Category Archives: Mathematics

1. For the following flows and a GCRA(100, 500), give the conformant and non-conformant cells. Times are in cell slots at the link rate.

(a) 0, 100, 110, 12, 130, 140, 150, 160, 170, 180, 1000, 1010

(b) 0, 100, 130, 160, 190, 220, 250, 280, 310, 1000, 1030

(c) 0, 10, 20, 300, 310, 320, 600, 610, 620, 800, 810, 820, 1000, 1010, 1020, 1200, 1210, 1220, 1400, 1410, 1420, 1600, 1610, 1620

2. Assume that a cell flow has a minimum spacing of γ time units between cell emission times (γ is the minimum time between the beginnings of two cell transmissions). What is the maximum burst size for GCRA(T,τ ) ? What is the minimum time between bursts of maximum size ?

3. Assume that….

What are the (P, B) parameters in b/s and bits for each case ?

For a CBR connection, here are some values from an ATM operator:

peak cell rate (cells/s) 100 1000 10000 100000

CDVT (microseconds) 2900 1200 400 135

1. What are the (P, B) parameters in b/s and bits for each case ? How does T compare to τ ?

2. If a connection requires a peak cell rate of 1000 cells per second and a cell delay variation of 1400 microseconds, what can be done ?

3. Assume the operator allocates the peak rate to every connection at one buffer. What is the amount of buffer required to assure absence of loss ? Numerical Application for each of the following cases, where a number N of identical connections with peak cell rate P is multiplexed.

case 1 2 ….

How many cells were sent by the source in time interval [0, 401] ?

In this problem, time is counted in slots. One slot is the duration to transmit one ATM cell on the link.

1. An ATM source S1 is constrained by GCRA(T = 50 slots, τ = 500 slots), The source sends cells according to the following algorithm.

• In a first phase, cells are sent at times t(1) = 0, t(2) = 10, t(3) = 20,…,t(n) = 10(n − 1) as long as all cells are conformant. In other words, the number n is the largest integer such that all cells sent at times t(i) = 10(i − 1), i ≤ n are conformant. The sending of cell n at time t(n) ends the first phase.

• Then the source enters the second phase. The subsequent cell n +….

What buffer size is required for the flow at the shaper ?

A flow with T-SPEC (p, M, r, b) traverses nodes 1 and 2. Node i offers a service curve ci (t) = Ri (t − Ti)+. A shaper is placed between nodes 1 and 2. The shaper forces the flow to the arrival curve z(t) = min(R2t, bt + m).

1. What buffer size is required for the flow at the shaper ?

2. What buffer size is required at node 2 ? What value do you find if T1 = T2 ?

3. Compare the sum of the preceding buffer sizes to the size that would be required if no re-shaping is performed.

4. Give an arrival curve for the output of node 2.

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….

What is now the condition on R which ensures that no bit is discarded by the policer (in other words, that the flow is conformant) ?

We consider a flow defined by its function R(t), with R(t) = the number of bits observed since time t = 0.

1. The flow is fed into a buffer, served at a rate r. Call q(t) the buffer content at time t. We do the same assumptions as in the lecture, namely, the buffer is large enough, and is initially empty. What is the expression of q(t) assuming we know R(t) ? We assume now that, unlike what we saw in the lecture, the initial buffer content (at time t = 0) is not 0, but some value q0 ≥ 0. What is now the expression for q(t) ?

2. The flow is put into a leaky bucket policer, with rate r and bucket size b…..

draw the acceptance region when K and P are the variables.

1. Assume K connections, each with peak rate p, sustainable rate m and burst tolerance b, are offered to a trunk with constant service rate P and FIFO buffer of capacity X. Find the conditions on K for the system to be loss-free.

2. If Km = P, what is the condition on X for K connections to be accepted ?

3. What is the maximum number of connection if p = 2 Mb/s, m = 0.2 Mb/s, X = 10MBytes, b = 1Mbyte and P = 0.1, 1, 2 or 10 Mb/s ?

4. For a fixed buffer size X, draw the acceptance region when K and….

Compute the effective bandwidth for a mix of VBR connections 1,…,I.

1. Compute the effective bandwidth for a mix of VBR connections 1,…,I.

2. Show how the homogeneous case can be derived from your formula

3. Assume K connections, each with peak rate p, sustainable rate m and burst tolerance b, are offered to a trunk with constant service rate P and FIFO buffer of capacity X. Find the conditions on K for the system to be loss-free.

4. Assume that there are two classes of connections, with Ki connections in class i, i = 1, 2, offered to a trunk with constant service rate P and FIFO buffer of infinite capacity X. The connections are accepted as long as their queuing delay does not exceed some value D. Draw the acceptance region, that is, the….

What are the conditions on d1 and d2 for the end-to-end delay bounds to be satisfied ?