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Memoryless Traffic Source
For many applications that relay data over the network, any additional time elapsed between data transmissions does not appear to depend on the time already elapsed since the previous data was transmitted. Such a source of data is referred to as a memoryless source. Let’s derive a mathematical model for such a memoryless source.
Consider a data source that produces data arrivals (i.e. data units) at any time starting at time t = 0. Such a source is referred to as a continuous source, since it continuously transmits data at periodic intervals.
Let the number of arrivals through time t be denoted A(t), where A(t) takes on integer values.
The time between the (i – 1)th and ith arrival is referred to as the interarrival time and is denoted τi. If we model each interarrival time τi as a RV, then A(t) is a random process (RP), and the arrival rate of the source is given as:
Suppose the interarrival times {τ1, τ2, …} are independent RVs and that for each i, τi has PDF such that:
where h(t) is the same function for all i. If this is true, then A(t) is the arrival process for a memoryless source. The remaining lifetime distribution for the RV τi is given by:
So, a source is memoryless if the interarrival times of its arrival process have a remaining lifetime distribution:
that does not depend on s. The arrival process of a memoryless source is an example of a continuous time Markov chain and a continuous time birth-death process.
Example 1
Assume the PDF and CDF of the above arrival process are as follows:
Suppose s = 1:
For example:
Suppose s = 2:
For example:
As indicated by the examples, the interarrival times depend on s. Thus, the source is not memoryless.
Example 2
Assume the PDF and CDF of the above arrival process are as follows:
Suppose s and t are >= 0:
For example:
For example:
As indicated by the examples, the interarrival times do not depend on s. Thus, the source is memoryless.
Deriving the PDF and CDF of a Memoryless Source
What does the memoryless property tell us about the PDF of τi?
Recall that:
So:
If we set s = 0, we find that:
This implies that:
which means that we can rewrite:
as:
So:
If the PDF of τi is:
then:
So:
So:
But:
So:
And:
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.
Example 1
Suppose a memoryless data source has an arrival rate of four data units per second. What is the probability that the time between the first and the second arrivals is more than one second?
So:
Example 2
A memoryless data source with arrival rate λ starts at time 0. At t = 0, what is the density function of possible times for the first arrival?
At t = t1, if no arrivals have occurred, what is the density of possible times for the first arrival?
Example 3
Suppose a memoryless source has an arrival rate of two per second. What is the probability that zero arrivals occur in the interval [t1, t2]?
If we assume that the (i - 1)th arrival is the last one before time t = t1 and that it occurs at t = t1 – t’. Then the answer is given by:
What is the probability that exactly one arrival occurs in [t1, t2]? We know that:
But this determines the probability that one or more arrivals occur in [t1, t2], which isn’t exactly what we’re after. We can show by solution of appropriate differential equations that:
Thus, the arrival rate of the memoryless source is:
which is a Poisson distribution.
Thus, it can be concluded that a continuous time, memoryless data source produces independent, exponentially distributed interarrival times. And, independent, identically distributed, exponentially distributed interarrival times result in a Poisson arrival process.
Example 4
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:
