A min, + system theory for constrained traffic regulation and dynamic service guarantees

By extending the system theory under the (min, +) algebra to the time-varying setting, we solve the problem of constrained traffic regulation and develop a calculus for dynamic service guarantees. For a constrained traffic-regulation problem with maximum tolerable delay d and maximum buffer size q, the optimal regulator that generates the output traffic conforming to a subadditive envelope f and minimizes the number of discarded packets is a concatenation of the g-clipper with g(t) = min[f(t+ d), f (t)+q] and the maximal f-regulator. The g-clipper is a bufferless device, which optimally drops packets as necessary in order that its output be conformant to an envelope g. The maximal f-regulator is a buffered device that delays packets as necessary in order that its output be conformant to an envelope f. The maximal f-regulator is a linear time-invariant filter with impulse response f, under the (min, +) algebra. To provide dynamic service guarantees in a network, we develop the concept of a dynamic server as a basic network element. Dynamic servers can be joined by concatenation, "filter bank summation," and feedback to form a composite dynamic server. We also show that dynamic service guarantees for multiple input streams sharing a work-conserving link can be achieved by a dynamic service curve earliest deadline scheduling algorithm, if an appropriate admission control is enforced.