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A single-server waiting model is one of the most fundamental queuing models studied in operations research. It helps analyze systems where there is a single service point (server) and customers arrive at random to receive a service. This model enables decision-makers to study various performance measures such as average queue length, waiting time, server utilization, and system capacity. These metrics are critical in optimizing operations across many domains such as transportation, healthcare, banking, telecommunications, and more. The classic single-server queuing model is often denoted by the Kendall notation: M/M/1, where: * M (Markovian arrival): Arrivals follow a Poisson process with exponential interarrival times. * M (Markovian service): Service times are exponentially distributed. * 1 (single server): There is only one service channel. ### Assumptions of the M/M/1 Model: 1. The arrival process is Poisson with mean arrival rate λ (customers per time unit). 2. Service times follow an exponential distribution with mean service rate μ (customers served per time unit). 3. The queue discipline is First-Come, First-Served (FCFS). 4. The system has infinite potential queue length and capacity. 5. There is only one server. 6. The system is in a steady-state (input rate is less than the service rate, i.e., λ < μ). ### Performance Measures: 1. Server utilization (ρ): ρ = λ / μ This represents the proportion of time the server is busy. It must be less than 1 for a steady-state to exist. 2. Average number of customers in the system (L): L = λ / (μ - λ) 3. Average number of customers in the queue (Lq): Lq = λ² / (μ(μ - λ)) 4. Average time a customer spends in the system (W): W = 1 / (μ - λ) 5. Average time a customer spends waiting ________ ____ __________ ________ _________ _______ _________.
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