mm1mn1

mm1mn1 is not a standard, universally defined term in queueing theory. In discussions that use Kendall’s notation, the string is sometimes encountered as a shorthand or a concatenation of related concepts, such as the classic M/M/1 model and its finite-capacity variant M/M/1/N, or as a reference to networks of M/M/1 queues. Because there is no single agreed definition, the exact meaning of mm1mn1 depends on context.

One common interpretation is M/M/1, a single-server queue with Poisson arrivals and exponential service times. Here arrival rate is lambda, service rate is mu, and the traffic intensity is rho = lambda/mu. If rho < 1, the system has a well-defined steady state with the stationary distribution P_n = (1 - rho) rho^n for n ≥ 0, a mean number in the system L = rho/(1 - rho), and a mean waiting time in the system W = 1/(mu - lambda). The mean waiting time in the queue W_q and the mean numbers in queue L_q follow from standard queueing formulas.

Another common interpretation is M/M/1/N, a finite-capacity version with maximum N customers in the system. The state probabilities are P_n = P_0 rho^n for n = 0,...,N, with P_0 = (1 - rho)/(1 - rho^{N+1}) if rho ≠ 1 (or P_0 = 1/(N+1) if rho = 1). The blocking probability is P_N, and the effective arrival rate is lambda_eff = lambda(1 - P_N). The mean number in the system is L = [rho(1 - (N+1)rho^N + N rho^{N+1})]/[(1 - rho)(1 - rho^{N+1})], with L_q = L - (1 - P_0). Throughput and delays can be expressed in terms of lambda_eff.

In other contexts, mm1mn1 might refer to tandem or networked configurations of M/M/1 queues. Clarification of the intended meaning would allow a more precise article.