Consider a network with three routers P, Q, R shown in the figure below. All the links have cost of unity. The routers exchange distance vector routing information and have converged on the routing tables, after which the link Q–R fails. Assume that P and Q send out routing updates at random times, each at the same average rate. The probability of a routing loop formation (rounded off to one decimal place) between P and Q, leading to count-to-infinity problem, is ________.

Before the link failure, the converged distances to destination $R$ are:

• Distance from $Q$ to $R = 1$ (via direct link to $R$)

• Distance from $P$ to $R = 2$ (via $Q$)

When link $Q\text{--}R$ fails:

1. $Q$ detects the failure and sets its direct cost to $R$ to $\infty$.

2. Since $P$ and $Q$ send out routing updates independently at random times with the same average rate, the probability that $P$ sends an update to $Q$ first is $0.5$, and the probability that $Q$ sends an update to $P$ first is $0.5$.

• Case 1: $P$ sends update to $Q$ before receiving an update from $Q$ (Probability = $0.5$): $P$ advertises its distance to $R$ as $2$. Router $Q$ receives this and updates its distance to $R$ via $P$ as $1 + 2 = 3$. This creates a routing loop between $P$ and $Q$, triggering the count-to-infinity problem.

• Case 2: $Q$ sends update to $P$ first (Probability = $0.5$): $Q$ advertises distance $\infty$ to $R$. $P$ updates its distance to $R$ as $\infty$, avoiding a routing loop.

Therefore, the probability of a routing loop formation is 0.5.