## 2 Background  
The literature on project duration estimation is vast and cannot be representatively outlined here. Some of the most relevant scheduling techniques were presented in _Introduction_, and more are commented on later in the _Discussions_. This section considers in detail why activity duration variability is the major source of inaccuracy of all deterministic scheduling techniques and what can be done to (partially) overcome that problem. Activities in all projects, construction projects included, suffer from some degree of duration variability [18,19]. Projects are unique, which means that, even when contractors have extensive experience, subsequent projects may differ regarding clients, contract type, location, regulations, labor, equipment, subcontractors, stakeholders, and weather [20]. All these (and many other) factors constitute potential causes of uncertainty and, eventually, of activity duration variability [21]. Deterministic scheduling techniques assume that activity durations are constant. In an attempt to draw up a representative execution plan, most project schedulers use the _average activity durations_, expecting that such an approach will produce the _most likely_ project duration [13]. However, such a schedule is often not reliable. When two or more activities are performed in parallel, the _average_ time it takes to complete all of them is _not_ the maximum of the activity averages, but generally more. Indeed, this situation is identical to the problem of calculating the maximum of two or more random variables. Imagine two activities whose duration is modeled by a normal distribution. Half of the time, each activity ends early, and, half of the time, it ends late. However, when performing both activities in parallel, only in one out of four occasions will both activities finish early. In the other three occasions, either one of the activitieswill finish late, or both will be late. Hence, the maximum time of completion of these two activities will be higher than the average of the activity durations. This phenomenon is known as the _merge event bias_, and it is the reason why deterministic scheduling techniques generally underestimate the actual project duration. The merge event bias is nothing but a manifestation of [PERSON]'s inequality [PERSON] (1969). More precisely, the maximum of several random variables (each representing the activity durations) is a convex function. This means that, _on average_, the maximum of several randomly generated activity durations will always be higher than, or at least equal to, the averages of those activity durations. Hence, the problem of obtaining the average project duration (also known as project makespan) is the same as obtaining the maximum of several random variables, and this was intensively studied in the field of stochastic network analysis (SNA). In SNA, activity durations are modeled with different statistical distributions. One randomly generated duration value is generated for each activity in each simulation run, and then the project duration is computed by following the activities precedence network. This operation is repeated thousands of times until the sample of project durations is deemed representative. Then, the statistical project duration distribution can be represented, and its first two moments (the project duration average and the standard deviation) can be easily calculated. However, as hinted above, there are _no analytical solutions_ to find (accurate) values of the average project duration in project network topologies that involve multiple parallel critical and subcritical paths. The only technique that provides reliable results when activities have variable durations is the Monte Carlo simulation, which is one of the most common in SNA Sna (1992); Sna and Sna (1993); Sna (1994). The main reason why there is no analytical solution in a schedule network is because there are no statistical distributions that are, at the same time, sum-stable and max-stable.