Number of Servers Arrival Rate 100 Service Rate 10.00 P(0), probability that there are no customers in the system 30% Lq: average length of the queue W: average time in the system 0.33 average number of customers in the system 2.33 Wq: average time in the queue 0.23 Utilization factor of the system 70% Table What is the probability that the service facility will be idle? Refer to the Table A) 0.23 B) 0.70 C) 0.33 D) 0.30 E) 0.233
The Correct Answer and Explanation is:
The correct answer is D) 0.30.
In queuing theory, the probability that a service facility will be idle corresponds to the probability that there are zero customers in the system, denoted as P(0). This value indicates the likelihood that the system is empty and the server is not occupied with any tasks. From the table, P(0) is given as 30%, or 0.30. This means that 30% of the time, there are no customers either waiting or being served, and the service facility is entirely idle.
To understand this more thoroughly, consider the structure of a single-server queuing system, where entities arrive at a certain rate and are served one at a time. The probability of the system being empty depends on the arrival rate (λ) and the service rate (μ). In this case, although the arrival rate is shown as 100, that appears inconsistent with other values, suggesting that this may either be an error or placeholder. However, since P(0) is explicitly listed as 0.30, we take it as a given.
The utilization factor (ρ) is another important concept, measuring the proportion of time the server is busy. In this table, utilization is given as 70%, meaning the server is occupied 70% of the time. The remaining 30% of the time, it is idle. This is precisely aligned with P(0) = 0.30, confirming that when the server is not utilized, it is in an idle state.
Options like 0.23 or 0.233 reflect average queue lengths or waiting times, and should not be confused with the probability of idleness. Hence, option D, 0.30, is the correct and most appropriate choice based on the table’s data.
