Final answer:
Operating characteristics of a waiting line system involve the queue discipline, system utilization, waiting line structure, and statistical distributions such as the Poisson and exponential distributions to model arrivals and services.
Step-by-step explanation:
Operating Characteristics of a Waiting Line System
Operating characteristics for a waiting line system include factors such as queue discipline, the system's utilization, the waiting line structure, and the statistical distributions that describe arrival and service processes, like the Poisson distribution and exponential distribution. To tackle the student's request, let's discuss these elements in the context of the examples given.
a. Queue discipline: This refers to the rule by which customers are served. Common disciplines include 'first come, first served' (FIFO) and 'last come, first served' (LIFO).
b. Utilization: This is the proportion of the system's capacity that is actually being used over time. It is a measure of how busy the system is.
c. Waiting line structure: This involves the physical layout and number of waiting lines, and service points, such as a single queue leading to multiple servers or multiple queues, each leading to its own server.
d. The Poisson distribution: This is often used to model the number of arrivals to a system within a given time period, and the exponential distribution models the time between these arrivals. If customers arrive at a store at an average rate of 30 per hour, and we assume that the time between arrivals is exponentially distributed, we can calculate probabilities related to these arrivals.
For example, to find out the probability that it takes less than one minute for the next customer to arrive, you would use the exponential distribution formula. Likewise, the probability of waiting more than five minutes for the next customer can also be calculated using the exponential distribution. On the other hand, when we are concerned with the number of arrivals over time rather than the time between arrivals, the Poisson distribution is used, as it deals with discrete events over a continuous time span. If calls come to a police station at an average rate of four calls per minute, you could determine, for example, the probability of receiving exactly five calls within a minute or more than 40 calls in an eight-minute period using the Poisson distribution.