The paper studies the problem of optimal adaptation of an M/M/1 queueing station, when the arrival rate λ0 of customers shifts at unknown epoch, τ, to a known value, λ1. The service intensity of the system starts at μ0 and can be increased at most N times to μ1 < μ2 < · · · < μN. The cost structure consists of the cost changing μi to μj (i + 1 ≦ j ≦ N); of maintaining service at rate μ (per unit of time) and of holding customers at the station (per unit of time). Adaptation policies are constrained by the fact that μ can be only increased. A Bayes solution is derived, under the prior assumption that τ has an exponential distribution. This solution minimizes the total expected discounted cost for the entire future.
