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 .