipates a longer waiting time. We have analyzed the system in the early arrival set up where the departure is having precedence over the arrivals. It

FIGURE 6.2 Impact of *a,* 6.0 and V on *E(L).*

**FIGURE 6.2 ***(Continued)*

is observed that the *P(0)* increases for increasing values of 0, the probability that the server is started successfully and decreases with increasing value of feedback probability v. Further, it is noticed that the expected system size increases with the increase in the value of v and decreases with the increase in the values of 0. The trends shown in the graphs are in line with our intuition, which ensures the correctness of the analytical results obtained. Our model can be used to study a flexible manufacturing system where the arrival of components and their service time are occurring at regularly spaced time epochs. The arrival of the components, the manufacturing unit, a pool of waiting components, the failure of the system, unsatisfied components processed again may correspond to the customers, the server, retrial customers, starting failure and feedback, respectively. The future work may include study of the same system in late arrival setup, waiting time distribution and cost analysis considering holding cost and the cost of losing a customer.

# ACKNOWLEDGEMENT

The authors thank the referees for their valuable input in improving the manuscript.

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