Читать книгу Queueing Theory 1 - Nikolaos Limnios - Страница 15

By differentiating x₀(b) with respect to b, we have

PROPOSITION 1.1.– By studying the derivative of x₀(b) w.r.t. b, we observe

 1) if f’(b) is negative, then an increase in b leads to an increase in the probability that the system is empty;

 2) if f’(b) is positive, then the probability that the system is empty could be increasing or decreasing, depending on whether respectively. Hence, an increase in the arrival rate as a function of the service rate does not necessarily mean a decrease in the emptiness probability.

We obtain additional information by considering the rate of increase in the probability of empty system as a rate of increase in the service completion probability, i.e.

PROPOSITION 1.2.– If f(b) is convex and f’(b) < 0, then x₀(b) is definitely concave. if f(b) is decreasing fast, then x₀(b) is increasing slowly, i.e. f(b) decreasing faster than increases. If, however, f(b) is convex and then x₀(b) is convex.