Deriving the Multi-Server Queue Model: Theory and Key Metrics
This article derives the theoretical model of a multi‑server queueing system, detailing Poisson arrivals, exponential service times, state balance equations, and formulas for average queue length, system size, waiting and sojourn times.
Multi-Server Queue System Theoretical Derivation
The standard model assumes customer arrivals follow a Poisson distribution with rate λ, service times are exponentially distributed with rate μ, there are s service stations, an infinite number of potential customers, unlimited system capacity, and arrivals and service times are independent. The queue discipline is first‑come‑first‑served.
The state balance equations lead to the steady‑state probability of each state, from which the following performance measures are obtained:
Average queue length
Average number of customers in the system
Average time a customer spends in the system
Average waiting time in the queue
Reference:
ThomsonRen GitHub – https://github.com/ThomsonRen/mathmodels
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Insights, knowledge, and enjoyment from a mathematical modeling researcher and educator. Hosted by Haihua Wang, a modeling instructor and author of "Clever Use of Chat for Mathematical Modeling", "Modeling: The Mathematics of Thinking", "Mathematical Modeling Practice: A Hands‑On Guide to Competitions", and co‑author of "Mathematical Modeling: Teaching Design and Cases".
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