Mathematical Analysis of Path MTU Discovery With New Generation Networks

In this paper we have presented the effects of path mtu discovery in IPv4 & IPv6 in mathematical, logical and graphical representation. We try to give a mathematical model to the working of path mtu discovery and calculated its behaviour using a transmission of a packet. We analysed the time consumed to transmit a single packet from source to destination in IPv6 network in the presence of PMTUD and similarly in IPv4 network with DF bit 1. Based on our analysis, we concluded that the communication time increases with the varying MTU of the intermediate nodes. Moreover, we formulated the mathematical model to determine the communication delay in a network. Our model shows that the asymptotic lower bound for time taken is $\Omega(n)$ and the asymptotic upper bound is $\Theta(n^2)$, using PMTUD. We have find that the packet drop frequency follows the Bernoulli's trials and which helps to define the success probability of the packet drop frequency, which shows that the probability is higher for packet drop rate for beginning $2\%$ of the total nodes in the path. We further found that $^{n}C_{a}$ possible number of a-combinations without repetitions that can be formed for a particular number of packet drop frequency. The relation between summation (acts as a coefficient in the time wastage equation) of each combination and their frequency resulted in symmetric graph and also mathematical and statistical structures to measure time wastage and its behaviour. This also helps in measuring the possible relative maximum, minimum and average time wastage. We also measured the probability of relative maximum, min and average summation for a given value of packet drop frequency and number of nodes in a path.