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Understanding exponential random variables is pivotal when studying Poisson processes, which are common in describing queuing systems like the one in our exercise. An exponential random variable is used to model the time between events in a Poisson process. This type of variable is characterized by a memoryless property, meaning the probability of an event occurring in the future is independent of any prior events.

For example, in the scenario of customer service at two-server stations, the service times are represented as exponential random variables with rates. These rates, four and six customers per hour for server 1 and server 2 respectively, indicate the average number of customers that can be serviced in a unit of time. Mathematically, if 'X' is an exponential random variable with rate \(\lambda\), the probability that 'X' exceeds a certain time 't' is given by \(P(X > t) = e^{-\lambda t}\), highlighting the continuous and decaying nature of this probability over time.

Grasping the concept of exponential random variables helps students solve part (b) of the exercise, where they need to calculate the average amount of time an entering customer spends in the system—key knowledge for nuanced fields such as operations research and network theory.

A state-transition diagram is a visual representation of a system that showcases possible states and the transitions between them. In the context of queuing theory, the states typically represent the number of entities (like people or data packets) within the system at any given time, and transitions represent the arrival or departure of these entities.

For step 1 of our solution, the state-transition diagram is drawn to illustrate two states - where state 0 is both servers being idle, state 1 is server 1 being busy and server 2 idle, and state 2 means both servers are busy. Arrows indicate customer transitions based on availability. Visualizing the problem in this manner aids to digest the system’s dynamics and sets the stage to derive the equilibrium equations necessary to solve the subsequent parts of the exercise. Students are usually encouraged to perfect this skill as it transcends many disciplines from computer science to economics.

Equilibrium equations, sometimes termed 'balance equations', are mathematical representations that describe the long-term behavior of a stochastic or random process. In our exercise, equilibrium equations are used to determine the steady-state probabilities of the system's states.

These equations ensure that, for each state in the system, the rate of entering the state equals the rate of leaving the state, suggesting a balanced system in the long run. Specifically, step 2 shows the equilibrium equations derived from the state-transition diagram. Solving these simultaneously with the normalization condition (the sum of all probabilities equals 1) provides us with the steady-state probabilities (\(P0, P1, P2\)).

By understanding how to formulate and solve equilibrium equations, students can answer questions regarding the fraction of customers not entering the system (part a) and the fraction receiving service from server 1 (part c). These solutions are fundamental to operations management, detailing how often resources are fully utilized—or which scarcities may lead to service denials, a crucial aspect for optimizing service delivery.