Boundedcapacity

Bounded capacity refers to a property of a system in which there is a finite upper limit on the amount of a resource, items in a queue, or throughput that can be present at any time. This contrasts with models that assume unlimited or unbounded capacity. Bounded-capacity models appear across fields such as queueing theory, telecommunications, manufacturing, and storage systems, where physical constraints or policy decisions enforce a maximum.

In queueing theory, a typical bounded-capacity model is the M/M/1/K system: a single-server queue with Poisson arrivals at rate lambda, exponential service at rate mu, and a total capacity of K, so the number in the system can be between 0 and K. When the system is full, new arrivals are blocked or lost. The process is a birth-death process with stationary distribution p_n for n = 0,...,K. If rho = lambda/mu != 1, then p_n = [(1 - rho) / (1 - rho^{K+1})] * rho^n; if rho = 1, p_n = 1/(K+1). The blocking probability is p_K, and the effective arrival rate is lambda_eff = lambda * (1 - p_K). Other metrics include the mean number in the system and throughput, obtainable via standard queueing formulas and Little’s law.

Applications include network routers with finite buffers, call centers with limited lines, and inventory or storage systems with fixed capacity. Design considerations balance the loss from blocking against underutilization from idle capacity, and control policies may regulate arrivals or adjust capacity where feasible. See also finite-capacity queue, blocking probability, and Little’s law.