## What is the maximum virtual delay if (P, B, R)=(P0, B0, R0) ?

A flow S(t) is constrained by an arrival curve α. The flow is fed into a shaper, with shaping curve σ. We assume that

α(s) = min(m + ps, b + rs)

and

σ(s) = min(P s, B + Rs)

We assume that p>r, m ≤ b and P ≥ R.

The shaper has a fixed buffer size equal to X ≥ m. We require that the buffer never overflows.

1. Assume that B = +∞. Find the smallest of P which guarantees that there is no buffer overflow. Let P0 be this value.

2. We do not assume that B = +∞ anymore, but we assume that P is set to the value P0 computed in the previous question. Find the value (B0, R0) of (B,R) which guarantees that there is no buffer overflow and minimizes the cost function c(B,R) = aB + R, where a is a positive constant. What is the maximum virtual delay if (P, B, R)=(P0, B0, R0) ?

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