how to solve using this N = λ*D

Consider a switch that uses time division multiplexing (rather than statistical multiplexing) to share a link between four concurrent connections (A, B, C, and D) whose packets arrive in bursts. The link’s data rate is 1 packet per time slot. Assume that the switch runs for a very long time.

The average packet arrival rates of the four connections (A through D), in packets per time slot, are 0.2, 0.2, 0.1, and 0.1 respectively. The average delays observed at the switch (in time slots) are 10, 10, 5, and 5. What are the average queue lengths of the four queues (A through D) at the switch?

With TDMA, each connection gets to send 1 packet every 4 time slots, or .25 packets/slot. And with TDMA, the behavior of each connection is independent of what’s happening on the other connections. All of the arrival rates are less than this number, so the queue lengths are bounded.

Using Little’s Law: N = λ*D, so

A: N = 0.2 * 10 = 2 packets
B: N = 0.2 * 10 = 2 packets
C: N = 0.1 * 5 = .5 packets
D: N = 0.1 * 5 = .5 packets

Connection A’s packet arrival rate now changes to 0.4 packets per time slot. All the other connections have the same arrival rates and the switch runs unchanged. What are the average queue lengths of the four queues (A through D) now?