## 3 Queueing Analysis
### Analytical Results  
Each significant section \((i,j)\), such as a section of a corridor between say two adjacent doors, where \(i\) and \(j\) are adjacent locations, has an associated average traversal delay from \(i\) to \(j\) that we call \(\frac{1}{\mu_{ij}}\), whose value can change with the conditions in the building. For any edge, the average traversal time will then include the congestion and be given by [17]:  
\[W_{ij}=\frac{\mu_{ij}^{-1}}{1-\rho_{ij}} \tag{2}\]  
where  
\[\rho_{ij}=\frac{\Lambda_{i}.p_{ij}}{\mu_{ij}} \tag{3}\]  
and \(p_{ij}\) is the fraction of evacuees that arrive at \(i\) and then take the section \((i,j)\) towards their goal destination \(s\):  
\[p_{ij}=\sum_{\pi(i,s)}p(\pi(i,s))\qquad\pi(i,s)=(i,j,\pi(j,s)) \tag{4}\]  
The average traversal time of a path, and the average evacuation time to any of the exit points will then be obtained in a straightforward manner by summing the appropriate traversal delays. The Expression (2) is a steady-state value, which is equivalent to assuming that the overall duration of an evacuation is much longer than the time it takes to traverse any of the individual sections of the graph model of the building. Furthermore this formula is based on a Jackson network representation [17] of steady-state, which assumes Poisson arrivals to the originating locations, exponentially distributed traversal delays, and that any directional section \((i,j)\) from \(i\) to \(j\) can only have at most one evacuee at a time. In order to compare the queuing analysis with the simulation results, we use identical \(\mu_{ij}\) rates and the same building graph. The set of L \"source\" nodes corresponds to the offices where users tend to reside in normal conditions (_i.e._, when there is no emergency) in the simulated environment. The \(\pi(L,i)\) values are chosen so that the evacuees are always being directed along the shortest paths to the evacuation points. Figure 4 shows the results of the queueing analysis and of the simulation. The scatter plot shows the time taken to reach the nearest exit (Y axis) based on the time at which a node on the first-floor lobby [Figure 4(a)] or at a second-floor hallway intersection [Figure 4(b)] is visited. The plot shows the output of 10 iterations of simulations involving 120 building users (which is the average occupancy of the real-life building modelled), with initial locations that have been randomized at each run. Both scatter plots clearly show a gradual increase in congestion at the beginning of the evacuation, until the network reaches its saturation point and the evacuation times become somewhat steady. The red line shows the corresponding steady-state node-to-exit delay that is predicted by the queueing model. In the examples of Figure 4 we see that the queueing analysis seems to over-estimate (worst case) the average traversal times when compared with the simulation results. This is not surprising because the way we compute the arrival rates in the queueing model can yield high worst case values. Furthermore the simulation assumes _deterministic_ traversal times and it is well known that service models with deterministic service times will underestimate the average response or traversal times [18, 19]. The deterministic section traversal times in the simulation are obviously unrealistic because different people will actually have different walking speeds, and furthermore the panic and other (e.g., fire, smoke) conditions in an emergency will render things far more random. Thus paradoxically, the analytical model may actually provide a more accurate rendering of what may be really happening during the latter parts (worst case) of an emergency when congestion starts to occur and the behaviour of the various agents  
Figure 4: Time needed to reach the exit based on the time at which the node of interest was visited. The red line shows the steady-state value predicted by the queueing network analysis. is far more random.