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A single-server waiting model is one of the most fundamental concepts in the field of Operations Research and is widely applied in analyzing queuing systems where a single service channel exists to cater to arriving customers or units. This model helps to understand the relationship between the arrival rate of customers, the service rate of the server, and the overall efficiency of the system. The idea is to model real-life service situations where entities such as people, machines, or jobs arrive at a service facility, wait if necessary, receive service from a single server, and then depart. The importance of this model lies in its ability to measure performance characteristics such as the average waiting time of customers, the average length of the queue, the probability that a customer has to wait, the server’s utilization rate, and the likelihood that the system is idle. In Operations Research, the single-server waiting model is usually denoted as M/M/1 in Kendall’s notation. Here, the first “M” refers to a Poisson arrival process or memoryless inter-arrival times, the second “M” refers to exponentially distributed service times, and “1” indicates a single server. The assumptions behind this model are simple yet powerful. It assumes that customers arrive randomly over time following a Poisson process with ___ ______ __________ _____ ________ ________ _______ _____ ________ _________.
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