I'm working on resolving a statistical problem and I came across some difficulties.
The content of the task:
A measurement of the traffic generated by the packet source indicates that the average traffic is $\lambda$ [packets/s] and maximum traffic is $\sigma$ [packets/s]. The classic exponential distribution is not appropriate for modeling such a source of traffic, because the exponential distribution contains only one parameter (and two parameters were measured).
To model such a source of motion, you can use the shifted exponential distribution which is described by two parameters ($\gamma, \delta > 0$): $$f(\tau) = \left\{\begin{matrix} 0 & \tau<d \\ \gamma e^{-\gamma (\tau-d)} & \tau\geq d \end{matrix}\right.$$ Random variable $\tau$ is the time interval between successive packets.
- Draw a distribution graph (1). What is the relationship between the maximum traffic and $\delta$, for the distribution (#)?
- Designate a mean value of the distribution (#).
- Based on measurements of traffic sources $(\lambda, \sigma)$ select firstly $\delta$ parameter for distribution (#), then $\gamma$ parameter for distribution (#).
So far I've managed to solve the subsection 2. and this is what I've came up with: my solution to subsection 2.
Part 1 is simple. If $\delta$ is the minimum interarrival time for consecutive packets, where time is measured in seconds, then at most $1/\delta$ packets can arrive in one second. For instance, if $\delta = 0.01$ seconds, meaning that we are assured that the time between packets is at least $0.01$ seconds, then the most packets we can observe in one second is $1/0.01 = 100$ packets, and this is $\sigma$. So $$\sigma \delta = 1.$$
For Part 3, use what you know from Parts 1 and 2 to establish a relationship between $\lambda$ and $\gamma$. The average traffic intensity $\lambda$ is simply the average you found in Part 2. Solve for $\gamma$ in terms of $\lambda$ and $\sigma$.