MM1ventekøs

MM1ventekøs is a single-server queuing model with a waiting line, described in Scandinavian queueing literature as a variant of the M/M/1 queue. It represents a system with one server, Poisson arrivals, and exponential service times, operating under a first-come, first-served discipline. The term emphasizes the presence of a waiting queue in the model.

The model is defined by standard assumptions: Arrivals follow a Poisson process with rate lambda, service times are exponentially distributed with rate mu, and the interarrival and service processes are independent. The system has unlimited waiting room and infinite capacity for those in the queue, except for the single server constraint. The state of the system is the number of customers in the system (in service plus in the queue), forming a birth-death process with birth rate lambda when the state is n and death rate mu when n > 0.

Stability requires that lambda is less than mu, often written as traffic intensity rho = lambda/mu < 1. Under this condition, the system reaches a steady state. The steady-state probabilities are P0 = 1 - rho and Pn = (1 - rho) rho^n for n >= 0.

Key performance measures include: L, the average number in the system, equal to rho/(1 - rho); Lq, the average number in the queue, equal to rho^2/(1 - rho); W, the average time a customer spends in the system, equal to 1/(mu - lambda); and Wq, the average time spent waiting in the queue, equal to lambda/(mu(mu - lambda)) or equivalently Lq/lambda. The probability that the server is busy is rho, and the probability of waiting in the queue is also rho.

Extensions of MM1ventekøs include finite-capacity variants such as M/M/1/K, which cap the total number in the system, and other queueing disciplines. Applications span telecommunications, customer service, and operations research where a single-server queue with a waiting line is a suitable abstraction.