Transportation Geography and Network Science/Reliability
The most commonly accepted definition declares reliability to be “the probability that [a] system can perform its desired function to an acceptable level of performance for some given period of time.” This definition is useful but general: any specific measure of reliability will depend on the nature of the desired function and what levels of performance are acceptable. As a result, a wide range of both definitions and measures of reliability have been proposed, each designed for application to a specific type of transportation network or a specific network function.
Measures of Reliability in Transportation Networks
Terminal reliability measures the probability that a transportation network can perform a very basic function: providing a path — any path, regardless of cost — from an origin to a destination. To determine this measure, each link in a network is assigned a functional probability x that represents the likelihood that the link will function at any given time. This will be 1 if the link always functions and 0 if it never functions; intermediate values of x denote links that function x% of the time.
These probabilities can be combined logically. In order for any single path to function, all of the links in that path must function. Since basic probability theory shows that P(A∩B)=P(A)×P(B), the probability that a single path will function is the product of the probabilities of all links in that path. In the case where the origin and destination are connected by more than one path, the network will function as long as any one path functions. Since P(A∪B)=P(A)+P(B)-P(A∩B), the probability that the origin and destination are connected can be calculated after establishing the functional probability of each possible path.
As an example, imagine a national rail network where each link is closed for maintenance two days out of every year. Scheduling of maintenance is handled separately by regional authorities, and a national coordinator wants to evaluate the chance that specific pairs of destinations might become disconnected at some point in a year due to maintenance closures. He assigns each link a functional probability of 363/365 = 99.45%, identifies the possible paths between the points of interest, and applies the logic above to evaluate the reliability of each path and then of the connection.
Travel Time Reliability
Travel time reliability refers to the probability that a trip from an origin to a destination can be made in a specific time, or within a given range of times. It is used most often on networks where travel demand fluctuations are to some degree random, such as road networks and other multi-user networks. Travel time reliability should not be confused with delays caused by predictable variations. For example, queues due to bottleneck congestion in the fixed demand case.
Most treatments of travel time reliability establish a baseline travel time through observation and then evaluate how different sources of uncertainty will affect actual travel times. The measured reliability is the probability that the baseline travel time plus uncertainty will be less than a desired value. Uncertainty is generated by two fundamental source: demand variation and capacity variation. Most attempts at describing travel time reliability have been based on the variation of travel time under normal daily (or other cyclical) usage variations. In these studies, each link is generally assumed to offer a normally-distributed range of travel times. These link travel time distributions are combined to compute the probability distribution of travel times for an entire path. 
Travel time reliability is analyzed from two approaches: Scheduling Delays with uncertainty, and Mean-Variance. The first assumes travelers have a preferred arrival time, and thus variations around this time time may be classified as schedule early or late delays. The second assumes an average travel time (mean) for the travelers, and variations around this time are classified as variance. Both approaches may be equal under some circumstances 
Capacity reliability measures the probability that the network can handle a specific amount of demand. It can also be viewed as the probability that a network use will be able to complete a trip without encountering a degraded or over-capacity link. In this sense it is a refinement of terminal reliability, which considers neither link capacity nor user demand and link selection.
Proposed calculations of capacity reliability begin estimating a network’s reserve capacity, which represents the volume of traffic the network can handle above some baseline without reaching its maximum capacity or exceeding some defined service level. Existing studies have modeled reserve capacity as a multiplier applied to the baseline traffic volume. Simulation techniques are then applied to model network users’ responses to degraded links, rerouting network flow as links begin to reach capacity.  
Vulnerability has been treated with varying degrees of complexity. In its simplest treatment, vulnerability is effectively the inverse of reliability: as a network’s reliability decreases, its vulnerability increases and vice versa.
A more complex theory of vulnerability connects the ideas of reliability and network resilience by considering how network structure mitigates or aggravates the effect of link degradation. Vulnerability is evaluated for each link in the network and reflects the degree to which network performance (by some chosen measure) is affected if a particular link is degraded. A link is highly vulnerable if a small degradation results in a large reduction of network performance.
This approach to vulnerability could be extended to encompass whole-network vulnerability by computing the ratios of links in a network that have various values of vulnerability. A network with a high proportion of highly vulnerable links would be described as highly vulnerable.
- Cassir, C., Bell, M., & Iida, Y. (2000). Introduction. In Bell, M. G. H., & Cassir, C. (Eds.), Reliability of transport networks (pp. vii-xi). Philadelphia: Research Studies Press.
- Nicholson, A. J., Schmöcker, J-D., Bell, M. G. H., & Iida, Y. (2003). Assessing transport reliability: Malevolence and user knowledge. In Bell, M. G. H., & Iida, Y. (Eds.), The network reliability of transport (pp. 1-22). Oxford: Elsevier Science.
- Berdica, K. (2002). An introduction to road vulnerability: what has been done, is done and should be done Transport Policy, 9 (2), 117-127 DOI: 10.1016/S0967-070X(02)00011-2
- Carrion, C. and Levinson, D. (2011), Value of Reliability: A review of current evidence
- D'Este, G. M., & Taylor, M. A. P. (2003). Network vulnerability: An approach to reliability analysis at the level of national strategic transport networks. In Bell, M. G. H., & Iida, Y. (Eds.), The network reliability of transport (pp. 1-22). Oxford: Elsevier Science.