I have a routing matrix with Node-$0$ (zero) being the source/sink (outside world) and there are service Nodes $1,2, \ldots k$ in the system.
The matrix has entries $R_{ij}$ = Probability of an arrival at Node-$i$ moving to Node-$j$ where $i$ and $j = 0,1,\ldots k$.
Let the entries $R_{ii} = 0~~\forall i$, namely, there is no routing to itself.
My problem is to define a metric that gives an estimate the proportion of arrivals that take a fixed path in the system, say $0 \to 1 \to 2 \to \ldots \to k \to 0$.
For example if all the arrivals take this specified path then the diagonal row $R_{i-1,~j}$ should all be equal to $1$. For example if none of the arrivals take this specified path then this diagonal row $R_{i-1,~j}$ should all be zeroes.
I first defined this metric as $R_{01} \cdot R_{12} \cdot R_{23} \cdot R_{34} \cdot \ldots \cdot R_{k-1,~k} \cdot R_{K,0}$ which is the probability that an arrival has taken the complete path (specified above) as observed from Node-$k$ and going to Node-$0$. However the problem with this metric is that it does not differentiate between the order of nodes, i.e. it yields the same result when $R_{12} \cdot R_{23} = R_{23} \cdot R_{12}$ and hence did not seem right for what I am trying to do !
I then defined this metric as $$\frac1k \left( R_{01} + R_{01} \cdot R_{12} + R_{01} \cdot R_{12} \cdot R_{23} + \cdots + R_{01} \cdot R_{12} \cdot R_{23} \cdot \ldots \cdot R_{k-1,K} \cdot R_{K0} \right)$$
Both the above metrics satisfy my two extremes i.e. when all arrivals pass through the specified path and none of the arrivals pass through the specified path. However I am not clear which of these metrics is a correct metric to measure what I am trying to measure. Both may be incorrect, and there may another metric also. Can anyone please help me find the metric ? Thanks a lot.