2.2. METRICS 15

Because of the randomness of the arrivals of packets and of the variability of the transmission

times, not all packets experience the same delay. With the M/M/1 model, it is, in principle, possible

for a packet to arrive at the buffer when it contains a very large number of packets. However, that is

not very likely. One can show that about 5% of the packets experience a delay larger than 12ms and

about 5% will face a delay less than 0.2ms. Thus, most of the packets have a delay between 0.2ms

and 12ms. One can then consider that the delay jitter is approximately 12ms, or three times the

average delay through the buffer.

To appreciate the effect of congestion, assume now that the packets arrive at the rate of 1,150

packets per second. One then ﬁnds that T = 10ms and L = 11.5. In this situation, the transmission

time is still 0.8ms but the average queuing time is equal to 11.5 transmission times, or 9.2ms. The

delay jitter for this system is now about 30ms. Thus, the average packet delay and the delay jitter

increase quickly as the arrival rate of packets λ approaches the transmission rate μ.

Now imagine that N computers share a link that can send Nμ packets per second. In this

case, replacing μ by Nμand λ by Nλ in the previous formulas, we ﬁnd that the average delay is now

T/N and that the average number of packets is still L. Thus, by sharing a faster link, the packets

face a smaller delay. This effect, which is not too surprising if we notice that the transmission time

of each packet is now much smaller, is another beneﬁt of having computers share fast links through

switches instead of having slower dedicated links.

Summarizing, the M/M/1 model enables to estimate the delay and backlog at a transmission

line.The average transmission time (in seconds) is the average packet length (in bits) divided by the

link rate (in bps). For a moderate utilization ρ (say ρ ≤ 80%), the average delay is a small multiple

of the average transmission time (say 3 to 5 times). Also, the delay jitter can be estimated as about

3 times the average delay through the buffer.

2.2.7 LITTLE’S RESULT

Another basic result helps understand some important aspects of networks: Little’s result,discovered

by John D.C.Little in 1961. Imagine a system where packets arrive with an average rate of λ packets

per second. Designate by L the average number of packets in the system and by W the average time

that a packet spends in the system.Then, under very weak assumptions on the arrivals and processing

of the packets, the following relation holds:

L = λW.

Note that this relation, called Little’s Result holds for the M/M/1 queue, as the formulas indicate.

However, this result holds much more generally.

To understand Little’s result, one can argue as follows. Imagine that a packet pays one dollar

per second it spends in the system. Accordingly, the average amount that each packet pays is W

dollars. Since packets go through the system at rate λ, the system gets paid at the average rate of λW

per second. Now, this average rate must be equal to the average number L of packets in the system

since they each pay at the rate of one dollar per second. Hence L = λW .

Get *Communication Networks* now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.