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CASE 1: Consider the case where f(b) is a decreasing function of b, say for example a = f(b) = 1 – b², with the strict condition that f(b) = 1 – b² < b. For this, we have

In this case, as b increases we see that E(X(b)) decreases as well. This is because the arrival probability decreases at a faster rate than service completion increases, i.e. f′(b) = – 2b.

REMARK 1.1.– For the case where a = f(b) = 1 – b², the arrival probability is a decreasing concave function of the service completion probability and the mean number in the system is decreasing in a convex form.

CASE 2: Next consider the case where f(b) is an increasing function of b, say for example a = f(b) = b², with the strict condition that f(b) = b² < b. In this case, we have

In this case, as b increases we see that E(X(b)) increases because the arrival probability increases faster than service completion increases, i.e. f′(b) = 2 b.

REMARK 1.2.– For the case where a = f(b) = b², with arrival probability increasing in a convex form of the service completion probability, the mean number in the system increases in a convex form as well.

CASE 3: Let us consider another case where f(b) is a linear decreasing function of b, say for example a = f(b) = 1 – b, with the strict condition that 1 – b < b. In this case, we have

In this case, as b increases we see that decreases because the arrival probability is decreasing in the same rate as the service completion rate increases, i.e.

REMARK 1.3.– However, for the case where a= f(b) = 1-b, the arrival probability decreasing in a linear form of the service completion probability, the mean number in the system decreases in a strict convex form.

We summarize the results of the mean number in the system for the three cases in Table 1.1, with **** representing infeasible situations because the traffic intensity is greater than 1.