...

with initial condition:

Thus:

So, the CDF of a memoryless source is defined by:

And the PDF of a memoryless source is defined by:

Thus, a memoryless source will have exponentially distributed interarrival times.

...

Let A(t₁) = the number of arrivals in the time interval [0, t₁] for a Poisson arrival process with an arrival rate of two per second. A(t₁) is a Poisson-distributed RV with PMF:

What is the probability that A(t₁) is greater than two?

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Fitting Models to Traffic Patterns

The arrival process of packet/frame can be modeled as a Poisson distribution given exponential interarrival times. If the interarrival times are fixed, such as in the audio/video example, a deterministic model is best suited. Lastly, an Erlang model best fits traffic such as the initiation of telephone calls. 

The lengths of packets/frames are commonly modeled as an exponential distribution given random traffic. However, if the traffic consists of fixed length packets/frames, then a deterministic model is more appropriate. On the other hand, a bimodal model is best suited for a mixture of short and long packets.

Poisson arrivals and exponential packet/frame lengths often provide a realistic model for interactive traffic in a computer network.

Measures of Network Performance

The performance of a computer network depends on:

• the initial state of the network
• the sequence of random inputs to the network (i.e. the arrivals)
• the random internal events within the network (i.e. data processing time, frame errors, retransmissions, etc.)
• the network architecture and protocols

Most analysis focuses on steady-state conditions, meaning the network has is exhibiting typical behavior. However, note that emphasis can also be placed on the network's response to a transient (i.e. one-time) event.

The type of performance measures to collect (i.e. means, percentiles, etc.) from the network depends on the goal of the analysis.

From a network operator's perspective, commonly sought after performance measures are throughput and utilization.

Throughput

The throughput of the network is the number of bits (or packets/frames) per second that were successfully transmitted at a given location (e.g. link or NIC) in the network. Note that any overhead for errors and retransmissions is not included in throughput. Throughput only includes the amount of data per second successfully transferred across the network as measured at a particular point. One can normalize throughput to a value between 0 and 1 by dividing by the transmission rate of the physical medium of the LAN.

Note that throughput can be computed for more than just a single physical point of a network. A "black box" network, which might consist of some unknown topology, can have its throughput measured by monitoring the total traffic per second that departs the network.

Furthermore, throughput can be computed from different perspectives in the network. If the goal is to compute the throughput of transmissions on the physical medium, then all bits from successfully transmitted frames are included in the calculation. On the other hand, if the goal is to compute the throughput of actual payload data across the network from end host to end host, the throughput computation will exclude the headers of the MAC protocol, as they would be considered overhead.

Utilization

The utilization of the network is the average fraction of time the physical medium of the LAN is busy. This includes all traffic, headers, unsuccessful transmissions, and retransmissions. Utilization can be determined by directly monitoring the physical medium of the LAN. If all transmissions on a LAN are successful, the utilization of the LAN is equal to the normalized throughput.

Related to utilization is channel efficiency, which is the utilization of the physical medium to carry actual data, excluding frame and MAC protocol overhead.